Finite Group Databasebeta

← C23×#3811 #11919166

Subgroups — C23×#3811 #11919166

Order 1152.

23929 conjugacy classes of subgroups; only the 4100 normal subgroups are shown.

normal subgroups only

This group has 23929 conjugacy classes of subgroups — too many to draw. Only its 4100 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C25×C3⋊D3C211yesmaximal, characteristic, complemented
2C24×D32C211yesmaximal, characteristic, complemented
3C23×C3×#67C212yesmaximal, complemented
4C22×#3811C2156yesmaximal, direct product factor
5C23×#626C212yesmaximal, complemented
6C23×#624C211yesmaximal, characteristic, complemented
7C23×D3×D4—32nomaximal
8C25×C32C2211yescharacteristic, complemented
9C24×C3⋊D3—216no—
10C24×C3⋊D3C2211yescharacteristic
11C24×C3⋊D3C22128yescomplemented
12C24×C3⋊D3C2211yescharacteristic, complemented
13C24×C3×D3C2212yescomplemented
14C23×D32—216no—
15C23×D32C22128yescomplemented
16C22×C3×#67C22156yescomplemented
17C23×C3×Dic3C2212yescomplemented
18C2×#3811C221448yesdirect product factor
19C22×#626C22156yescomplemented
20C22×#624C22128yescomplemented
21C25×D3—32no—
22C25×D3—32no—
23C25×D3—61no—
24C23×C3×D4—32no—
25C23×#67—32no—
26C23×#67D312yescomplemented
27C22×D3×D4—3112no—
28C23×D12—32no—
29C23×C4×D3—32no—
30C24×C32—28no—
31C24×C32C2311yescharacteristic
32C24×C32C23114yescomplemented
33C23×C3⋊D3—28no—
34C23×C3⋊D3—2224no—
35C23×C3⋊D3D4116yescomplemented
36C23×C3⋊D3C23114yes—
37C23×C3⋊D3C231112yescomplemented
38C23×C3⋊D3C23114yescomplemented
39C23×C3×D3—216no—
40C23×C3×D3C23128yescomplemented
41C22×D32—2224no—
42C22×D32C231112yescomplemented
43C2×C3×#67C231224yescomplemented
44C22×C3×Dic3C23128yescomplemented
45#3811C231512yesdirect product factor
46C2×#626C231224yescomplemented
47C2×#624C231112yescomplemented
48C24×D4—91noSylow
49C25×C3—21no—
50C25×C3—32no—
51C25×C3C2×D312yescomplemented
52C24×D3—32no—
53C24×D3—32no—
54C24×D3—356no—
55C24×D3—356no—
56C24×D3—61no—
57C24×D3—61no—
58C24×D3—628no—
59C24×D3—632no—
60C24×D3—632no—
61C24×D3—632no—
62C24×D3C2×D312yescomplemented
63C22×C3×D4—356no—
64C23×C3×C4—32no—
65C22×#67—356no—
66C22×#67C2×D3156yescomplemented
67C23×Dic3C2×D312yescomplemented
68C2×D3×D4—3896no—
69C22×D12—356no—
70C22×C4×D3—356no—
71C23×C32—256no—
72C23×C32C2×D418yescomplemented
73C23×C32C2417yes—
74C23×C32C24128yescomplemented
75C22×C3⋊D3—256no—
76C22×C3⋊D3—2448no—
77C22×C3⋊D3C2×D41112yescomplemented
78C22×C3⋊D3C24128yes—
79C22×C3⋊D3C24164yescomplemented
80C22×C3⋊D3C24128yescomplemented
81C22×C3×D3—2112no—
82C22×C3×D3C24156yescomplemented
83C2×D32—2448no—
84C2×D32C24164yescomplemented
85C3×#67C241128yescomplemented
86C2×C3×Dic3C24156yescomplemented
87#626C241128yescomplemented
88#624C24164yescomplemented
89C26—91no—
90C26—91no—
91C23×D4—32no—
92C23×D4—92no—
93C23×D4—956no—
94C24×C4—91no—
95C24×C3—216no—
96C24×C3—216no—
97C24×C3—21no—
98C24×C3—214no—
99C24×C3—328no—
100C24×C3—616no—
101C24×C3C22×D312yes—
102C24×C3C22×D3128yescomplemented
103C23×D3—216no—
104C23×D3—332no—
105C23×D3—328no—
106C23×D3—328no—
107C23×D3—3224no—
108C23×D3—3224no—
109C23×D3—616no—
110C23×D3—614no—
111C23×D3—614no—
112C23×D3—6112no—
113C23×D3—6448no—
114C23×D3—6448no—
115C23×D3—6448no—
116C23×D3—616no—
117C23×D3—616no—
118C23×D3C22×D3128yescomplemented
119C2×C3×D4—3224no—
120C22×C3×C4—328no—
121C2×#67—3224no—
122C2×#67C22×D31224yescomplemented
123C22×Dic3C22×D3128yescomplemented
124D3×D4—31024no—
125C2×D12—3224no—
