List of space groups

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There are 230 space groups in three dimensions, given by a number index, and a full name in Hermann–Mauguin notation, and a short name (international short symbol). The long names are given with spaces for readability. The groups each have a point groups of the unit cell.

Symbols[edit]

In Hermann–Mauguin notation, space groups are named by a symbol combining the point group identifier with the uppercase letters describing the lattice type. Translations within the lattice in the form of screw axes and glide planes are also noted, giving a complete crystallographic space group.

These are the Bravais lattices in three dimensions:

  • P primitive
  • I body centered (from the German "Innenzentriert")
  • F face centered (from the German "Flächenzentriert")
  • A centered on A faces only
  • B centered on B faces only
  • C centered on C faces only
  • R rhombohedral

A reflection plane m within the point groups can be replaced by a glide plane, labeled a a, b, or c depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a quarter of either a face or space diagonal of the unit cell. The d glide is often called the diamond glide plane as it features in the diamond structure.

  • a, b, or c glide translation along half the lattice vector of this face
  • n glide translation along with half a face diagonal
  • d glide planes with translation along a quarter of a face diagonal.
  • e two glides with the same glide plane and translation along two (different) half-lattice vectors.

A gyration point can be replaced by a screw axis is noted by a number, n, where the angle of rotation is \color{Black}\tfrac{360^\circ}{n}. The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 21 is a 180° (twofold) rotation followed by a translation of ½ of the lattice vector. 31 is a 120° (threefold) rotation followed by a translation of ⅓ of the lattice vector.

The possible screw axis are: 21, 31, 32, 41, 42, 43, 61, 62, 63, 64, and 65.

List of Triclinic[edit]

Triclinic (a ≠ b ≠ c and α ≠ β ≠ γ )
Triclinic crystal system
Number Point group Short name Full name Schoenflies Fibrifold
1 1 P1 P 1 C_1^1
2 1 P1 P 1 C_i^1

List of Monoclinic[edit]

Monoclinic Bravais lattice
Simple
(P)
Base
(C)
Monoclinic.svg Monoclinic-base-centered.svg
Monoclinic crystal system
Number Point group Short name Full name(s) Schoenflies Fibrifold
3 2 P2 P 1 2 1 P 1 1 2 C_2^1
4 2 P21 P 1 21 1 P 1 1 21 C_2^2
5 2 C2 C 1 2 1 B 1 1 2 C_2^3
6 m Pm P 1 m 1 P 1 1 m C_s^1
7 m Pc P 1 c 1 P 1 1 b C_s^2
8 m Cm C 1 m 1 B 1 1 m C_s^3
9 m Cc C 1 c 1 B 1 1 b C_s^4
10 2/m P2/m P 1 2/m 1 P 1 1 2/m C_{2h}^1
11 2/m P21/m P 1 21/m 1 P 1 1 21/m C_{2h}^2
12 2/m C2/m C 1 2/m 1 B 1 1 2/m C_{2h}^3
13 2/m P2/c P 1 2/c 1 P 1 1 2/b C_{2h}^4
14 2/m P21/c P 1 21/c 1 P 1 1 21/b C_{2h}^5
15 2/m C2/c C 1 2/c 1 B 1 1 2/b C_{2h}^6

List of Orthorhombic[edit]

