Finite Group Databasebeta

← #15676380

Subgroups — #15676380

Order 2944.

928 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C2×#10625108C211yesmaximal, characteristic
2C2×C4×C22⋊D23C212yesmaximal, complemented
3C2×C22⋊C4×D23C212yesmaximal, complemented
4#13092320C211yesmaximal, characteristic, complemented
5#13092311C212yesmaximal, complemented
6#13092293C212yesmaximal, complemented
7#13092291C212yesmaximal, complemented
8#13092264C212yesmaximal, complemented
9C23×#386C211yesmaximal, characteristic, complemented
10C24×D23C2211yescharacteristic
11C22×D4×C23C2211yescharacteristic
12C23×C4×C23C2211yescharacteristic, complemented
13C22×C22⋊D23C2212yes—
14C22×C4×D23C2212yes—
15C22×C4×D23C2212yes—
16C2×C22⋊C4×C23C2212yes—
17C2×C22⋊C4×C23C2211yescharacteristic, complemented
18C2×C22⋊Dic23C2212yes—
19C2×C22⋊Dic23C2211yescharacteristic
20C2×C22⋊Dic23C2212yescomplemented
21#10625108C418yescomplemented
22C4×C22⋊D23—28no—
23C2×(C2×D23)⋊C4C2212yes—
24C2×(C2×D23)⋊C4C2212yes—
25C2×(C2×D23)⋊C4C2212yes—
26C2×C4⋊Dic23C2211yescharacteristic, complemented
27C2×Dic23⋊C4C2212yescomplemented
28C2×C4×Dic23C2212yescomplemented
29C2×C4×Dic23C2211yescharacteristic, complemented
30C22⋊C4×D23—28no—
31C23×#95C2212yescomplemented
32C23⋊#95C2212yescomplemented
33C23⋊#95C2212yescomplemented
34C24×C23C2311yescharacteristic
35C24×C23C2311yescharacteristic
36C23×D23—24no—
37C23×D23—24no—
38C23×D23—41no—
39C23×D23—41no—
40C23×D23C2×C414yescomplemented
41C23×D23C2312yes—
42C2×D4×C23—41no—
43C2×D4×C23—41no—
44C2×D4×C23C2×C414yescomplemented
45C22×C4×C23—24no—
46C22×C4×C23—41no—
47C22×C4×C23C2312yes—
48C22×C4×C23C2311yescharacteristic
49C22×C4×C23C2312yes—
50C2×C22⋊D23—24no—
51C2×C22⋊D23—24no—
52C2×C22⋊D23C2×C418yescomplemented
53C22×Dic23C2312yes—
54C22×Dic23C2312yes—
55C22×Dic23C2312yes—
56C2×C4×D23—24no—
57C2×C4×D23—24no—
58C2×C4×D23—24no—
59C2×C4×D23—24no—
60C2×C4×D23—42no—
61C2×C4×D23—42no—
62C22⋊C4×C23—24no—
63C22⋊C4×C23—41no—
64C22⋊Dic23—24no—
65C22⋊Dic23—24no—
66C22⋊Dic23C2×C414yescomplemented
67(C2×D23)⋊C4—24no—
68(C2×D23)⋊C4C2×C418yescomplemented
69(C2×D23)⋊C4D418yescomplemented
70C4⋊Dic23—41no—
71Dic23⋊C4—24no—
72C4×Dic23—24no—
73C4×Dic23—41no—
74C23×C23—22no—
75C23×C23—22no—
76C23×C23—22no—
77C23×C23—22no—
78C23×C23—21no—
79C23×C23—21no—
80C23×C23—21no—
81C23×C23—21no—
82C23×C23C22×C412yes—
83C23×C23C22×C412yes—
84C23×C23C2411yescharacteristic
85C22×D23—22no—
86C22×D23—22no—
87C22×D23—22no—
88C22×D23—22no—
89C22×D23—216no—
90C22×D23—44no—
91C22×D23—44no—
92C22×D23—48no—
93C22×D23—44no—
94C22×D23C22×C414yes—
95C22×D23C2×D414yes—
96C22×D23#4114yes—
97D4×C23—44no—
98C2×C4×C23—22no—
