List of small groups

The following list in mathematics contains the finite groups of small order up to group isomorphism.

Counts

(sequence A000001 in the OEIS)

Total number of nonisomorphic groups by order
+	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
0	0	1	1	1	2	1	2	1	5	2	2	1	5	1	2	1
16	14	1	5	1	5	2	2	1	15	2	2	5	4	1	4	1
32	51	1	2	1	14	1	2	2	14	1	6	1	4	2	2	1
48	52	2	5	1	5	1	15	2	13	2	2	1	13	1	2	4
64	267	1	4	1	5	1	4	1	50	1	2	3	4	1	6	1
80	52	15	2	1	15	1	2	1	12	1	10	1	4	2	2	1
96	231	1	5	2	16	1	4	1	14	2	2	1	45	1	6	2
112	43	1	6	1	5	4	2	1	47	2	2	1	4	5	16	1
128	2328	2	4	1	10	1	2	5	15	1	4	1	11	1	2	1

For labeled groups, see OEIS: A034383.

Glossary

Each group is named by their Small Groups library index as G_oⁱ, where o is the order of the group, and i is the index of the group within that order.

Common group names:

Z_n: the cyclic group of order n (the notation C_n is also used; it is isomorphic to the additive group of Z/nZ).
- K₄: the Klein four-group of order 4, same as Z₂ × Z₂ or Dih₂.
Dih_n: the dihedral group of order 2n (often the notation D_n or D_2n is used )
S_n: the symmetric group of degree n, containing the n! permutations of n elements.
A_n: the alternating group of degree n, containing the n!/2 even permutations of n elements.
Dic_n or Q_4n: the dicyclic group of order 4n.
- Q₈: the quaternion group of order 8, also Dic₂.

The notations Z_n and Dih_n have the advantage that point groups in three dimensions C_n and D_n do not have the same notation. There are more isometry groups than these two, of the same abstract group type.

The notation G × H denotes the direct product of the two groups; Gⁿ denotes the direct product of a group with itself n times. G ⋊ H denotes a semidirect product where H acts on G; this may also depend on the choice of action of H on G

Abelian and simple groups are noted. (For groups of order n < 60, the simple groups are precisely the cyclic groups Z_n, for prime n.) The equality sign ("=") denotes isomorphism.

The identity element in the cycle graphs is represented by the black circle. The lowest order for which the cycle graph does not uniquely represent a group is order 16.

In the lists of subgroups, the trivial group and the group itself are not listed. Where there are multiple isomorphic subgroups, their number is indicated in parentheses.

List of small abelian groups

The finite abelian groups are either cyclic groups, or direct products thereof; see abelian groups.

(sequence A000688 in the OEIS)

Number of nonisomorphic abelian groups by order
+	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
0	0	1	1	1	2	1	1	1	3	2	1	1	2	1	1	1
16	5	1	2	1	2	1	1	1	3	2	1	3	2	1	1	1
32	7	1	1	1	4	1	1	1	3	1	1	1	2	2	1	1
48	5	2	2	1	2	1	3	1	3	1	1	1	2	1	1	2
64	11	1	1	1	2	1	1	1	6	1	1	2	2	1	1	1
80	5	5	1	1	2	1	1	1	3	1	2	1	2	1	1	1
96	7	1	2	2	4	1	1	1	3	1	1	1	6	1	1	1
112	5	1	1	1	2	2	1	1	3	2	1	1	2	3	2	1
128	15	1	1	1	2	1	1	3	3	1	1	1	2	1	1	1

For labeled Abelian groups, see OEIS: A034382.

