Finite Group Databasebeta

← C23×#3804 #11919202

Subgroups — C23×#3804 #11919202

Order 1152.

13721 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#638C211yesmaximal, characteristic, complemented
2C23×C3×#67C211yesmaximal, characteristic, complemented
3C24×C3×Dic3C211yesmaximal, characteristic, complemented
4C23×#627C211yesmaximal, characteristic, complemented
5C23×#626C211yesmaximal, characteristic, complemented
6C23×#624C211yesmaximal, characteristic, complemented
7C22×#3804C2156yesmaximal, direct product factor
8C23×D3×Dic3C211yesmaximal, characteristic, complemented
9C23×#237—31nomaximal
10C23×#235—31nomaximal
11C25×C32C2211yescharacteristic
12C24×C3⋊D3C2211yescharacteristic
13C24×C3×D3C2211yescharacteristic
14C22×#638C22128yescomplemented
15C23×C3⋊Dic3C2211yescharacteristic
16C22×C3×#67C22128yescomplemented
17C23×C3×Dic3C2211yescharacteristic
18C23×C3×Dic3C2211yescharacteristic
19C23×C3×Dic3C2211yescharacteristic
20C23×C3×Dic3C22128yescomplemented
21C22×#627C22128yescomplemented
22C22×#626C22128yescomplemented
23C22×#624C22128yescomplemented
24C2×#3804C221448yesdirect product factor
25C22×D3×Dic3C22128yescomplemented
26C23×C3×D4—31no—
27C24×C3×C4—31no—
28C23×#67—31no—
29C23×#67—31no—
30C23×#67—31no—
31C23×#67—61no—
32C23×#67D311yescharacteristic, complemented
33C24×Dic3—31no—
34C24×Dic3D311yescharacteristic, complemented
35C22×#237—356no—
36C22×#235—356no—
37C23×D12—31no—
38C23×C4×D3—31no—
39C23×C4×D3—31no—
40C23×C4×D3—31no—
41C23×Dic6—31no—
42C23×Dic6—31no—
43C24×C32—28no—
44C24×C32C23114yes—
45C24×C32C2311yescharacteristic
46C23×C3⋊D3—28no—
47C23×C3⋊D3C23114yes—
48C23×C3×D3—28no—
49C23×C3×D3C23114yes—
50C2×#638C231112yescomplemented
51C22×C3⋊Dic3C23114yes—
52C2×C3×#67C231112yescomplemented
53C22×C3×Dic3C23114yes—
54C22×C3×Dic3C23114yes—
55C22×C3×Dic3C23114yes—
56C22×C3×Dic3C231112yescomplemented
57C2×#627C231112yescomplemented
58C2×#626C231112yescomplemented
59C2×#624C231112yescomplemented
60#3804C231512yesdirect product factor
61C2×D3×Dic3C231112yescomplemented
62C23×#41—91noSylow
63C25×C3—21no—
64C25×C3—31no—
65C25×C3C2×D311yescharacteristic
66C25×C3C2×D311yescharacteristic
67C24×D3—31no—
68C24×D3—31no—
69C24×D3—61no—
70C24×D3C2×D311yescharacteristic
71C22×C3×D4—328no—
72C23×C3×C4—31no—
73C23×C3×C4—31no—
74C23×C3×C4—31no—
75C23×C3×C4—328no—
76C22×#67—328no—
77C22×#67—328no—
78C22×#67—328no—
79C22×#67—628no—
80C22×#67C2×D3128yescomplemented
81C23×Dic3—31no—
82C23×Dic3—31no—
83C23×Dic3—328no—
84C23×Dic3—61no—
85C23×Dic3C2×D311yescharacteristic
86C23×Dic3C2×D311yescharacteristic
87C23×Dic3C2×D311yescharacteristic
88C23×Dic3C2×D3128yescomplemented
89C2×#237—3448no—
90C2×#235—3448no—
91C22×D12—328no—
92C22×C4×D3—328no—
93C22×C4×D3—328no—
94C22×C4×D3—328no—
95C22×Dic6—328no—
96C22×Dic6—328no—
97C23×C32—256no—
98C23×C32#4118yescomplemented
99C23×C32C24128yes—
100C23×C32C2417yes—
101C22×C3⋊D3—256no—
102C22×C3⋊D3C24128yes—
103C22×C3×D3—256no—
104C22×C3×D3C24128yes—
105#638C24164yescomplemented
106C2×C3⋊Dic3C24128yes—
107C3×#67C24164yescomplemented
108C2×C3×Dic3C24128yes—
109C2×C3×Dic3C24128yes—
110C2×C3×Dic3C24128yes—
111C2×C3×Dic3C24164yescomplemented
112#627C24164yescomplemented
113#626C24164yescomplemented
114#624C24164yescomplemented
115D3×Dic3C24164yescomplemented
116C22×#41—956no—
117C23×Q8—91no—
118C23×D4—31no—
119C23×D4—91no—
120C23×D4—91no—
121C24×C4—31no—
122C24×C4—91no—
123C24×C4—91no—
124C24×C3—28no—
125C24×C3—28no—
126C24×C3—214no—
127C24×C3—21no—
128C24×C3—314no—
129C24×C3—48no—
130C24×C3—68no—
131C24×C3C22×D3114yes—
132C24×C3C22×D3114yes—
133C24×C3C22×D311yescharacteristic
134C24×C3C22×D311yescharacteristic
135C23×D3—28no—
136C23×D3—314no—
137C23×D3—314no—
138C23×D3—614no—
139C23×D3—68no—
140C23×D3—68no—
141C23×D3—128no—
142C23×D3C22×D3114yes—
143C2×C3×D4—3112no—
144C22×C3×C4—314no—
145C22×C3×C4—314no—
146C22×C3×C4—314no—
147C22×C3×C4—3112no—
148C2×#67—3112no—
149C2×#67—3112no—
150C2×#67—3112no—
151C2×#67—6112no—
152C2×#67C22×D31112yescomplemented
153C22×Dic3—314no—
154C22×Dic3—314no—
