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 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 is noted 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 axis 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. It is related to the order in which Shoenflies derived 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 Short name Full name Schoenflies Fedorov Shubnikov
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 Short name Full name(s) Schoenflies Fedorov Shubnikov
3 2 P2 P 1 2 1 P 1 1 2 3s
4 2 P21 P 1 21 1 P 1 1 21 1a
5 2 C2 C 1 2 1 B 1 1 2 4s
6 m Pm P 1 m 1 P 1 1 m 5s
7 m Pc P 1 c 1 P 1 1 b 1h
8 m Cm C 1 m 1 B 1 1 m 6s
9 m 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 2/m P21/m P 1 21/m 1 P 1 1 21/m 2a
12 2/m C2/m C 1 2/m 1 B 1 1 2/m 8s
13 2/m P2/c P 1 2/c 1 P 1 1 2/b 3h
14 2/m P21/c P 1 21/c 1 P 1 1 21/b 3a
15 2/m C2/c C 1 2/c 1 B 1 1 2/b 4h

List of Orthorhombic[edit]

Orthorhombic crystal system
Number Point group Short name Full name Schoenflies Fedorov Shubnikov
16 222 P222 P 2 2 2 9s
17 222 P2221 P 2 2 21 4a
18 222 P21212 P 21 21 2 7a Circled colon.png
19 222 P212121 P 21 21 21 8a Circled colon.png
20 222 C2221 C 2 2 21 5a
21 222 C222 C 2 2 2 10s
22 222 F222 F 2 2 2 12s
23 222 I222 I 2 2 2 11s
24 222 I212121 I 21 21 21 6a
25 mm2 Pmm2 P m m 2 13s
26 mm2 Pmc21 P m c 21 9a
27 mm2 Pcc2 P c c 2 5h
28 mm2 Pma2 P m a 2 6h
29 mm2 Pca21 P c a 21 11a
30 mm2 Pnc2 P n c 2 7h
31 mm2 Pmn21 P m n 21 10a
32 mm2 Pba2 P b a 2 9h
33 mm2 Pna21 P n a 21 12a
34 mm2 Pnn2 P n n 2 8h
35 mm2 Cmm2 C m m 2 14s
36 mm2 Cmc21 C m c 21 13a
37 mm2 Ccc2 C c c 2 10h
38 mm2 Amm2 A m m 2 15s
39 mm2 Aem2 A b m 2 11h
40 mm2 Ama2 A m a 2 12h
41 mm2 Aea2 A b a 2 13h
42 mm2 Fmm2 F m m 2 17s
43 mm2 Fdd2 F dd2 16h
44 mm2 Imm2 I m m 2 16s
45 mm2 Iba2 I b a 2 15h
46 mm2 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 Short name Full name Schoenflies Fedorov Shubnikov
75 4 P4 P 4 22s
76 4 P41 P 41 30a
77 4 P42 P 42 33a
78 4 P43 P 43 31a
79 4 I4 I 4 23s
80 4 I41 I 41 32a
81 4 P4 P 4 26s
82 4 I4 I 4 27s
83 4/m P4/m P 4/m 28s
84 4/m P42/m P 42/m 41a
85 4/m P4/n P 4/n 29h
86 4/m P42/n P 42/n 42a
87 4/m I4/m I 4/m 29s
88 4/m I41/a I 41/a 40a
89 422 P422 P 4 2 2 30s
90 422 P4212 P4212 43a Circled colon.png
91 422 P4122 P 41 2 2 44a
92 422 P41212 P 41 21 2 48a Circled colon.png
