Finite Group Databasebeta

← #27155127

Subgroups — #27155127

Order 12416.

1040 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261358C212yesmaximal, complemented
2#20261357C212yesmaximal, complemented
3#20261346C211yesmaximal, characteristic, complemented
4#20261341C211yesmaximal, characteristic, complemented
5#20261340C211yesmaximal, characteristic, complemented
6#20261333C214yesmaximal, complemented
7#20261324C214yesmaximal, complemented
8#20261291C211yesmaximal, characteristic, complemented
9#20261286C211yesmaximal, characteristic, complemented
10#20261270C214yesmaximal, complemented
11#20261268C214yesmaximal, complemented
12#20261248C212yesmaximal, complemented
13C4×#13095512C212yesmaximal, complemented
14#117×D97C211yesmaximal, characteristic, complemented
15#541×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2212yes—
17C22×C4×D97C2211yescharacteristic
18#15690809C2212yescomplemented
19#15690807C2211yescharacteristic, complemented
20#15690805C2214yes—
21#15690805C2214yescomplemented
22#15690804C2214yescomplemented
23#15690802C2211yescharacteristic, complemented
24#15690800C2214yescomplemented
25#15690799C2212yescomplemented
26#15690798C2214yes—
27#15690796C2214yes—
28#15690795C2214yes—
29#15690795C2214yes—
30#15690795C2214yescomplemented
31#15690795C2214yescomplemented
32#15690792C2214yes—
33#15690790C2214yescomplemented
34#15690788C2214yes—
35#15690788C2214yescomplemented
36#15690786C2212yes—
37#15690785C2212yes—
38#15690785C2214yes—
39#15690785C2214yes—
40#15690784C2212yes—
41#15690783C2214yes—
42#15690780C2214yes—
43Q8×Dic97C2212yescomplemented
44#15690775C2214yescomplemented
45C4×D388C2214yes—
46C4×D388C2212yescomplemented
47#15690772C2214yes—
48#15690772C2214yes—
49#15690772C2212yescomplemented
50#15690772C2214yescomplemented
51#15690770C2214yes—
52#15690769C2211yescharacteristic
53#15690768C2212yes—
54#15690768C2212yes—
55#15690768C2214yescomplemented
56#15690767C2211yescharacteristic
57#15690767C2211yescharacteristic
58#15690766C2212yes—
59C4⋊C4×D97C2212yes—
60C22⋊C4×D97C2212yes—
61C42×D97C2211yescharacteristic
62C42×D97C2211yescharacteristic
63C42×D97C2214yescomplemented
64C4⋊Q8×C97C2211yescharacteristic, complemented
65#126×C97C2214yes—
66#125×C97C2214yescomplemented
67#125×C97C2211yescharacteristic, complemented
68C22⋊Q8×C97C2212yes—
69C4×Q8×C97C2212yescomplemented
70#117×C97C2211yescharacteristic
71C23×D97C2311yescharacteristic
72#13095512—28no—
73C2×Dic194C2314yes—
74C2×D388C2314yes—
75C2×D388C2312yes—
76C2×D388C2314yes—
77C2×D388C2314yes—
78C2×C22⋊D97C2314yes—
79C22×Dic97C2311yescharacteristic
80C2×C4×D97—24no—
81C2×C4×D97C2314yes—
82C2×C4×D97C2311yescharacteristic
83C2×C4×D97C2311yescharacteristic
84C2×C4×D97C2311yescharacteristic
85C2×C4×D97C2312yes—
86C2×C4×D97C2312yes—
87C2×C4×D97C2314yes—
88C2×C4×D97C2314yes—
89C2×C4×D97C2311yescharacteristic
90C4⋊Dic97C2312yes—
91C4⋊Dic97C2312yes—
92C4⋊Dic97C2314yes—
93#13095490C2312yes—
94#13095490C2312yes—
95#13095490C2314yes—
96#13095490C2314yes—
97#13095490C2314yes—
98C22⋊Dic97C2312yes—
99#13095488C2312yes—
100#13095488C2312yes—
101#13095488C2314yes—
102#13095488C2314yes—
103#13095488C2314yes—
104#13095488C2314yes—
105#13095488C2314yes—
106#13095488C2314yes—
107#13095488C2314yes—
108#13095488C2314yes—
109C4×Dic97C2312yes—
110C4×Dic97C2312yes—
111C4×Dic97C2311yescharacteristic
112C4×Dic97C2311yescharacteristic
113C4×Dic97C2314yes—
114C4×Dic97C2314yes—
115C2×Q8×C97C2312yes—
116C22×C4×C97C2311yescharacteristic
117C4⋊C4×C97C2312yes—
118C4⋊C4×C97C2312yes—
119C4⋊C4×C97C2314yes—
120C4⋊C4×C97C2314yes—
121C4⋊C4×C97C2314yes—
122C4⋊C4×C97C2314yes—
123C22⋊C4×C97C2312yes—
124C22⋊C4×C97C2314yes—
125C42×C97C2311yescharacteristic
126C42×C97C2311yescharacteristic
127C42×C97C2314yes—
128C23×C97C2411yescharacteristic
129C22×D97—22no—
130C22×D97—42no—
131C22×D97C2411yescharacteristic
132C22×D97C2414yes—
133C22×D97C2411yescharacteristic
134Q8×C97—24no—
135C2×C4×C97—22no—
136C2×C4×C97C2412yes—
137C2×C4×C97C2414yes—
138C2×C4×C97C2412yes—
139C2×C4×C97C2414yes—
140C2×C4×C97C2411yescharacteristic
141C2×C4×C97C2411yescharacteristic
142C22⋊D97—44no—
143C2×Dic97—22no—
144C2×Dic97C2412yes—
145C2×Dic97C2412yes—
146C2×Dic97C2414yes—
147C2×Dic97C2411yescharacteristic
148C2×Dic97C2411yescharacteristic
149D388—28no—
150D388—24no—
151D388—44no—