Orthorhombic Bravais lattices
Primitive
(P)
Body-centered
(I)
Orthohombic, simple Orthohombic, body-centered
Base-centered
(A, B or C)
Face-centered
(F)
Orthohombic, base-centered Orthohombic, face-centered
Orthorhombic crystal system
Number Point group Short name Full name Schoenflies Fibrifold
16 222 P222 P 2 2 2 D_2^1
17 222 P2221 P 2 2 21 D_2^2
18 222 P21212 P 21 21 2 D_2^3
19 222 P212121 P 21 21 21 D_2^4
20 222 C2221 C 2 2 21 D_2^5
21 222 C222 C 2 2 2 D_2^6
22 222 F222 F 2 2 2 D_2^7
23 222 I222 I 2 2 2 D_2^8
24 222 I212121 I 21 21 21 D_2^9
25 mm2 Pmm2 P m m 2 C_{2v}^1
26 mm2 Pmc21 P m c 21 C_{2v}^2
27 mm2 Pcc2 P c c 2 C_{2v}^3
28 mm2 Pma2 P m a 2 C_{2v}^4
29 mm2 Pca21 P c a 21 C_{2v}^5
30 mm2 Pnc2 P n c 2 C_{2v}^6
31 mm2 Pmn21 P m n 21 C_{2v}^7
32 mm2 Pba2 P b a 2 C_{2v}^8
33 mm2 Pna21 P n a 21 C_{2v}^9
34 mm2 Pnn2 P n n 2 C_{2v}^{10}
35 mm2 Cmm2 C m m 2 C_{2v}^{11}
36 mm2 Cmc21 C m c 21 C_{2v}^{12}
37 mm2 Ccc2 C c c 2 C_{2v}^{13}
38 mm2 Amm2 A m m 2 C_{2v}^{14}
39 mm2 Aem2 A b m 2 C_{2v}^{15}
40 mm2 Ama2 A m a 2 C_{2v}^{16}
41 mm2 Aea2 A b a 2 C_{2v}^{17}
42 mm2 Fmm2 F m m 2 C_{2v}^{18}
43 mm2 Fdd2 F dd2 C_{2v}^{19}
44 mm2 Imm2 I m m 2 C_{2v}^{20}
45 mm2 Iba2 I b a 2 C_{2v}^{21}
46 mm2 Ima2 I m a 2 C_{2v}^{22}
47 2/m 2/m 2/m Pmmm P 2/m 2/m 2/m D_{2h}^1
48 2/m 2/m 2/m Pnnn P 2/n 2/n 2/n D_{2h}^2
49 2/m 2/m 2/m Pccm P 2/c 2/c 2/m D_{2h}^3
50 2/m 2/m 2/m Pban P 2/b 2/a 2/n D_{2h}^4
51 2/m 2/m 2/m Pmma P 21/m 2/m 2/a D_{2h}^5
52 2/m 2/m 2/m Pnna P 2/n 21/n 2/a D_{2h}^6
53 2/m 2/m 2/m Pmna P 2/m 2/n 21/a D_{2h}^7
54 2/m 2/m 2/m Pcca P 21/c 2/c 2/a D_{2h}^8
55 2/m 2/m 2/m Pbam P 21/b 21/a 2/m D_{2h}^9
56 2/m 2/m 2/m Pccn P 21/c 21/c 2/n D_{2h}^{10}
57 2/m 2/m 2/m Pbcm P 2/b 21/c 21/m D_{2h}^{11}
58 2/m 2/m 2/m Pnnm P 21/n 21/n 2/m D_{2h}^{12}
59 2/m 2/m 2/m Pmmn P 21/m 21/m 2/n D_{2h}^{13}
60 2/m 2/m 2/m Pbcn P 21/b 2/c 21/n D_{2h}^{14}
61 2/m 2/m 2/m Pbca P 21/b 21/c 21/a D_{2h}^{15}
62 2/m 2/m 2/m Pnma P 21/n 21/m 21/a D_{2h}^{16}
63 2/m 2/m 2/m Cmcm C 2/m 2/c 21/m D_{2h}^{17}
64 2/m 2/m 2/m Cmce C 2/m 2/c 21/a D_{2h}^{18}
65 2/m 2/m 2/m Cmmm C 2/m 2/m 2/m D_{2h}^{19}
66 2/m 2/m 2/m Cccm C 2/c 2/c 2/m D_{2h}^{20}
67 2/m 2/m 2/m Cmme C 2/m 2/m 2/e D_{2h}^{21}
68 2/m 2/m 2/m Ccce C 2/c 2/c 2/e D_{2h}^{22}
69 2/m 2/m 2/m Fmmm F 2/m 2/m 2/m D_{2h}^{23}
70 2/m 2/m 2/m Fddd F 2/d 2/d 2/d D_{2h}^{24}
71 2/m 2/m 2/m Immm I 2/m 2/m 2/m D_{2h}^{25}
72 2/m 2/m 2/m Ibam I 2/b 2/a 2/m D_{2h}^{26}
73 2/m 2/m 2/m Ibca I 2/b 2/c 2/a D_{2h}^{27}
74 2/m 2/m 2/m Imma I 2/m 2/m 2/a D_{2h}^{28}