93 422 P4222 P 42 2 2 47a
94 422 P42212 P 42 21 2 50a Circled colon.png
95 422 P4322 P 43 2 2 45a
96 422 P43212 P 43 21 2 49a Circled colon.png
97 422 I422 I 4 2 2 31s
98 422 I4122 I 41 2 2 46a
99 4mm P4mm P 4 m m 24s
100 4mm P4bm P 4 b m 26h
101 4mm P42cm P 42 c m 37a
102 4mm P42nm P 42 n m 38a
103 4mm P4cc P 4 c c 25h
104 4mm P4nc P 4 n c 27h
105 4mm P42mc P 42 m c 36a
106 4mm P42bc P 42 b c 39a
107 4mm I4mm I 4 m m 25s
108 4mm I4cm I 4 c m 28h
109 4mm I41md I 41 m d 34a
110 4mm I41cd I 41 c d 35a
111 42m P42m P 4 2 m 32s
112 42m P42c P 4 2 c 30h Circled colon.png
113 42m P421m P 4 21 m 52a
114 42m P421c P 4 21 c 53a
115 42m P4m2 P 4 m 2 33s
116 42m P4c2 P 4 c 2 31h
117 42m P4b2 P 4 b 2 32h
118 42m P4n2 P 4 n 2 33h
119 42m I4m2 I 4 m 2 35s
120 42m I4c2 I 4 c 2 34h
121 42m I42m I 4 2 m 34s
122 42m I42d I 4 2 d 51a
123 4/m 2/m 2/m P4/mmm P 4/m 2/m 2/m 36s
124 4/m 2/m 2/m P4/mcc P 4/m 2/c 2/c 35h
125 4/m 2/m 2/m P4/nbm P 4/n 2/b 2/m 36h
126 4/m 2/m 2/m P4/nnc P 4/n 2/n 2/c 37h
127 4/m 2/m 2/m P4/mbm P 4/m 21/b 2/m 54a
128 4/m 2/m 2/m P4/mnc P 4/m 21/n 2/c 56a
129 4/m 2/m 2/m P4/nmm P 4/n 21/m 2/m 55a
130 4/m 2/m 2/m P4/ncc P 4/n 21/c 2/c 57a
131 4/m 2/m 2/m P42/mmc P 42/m 2/m 2/c 60a
132 4/m 2/m 2/m P42/mcm P 42/m 2/c 2/m 61a
133 4/m 2/m 2/m P42/nbc P 42/n 2/b 2/c 63a
134 4/m 2/m 2/m P42/nnm P 42/n 2/n 2/m 62a
135 4/m 2/m 2/m P42/mbc P 42/m 21/b 2/c 66a
136 4/m 2/m 2/m P42/mnm P 42/m 21/n 2/m 65a
137 4/m 2/m 2/m P42/nmc P 42/n 21/m 2/c 67a
138 4/m 2/m 2/m P42/ncm P 42/n 21/c 2/m 65a
139 4/m 2/m 2/m I4/mmm I 4/m 2/m 2/m 37s
140 4/m 2/m 2/m I4/mcm I 4/m 2/c 2/m 38h
141 4/m 2/m 2/m I41/amd I 41/a 2/m 2/d 59a
142 4/m 2/m 2/m 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 Short name Full name Schoenflies Fedorov Shubnikov
143 3 P3 P 3 38s
144 3 P31 P 31 68a
145 3 P32 P 32 69a
146 3 R3 R 3 39s
147 3 P3 P 3 51s
148 3 R3 R 3 52s
149 32 P312 P 3 1 2 45s
150 32 P321 P 3 2 1 44s
151 32 P3112 P 31 1 2 72a
152 32 P3121 P 31 2 1 70a
153 32 P3212 P 32 1 2 73a
154 32 P3221 P 32 2 1 71a
155 32 R32 R 3 2 46s
156 3m P3m1 P 3 m 1 40s
157 3m P31m P 3 1 m 41s
158 3m P3c1 P 3 c 1 39h
159 3m P31c P 3 1 c 40h
160 3m R3m R 3 m 42s
161 3m R3c R 3 c 41h
162 3 2/m P31m P 3 1 2/m 56s
163 3 2/m P31c P 3 1 2/c 46h
164 3 2/m P3m1 P 3 2/m 1 55s
165 3 2/m P3c1 P 3 2/c 1 45h
166 3 2/m R3m R 3 2/m 57s
167 3 2/m R3c R 3 2/c 47h

