Finite Group Databasebeta

← #26025404

Subgroups — #26025404

Order 11392.

672 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 139 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×#15667992C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D89C211yesmaximal, characteristic
3#19151524C211yesmaximal, characteristic, complemented
4#19151511C211yesmaximal, characteristic, complemented
5#19151510C211yesmaximal, characteristic, complemented
6#19151491C212yesmaximal, complemented
7#19151490C212yesmaximal, complemented
8#19151460C212yesmaximal, complemented
9#19151453C212yesmaximal
10#19151445C211yesmaximal, characteristic, complemented
11#397×C89C211yesmaximal, characteristic, complemented
12C24×D89C2211yescharacteristic
13C22×Dic178C2211yescharacteristic, complemented
14C22×C4×D89C2211yescharacteristic
15C22×C4×D89C2211yescharacteristic
16#15667992—24no—
17C2×C4⋊Dic89C2212yescomplemented
18C2×C22⋊Dic89C2211yescharacteristic
19C2×C22⋊Dic89C2212yes—
20C2×#13083551C2212yes—
21C2×#13083551C2212yes—
22C2×#13083551C2212yes—
23C2×#13083551C2211yescharacteristic
24C2×#13083551C2211yescharacteristic
25C2×#13083551C2212yes—
26C22⋊C4×D89—24no—
27C2×C4×Dic89C2212yescomplemented
28#15667919C2211yescharacteristic, complemented
29#15667919C2212yescomplemented
30#15667919C2212yescomplemented
31#15667919C2212yescomplemented
32C2×C4⋊C4×C89C2211yescharacteristic, complemented
33C2×C22⋊C4×C89C2212yes—
34C2×C22⋊C4×C89C2211yescharacteristic
35#95×C89C2211yescharacteristic, complemented
36#95×C89C2212yescomplemented
37C23×D89—22no—
38C23×D89—22no—
39C23×D89—22no—
40C23×D89—24no—
41C23×D89—42no—
42C23×D89C2311yescharacteristic
43C23×D89C2311yescharacteristic
44C2×Dic178—22no—
45C2×Dic178—42no—
46C22×Dic89C2311yescharacteristic
47C22×Dic89C2312yes—
48C22×Dic89C2312yes—
49C22×Dic89C2311yescharacteristic
50C2×C4×D89—22no—
51C2×C4×D89—24no—
52C2×C4×D89—42no—
53C2×C4×D89D414yes—
54C4⋊Dic89—24no—
55C22⋊Dic89—22no—
56C22⋊Dic89—42no—
57#13083551—24no—
58#13083551—24no—
59#13083551—24no—
60#13083551—22no—
61#13083551—22no—
62#13083551—42no—
63C4×Dic89—42no—
64C24×C89C2311yescharacteristic
65C22×C4×C89C2311yescharacteristic
66C22×C4×C89C2312yes—
67C22×C4×C89C2312yes—
68C22×C4×C89C2311yescharacteristic
69C4⋊C4×C89—22no—
70C22⋊C4×C89—24no—
71C22⋊C4×C89—22no—
72C23×C89—21no—
73C23×C89—21no—
74C23×C89—22no—
75C23×C89—22no—
76C23×C89#4112yes—
77C23×C89C2411yescharacteristic
78C22×D89—21no—
79C22×D89—21no—
80C22×D89—22no—
81C22×D89—28no—
82C22×D89—21no—
83C22×D89—22no—
84C22×D89—21no—
85C22×D89—22no—
86C22×D89—48no—
87C22×D89—48no—
88C22×D89—44no—
89C22×D89—44no—
90C22×D89C2×D412yes—
91C22×D89#4112yes—
92C22×D89#4114yes—
93C2×C4×C89—22no—
94C2×C4×C89—21no—
95C2×C4×C89—22no—
96C2×C4×C89—22no—
97C2×C4×C89—22no—
98C2×C4×C89—22no—
99C2×C4×C89—22no—
100C2×C4×C89—22no—
101C2×C4×C89—22no—
102C2×C4×C89C2×D412yes—
103C2×Dic89—22no—
104C2×Dic89—21no—
105C2×Dic89—22no—
106C2×Dic89—22no—
107C2×Dic89—22no—
108C2×Dic89—22no—
109C2×Dic89—22no—
110C2×Dic89—22no—
111C2×Dic89—22no—
112C2×Dic89C2×D412yes—
113C4×D89—44no—
114C4×D89—44no—
115Dic178—44no—
116C22×C89—22no—
117C22×C89—24no—
118C22×C89—41no—
119C22×C89—41no—
120C22×C89—42no—
121C22×C89C22×D411yescharacteristic
122C22×C89C2×#4111yescharacteristic
123C22×C89C2×#4111yescharacteristic
124C22×C89C2×#4112yes—
125C22×C89H(5,𝔽2)12yes—
126C2×D89—22no—
127C2×D89—22no—
128C2×D89—24no—
129C2×D89—24no—
130C2×D89—22no—
131C2×D89—24no—
132C2×D89—416no—
133C2×D89—41no—
134C2×D89—41no—
135C2×D89—42no—
136C2×D89#12414yes—
137C4×C89—22no—
138C4×C89—42no—
139C4×C89—41no—
140C4×C89—42no—
141Dic89—22no—
142Dic89—41no—
143Dic89—42no—
144Dic89—42no—
145C2×C89—42no—
146C2×C89#52511yescharacteristic
147C2×C89C2×#12411yescharacteristic
148C2×C89#53411yescharacteristic
149C2×C89#55112yes—
150C2×C89#55112yes—
151D89—24no—
152D89—42no—
153#2561—891nomaximal, Sylow
