Finite Group Databasebeta

← #26031006

Subgroups — #26031006

Order 11392.

468 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D89C211yesmaximal, characteristic, complemented
2C2×#15667983C211yesmaximal, characteristic
3#19151929C211yesmaximal, characteristic, complemented
4#19151926C211yesmaximal, characteristic, complemented
5#19151915C212yesmaximal, complemented
6#19151826C212yesmaximal, complemented
7#19151824C212yesmaximal, complemented
8#19151749C211yesmaximal, characteristic, complemented
9#19151735C212yesmaximal, complemented
10C22⋊C8×D89C211yesmaximal, characteristic, complemented
11#450×C89C211yesmaximal, characteristic, complemented
12C2×Q8×D89C2211yescharacteristic, complemented
13C22×C22⋊D89—21no—
14C22×C4×D89C2211yescharacteristic
15C2×#13083562C2212yescomplemented
16C2×C8×D89C2212yescomplemented
17#15667992C2211yescharacteristic, complemented
18#15667991C2211yescharacteristic
19#15667990—21no—
20#15667989—21no—
21#15667988C2211yescharacteristic, complemented
22#15667983—21no—
23#15667983—21no—
24#15667983C2212yes—
25#15667983C2212yes—
26C2×C4⋊Dic89C2211yescharacteristic
27C4⋊C4×D89—21no—
28C4⋊C4×D89C2211yescharacteristic, complemented
29C2×C22⋊Dic89—21no—
30C22⋊C4×D89—21no—
31#15667944C2212yescomplemented
32#15667943C2212yescomplemented
33#15667937C2212yescomplemented
34#15667936C2212yescomplemented
35#15667933C2212yescomplemented
36C22⋊#10623494C2211yescharacteristic
37#15667924C2212yescomplemented
38C22×D4×C89C2211yescharacteristic
39C2×SD16×C89C2212yescomplemented
40C22⋊Q8×C89C2211yescharacteristic
41#102×C89C2212yescomplemented
42C22⋊C8×C89C2211yescharacteristic
43C23×D89D411yescharacteristic
44C2×Dic178—21no—
45C2×Dic178C2311yescharacteristic
46Q8×D89—42no—
47C2×C22⋊D89—22no—
48C2×C22⋊D89—22no—
49C2×C22⋊D89—44no—
50C22×Dic89—21no—
51C22×Dic89D411yescharacteristic
52C2×C4×D89—21no—
53C2×C4×D89—21no—
54C2×C4×D89—42no—
55C2×C4×D89D411yescharacteristic
56C2×C4×D89D411yescharacteristic
57C2×C4×D89C2311yescharacteristic
58C2×C4×D89C2311yescharacteristic
59#13083562—42no—
60C2×#10623494C2312yes—
61C8×D89—42no—
62C4⋊Dic89—21no—
63C4⋊Dic89—21no—
64C4⋊Dic89C2312yes—
65C4⋊Dic89C2311yescharacteristic
66#13083553—21no—
67#13083553—21no—
68#13083553—21no—
69C22⋊Dic89—21no—
70C22⋊Dic89—22no—
71C22⋊Dic89—22no—
72#13083551—21no—
73#13083551—21no—
74C24×C89—21no—
75C2×Q8×C89C2311yescharacteristic
76C2×D4×C89—21no—
77C2×D4×C89—42no—
78C2×D4×C89C2312yes—
79C22×C4×C89C2311yescharacteristic
80SD16×C89—42no—
81C2×C8×C89C2312yes—
82C4⋊C4×C89—21no—
83C4⋊C4×C89C2311yescharacteristic
84C22⋊C4×C89—21no—
85C23×C89—22no—
86C23×C89—42no—
87C23×C89—42no—
88C23×C89—42no—
89C23×C89C2×D411yescharacteristic
90C22×D89—41no—
91C22×D89—41no—
92C22×D89SD1614yescomplemented
93C22×D89C2×D411yescharacteristic
94C22×D89C2×D411yescharacteristic
95Q8×C89—22no—
96D4×C89—22no—
97D4×C89—22no—
98D4×C89—42no—
99C2×C4×C89—22no—
100C2×C4×C89—21no—
101C2×C4×C89—21no—
102C2×C4×C89C2×D411yescharacteristic
103C2×C4×C89C2411yescharacteristic
104C22⋊D89—44no—
105C22⋊D89—44no—
106C2×Dic89—21no—
107C2×Dic89—21no—
108C2×Dic89—22no—
109C2×Dic89—41no—
110C2×Dic89—42no—
111C2×Dic89C2×D411yescharacteristic
112C2×Dic89C2×D411yescharacteristic
113C4×D89—22no—
114C4×D89—22no—
115C4×D89—41no—
116C4×D89—41no—
117C4×D89—42no—
118C4×D89—42no—
119Dic178—22no—
120Dic178—42no—
121C8×C89—22no—
122#10623494—22no—
123C22×C89—21no—
124C22×C89—21no—
125C22×C89—42no—
126C22×C89—42no—
127C22×C89—42no—
128C22×C89—84no—
129C22×C89C2×SD1612yes—
130C22×C89C22×D411yescharacteristic
131C2×D89—21no—
132C2×D89—21no—
133C2×D89—21no—
134C2×D89—21no—
135C2×D89—44no—
136C2×D89C2×SD1612yes—
137C2×D89C2×SD1612yes—
138C4×C89—21no—
139C4×C89—41no—
140C4×C89—41no—
141C4×C89H(5,𝔽2)12yes—
142Dic89—21no—
143Dic89—21no—
144Dic89—41no—
145Dic89—42no—
146Dic89—41no—
147C2×C89—22no—
148C2×C89—82no—
149C2×C89#53411yescharacteristic
150C2×C89C22×SD1611yescharacteristic
151C2×C89#57511yescharacteristic
152D89—22no—
153D89—22no—
154#3187—891nomaximal, Sylow