List of Tetragonal[edit]

Tetragonal Bravais lattices
Primitive
(P)
Body-centered
(I)
Tetragonal.svg Tetragonal-body-centered.svg
Tetragonal crystal system
Number Point group Short name Full name Schoenflies Fibrifold
75 4 P4 P 4 C_4^1
76 4 P41 P 41 C_4^2
77 4 P42 P 42 C_4^3
78 4 P43 P 43 C_4^4
79 4 I4 I 4 C_4^5
80 4 I41 I 41 C_4^6
81 4 P4 P 4 S_4^1
82 4 I4 I 4 S_4^2
83 4/m P4/m P 4/m C_{4h}^1
84 4/m P42/m P 42/m C_{4h}^2
85 4/m P4/n P 4/n C_{4h}^3
86 4/m P42/n P 42/n C_{4h}^4
87 4/m I4/m I 4/m C_{4h}^5
88 4/m I41/a I 41/a C_{4h}^6
89 422 P422 P 4 2 2 D_4^1
90 422 P4212 P 42 1 2 D_4^2
91 422 P4122 P 41 2 2 D_4^3
92 422 P41212 P 41 21 2 D_4^4
93 422 P4222 P 42 2 2 D_4^5
94 422 P42212 P 42 21 2 D_4^6
95 422 P4322 P 43 2 2 D_4^7
96 422 P43212 P 43 21 2 D_4^8
97 422 I422 I 4 2 2 D_4^9
98 422 I4122 I 41 2 2 D_4^{10}
99 4mm P4mm P 4 m m C_{4v}^1
100 4mm P4bm P 4 b m C_{4v}^2
101 4mm P42cm P 42 c m C_{4v}^3
102 4mm P42nm P 42 n m C_{4v}^4
103 4mm P4cc P 4 c c C_{4v}^5
104 4mm P4nc P 4 n c C_{4v}^6
105 4mm P42mc P 42 m c C_{4v}^7
106 4mm P42bc P 42 b c C_{4v}^8
107 4mm I4mm I 4 m m C_{4v}^9
108 4mm I4cm I 4 c m C_{4v}^{10}
109 4mm I41md I 41 m d C_{4v}^{11}
110 4mm I41cd I 41 c d C_{4v}^{12}
111 42m P42m P 4 2 m D_{2d}^1
112 42m P42c P 4 2 c D_{2d}^2
113 42m P421m P 4 21 m D_{2d}^3
114 42m P421c P 4 21 c D_{2d}^4
115 42m P4m2 P 4 m 2 D_{2d}^5
116 42m P4c2 P 4 c 2 D_{2d}^6
117 42m P4b2 P 4 b 2 D_{2d}^7
118 42m P4n2 P 4 n 2 D_{2d}^8
119 42m I4m2 I 4 m 2 D_{2d}^9
120 42m I4c2 I 4 c 2 D_{2d}^{10}
121 42m I42m I 4 2 m D_{2d}^{11}
122 42m I42d I 4 2 d D_{2d}^{12}
123 4/m 2/m 2/m P4/mmm P 4/m 2/m 2/m D_{4h}^1
124 4/m 2/m 2/m P4/mcc P 4/m 2/c 2/c D_{4h}^2
125 4/m 2/m 2/m P4/nbm P 4/n 2/b 2/m D_{4h}^3
126 4/m 2/m 2/m P4/nnc P 4/n 2/n 2/c D_{4h}^4
127 4/m 2/m 2/m P4/mbm P 4/m 21/b 2/m D_{4h}^5
128 4/m 2/m 2/m P4/mnc P 4/m 21/n 2/c D_{4h}^6
129 4/m 2/m 2/m P4/nmm P 4/n 21/m 2/m D_{4h}^7
130 4/m 2/m 2/m P4/ncc P 4/n 21/c 2/c D_{4h}^8
131 4/m 2/m 2/m P42/mmc P 42/m 2/m 2/c D_{4h}^9
132 4/m 2/m 2/m P42/mcm P 42/m 2/c 2/m D_{4h}^{10}
133 4/m 2/m 2/m P42/nbc P 42/n 2/b 2/c D_{4h}^{11}
134 4/m 2/m 2/m P42/nnm P 42/n 2/n 2/m D_{4h}^{12}
135 4/m 2/m 2/m P42/mbc P 42/m 21/b 2/c D_{4h}^{13}
136 4/m 2/m 2/m P42/mnm P 42/m 21/n 2/m D_{4h}^{14}
137 4/m 2/m 2/m P42/nmc P 42/n 21/m 2/c D_{4h}^{15}
138 4/m 2/m 2/m P42/ncm P 42/n 21/c 2/m D_{4h}^{16}
139 4/m 2/m 2/m I4/mmm I 4/m 2/m 2/m D_{4h}^{17}
140 4/m 2/m 2/m I4/mcm I 4/m 2/c 2/m D_{4h}^{18}
141 4/m 2/m 2/m I41/amd I 41/a 2/m 2/d D_{4h}^{19}
142 4/m 2/m 2/m I41/acd I 41/a 2/c 2/d D_{4h}^{20}