152D388—44no—
153C4×D97—22no—
154C4×D97—22no—
155C4×D97—22no—
156C4×D97—28no—
157C4×D97—42no—
158C4×D97—44no—
159C4×D97—44no—
160C4×D97—42no—
161C4×D97#4114yes—
162Dic194—44no—
163C22×C97—21no—
164C22×C97—22no—
165C22×C97C2511yescharacteristic
166C2×D97—22no—
167C2×D97—21no—
168C2×D97—24no—
169C2×D97—24no—
170C2×D97—24no—
171C2×D97—22no—
172C2×D97—44no—
173C2×D97C2×#4112yes—
174C4×C97—21no—
175C4×C97—24no—
176C4×C97—24no—
177C4×C97—22no—
178C4×C97—22no—
179C4×C97C2×#4112yes—
180Dic97—21no—
181Dic97—22no—
182Dic97—24no—
183Dic97—22no—
184Dic97C2×#4112yes—
185C2×C97—41no—
186C2×C97C22×#4111yescharacteristic
187C2×C97#58512yes—
188D97—22no—
189D97—44no—
190D97—41no—
191#3455—971nomaximal, Sylow
192C97#345511yescharacteristic, complemented, Sylow
193#542—972no—
194#542—972no—
195#541—971no—
196#541D9711yescharacteristic, complemented
197#540—971no—
198#540—971no—
199#539—974no—
200#538—974no—
201#533—971no—
202#532—971no—
203#529—974no—
204#529—974no—
205#525—972no—
206C4×#41—972no—
207C2×#117—971no—
208C2×#41—972no—
209C23×C4—971no—
210C4⋊Q8—972no—
211C4⋊Q8C2×D9711yescharacteristic, complemented
212#127—971no—
213#126—974no—
214#126—974no—
215#126—974no—
216#126C2×D9714yes—
217#125—974no—
218#125—971no—
219#125C2×D9714yescomplemented
220#125C2×D9711yescharacteristic, complemented
221#124—972no—
222#124—974no—
223#124—974no—
224#123—974no—
225#123—974no—
226#123—974no—
227#123—974no—
228#123—974no—
229C22⋊Q8—972no—
230C22⋊Q8—974no—
231C22⋊Q8—974no—
232C22⋊Q8—974no—
233C22⋊Q8C2×D9712yes—
234#121—972no—
235#121—972no—
236#121—974no—
237#121—974no—
238#121—974no—
239#120—974no—
240C4×Q8—972no—
241C4×Q8—974no—
242C4×Q8C2×D9712yescomplemented
243C4×D4—974no—
244C4×D4—972no—
245C4×D4—974no—
246C4×D4—974no—
247C4×D4—974no—
248C4×D4—974no—
249C4×D4—972no—
250#117—971no—
251#117—972no—
252#117—972no—
253#117—974no—
254#117—972no—
255#117—971no—
256#117—971no—
257#117C2×D9711yescharacteristic
258C2×C4⋊C4—972no—
259C2×C22⋊C4—972no—
260C2×C42—971no—
261C2×C42—971no—
262C2×C42—974no—
263C24—971no—
264#41—1948no—
265C2×Q8—974no—
266C2×Q8C22×D9712yes—
267C2×D4—974no—
268C2×D4—974no—
269C2×D4—974no—
270C2×D4—974no—
271C2×D4—972no—
272C22×C4—971no—
273C22×C4—971no—
274C22×C4—972no—
275C22×C4—972no—
276C22×C4—974no—
277C22×C4—974no—
278C22×C4—974no—
279C22×C4—971no—
280C22×C4—971no—
281C22×C4—971no—
282C22×C4—1944no—
283C22×C4C22×D9711yescharacteristic
284C4⋊C4—972no—
285C4⋊C4—972no—
286C4⋊C4—974no—
287C4⋊C4—974no—
288C4⋊C4—974no—
289C4⋊C4—972no—
290C4⋊C4—972no—
291C4⋊C4—974no—
292C4⋊C4C22×D9712yes—
293C4⋊C4C22×D9712yes—
294C4⋊C4C22×D9714yes—
295C4⋊C4C22×D9714yes—
296C4⋊C4C22×D9714yes—
297C4⋊C4C22×D9714yes—
298C22⋊C4—972no—
299C22⋊C4—972no—
300C22⋊C4—972no—
301C22⋊C4—974no—
302C22⋊C4—974no—
303C22⋊C4—974no—
304C22⋊C4—974no—
305C22⋊C4—974no—
306C22⋊C4—974no—
307C22⋊C4—974no—
308C22⋊C4—974no—
309C22⋊C4C22×D9712yes—
310C22⋊C4C22×D9714yes—
311C42—972no—
312C42—972no—
313C42—974no—
314C42—971no—
315C42—971no—
316C42—974no—
317C42C22×D9711yescharacteristic
318C42C22×D9711yescharacteristic
319C42C22×D9714yes—
320C23—971no—
321C23—971no—
322C23—974no—
323C23—3882no—
324C23—1942no—
325C23C23×D9711yescharacteristic
326Q8—24no—
327Q8—3884no—
328D4—3884no—
329D4—3884no—
330D4—3884no—
331D4—1948no—
332D4—1944no—
333C2×C4—22no—
334C2×C4—971no—
335C2×C4—974no—
336C2×C4—972no—
337C2×C4—972no—
338C2×C4—974no—
339C2×C4—971no—
340C2×C4—3882no—
341C2×C4—3884no—
342C2×C4—3882no—
343C2×C4—3884no—
344C2×C4—1942no—
345C2×C4—1948no—
346C2×C4—1942no—
347C2×C4—1942no—
348C2×C4—1942no—
349C2×C4C23×D9712yes—
350C2×C4C23×D9714yes—
351C2×C4C23×D9714yes—
352C2×C4C23×D9712yes—
353C2×C4C23×D9711yescharacteristic
354C2×C4C23×D9711yescharacteristic
355C22—21no—
356C22—22no—
357C22—972no—
358C22—3884no—
359C22—1941no—
360C22—1944no—
361C22—1944no—
362C22—1944no—
363C22—1942no—
364C22—1942no—
365C22C24×D9711yescharacteristic
366C4—21no—
367C4—24no—
368C4—22no—
369C4—22no—
370C4—24no—
371C4—972no—
372C4—1941no—
373C4—1942no—
374C4—1944no—
375C4—1942no—
376C4#41×D9712yes—
377C2—41no—
378C2—3881no—
379C2—3884no—
380C2—1942no—
381C2C2×#41×D9711yescharacteristic
382C2#2026157212yes—
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#541×C97C2
hypercenter#541D97
hypocenterC97#3455
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #541

