Finite Group Databasebeta

← #27155120

Subgroups — #27155120

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#20261352C212yesmaximal, complemented
2#20261350C212yesmaximal, complemented
3#20261348C211yesmaximal, characteristic, complemented
4#20261347C211yesmaximal, characteristic, complemented
5#20261339C211yesmaximal, characteristic, complemented
6#20261328C214yesmaximal, complemented
7#20261317C214yesmaximal, complemented
8#20261291C211yesmaximal, characteristic, complemented
9#20261283C211yesmaximal, characteristic, complemented
10#20261266C214yesmaximal, complemented
11#20261262C214yesmaximal, complemented
12#20261245C212yesmaximal, complemented
13C4×#13095510C212yesmaximal, complemented
14#117×D97C211yesmaximal, characteristic, complemented
15#540×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2212yes—
17C22×C4×D97C2211yescharacteristic
18#15690810C2211yescharacteristic, complemented
19#15690809C2212yescomplemented
20#15690805C2214yes—
21#15690803C2214yes—
22#15690803C2214yescomplemented
23#15690803C2214yescomplemented
24#15690802C2211yescharacteristic, complemented
25#15690801C2214yescomplemented
26#15690800C2214yescomplemented
27#15690797C2214yes—
28#15690797C2212yescomplemented
29#15690796C2214yes—
30#15690793C2214yes—
31#15690792C2214yes—
32#15690791C2214yes—
33#15690791C2214yescomplemented
34#15690790C2212yes—
35#15690789C2212yes—
36#15690789C2214yescomplemented
37#15690788C2214yes—
38#15690787C2214yescomplemented
39#15690782C2214yes—
40#15690781C2212yes—
41#15690781C2214yes—
42#15690778C2214yes—
43C4×Dic194C2212yescomplemented
44#15690775C2212yescomplemented
45#15690775C2214yescomplemented
46D4×Dic97C2214yes—
47D4×Dic97C2212yescomplemented
48D4×Dic97C2214yescomplemented
49C4×C22⋊D97C2214yes—
50#15690770C2214yes—
51#15690770C2214yes—
52#15690769C2211yescharacteristic
53#15690768C2212yes—
54#15690768C2212yes—
55#15690767C2211yescharacteristic
56#15690767C2211yescharacteristic
57#15690766C2212yes—
58#15690766C2214yescomplemented
59C4⋊C4×D97C2212yes—
60C22⋊C4×D97C2212yes—
61C2×C4×Dic97C2214yescomplemented
62C42×D97C2211yescharacteristic
63C42×D97C2211yescharacteristic
64#127×C97C2211yescharacteristic, complemented
65#125×C97C2211yescharacteristic, complemented
66#123×C97C2214yescomplemented
67#121×C97C2212yes—
68#121×C97C2214yes—
69C4×D4×C97C2212yescomplemented
70#117×C97C2211yescharacteristic
71C23×D97C2311yescharacteristic
72#13095510—28no—
73C2×Dic194C2312yes—
74C2×Dic194C2314yes—
75C2×C22⋊D97C2314yes—
76C2×C22⋊D97C2314yes—
77C22×Dic97C2314yes—
78C22×Dic97C2311yescharacteristic
79C22×Dic97C2314yes—
80C2×C4×D97—24no—
81C2×C4×D97C2311yescharacteristic
82C2×C4×D97C2311yescharacteristic
83C2×C4×D97C2311yescharacteristic
84C2×C4×D97C2312yes—
85C2×C4×D97C2312yes—
86C2×C4×D97C2311yescharacteristic
87C4⋊Dic97C2312yes—
88C4⋊Dic97C2312yes—
89C4⋊Dic97C2314yes—
90C4⋊Dic97C2314yes—
91#13095490C2312yes—
92#13095490C2312yes—
93#13095490C2314yes—
94#13095490C2314yes—
95#13095490C2314yes—
96#13095490C2314yes—
97#13095490C2314yes—
98C22⋊Dic97C2312yes—
99C22⋊Dic97C2314yes—
100C22⋊Dic97C2314yes—
101C22⋊Dic97C2314yes—
102C22⋊Dic97C2314yes—
103C22⋊Dic97C2314yes—
104#13095488C2312yes—
105#13095488C2312yes—
106#13095488C2314yes—
107#13095488C2314yes—
108C4×Dic97C2312yes—
109C4×Dic97C2312yes—
110C4×Dic97C2311yescharacteristic
111C4×Dic97C2311yescharacteristic
112C4×Dic97C2314yes—
113C4×Dic97C2314yes—
114C4×Dic97C2314yes—
115C2×D4×C97C2312yes—
116C2×D4×C97C2314yes—
117C2×D4×C97C2314yes—
118C22×C4×C97C2311yescharacteristic
119C22×C4×C97C2314yes—
120C4⋊C4×C97C2312yes—
121C4⋊C4×C97C2312yes—
122C4⋊C4×C97C2314yes—
123C22⋊C4×C97C2312yes—
124C22⋊C4×C97C2314yes—
125C22⋊C4×C97C2314yes—
126C42×C97C2311yescharacteristic
127C42×C97C2311yescharacteristic
128C23×C97C2411yescharacteristic
129C23×C97C2414yes—
130C22×D97—22no—
131C22×D97—42no—
132C22×D97C2411yescharacteristic
