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 group 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 as 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.

  • , , or glide translation along half the lattice vector of this face
  • glide translation along with half a face diagonal
  • glide planes with translation along a quarter of a face diagonal.
  • 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 denoted by a number, n, where the angle of rotation is . 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 axes are: 21, 31, 32, 41, 42, 43, 61, 62, 63, 64, and 65.

In Schoenflies notation, the symbol of a space group is represented by the symbol of corresponding point group with additional superscript. The superscript doesn't give any additional information about symmetry elements of the space group, but is instead related to the order in which Schoenflies derived the space groups.

In Fedorov symbol, the type of space group is denoted as s (symmorphic ), h (hemisymmorphic), or a (asymmorphic). The number is related to the order in which Fedorov derived space groups. There are 73 symmorphic, 54 hemisymmorphic, and 103 asymmorphic space groups. Symmorphic space groups can be obtained as combination of Bravais lattices with corresponding point group. These groups contain the same symmetry elements as the corresponding point groups. Hemisymmorphic space groups contain only axial combination of symmetry elements from the corresponding point groups. All the other space groups are asymmorphic. Example for point group 4/mmm (): the symmorphic space groups are P4/mmm (, 36s) and I4/mmm (, 37s); hemisymmorphic space groups should contain axial combination 422, these are P4/mcc (, 35h), P4/nbm (, 36h), P4/nnc (, 37h), and I4/mcm (, 38h).

List of Triclinic[edit]

Triclinic Bravais lattice
Triclinic.svg
Triclinic crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
1 1 P1 P 1 1s
2 1 P1 P 1 2s

List of Monoclinic[edit]

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

List of Orthorhombic[edit]

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

List of Tetragonal[edit]

Tetragonal crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
75 4 P4 P 4 22s
76 P41 P 41 30a
77 P42 P 42 33a
78 P43 P 43 31a
79 I4 I 4 23s
80 I41 I 41 32a
81 4 P4 P 4 26s
82 I4 I 4 27s
83 4/m P4/m P 4/m 28s
84 P42/m P 42/m 41a
85 P4/n P 4/n 29h
86 P42/n P 42/n 42a
87 I4/m I 4/m 29s
88 I41/a I 41/a 40a
89 422 P422 P 4 2 2 30s
90 P4212 P4212 43a Circled colon.png
91 P4122 P 41 2 2 44a
92 P41212 P 41 21 2 48a Circled colon.png
93 P4222 P 42 2 2 47a
94 P42212 P 42 21 2 50a Circled colon.png
95 P4322 P 43 2 2 45a
96 P43212 P 43 21 2 49a Circled colon.png
97 I422 I 4 2 2 31s
98 I4122 I 41 2 2 46a
99 4mm P4mm P 4 m m 24s
100 P4bm P 4 b m 26h
101 P42cm P 42 c m 37a
102 P42nm P 42 n m 38a
103 P4cc P 4 c c 25h
104 P4nc P 4 n c 27h
105 P42mc P 42 m c 36a
106 P42bc P 42 b c 39a
107 I4mm I 4 m m 25s
108 I4cm I 4 c m 28h
109 I41md I 41 m d 34a
110 I41cd I 41 c d 35a
111 42m P42m P 4 2 m 32s
112 P42c P 4 2 c 30h Circled colon.png
113 P421m P 4 21 m 52a
114 P421c P 4 21 c 53a
115 P4m2 P 4 m 2 33s
116 P4c2 P 4 c 2 31h
117 P4b2 P 4 b 2 32h
118 P4n2 P 4 n 2 33h
119 I4m2 I 4 m 2 35s
120 I4c2 I 4 c 2 34h
121 I42m I 4 2 m 34s
122 I42d I 4 2 d 51a
123 4/m 2/m 2/m P4/mmm P 4/m 2/m 2/m 36s
124 P4/mcc P 4/m 2/c 2/c 35h
125 P4/nbm P 4/n 2/b 2/m 36h
126 P4/nnc P 4/n 2/n 2/c 37h
127 P4/mbm P 4/m 21/b 2/m 54a
128 P4/mnc P 4/m 21/n 2/c 56a
129 P4/nmm P 4/n 21/m 2/m 55a
130 P4/ncc P 4/n 21/c 2/c 57a
131 P42/mmc P 42/m 2/m 2/c 60a
132 P42/mcm P 42/m 2/c 2/m 61a
133 P42/nbc P 42/n 2/b 2/c 63a
134 P42/nnm P 42/n 2/n 2/m 62a
135 P42/mbc P 42/m 21/b 2/c 66a
136 P42/mnm P 42/m 21/n 2/m 65a
137 P42/nmc P 42/n 21/m 2/c 67a
138 P42/ncm P 42/n 21/c 2/m 65a
139 I4/mmm I 4/m 2/m 2/m 37s
140 I4/mcm I 4/m 2/c 2/m 38h
141 I41/amd I 41/a 2/m 2/d 59a
142 I41/acd I 41/a 2/c 2/d 58a