List of Hexagonal[edit]

Hexagonal lattice cell
(P)
Hexagonal crystal system
Number Point group Short name Full name Schoenflies Fedorov Shubnikov
168 6 P6 P 6 49s
169 6 P61 P 61 74a
170 6 P65 P 65 75a
171 6 P62 P 62 76a
172 6 P64 P 64 77a
173 6 P63 P 63 78a
174 6 P6 P 6 43s
175 6/m P6/m P 6/m 53s
176 6/m P63/m P 63/m 81a
177 622 P622 P 6 2 2 54s
178 622 P6122 P 61 2 2 82a
179 622 P6522 P 65 2 2 83a
180 622 P6222 P 62 2 2 84a
181 622 P6422 P 64 2 2 85a
182 622 P6322 P 63 2 2 86a
183 6mm P6mm P 6 m m 50s
184 6mm P6cc P 6 c c 44h
185 6mm P63cm P 63 c m 80a
186 6mm P63mc P 63 m c 79a
187 6m2 P6m2 P 6 m 2 48s
188 6m2 P6c2 P 6 c 2 43h
189 6m2 P62m P 6 2 m 47s
190 6m2 P62c P 6 2 c 42h
191 6/m 2/m 2/m P6/mmm P 6/m 2/m 2/m 58s
192 6/m 2/m 2/m P6/mcc P 6/m 2/c 2/c 48h
193 6/m 2/m 2/m P63/mcm P 63/m 2/c 2/m 87a
194 6/m 2/m 2/m 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.
(216) Sphalerite
Cubic crystal system
Number Point group Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
195 23 P23 P 2 3 59s 2o
196 23 F23 F 2 3 61s 1o
197 23 I23 I 2 3 60s 4oo
198 23 P213 P 21 3 89a 1o/4
199 23 I213 I 21 3 90a 2o/4
200 2/m 3 Pm3 P 2/m 3 62s 4
201 2/m 3 Pn3 P 2/n 3 49h 4+o
202 2/m 3 Fm3 F 2/m 3 64s 2
203 2/m 3 Fd3 F 2/d 3 50h 2+o
204 2/m 3 Im3 I 2/m 3 63s 8−o
205 2/m 3 Pa3 P 21/a 3 91a 2/4
206 2/m 3 Ia3 I 21/a 3 92a 4/4
207 432 P432 P 4 3 2 68s 4−o
208 432 P4232 P 42 3 2 98a 4+
209 432 F432 F 4 3 2 70s 2−o
210 432 F4132 F 41 3 2 97a 2+
211 432 I432 I 4 3 2 69s 8+o
212 432 P4332 P 43 3 2 94a 2+/4
213 432 P4132 P 41 3 2 95a 2+/4
214 432 I4132 I 41 3 2 96a 4+/4
215 43m P43m P 4 3 m 65s 2o:2
216 43m F43m F 4 3 m 67s 1o:2
217 43m I43m I 4 3 m 66s 4o:2
218 43m P43n P 4 3 n 51h 4o
219 43m F43c F 4 3 c 52h 2oo
220 43m I43d I 4 3 d 93a 4o/4
221 4/m 3 2/m Pm3m P 4/m 3 2/m 71s 4:2
222 4/m 3 2/m Pn3n P 4/n 3 2/n 53h 8oo
223 4/m 3 2/m Pm3n P 42/m 3 2/n 102a 8o
224 4/m 3 2/m Pn3m P 42/n 3 2/m 103a 4+:2
225 4/m 3 2/m Fm3m F 4/m 3 2/m 73s 2:2
226 4/m 3 2/m Fm3c F 4/m 3 2/c 54h 4−−
227 4/m 3 2/m Fd3m F 41/d 3 2/m 100a 2+:2
228 4/m 3 2/m Fd3c F 41/d 3 2/c 101a 4++
229 4/m 3 2/m Im3m I 4/m 3 2/m 72s 8o:2
230 4/m 3 2/m Ia3d I 41/a 3 2/d 99a 8o/4

External links[edit]