99C2×C4×C23—21no—
100C2×C4×C23—21no—
101C2×C4×C23—22no—
102C2×C4×C23—22no—
103C2×C4×C23—22no—
104C2×C4×C23—22no—
105C2×C4×C23—44no—
106C2×C4×C23C22×C412yes—
107C2×C4×C23C2×D414yes—
108C22⋊D23—216no—
109C2×Dic23—22no—
110C2×Dic23—22no—
111C2×Dic23—22no—
112C2×Dic23—22no—
113C2×Dic23—22no—
114C2×Dic23—22no—
115C2×Dic23—22no—
116C2×Dic23C22×C414yes—
117C2×Dic23C2×D414yes—
118C4×D23—48no—
119C4×D23—48no—
120#2358—231nomaximal, Sylow
121C22×C23—24no—
122C22×C23—24no—
123C22×C23—22no—
124C22×C23—22no—
125C22×C23—41no—
126C22×C23—41no—
127C22×C23—42no—
128C22×C23—42no—
129C22×C23—41no—
130C22×C23—41no—
131C22×C23C23×C411yescharacteristic
132C22×C23C22×D412yes—
133C22×C23C2×#4112yes—
134C22×C23H(5,𝔽2)11yescharacteristic
135C22×C23H(5,𝔽2)11yescharacteristic
136C2×D23—24no—
137C2×D23—24no—
138C2×D23—24no—
139C2×D23—24no—
140C2×D23—24no—
141C2×D23—24no—
142C2×D23—416no—
143C2×D23C4×D418yescomplemented
144C4×C23—22no—
145C4×C23—24no—
146C4×C23—42no—
147Dic23—24no—
148Dic23—24no—
149Dic23—42no—
150C2×#120—231no—
151C2×C4×D4—232no—
152C22×C22⋊C4—232no—
153#390—231no—
154#388—232no—
155#386—232no—
156#386D2311yescharacteristic, complemented
157#385—232no—
158C4×C22⋊C4—232no—
159C2×C23—42no—
160C2×C23—42no—
161C2×C23C2×C4×D412yes—
162C2×C23#51811yescharacteristic
163C2×C23D4211yescharacteristic
164C2×C23#54612yes—
165C2×C23#55111yescharacteristic
166D23—28no—
167C25—231no—
168C22×D4—232no—
169C22×D4C2×D2311yescharacteristic
170C23×C4—232no—
171C23×C4—232no—
172C23×C4C2×D2311yescharacteristic, complemented
173#120—238no—
174C4×D4—468no—
175C2×C4⋊C4—232no—
176C2×C4⋊C4—231no—
177C2×C22⋊C4—231no—
178C2×C22⋊C4—232no—
179C2×C22⋊C4—232no—
180C2×C22⋊C4—232no—
181C2×C22⋊C4—232no—
182C2×C22⋊C4—232no—
183C2×C22⋊C4—468no—
184C2×C22⋊C4C2×D2312yes—
185C2×C22⋊C4C2×D2311yescharacteristic, complemented
186C2×C42—232no—
187C2×C42—231no—
188#95—232no—
189#95—232no—
190#95C2×D2312yescomplemented
191C23#235811yescharacteristic, complemented, Sylow
192C24—232no—
193C24—234no—
194C24—464no—
195C24—464no—
196C24—921no—
197C24—921no—
198C24C22×D2311yescharacteristic
199C24C22×D2311yescharacteristic
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—238no—
203C2×D4—464no—
204C2×D4—464no—
205C2×D4C4×D2314yescomplemented
206C22×C4—24no—
207C22×C4—41no—
208C22×C4—232no—
209C22×C4—232no—
210C22×C4—232no—
211C22×C4—464no—
212C22×C4—464no—
213C22×C4—464no—
214C22×C4—464no—
215C22×C4—922no—
216C22×C4—922no—
217C22×C4C22×D2312yes—
218C22×C4C22×D2311yescharacteristic
219C22×C4C22×D2312yes—
220C4⋊C4—464no—
221C4⋊C4—921no—
222C22⋊C4—24no—
223C22⋊C4—41no—
224C22⋊C4—234no—
225C22⋊C4—238no—
226C22⋊C4—238no—
227C22⋊C4—464no—