List of all abelian groups up to order 30
Order	G_oⁱ	Group	Nontrivial proper Subgroups	Properties
1 ^[1]	G₁¹	Z₁^[2] = S₁ = A₂	–	Trivial group. Cyclic, alternating group, symmetric group. Elementary.
2 ^[3]	G₂¹	Z₂^[4] = S₂ = Dih₁	–	simple, the smallest non-trivial group. Symmetric group. Cyclic. Elementary.
3 ^[5]	G₃¹	Z₃^[6] = A₃	–	simple. Alternating group. Cyclic. Elementary.
4^[7]	G₄¹	Z₄^[8] = Dic₁	Z₂	cyclic.
4^[7]	G₄²	Z₂² = K₄^[9] = Dih₂	Z₂ (3)	Klein four-group, the smallest non-cyclic group. Elementary. Product.
5^[10]	G₅¹	Z₅^[11]	–	Simple. Cyclic. Elementary.
6^[12]	G₆²	Z₆^[13] = Z₃ × Z₂^[14]	Z₃, Z₂	Cyclic. Product.
7^[15]	G₇¹	Z₇^[16]	–	Simple. Cyclic. Elementary.
8^[17]	G₈¹	Z₈^[18]	Z₄, Z₂	Cyclic.
	G₈²	Z₄ × Z₂^[19]	Z₂², Z₄ (2), Z₂ (3)	Product.
	G₈⁵	Z₂³^[20]	Z₂² (7), Z₂ (7)	The non-identity elements correspond to the points in the Fano plane, the Z₂ × Z₂ subgroups to the lines. Product. Elementary.
9^[21]	G₉¹	Z₉^[22]	Z₃	Cyclic.
9^[21]	G₉²	Z₃²^[23]	Z₃ (4)	Elementary. Product.
10^[24]	G₁₀²	Z₁₀^[25] = Z₅ × Z₂	Z₅, Z₂	Cyclic. Product.
11	G₁₁¹	Z₁₁^[26]	–	Simple. Cyclic. Elementary.
12^[27]	G₁₂²	Z₁₂^[28] = Z₄ × Z₃	Z₆, Z₄, Z₃, Z₂	Cyclic. Product.
12^[27]	G₁₂⁵	Z₆ × Z₂^[29] = Z₃ × Z₂²	Z₆ (3), Z₃, Z₂ (3), Z₂²	Product.
13	G₁₃¹	Z₁₃^[30]	–	Simple. Cyclic. Elementary.
14^[31]	G₁₄²	Z₁₄^[32] = Z₇ × Z₂	Z₇, Z₂	Cyclic. Product.
15^[33]	G₁₅¹	Z₁₅^[34] = Z₅ × Z₃	Z₅, Z₃	Cyclic. Product.
16^[35]	G₁₆¹	Z₁₆^[36]	Z₈, Z₄, Z₂	Cyclic.
	G₁₆²	Z₄²^[37]	Z₂ (3), Z₄ (6), Z₂², Z₄ × Z₂ (3)	Product.
	G₁₆⁵	Z₈ × Z₂^[38]	Z₂ (3), Z₄ (2), Z₂², Z₈ (2), Z₄ × Z₂	Product.
	G₁₆¹⁰	Z₄ × Z₂²^[39]	Z₂ (7), Z₄ (4), Z₂² (7), Z₂³, Z₄ × Z₂ (6)	Product.
	G₁₆¹⁴	Z₂⁴^[19] = K₄²	Z₂ (15), Z₂² (35), Z₂³ (15)	Product. Elementary.
17	G₁₇¹	Z₁₇^[40]	–	Simple. Cyclic. Elementary.
18^[41]	G₁₈²	Z₁₈^[42] = Z₉ × Z₂	Z₉, Z₆, Z₃, Z₂	Cyclic. Product.
18^[41]	G₁₈⁵	Z₆ × Z₃^[43] = Z₃² × Z₂	Z₆, Z₃, Z₂	Product.
19	G₁₉¹	Z₁₉^[44]	–	Simple. Cyclic. Elementary.
20^[45]	G₂₀²	Z₂₀^[46] = Z₅ × Z₄	Z₂₀, Z₁₀, Z₅, Z₄, Z₂	Cyclic. Product.
20^[45]	G₂₀⁵	Z₁₀ × Z₂^[47] = Z₅ × Z₂²	Z₅, Z₂	Product.
21	G₂₁²	Z₂₁^[48] = Z₇ × Z₃	Z₇, Z₃	Cyclic. Product.
22	G₂₂²	Z₂₂^[49] = Z₁₁ × Z₂	Z₁₁, Z₂	Cyclic. Product.
23	G₂₃¹	Z₂₃^[50]	–	Simple. Cyclic. Elementary.
24^[51]	G₂₄²	Z₂₄^[52] = Z₈ × Z₃	Z₁₂, Z₈, Z₆, Z₄, Z₃, Z₂	Cyclic. Product.
	G₂₄⁹	Z₁₂ × Z₂^[53] = Z₆ × Z₄ = Z₄ × Z₃ × Z₂	Z₁₂, Z₆, Z₄, Z₃, Z₂	Product.
	G₂₄¹⁵	Z₆ × Z₂²^[39] = Z₃ × Z₂³	Z₆, Z₃, Z₂	Product.
25	G₂₅¹	Z₂₅	Z₅	Cyclic.
25	G₂₅²	Z₅²	Z₅	Product. Elementary.
26	G₂₆¹	Z₂₆ = Z₁₃ × Z₂	Z₁₃, Z₂	Cyclic. Product.
27^[54]	G₂₇¹	Z₂₇	Z₉, Z₃	Cyclic.
	G₂₇²	Z₉ × Z₃	Z₉, Z₃	Product.
	G₂₇	Z₃³	Z₃	Product. Elementary.
28	G₂₈²	Z₂₈ = Z₇ × Z₄	Z₁₄, Z₇, Z₄, Z₂	Cyclic. Product.
28	G₂₈⁴	Z₁₄ × Z₂ = Z₇ × Z₂²	Z₁₄, Z₇, Z₄, Z₂	Product.
29	G₂₉¹	Z₂₉	–	Simple. Cyclic. Elementary.
30^[55]	G₃₀⁴	Z₃₀ = Z₁₅ × Z₂ = Z₁₀ × Z₃ = Z₆ × Z₅ = Z₅ × Z₃ × Z₂	Z₁₅, Z₁₀, Z₆, Z₅, Z₃, Z₂	Cyclic. Product.