154C89#256111yescharacteristic, complemented, Sylow
155C2×C22⋊Q8—891no—
156C22×C22⋊C4—891no—
157#397—891no—
158#397D8911yescharacteristic, complemented
159#393—891no—
160#393—891no—
161#388—892no—
162#388—892no—
163#381—892no—
164#379—892no—
165C4×C22⋊C4—891no—
166C25—891no—
167C22×Q8—891no—
168C23×C4—891no—
169C23×C4—891no—
170C22⋊Q8—1784no—
171C2×C4⋊C4—892no—
172C2×C4⋊C4C2×D8911yescharacteristic, complemented
173C2×C22⋊C4—892no—
174C2×C22⋊C4—892no—
175C2×C22⋊C4—891no—
176C2×C22⋊C4—891no—
177C2×C22⋊C4—891no—
178C2×C22⋊C4—892no—
179C2×C22⋊C4—892no—
180C2×C22⋊C4—892no—
181C2×C22⋊C4—1784no—
182C2×C22⋊C4C2×D8911yescharacteristic
183C2×C22⋊C4C2×D8912yes—
184C2×C42—892no—
185#95—891no—
186#95—892no—
187#95—892no—
188#95—892no—
189#95C2×D8911yescharacteristic, complemented
190#95C2×D8912yescomplemented
191C24—891no—
192C24—891no—
193C24—3562no—
194C24—1784no—
195C24—1782no—
196C24—1782no—
197C24—1782no—
198C24C22×D8911yescharacteristic
199C2×Q8—3562no—
200C2×Q8—1782no—
201C22×C4—891no—
202C22×C4—894no—
203C22×C4—891no—
204C22×C4—892no—
205C22×C4—892no—
206C22×C4—3562no—
207C22×C4—1782no—
208C22×C4—1784no—
209C22×C4C22×D8911yescharacteristic
210C22×C4C22×D8912yes—
211C22×C4C22×D8912yes—
212C22×C4C22×D8911yescharacteristic
213C4⋊C4—22no—
214C4⋊C4—1784no—
215C22⋊C4—24no—
216C22⋊C4—22no—
217C22⋊C4—3562no—
218C22⋊C4—3562no—
219C22⋊C4—1782no—
220C22⋊C4—1784no—
221C22⋊C4—1784no—
222C22⋊C4—1784no—
223C22⋊C4—1782no—
224C22⋊C4—1782no—
225C42—3562no—
226C23—21no—
227C23—22no—
228C23—21no—
229C23—22no—
230C23—892no—
231C23—894no—
232C23—892no—
233C23—3568no—
234C23—3564no—
235C23—3564no—
236C23—3568no—
237C23—1788no—
238C23—1781no—
239C23—1782no—
240C23—1781no—
241C23—1782no—
242C23—1781no—
243C23—1781no—
244C23—1782no—
245C23#1308357312yes—
246C23C23×D8911yescharacteristic
247Q8—3564no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—892no—
258C2×C4—3564no—
259C2×C4—3564no—
260C2×C4—1781no—
261C2×C4—1782no—
262C2×C4—1782no—
263C2×C4—1782no—
264C2×C4—1782no—
265C2×C4—1782no—
266C2×C4—1782no—
267C2×C4—1782no—
268C2×C4—1782no—
269C2×C4D4×D8912yes—
270C22—24no—
271C22—22no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—894no—
276C22—35616no—
277C22—3561no—
278C22—3562no—
279C22—3561no—
280C22—1784no—
281C22—1782no—
282C22—1782no—
283C22—1784no—
284C22—1784no—
285C22—1782no—
286C22C2×D4×D8911yescharacteristic
287C22#41×D8912yes—
288C22#41×D8911yescharacteristic
289C22C2×#1308357311yescharacteristic
290C22#1566805812yes—
291C4—22no—
292C4—42no—
293C4—42no—
294C4—41no—
295C4—3562no—
296C4—3562no—
297C4—3561no—
298C4—1782no—
299C2—42no—
300C2—3562no—
301C2—1784no—
302C2#1915210811yescharacteristic
303C2#124×D8911yescharacteristic
304C2#1915215911yescharacteristic
305C2#1915227112yes—
306C2#1915227312yes—
subgroupG/H
center, FrattiniC23C23×D89
commutator, socle, cosocleC23×C89C24
Fitting#397×C89C2
prime radicalC24×D89C22
hypercenter#397D89
hypocenterC89#2561
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#15667992C2semidirect
3. #19151524C2semidirect
4. #19151511C2semidirect
5. #19151510C2semidirect
6. #19151491C2semidirect
7. #19151490C2semidirect
8. #19151460C2semidirect
10. #19151445C2semidirect
11. #397×C89C2semidirect
13. C22×Dic178C22semidirect
17. C2×C4⋊Dic89C22semidirect
27. C2×C4×Dic89C22semidirect
28. #15667919C22semidirect
29. #15667919C22semidirect
30. #15667919C22semidirect
31. #15667919C22semidirect
32. C2×C4⋊C4×C89C22semidirect
35. #95×C89C22semidirect
36. #95×C89C22semidirect
154. C89#2561semidirect
158. #397D89semidirect
172. C2×C4⋊C4C2×D89semidirect
189. #95C2×D89semidirect
190. #95C2×D89semidirect

Maximal subgroups

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

subgroupindex
C2×#156679922
C2×C22⋊C4×D892
#191515242
#191515112
#191515102
#191514912
#191514902
#191514602
#191514532
#191514452
#397×C892
#256189