155C22×Dic3—3112no—
156C22×Dic3—614no—
157C22×Dic3C22×D3114yes—
158C22×Dic3C22×D3114yes—
159C22×Dic3C22×D3114yes—
160C22×Dic3C22×D31112yescomplemented
161#237—3512no—
162#235—3512no—
163C2×D12—3112no—
164C2×C4×D3—3112no—
165C2×C4×D3—3112no—
166C2×C4×D3—3112no—
167C2×Dic6—3112no—
168C2×Dic6—3112no—
169C22×C32—256no—
170C22×C32C2×#41128yescomplemented
171C22×C32C2518yes—
172C22×C32C2517yes—
173C2×C3⋊D3—256no—
174C2×C3⋊D3C2518yes—
175C2×C3×D3—256no—
176C2×C3×D3C2518yes—
177C3⋊Dic3C2518yes—
178C3×Dic3C2518yes—
179C3×Dic3C2518yes—
180C3×Dic3C2518yes—
181C25—31no—
182C25—91no—
183C25D3211yescharacteristic
184C2×#41—9448no—
185C22×Q8—928no—
186C22×D4—328no—
187C22×D4—928no—
188C22×D4—928no—
189C23×C4—328no—
190C23×C4—31no—
191C23×C4—31no—
192C23×C4—31no—
193C23×C4—928no—
194C23×C4—928no—
195C23×C4—91no—
196C23×C3—256no—
197C23×C3—256no—
198C23×C3—27no—
199C23×C3—228no—
200C23×C3—28no—
201C23×C3—328no—
202C23×C3—456no—
203C23×C3—656no—
204C23×C3#23518yescomplemented
205C23×C3#23718yescomplemented
206C23×C3C23×D3128yes—
207C23×C3C23×D3128yes—
208C23×C3C23×D317yes—
209C23×C3C23×D317yes—
210C22×D3—256no—
211C22×D3—328no—
212C22×D3—328no—
213C22×D3—628no—
214C22×D3—656no—
215C22×D3—656no—
216C22×D3—1256no—
217C22×D3C23×D3128yes—
218C3×D4—364no—
219C2×C3×C4—328no—
220C2×C3×C4—328no—
221C2×C3×C4—328no—
222C2×C3×C4—364no—
223#67—364no—
224#67—364no—
225#67—364no—
226#67—664no—
227#67C23×D3164yescomplemented
228C2×Dic3—328no—
229C2×Dic3—328no—
230C2×Dic3—364no—
231C2×Dic3—628no—
232C2×Dic3C23×D3128yes—
233C2×Dic3C23×D3128yes—
234C2×Dic3C23×D3128yes—
235C2×Dic3C23×D3164yescomplemented
236D12—364no—
237C4×D3—364no—
238C4×D3—364no—
239C4×D3—364no—
240Dic6—364no—
241Dic6—364no—
242C2×C32—28no—
243C2×C32C22×#41114yescomplemented
244C2×C32C2611yescharacteristic
245C3⋊D3—28no—
246C3×D3—28no—
247C24—28no—
248C24—314no—
249C24—68no—
250C24—914no—
251C24—188no—
252C24C2×D32114yes—
253C24C2×D3211yescharacteristic
254#41—9512no—
255C2×Q8—9112no—
256C2×D4—3112no—
257C2×D4—9112no—
258C2×D4—9112no—
259C22×C4—314no—
260C22×C4—314no—
261C22×C4—3112no—
262C22×C4—314no—
263C22×C4—9112no—
264C22×C4—9112no—
265C22×C4—914no—
266C22×C3—256no—
267C22×C3—256no—
268C22×C3—228no—
269C22×C3—27no—
270C22×C3—28no—
271C22×C3—38no—
272C22×C3—456no—
273C22×C3—656no—
274C22×C3C2×#235128yescomplemented
275C22×C3C2×#237128yescomplemented
276C22×C3C24×D318yes—
277C22×C3C24×D318yes—
278C22×C3C24×D317yes—
279C22×C3C24×D317yes—
280C2×D3—256no—
281C2×D3—38no—
282C2×D3—38no—
283C2×D3—68no—
284C2×D3—656no—
285C2×D3—656no—
286C2×D3—1256no—
287C2×D3C24×D318yes—
288C3×C4—38no—
289C3×C4—38no—
290C3×C4—38no—
291Dic3—38no—
292Dic3—38no—
293Dic3—68no—
294Dic3C24×D318yes—
295Dic3C24×D318yes—
296Dic3C24×D318yes—
297C32C23×#4111yescharacteristic, complemented, Sylow
298C23—256no—
299C23—328no—
300C23—656no—
301C23—928no—
302C23—1856no—
303C23#380418yesdirect product factor
304C23C22×D32128yes—
305C23C22×D3217yes—
306Q8—964no—
307D4—364no—
308D4—964no—
309D4—964no—
310C2×C4—328no—
311C2×C4—328no—
312C2×C4—364no—
313C2×C4—328no—
314C2×C4—964no—
315C2×C4—964no—
316C2×C4—928no—
317C2×C3—28no—
318C2×C3—28no—
319C2×C3—214no—
320C2×C3—21no—
321C2×C3—48no—
322C2×C3—68no—
323C2×C3C22×#235114yescomplemented
324C2×C3C22×#237114yescomplemented
325C2×C3C25×D311yescharacteristic
326C2×C3C25×D311yescharacteristic
327D3—28no—
328D3—68no—
329D3—68no—
330D3—128no—
331C22—256no—
332C22—38no—
333C22—656no—
334C22—98no—
335C22—1856no—
336C22C2×#3804128yesdirect product factor
337C22C23×D3218yes—
338C22C23×D3217yes—
339C4—38no—
340C4—38no—
341C4—38no—
342C4—98no—
343C3—21no—
344C3C23×#23511yescharacteristic, complemented
345C3C23×#23711yescharacteristic, complemented
346C2—28no—
347C2—68no—
348C2—188no—
349C2C22×#3804114yesdirect product factor
350C2C24×D3211yescharacteristic
subgroupG/H
centerC24C2×D32
commutator, cosocleC2×C32C26
FrattiniC2C24×D32
FittingC25×C32C22
socleC24×C32C23
hypercenterC25D32
hypocenterC32C23×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C24 ◁ C25