List of small non-abelian groups

(sequence A060689 in the OEIS)

Number of nonisomorphic nonabelian groups by order
+	0	1	2	4	5	6	7	8	9	10	11	12	13	14	15
0	0	0	0	0	0	1	0	2	0	1	0	3	0	1	0
16	9	0	3	3	1	1	0	12	0	1	2	2	0	3	0
32	44	0	1	10	0	1	1	11	0	5	0	2	0	1	0
48	47	0	3	3	0	12	1	10	1	1	0	11	0	1	2
64	256	0	3	3	0	3	0	44	0	1	1	2	0	5	0
80	47	10	1	13	0	1	0	9	0	8	0	2	1	1	0
96	224	0	3	12	0	3	0	11	1	1	0	39	0	5	1
112	38	0	5	3	2	1	0	44	0	1	0	2	2	14	0
128	2313	1	3	8	0	1	2	12	0	3	0	9	0	1	0

Order of non-abelian groups are

6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 117, 118, 120, 122, 124, 125, 126, 128, 129, 130, 132, 134, 135, 136, 138, 140, 142, ... (sequence A060652 in the OEIS)

List of all nonabelian groups up to order 30
Order	G_oⁱ	Group	Nontrivial proper Subgroups	Properties
6^[12]	G₆¹	Dih₃ = S₃^[56] = D₆	Z₃, Z₂ (3)	Dihedral group, the smallest non-abelian group, symmetric group, Frobenius group
8^[17]	G₈³	Dih₄ = D₈^[57]	Z₄, Z₂² (2), Z₂ (5)	Dihedral group. Extraspecial group. Nilpotent.
8^[17]	G₈⁴	Q₈^[58] = Dic₂ = <2,2,2>	Z₄ (3), Z₂	Quaternion group, Hamiltonian group. all subgroups are normal without the group being abelian. The smallest group G demonstrating that for a normal subgroup H the quotient group G/H need not be isomorphic to a subgroup of G. Extraspecial group Binary dihedral group. Nilpotent.
10^[24]	G₁₀¹	Dih₅ = D₁₀^[59]	Z₅, Z₂ (5)	Dihedral group, Frobenius group
12^[27]	G₁₂¹	Q₁₂^[60] = Dic₃ = <3,2,2> = Z₃ ⋊ Z₄	Z₂, Z₃, Z₄ (3), Z₆	Binary dihedral group
	G₁₂³	A₄^[61]	Z₂², Z₃ (4), Z₂ (3)	Alternating group. No subgroups of order 6, although 6 divides its order. Frobenius group
	G₁₂⁴	Dih₆ = D₁₂^[62] = Dih₃ × Z₂	Z₆, Dih₃ (2), Z₂² (3), Z₃, Z₂ (7)	Dihedral group, product
14^[31]	G₁₄¹	Dih₇	Z₇, Z₂ (7)	Dihedral group, Frobenius group
16^[35]^[63]	G₁₆³	G_4,4 = K₄ ⋊ Z₄ (Z₄×Z₂) ⋊ Z₂		Has the same number of elements of every order as the Pauli group. Nilpotent.
	G₁₆⁴	Z₄ ⋊ Z₄		The squares of elements do not form a subgroup. Has the same number of elements of every order as Q₈ × Z₂. Nilpotent.
	G₁₆⁶	Z₈ ⋊ Z₂		Sometimes called the modular group of order 16, though this is misleading as abelian groups and Q₈ × Z₂ are also modular. Nilpotent.
	G₁₆⁷	Dih₈	Z₈, Dih₄ (2), Z₂² (4), Z₄, Z₂ (9)	Dihedral group. Nilpotent.
	G₁₆⁸	QD₁₆		The order 16 quasidihedral group. Nilpotent.
	G₁₆⁹	Q₁₆ = Dic₄ = <4,2,2>		generalized quaternion group, binary dihedral group. Nilpotent.
	G₁₆¹¹	Dih₄ × Z₂	Dih₄ (2), Z₄ × Z₂, Z₂³ (2), Z₂² (11), Z₄ (2), Z₂ (11)	Product. Nilpotent.
	G₁₆¹²	Q₈ × Z₂		Hamiltonian, product. Nilpotent.
	G₁₆¹³	(Z₄ × Z₂) ⋊ Z₂		The Pauli group generated by the Pauli matrices. Nilpotent.
18^[41]	G₁₈¹	Dih₉		Dihedral group, Frobenius group
	G₁₈³	S₃×Z₃		Product
	G₁₈⁴	(Z₃ × Z₃)⋊ Z₂		Frobenius group
20^[45]	G₂₀¹	Q₂₀ = Dic₅ = <5,2,2>		Binary dihedral group
	G₂₀³	Z₅ ⋊ Z₄		Frobenius group