Complements

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

NH ≅ G/Ntype
1. #20261358C2semidirect
2. #20261357C2semidirect
3. #20261346C2semidirect
4. #20261341C2semidirect
5. #20261340C2semidirect
6. #20261333C2semidirect
7. #20261324C2semidirect
8. #20261291C2semidirect
9. #20261286C2semidirect
10. #20261270C2semidirect
11. #20261268C2semidirect
12. #20261248C2semidirect
13. C4×#13095512C2semidirect
14. #117×D97C2semidirect
15. #541×C97C2semidirect
18. #15690809C22semidirect
19. #15690807C22semidirect
21. #15690805C22semidirect
22. #15690804C22semidirect
23. #15690802C22semidirect
24. #15690800C22semidirect
25. #15690799C22semidirect
30. #15690795C22semidirect
31. #15690795C22semidirect
33. #15690790C22semidirect
35. #15690788C22semidirect
43. Q8×Dic97C22semidirect
44. #15690775C22semidirect
46. C4×D388C22semidirect
49. #15690772C22semidirect
50. #15690772C22semidirect
55. #15690768C22semidirect
63. C42×D97C22semidirect
64. C4⋊Q8×C97C22semidirect
66. #125×C97C22semidirect
67. #125×C97C22semidirect
69. C4×Q8×C97C22semidirect
192. C97#3455semidirect
196. #541D97semidirect
211. C4⋊Q8C2×D97semidirect
219. #125C2×D97semidirect
220. #125C2×D97semidirect
242. C4×Q8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613582
#202613572
#202613462
#202613412
#202613402
#202613332
#202613242
#202612912
#202612862
#202612702
#202612682
#202612482
C4×#130955122
#117×D972
#541×C972
#345597