Complements

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

NH ≅ G/Ntype
1. C23×#638C2semidirect
2. C23×C3×#67C2semidirect
3. C24×C3×Dic3C2semidirect
4. C23×#627C2semidirect
5. C23×#626C2semidirect
6. C23×#624C2semidirect
7. C22×#3804C2direct (N × H)
8. C23×D3×Dic3C2semidirect
14. C22×#638C22semidirect
16. C22×C3×#67C22semidirect
20. C23×C3×Dic3C22semidirect
21. C22×#627C22semidirect
22. C22×#626C22semidirect
23. C22×#624C22semidirect
24. C2×#3804C22direct (N × H)
25. C22×D3×Dic3C22semidirect
32. C23×#67D3semidirect
34. C24×Dic3D3semidirect
50. C2×#638C23semidirect
52. C2×C3×#67C23semidirect
56. C22×C3×Dic3C23semidirect
57. C2×#627C23semidirect
58. C2×#626C23semidirect
59. C2×#624C23semidirect
60. #3804C23direct (N × H)
61. C2×D3×Dic3C23semidirect
80. C22×#67C2×D3semidirect
88. C23×Dic3C2×D3semidirect
98. C23×C32#41semidirect
105. #638C24semidirect
107. C3×#67C24semidirect
111. C2×C3×Dic3C24semidirect
112. #627C24semidirect
113. #626C24semidirect
114. #624C24semidirect
115. D3×Dic3C24semidirect
152. C2×#67C22×D3semidirect
160. C22×Dic3C22×D3semidirect
170. C22×C32C2×#41semidirect
204. C23×C3#235semidirect
205. C23×C3#237semidirect
227. #67C23×D3semidirect
235. C2×Dic3C23×D3semidirect
243. C2×C32C22×#41semidirect
274. C22×C3C2×#235semidirect
275. C22×C3C2×#237semidirect
297. C32C23×#41semidirect
303. C23#3804direct (N × H)
323. C2×C3C22×#235semidirect
324. C2×C3C22×#237semidirect
336. C22C2×#3804direct (N × H)
344. C3C23×#235semidirect
345. C3C23×#237semidirect
349. C2C22×#3804direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#6382
C23×C3×#672
C24×C3×Dic32
C23×#6272
C23×#6262
C23×#6242
C22×#38042
C23×D3×Dic32
C23×#2373
C23×#2353