List of Trigonal[edit]

Unit cells for trigonal crystal system
Rhombohedral
(R)
Hexagonal
(P)
Hexagonal latticeR.svg Hexagonal latticeFRONT.svg
Trigonal crystal system
Number Point group Short name Full name Schoenflies Fibrifold
143 3 P3 P 3 C_3^1
144 3 P31 P 31 C_3^2
145 3 P32 P 32 C_3^3
146 3 R3 R 3 C_3^4
147 3 P3 P 3 C_{3i}^1
148 3 R3 R 3 C_{3i}^2
149 32 P312 P 3 1 2 D_3^1
150 32 P321 P 3 2 1 D_3^2
151 32 P3112 P 31 1 2 D_3^3
152 32 P3121 P 31 2 1 D_3^4
153 32 P3212 P 32 1 2 D_3^5
154 32 P3221 P 32 2 1 D_3^6
155 32 R32 R 3 2 D_3^7
156 3m P3m1 P 3 m 1 C_{3v}^1
157 3m P31m P 3 1 m C_{3v}^2
158 3m P3c1 P 3 c 1 C_{3v}^3
159 3m P31c P 3 1 c C_{3v}^4
160 3m R3m R 3 m C_{3v}^5
161 3m R3c R 3 c C_{3v}^6
162 3 2/m P31m P 3 1 2/m D_{3d}^1
163 3 2/m P31c P 3 1 2/c D_{3d}^2
164 3 2/m P3m1 P 3 2/m 1 D_{3d}^3
165 3 2/m P3c1 P 3 2/c 1 D_{3d}^4
166 3 2/m R3m R 3 2/m D_{3d}^5
167 3 2/m R3c R 3 2/c D_{3d}^6

List of Hexagonal[edit]