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 Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
143 3 P3 P 3 38s
144 P31 P 31 68a
145 P32 P 32 69a
146 R3 R 3 39s
147 3 P3 P 3 51s
148 R3 R 3 52s
149 32 P312 P 3 1 2 45s
150 P321 P 3 2 1 44s
151 P3112 P 31 1 2 72a
152 P3121 P 31 2 1 70a
153 P3212 P 32 1 2 73a
154 P3221 P 32 2 1 71a
155 R32 R 3 2 46s
156 3m P3m1 P 3 m 1 40s
157 P31m P 3 1 m 41s
158 P3c1 P 3 c 1 39h
159 P31c P 3 1 c 40h
160 R3m R 3 m 42s
161 R3c R 3 c 41h
162 3 2/m P31m P 3 1 2/m 56s
163 P31c P 3 1 2/c 46h
164 P3m1 P 3 2/m 1 55s
165 P3c1 P 3 2/c 1 45h
166 R3m R 3 2/m 57s
167 R3c R 3 2/c 47h

List of Hexagonal[edit]

Hexagonal lattice cell
(P)
Hexagonal crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
168 6 P6 P 6 49s
169 P61 P 61 74a
170 P65 P 65 75a
171 P62 P 62 76a
172 P64 P 64 77a
173 P63 P 63 78a
174 6 P6 P 6 43s
175 6/m P6/m P 6/m 53s
176 P63/m P 63/m 81a
177 622 P622 P 6 2 2 54s
178 P6122 P 61 2 2 82a
179 P6522 P 65 2 2 83a
180 P6222 P 62 2 2 84a
181 P6422 P 64 2 2 85a
182 P6322 P 63 2 2 86a
183 6mm P6mm P 6 m m 50s
184 P6cc P 6 c c 44h
185 P63cm P 63 c m 80a
186 P63mc P 63 m c 79a
187 6m2 P6m2 P 6 m 2 48s
188 P6c2 P 6 c 2 43h
189 P62m P 6 2 m 47s
190 P62c P 6 2 c 42h
191 6/m 2/m 2/m P6/mmm P 6/m 2/m 2/m 58s
192 P6/mcc P 6/m 2/c 2/c 48h
193 P63/mcm P 63/m 2/c 2/m 87a
194 P63/mmc P 63/m 2/m 2/c 88a

List of Cubic[edit]

Cubic Bravais lattice
Simple
(P)
Body centered
(I)
Face centered
(F)
Lattic simple cubic.svg Lattice body centered cubic.svg Lattice face centered cubic.svg
(221) Caesium chloride. Different colors for the two atom types.
Cubic crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Conway Fibrifold (preserving ) Fibrifold (preserving , , )
195 23 P23 P 2 3 59s
196 F23 F 2 3 61s
197 I23 I 2 3 60s
198 P213 P 21 3 89a
199 I213 I 21 3 90a
200 2/m 3 Pm3 P 2/m 3 62s
201 Pn3 P 2/n 3 49h
202 Fm3 F 2/m 3 64s
203 Fd3 F 2/d 3 50h
204 Im3 I 2/m 3 63s
205 Pa3 P 21/a 3 91a
206 Ia3 I 21/a 3 92a
207 432 P432 P 4 3 2 68s
208 P4232 P 42 3 2 98a
209 F432 F 4 3 2 70s
210 F4132 F 41 3 2 97a
211 I432 I 4 3 2 69s
212 P4332 P 43 3 2 94a
213 P4132 P 41 3 2