126C2×C4×D3—3224no—
127C22×C32—256no—
128C22×C32C22×D4128yescomplemented
129C22×C32C2517yes—
130C22×C32C2518yescomplemented
131C2×C3⋊D3—256no—
132C2×C3⋊D3—2128no—
133C2×C3⋊D3C22×D41112yescomplemented
134C2×C3⋊D3C2518yes—
135C2×C3⋊D3C2518yescomplemented
136C2×C3×D3—2112no—
137C2×C3×D3C25116yescomplemented
138D32—2128no—
139C3×Dic3C25116yescomplemented
140C25—32no—
141C25—928no—
142C25—91no—
143C25—91no—
144C25—928no—
145C25—1816no—
146C25—1816no—
147C25D3211yescharacteristic, complemented
148C22×D4—356no—
149C22×D4—956no—
150C22×D4—9448no—
151C23×C4—32no—
152C23×C4—928no—
153C23×C3—2112no—
154C23×C3—2112no—
155C23×C3—27no—
156C23×C3—228no—
157C23×C3—28no—
158C23×C3—356no—
159C23×C3—6112no—
160C23×C3D3×D4116yescomplemented
161C23×C3C23×D3114yes—
162C23×C3C23×D3156yescomplemented
163C22×D3—2112no—
164C22×D3—3224no—
165C22×D3—356no—
166C22×D3—356no—
167C22×D3—3128no—
168C22×D3—3128no—
169C22×D3—6112no—
170C22×D3—628no—
171C22×D3—628no—
172C22×D3—664no—
173C22×D3—6896no—
174C22×D3—6896no—
175C22×D3—6896no—
176C22×D3—6112no—
177C22×D3—6112no—
178C22×D3C23×D3156yescomplemented
179C3×D4—3128no—
180C2×C3×C4—356no—
181#67—3128no—
182#67C23×D31128yescomplemented
183C2×Dic3C23×D3156yescomplemented
184D12—3128no—
185C4×D3—3128no—
186C2×C32—28no—
187C2×C32C23×D4114yescomplemented
188C2×C32C2611yescharacteristic
189C3⋊D3—28no—
190C3⋊D3C23×D4116yescomplemented
191C3×D3—216no—
192C24—28no—
193C24—328no—
194C24—616no—
195C24—9112no—
196C24—914no—
197C24—916no—
198C24—914no—
199C24—9112no—
200C24—18224no—
201C24—18224no—
202C24—188no—
203C24C2×D3211yescharacteristic
204C24C2×D32114yescomplemented
205C2×D4—3224no—
206C2×D4—9224no—
207C2×D4—9512no—
208C22×C4—328no—
209C22×C4—9112no—
210C22×C3—2112no—
211C22×C3—2112no—
212C22×C3—228no—
213C22×C3—27no—
214C22×C3—28no—
215C22×C3—316no—
216C22×C3—6112no—
217C22×C3C2×D3×D4156yescomplemented
218C22×C3C24×D3114yes—
219C22×C3C24×D3116yescomplemented
220C2×D3—2112no—
221C2×D3—3224no—
222C2×D3—316no—
223C2×D3—316no—
224C2×D3—6112no—
225C2×D3—68no—
226C2×D3—68no—
227C2×D3—6256no—
228C2×D3—6256no—
229C2×D3—6256no—
230C2×D3—6112no—
231C2×D3—6112no—
232C2×D3C24×D3116yescomplemented
233C3×C4—316no—
234Dic3C24×D3116yescomplemented
235C32C24×D411yescharacteristic, complemented, Sylow
236C23—256no—
237C23—356no—
238C23—6112no—
239C23—964no—
240C23—9112no—
241C23—928no—
242C23—928no—
243C23—964no—
244C23—18448no—
245C23—18448no—
246C23—1856no—
247C23#381118yesdirect product factor
248C23C22×D3217yes—
249C23C22×D32128yescomplemented
250D4—3128no—
251D4—9128no—
252C2×C4—356no—
253C2×C4—964no—
254C2×C3—216no—
255C2×C3—216no—
256C2×C3—214no—
257C2×C3—21no—
258C2×C3—616no—
259C2×C3C22×D3×D4128yescomplemented
260C2×C3C25×D312yes—
261D3—216no—
262D3—332no—
263D3—616no—
264D3—616no—
265D3—616no—
266C22—256no—
267C22—316no—
268C22—6112no—
269C22—9112no—
270C22—98no—
271C22—98no—
272C22—18128no—
273C22—18128no—
274C22—1856no—
275C22C2×#3811128yesdirect product factor
276C22C23×D3217yes—
277C22C23×D3218yescomplemented
278C4—316no—
279C3—21no—
280C3C23×D3×D412yescomplemented
281C2—28no—
282C2—616no—
283C2—916no—
284C2—188no—
285C2C22×#3811114yesdirect product factor
286C2C24×D3211yescharacteristic
subgroupG/H
centerC24C2×D32
commutator, cosocleC2×C32C26
FrattiniC2C24×D32
FittingC25×C32C22
socleC24×C32C23
hypercenterC25D32
hypocenterC32C24×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C24 ◁ C25