155C89#318711yescharacteristic, complemented, Sylow
156C2×C22⋊Q8—891no—
157C2×#121—891no—
158#483—891no—
159#483—891no—
160#481—892no—
161#465—892no—
162#465—892no—
163#450—891no—
164#450D8911yescharacteristic, complemented
165#448—892no—
166C2×C22⋊C8—891no—
167C22×Q8—891no—
168C22×D4—1781no—
169C22×D4C2×D8911yescharacteristic
170C23×C4—891no—
171C2×SD16—892no—
172C2×SD16C2×D8912yescomplemented
173C22×C8—892no—
174C22⋊Q8—891no—
175C22⋊Q8—891no—
176C22⋊Q8—891no—
177C22⋊Q8—1781no—
178C22⋊Q8—1781no—
179C22⋊Q8C2×D8911yescharacteristic
180#121—892no—
181#121—892no—
182#121—1781no—
183#121—1781no—
184C2×C4⋊C4—891no—
185C2×C4⋊C4—891no—
186C2×C4⋊C4—1781no—
187C2×C22⋊C4—1781no—
188C2×C22⋊C4—1781no—
189#106—892no—
190#106—892no—
191#103—892no—
192#103—892no—
193#102—892no—
194#102C2×D8912yescomplemented
195C22⋊C8—891no—
196C22⋊C8—892no—
197C22⋊C8C2×D8911yescharacteristic
198C24—21no—
199C24—891no—
200C2×Q8—891no—
201C2×Q8—3562no—
202C2×Q8—1781no—
203C2×Q8C22×D8911yescharacteristic
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—3564no—
207C2×D4—1782no—
208C2×D4—1782no—
209C2×D4C22×D8912yes—
210C22×C4—891no—
211C22×C4—891no—
212C22×C4—891no—
213C22×C4—891no—
214C22×C4—891no—
215C22×C4—3562no—
216C22×C4—1781no—
217C22×C4—1781no—
218C22×C4—1781no—
219C22×C4C22×D8911yescharacteristic
220SD16—42no—
221SD16—3562no—
222C2×C8—892no—
223C2×C8—3562no—
224C2×C8C22×D8912yes—
225C4⋊C4—21no—
226C4⋊C4—891no—
227C4⋊C4—892no—
228C4⋊C4—1781no—
229C4⋊C4—1781no—
230C4⋊C4—1781no—
231C4⋊C4—1781no—
232C4⋊C4—1781no—
233C4⋊C4C22×D8911yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—1781no—
236C22⋊C4—1781no—
237C22⋊C4—1782no—
238C22⋊C4—1782no—
239C22⋊C4—1781no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—894no—
245C23—891no—
246C23—891no—
247C23—3561no—
248C23—3561no—
249C23D4×D8911yescharacteristic
250Q8—22no—
251Q8—3562no—
252Q8—1782no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—3564no—
257D4—3564no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—891no—
262C2×C4—891no—
263C2×C4—3561no—
264C2×C4—3561no—
265C2×C4—3562no—
266C2×C4—3562no—
267C2×C4—3562no—
268C2×C4—3561no—
269C2×C4—1781no—
270C2×C4—1782no—
271C2×C4—1781no—
272C2×C4—1782no—
273C2×C4—1782no—
274C2×C4D4×D8911yescharacteristic
275C2×C4C23×D8911yescharacteristic
276C8—22no—
277C8—1782no—
278C22—21no—
279C22—21no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—84no—
284C22—892no—
285C22—892no—
286C22—3564no—
287C22—1781no—
288C22—1781no—
289C22—1781no—
290C22—1781no—
291C22SD16×D8912yes—
292C22C2×D4×D8911yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—3561no—
297C4—3561no—
298C4—3562no—
299C4—1781no—
300C4—1781no—
301C4#1566805712yes—
302C2—22no—
303C2—82no—
304C2—1782no—
305C2—1782no—
306C2#1915215811yescharacteristic
307C2C2×SD16×D8911yescharacteristic
308C2#1915239411yescharacteristic
subgroupG/H
centerC22C2×D4×D89
commutator, cosocleC2×C4×C89C24
FrattiniC2×C4C23×D89
Fitting#450×C89C2
socleC22×C89C22×D4
prime radicalC2×#15667983C2
hypercenter#450D89
hypocenterC89#3187
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D89C2semidirect
3. #19151929C2semidirect
4. #19151926C2semidirect
5. #19151915C2semidirect
6. #19151826C2semidirect
7. #19151824C2semidirect
8. #19151749C2semidirect
9. #19151735C2semidirect
10. C22⋊C8×D89C2semidirect
11. #450×C89C2semidirect
12. C2×Q8×D89C22semidirect
15. C2×#13083562C22semidirect
16. C2×C8×D89C22semidirect
17. #15667992C22semidirect
21. #15667988C22semidirect
28. C4⋊C4×D89C22semidirect
31. #15667944C22semidirect
32. #15667943C22semidirect
33. #15667937C22semidirect
34. #15667936C22semidirect
35. #15667933C22semidirect
37. #15667924C22semidirect
39. C2×SD16×C89C22semidirect
41. #102×C89C22semidirect
92. C22×D89SD16semidirect
155. C89#3187semidirect
164. #450D89semidirect
172. C2×SD16C2×D89semidirect
194. #102C2×D89semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D892
C2×#156679832
#191519292
#191519262
#191519152
#191518262
#191518242
#191517492
#191517352
C22⋊C8×D892
#450×C892
#318789