228C22⋊C4—464no—
229C22⋊C4—464no—
230C42—464no—
231C42—921no—
232C23—22no—
233C23—22no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—22no—
239C23—22no—
240C23—234no—
241C23—234no—
242C23—234no—
243C23—4616no—
244C23—462no—
245C23—462no—
246C23—462no—
247C23—462no—
248C23—924no—
249C23—924no—
250C23—928no—
251C23—924no—
252C23C2×C4×D2312yes—
253C23C2×C4×D2312yes—
254C23C23×D2311yescharacteristic
255D4—44no—
256D4—4616no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—44no—
265C2×C4—234no—
266C2×C4—234no—
267C2×C4—462no—
268C2×C4—462no—
269C2×C4—462no—
270C2×C4—462no—
271C2×C4—462no—
272C2×C4—462no—
273C2×C4—462no—
274C2×C4—928no—
275C2×C4—928no—
276C2×C4C2×C4×D2312yes—
277C2×C4D4×D2314yes—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—22no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—238no—
289C22—464no—
290C22—464no—
291C22—464no—
292C22—464no—
293C22—464no—
294C22—464no—
295C22—9216no—
296C22C22×C4×D2311yescharacteristic
297C22C2×D4×D2312yes—
298C22#1062515511yescharacteristic
299C22#1062515511yescharacteristic
300C22#41×D2312yes—
301C4—22no—
302C4—24no—
303C4—42no—
304C4—464no—
305C4—464no—
306C4—922no—
307C2—42no—
308C2—42no—
309C2—468no—
310C2C4×D4×D2312yes—
311C2#1309289011yescharacteristic
312C2#1309304911yescharacteristic
313C2#1309306412yes—
314C2#1309309511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D23
commutatorC22×C23C23×C4
FittingC23×#386C2
socle, cosocleC23×C23C24
prime radicalC2×#10625108C2
hypercenter#386D23
hypocenterC23#2358
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D23C2semidirect
3. C2×C22⋊C4×D23C2semidirect
4. #13092320C2semidirect
5. #13092311C2semidirect
6. #13092293C2semidirect
7. #13092291C2semidirect
8. #13092264C2semidirect
9. C23×#386C2semidirect
12. C23×C4×C23C22semidirect
17. C2×C22⋊C4×C23C22semidirect
20. C2×C22⋊Dic23C22semidirect
21. #10625108C4semidirect
26. C2×C4⋊Dic23C22semidirect
27. C2×Dic23⋊C4C22semidirect
28. C2×C4×Dic23C22semidirect
29. C2×C4×Dic23C22semidirect
31. C23×#95C22semidirect
32. C23⋊#95C22semidirect
33. C23⋊#95C22semidirect
40. C23×D23C2×C4semidirect
44. C2×D4×C23C2×C4semidirect
52. C2×C22⋊D23C2×C4semidirect
66. C22⋊Dic23C2×C4semidirect
68. (C2×D23)⋊C4C2×C4semidirect
69. (C2×D23)⋊C4D4semidirect
143. C2×D23C4×D4semidirect
156. #386D23semidirect
172. C23×C4C2×D23semidirect
185. C2×C22⋊C4C2×D23semidirect
190. #95C2×D23semidirect
191. C23#2358semidirect
205. C2×D4C4×D23semidirect

Maximal subgroups

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

subgroupindex
C2×#106251082
C2×C4×C22⋊D232
C2×C22⋊C4×D232
#130923202
#130923112
#130922932
#130922912
#130922642
C23×#3862
#235823