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C25×C3⋊D3C2semidirect
2. C24×D32C2semidirect
3. C23×C3×#67C2semidirect
4. C22×#3811C2direct (N × H)
5. C23×#626C2semidirect
6. C23×#624C2semidirect
8. C25×C32C22semidirect
11. C24×C3⋊D3C22semidirect
12. C24×C3⋊D3C22semidirect
13. C24×C3×D3C22semidirect
15. C23×D32C22semidirect
16. C22×C3×#67C22semidirect
17. C23×C3×Dic3C22semidirect
18. C2×#3811C22direct (N × H)
19. C22×#626C22semidirect
20. C22×#624C22semidirect
26. C23×#67D3semidirect
32. C24×C32C23semidirect
35. C23×C3⋊D3D4semidirect
37. C23×C3⋊D3C23semidirect
38. C23×C3⋊D3C23semidirect
40. C23×C3×D3C23semidirect
42. C22×D32C23semidirect
43. C2×C3×#67C23semidirect
44. C22×C3×Dic3C23semidirect
45. #3811C23direct (N × H)
46. C2×#626C23semidirect
47. C2×#624C23semidirect
51. C25×C3C2×D3semidirect
62. C24×D3C2×D3semidirect
66. C22×#67C2×D3semidirect
67. C23×Dic3C2×D3semidirect
72. C23×C32C2×D4semidirect
74. C23×C32C24semidirect
77. C22×C3⋊D3C2×D4semidirect
79. C22×C3⋊D3C24semidirect
80. C22×C3⋊D3C24semidirect
82. C22×C3×D3C24semidirect
84. C2×D32C24semidirect
85. C3×#67C24semidirect
86. C2×C3×Dic3C24semidirect
87. #626C24semidirect
88. #624C24semidirect
102. C24×C3C22×D3semidirect
118. C23×D3C22×D3semidirect
122. C2×#67C22×D3semidirect
123. C22×Dic3C22×D3semidirect
128. C22×C32C22×D4semidirect
130. C22×C32C25semidirect
133. C2×C3⋊D3C22×D4semidirect
135. C2×C3⋊D3C25semidirect
137. C2×C3×D3C25semidirect
139. C3×Dic3C25semidirect
147. C25D32semidirect
160. C23×C3D3×D4semidirect
162. C23×C3C23×D3semidirect
178. C22×D3C23×D3semidirect
182. #67C23×D3semidirect
183. C2×Dic3C23×D3semidirect
187. C2×C32C23×D4semidirect
190. C3⋊D3C23×D4semidirect
204. C24C2×D32semidirect
217. C22×C3C2×D3×D4semidirect
219. C22×C3C24×D3semidirect
232. C2×D3C24×D3semidirect
234. Dic3C24×D3semidirect
235. C32C24×D4semidirect
247. C23#3811direct (N × H)
249. C23C22×D32semidirect
259. C2×C3C22×D3×D4semidirect
275. C22C2×#3811direct (N × H)
277. C22C23×D32semidirect
280. C3C23×D3×D4semidirect
285. C2C22×#3811direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C25×C3⋊D32
C24×D322
C23×C3×#672
C22×#38112
C23×#6262
C23×#6242
C23×D3×D43