	G₂₀⁴	Dih₁₀ = Dih₅ × Z₂		Dihedral group, product
21	G₂₁¹	Z₇ ⋊ Z₃		Smallest non-abelian group of odd order. Frobenius group
22	G₂₂¹	Dih₁₁		Dihedral group, Frobenius group
24^[51]	G₂₄¹	Z₃ ⋊ Z₈		Central extension of S₃
	G₂₄³	SL(2,3) = 2T = Q₈ ⋊ Z₃		Binary tetrahedral group
	G₂₄⁴	Q₂₄ = Dic₆ = <6,2,2> = Z₃ ⋊ Q₈		Binary dihedral
	G₂₄⁵	Z₄ × S₃		Product
	G₂₄⁶	Dih₁₂		Dihedral group
	G₂₄⁷	Dic₃ × Z₂ = Z₂ × (Z₃ × Z₄)		Product
	G₂₄⁸	(Z₆ × Z₂)⋊ Z₂ = Z₃ ⋊ Dih₄		Double cover of dihedral group
	G₂₄¹⁰	Dih₄ × Z₃		Product. Nilpotent.
	G₂₄¹¹	Q₈ × Z₃		Product. Nilpotent.
	G₂₄¹²	S₄		Symmetric group. Has no normal Sylow subgroups.
	G₂₄¹³	A₄ × Z₂		Product
	G₂₄¹⁴	D₁₂× Z₂		Product
26	G₂₆¹	Dih₁₃		Dihedral group, Frobenius group
27^[54]	G₂₇³	Z₃² ⋊ Z₃		All non-trivial elements have order 3. Extraspecial group. Nilpotent.
27^[54]	G₂₇⁴	Z₉ ⋊ Z₃		Extraspecial group. Nilpotent.
28	G₂₈¹	Z₇ ⋊ Z₄		Binary dihedral group
28	G₂₈³	Dih₁₄		Dihedral group, product
30^[55]	G₃₀¹	Z₅ × S₃		Product
	G₃₀³	Dih₁₅		Dihedral group, Frobenius group
	G₃₀⁴	Z₃ × Dih₅		Product

Classifying groups of small order

Small groups of prime power order pⁿ are given as follows:

Order p: The only group is cyclic.
Order p²: There are just two groups, both abelian.
Order p³: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p² by a cyclic group of order p. The other is the quaternion group for p=2 and a group of exponent p for p'>2.
Order p⁴: The classification is complicated, and gets much harder as the exponent of p increases.

Most groups of small order have a Sylow p subgroup P with a normal p-complement N for some prime p dividing the order, so can be classified in terms of the possible primes p, p-groups P, groups N, and actions of P on N. In some sense this reduces the classification of these groups to the classification of p-groups. Some of the small groups that do not have a normal p complement include:

Order 24: The symmetric group S₄
Order 48: The binary octahedral group and the product S₄ × Z/2Z
Order 60: The alternating group A₅.

Small groups library

The group theoretical computer algebra system GAP contains the "Small Groups library" which provides access to descriptions of small order groups. The groups are listed up to isomorphism. At present, the library contains the following groups:^[64]

those of order at most 2000, except for order 1024 (423164062 groups in the library; the ones of order 1024 had to be skipped, as there are an additional 49487365422 nonisomorphic 2-groups of order 1024.);
those of cubefree order at most 50000 (395 703 groups);
those of squarefree order;
those of order pⁿ for n at most 6 and p prime;
those of order p⁷ for p = 3, 5, 7, 11 (907 489 groups);
those of order pqⁿ where qⁿ divides 2⁸, 3⁶, 5⁵ or 7⁴ and p is an arbitrary prime which differs from q;
those whose orders factorise into at most 3 primes (count with multiplicity).