Hexagonal lattice cell
(P)
Hexagonal crystal system
Number Point group Short name Full name Schoenflies Fibrifold
168 6 P6 P 6 C_6^1
169 6 P61 P 61 C_6^2
170 6 P65 P 65 C_6^3
171 6 P62 P 62 C_6^4
172 6 P64 P 64 C_6^5
173 6 P63 P 63 C_6^6
174 6 P6 P 6 C_{3h}^1
175 6/m P6/m P 6/m C_{6h}^1
176 6/m P63/m P 63/m C_{6h}^2
177 622 P622 P 622 D_6^1
178 622 P6122 P 61 2 2 D_6^2
179 622 P6522 P 65 2 2 D_6^3
180 622 P6222 P 62 2 2 D_6^4
181 622 P6422 P 64 2 2 D_6^5
182 622 P6322 P 63 2 2 D_6^6
183 6mm P6mm P 6 m m C_{6v}^1
184 6mm P6cc P 6 c c C_{6v}^2
185 6mm P63cm P 63 c m C_{6v}^3
186 6mm P63mc P 63 m c C_{6v}^4
187 6m2 P6m2 P 6 m 2 D_{3h}^1
188 6m2 P6c2 P 6 c 2 D_{3h}^2
189 6m2 P62m P 6 2 m D_{3h}^3
190 6m2 P62c P 6 2 c D_{3h}^4
191 6/m 2/m 2/m P6/mmm P 6/m 2/m 2/m D_{6h}^1
192 6/m 2/m 2/m P6/mcc P 6/m 2/c 2/c D_{6h}^2
193 6/m 2/m 2/m P63/mcm P 63/m 2/c 2/m D_{6h}^3
194 6/m 2/m 2/m P63/mmc P 63/m 2/m 2/c D_{6h}^4

List of Cubic[edit]

Cubic Bravais lattices
Primitive
(P)
Body-centered
(I)
Face-centered
(F)
Lattic simple cubic.svg Lattice body centered cubic.svg Lattice face centered cubic.svg
(221) caesium chloride unit cell. The two colors of spheres represent the two types of atoms.
(216) zincblende unit cell
Cubic crystal system
Number Point group Short name Full name( Schoenflies Fibrifold
195 23 P23 P 2 3 T^1 2o
196 23 F23 F 2 3 T^2 1o
197 23 I23 I 2 3 T^3 4oo
198 23 P213 P 21 3 T^4 1o/4
199 23 I213 I 21 3 T^5 2o/4
200 2/m 3 Pm3 P 2/m 3 T_h^1 4
201 2/m 3 Pn3 P 2/n 3 T_h^2 4+o
202 2/m 3 Fm3 F 2/m 3 T_h^3 2
203 2/m 3 Fd3 F 2/d 3 T_h^4 2+o
204 2/m 3 Im3 I 2/m 3 T_h^5 8−o
205 2/m 3 Pa3 P 21/a 3 T_h^6 2/4
206 2/m 3 Ia3 I 21/a 3 T_h^7 4/4
207 432 P432 P 4 3 2 O^1 4−o
208 432 P4232 P 42 3 2 O^2 4+
209 432 F432 F 4 3 2 O^2 2−o
210 432 F4132 F 41 3 2 O^4 2+
211 432 I432 I 4 3 2 O^5 8+o
212 432 P4332 P 43 3 2 O^6 2+/4
213 432 P4132 P 41 3 2 O^7 2+/4
214 432 I4132 I 41 3 2 O^8 4+/4
215 43m P43m P 4 3 m T_d^1 2o:2
216 43m F43m F 4 3 m T_d^2 1o:2
217 43m I43m I 4 3 m T_d^3 4o:2
218 43m P43n P 4 3 n T_d^4 4o
219 43m F43c F 4 3 c T_d^5 2oo
220 43m I43d I 4 3 d T_d^6 4o/4
221 4/m 3 2/m Pm3m P 4/m 3 2/m O_h^1 4:2
222 4/m 3 2/m Pn3n P 4/n 3 2/n O_h^2 8oo
223 4/m 3 2/m Pm3n P 42/m 3 2/n O_h^3 8o
224 4/m 3 2/m Pn3m P 42/n 3 2/m O_h^4 4+:2
225 4/m 3 2/m Fm3m F 4/m 3 2/m O_h^5 2:2
226 4/m 3 2/m Fm3c F 4/m 3 2/c O_h^6 4−−
227 4/m 3 2/m Fd3m F 41/d 3 2/m O_h^7 2+:2
228 4/m 3 2/m Fd3c F 41/d 3 2/c O_h^8 4++
229 4/m 3 2/m Im3m I 4/m 3 2/m O_h^9 8o:2
230 4/m 3 2/m Ia3d I 41/a 3 2/d O_h^{10} 8o/4

External links[edit]