It contains explicit descriptions of the available groups in computer readable format.

The smallest 10 orders for which the SmallGroups library does not have information are 1024, 2016, 2024, 2025, 2040, 2048, 2052, 2058, 2064, 2072.

Notes

^ Groups of order 1
^ Z1
^ Groups of order 2
^ Z2
^ Groups of order 3
^ Z3
^ Groups of order 4
^ Z4
^ Klein group
^ Groups of order 5
^ Z5
^ ^a ^b Groups of order 6
^ Z6
^ See a worked example showing the isomorphism Z₆ = Z₃ × Z₂.
^ Groups of order 7
^ Z7
^ ^a ^b Groups of order 8
^ Z8
^ ^a ^b Z4×Z2
^ Elementary abelian group:E8
^ Groups of order 9
^ Z9
^ Z3×Z3
^ ^a ^b Groups of order 10
^ Z10
^ Z11
^ ^a ^b Groups of order 12
^ Z12
^ Z6×Z2
^ Z13
^ ^a ^b Groups of order 14
^ Z14
^ Groups of order 15
^ Z15
^ ^a ^b Groups of order 16
^ Z16
^ Z4×Z4
^ Z8×Z2
^ ^a ^b Z4×Z2×Z2
^ Z17
^ ^a ^b Groups of order 18
^ Z18
^ Z6×Z3
^ Z19
^ ^a ^b Groups of order 20
^ Z20
^ Z10×Z2
^ Z21
^ Z22
^ Z23
^ ^a ^b Groups of order 24
^ Z24
^ Z12×Z2
^ ^a ^b Groups of order 27
^ ^a ^b Groups of order 30
^ S3
^ D8
^ Q8
^ D10
^ Q12
^ A4
^ D12
^ Wild, Marcel. "The Groups of Order Sixteen Made Easy", American Mathematical Monthly, Jan 2005
^ Hans Ulrich Besche The Small Groups library

References

Coxeter, H. S. M. and Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9.{{cite book}}: CS1 maint: multiple names: authors list (link), Table 1, Nonabelian groups order<32.
Hall, Jr., Marshall; Senior, James K. (1964). "The Groups of Order 2ⁿ (n ≤ 6)" (Document). Macmillan. MR 0168631. A catalog of the 340 groups of order dividing 64 with tables of defining relations, constants, and lattice of subgroups of each group.{{cite document}}: CS1 maint: postscript (link)

External links

Particular groups in the Group Properties Wiki
Number of groups of given order
Groups of given order
Besche, H. U.; Eick, B.; O'Brien, E. "small group library".
R. J. Mathar (2014). "Plots of cycle graphs of the finite groups up to order 36".

[g1-1] Groups of order 1

[2] Z1

[g2-3] Groups of order 2

[4] Z2

[g3-5] Groups of order 3

[6] Z3

[g4-7] Groups of order 4

[8] Z4

[9] Klein group

[g5-10] Groups of order 5

[11] Z5

[g6-12] Groups of order 6

[13] Z6

[14] See a worked example showing the isomorphism Z₆ = Z₃ × Z₂.

[g7-15] Groups of order 7

[16] Z7

[g8-17] Groups of order 8

[18] Z8

[Z4×Z2-19] Z4×Z2

[20] Elementary abelian group:E8

[g9-21] Groups of order 9

[22] Z9

[23] Z3×Z3

[g10-24] Groups of order 10

[25] Z10

[26] Z11

[g12-27] Groups of order 12

[28] Z12

[29] Z6×Z2

[30] Z13

[g14-31] Groups of order 14

[32] Z14

[g15-33] Groups of order 15

[34] Z15

[g16-35] Groups of order 16

[36] Z16

[37] Z4×Z4

[38] Z8×Z2

[Z4×Z2×Z2-39] Z4×Z2×Z2

[40] Z17

[g18-41] Groups of order 18

[42] Z18

[43] Z6×Z3

[44] Z19

[g20-45] Groups of order 20

[46] Z20

[47] Z10×Z2

[48] Z21

[49] Z22

[50] Z23

[g24-51] Groups of order 24

[52] Z24

[53] Z12×Z2

[g27-54] Groups of order 27

[g30-55] Groups of order 30

[56] S3

[57] D8

[58] Q8

[59] D10

[60] Q12

[61] A4

[62] D12

[63] Wild, Marcel. "The Groups of Order Sixteen Made Easy", American Mathematical Monthly, Jan 2005

[gap-64] Hans Ulrich Besche The Small Groups library

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