133C22×D97C2411yescharacteristic
134D4×C97—24no—
135D4×C97—44no—
136D4×C97—44no—
137C2×C4×C97—22no—
138C2×C4×C97—44no—
139C2×C4×C97C2412yes—
140C2×C4×C97C2412yes—
141C2×C4×C97C2411yescharacteristic
142C2×C4×C97C2414yes—
143C2×C4×C97C2411yescharacteristic
144C22⋊D97—28no—
145C22⋊D97—44no—
146C2×Dic97—22no—
147C2×Dic97—28no—
148C2×Dic97—44no—
149C2×Dic97C2412yes—
150C2×Dic97C2412yes—
151C2×Dic97C2414yes—
152C2×Dic97C2414yes—
153C2×Dic97C2411yescharacteristic
154C2×Dic97C2411yescharacteristic
155C4×D97—22no—
156C4×D97—22no—
157C4×D97—22no—
158C4×D97—42no—
159C4×D97—42no—
160C4×D97#4114yes—
161Dic194—24no—
162Dic194—44no—
163C22×C97—24no—
164C22×C97—21no—
165C22×C97—24no—
166C22×C97—24no—
167C22×C97—22no—
168C22×C97C2511yescharacteristic
169C2×D97—22no—
170C2×D97—21no—
171C2×D97—22no—
172C2×D97—44no—
173C2×D97C2×#4112yes—
174C4×C97—21no—
175C4×C97—24no—
176C4×C97—22no—
177C4×C97—22no—
178C4×C97C2×#4112yes—
179Dic97—24no—
180Dic97—21no—
181Dic97—22no—
182Dic97—24no—
183Dic97—22no—
184Dic97C2×#4112yes—
185C2×C97—41no—
186C2×C97—44no—
187C2×C97C22×#4111yescharacteristic
188C2×C97#58512yes—
189D97—22no—
190D97—41no—
191#3455—971nomaximal, Sylow
192C97#345511yescharacteristic, complemented, Sylow
193#542—972no—
194#542—972no—
195#541—971no—
196#541—971no—
197#540—971no—
198#540D9711yescharacteristic, complemented
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⋊Q8—971no—
212#127C2×D9711yescharacteristic, complemented
213#126—974no—
214#126—974no—
215#126—974no—
216#126—974no—
217#125—974no—
218#125—974no—
219#125—971no—
220#125C2×D9711yescharacteristic, complemented
221#124—974no—
222#124—972no—
223#124—974no—
224#123—974no—
225#123—974no—
226#123—974no—
227#123—974no—
228#123C2×D9714yescomplemented
229C22⋊Q8—972no—
230C22⋊Q8—974no—
231C22⋊Q8—972no—
232C22⋊Q8—974no—
233C22⋊Q8—974no—
234#121—972no—
235#121—974no—
236#121—974no—
237#121C2×D9712yes—
238#121C2×D9714yes—
239#120—974no—
240C4×Q8—972no—
241C4×Q8—974no—
242C4×Q8—972no—
243C4×D4—972no—
244C4×D4—974no—
245C4×D4—974no—
246C4×D4—974no—
247C4×D4—974no—
248C4×D4—974no—
249C4×D4C2×D9712yescomplemented
250#117—971no—
251#117—972no—
252#117—972no—
253#117—972no—
254#117—974no—
255#117—971no—
256#117—971no—
257#117C2×D9711yescharacteristic
258C2×C4⋊C4—972no—
259C2×C22⋊C4—972no—
260C2×C42—974no—
261C2×C42—971no—
262C2×C42—971no—
263C24—971no—
264#41—1948no—
265C2×Q8—974no—
266C2×Q8—972no—
267C2×D4—974no—
268C2×D4—974no—
269C2×D4C22×D9712yes—
270C2×D4C22×D9714yes—
271C2×D4C22×D9714yes—
272C22×C4—971no—
273C22×C4—974no—
274C22×C4—974no—
275C22×C4—971no—
276C22×C4—972no—
277C22×C4—972no—
278C22×C4—971no—
279C22×C4—971no—
280C22×C4—971no—
281C22×C4—1944no—
282C22×C4C22×D9711yescharacteristic
283C22×C4C22×D9714yes—
284C4⋊C4—972no—
285C4⋊C4—972no—
286C4⋊C4—974no—
287C4⋊C4—974no—
288C4⋊C4—974no—
289C4⋊C4—974no—
290C4⋊C4—974no—
291C4⋊C4—972no—
292C4⋊C4—972no—
293C4⋊C4—974no—
294C4⋊C4—974no—
295C4⋊C4C22×D9712yes—
296C4⋊C4C22×D9712yes—
297C4⋊C4C22×D9714yes—
298C22⋊C4—972no—
299C22⋊C4—974no—
300C22⋊C4—974no—
301C22⋊C4—974no—
302C22⋊C4—974no—
303C22⋊C4—974no—
304C22⋊C4—972no—
305C22⋊C4—972no—
306C22⋊C4—974no—
307C22⋊C4—974no—
308C22⋊C4C22×D9712yes—
309C22⋊C4C22×D9714yes—
310C22⋊C4C22×D9714yes—
311C42—972no—
312C42—972no—
313C42—974no—
314C42—971no—
315C42—971no—
316C42—974no—
317C42—974no—
318C42C22×D9711yescharacteristic
319C42C22×D9711yescharacteristic
320C23—971no—
321C23—971no—
322C23—3882no—
323C23—1942no—
324C23C23×D9711yescharacteristic
325C23C23×D9714yes—
326Q8—3884no—
327Q8—1944no—
328D4—24no—
329D4—44no—
330D4—44no—
331D4—3884no—
332D4—1948no—
333C2×C4—22no—
334C2×C4—44no—
335C2×C4—971no—
336C2×C4—974no—
337C2×C4—972no—
338C2×C4—972no—
339C2×C4—974no—
340C2×C4—974no—
341C2×C4—971no—
342C2×C4—3882no—
343C2×C4—3882no—
344C2×C4—3884no—
345C2×C4—1942no—
346C2×C4—1942no—
347C2×C4—1942no—
348C2×C4—1948no—
349C2×C4—1942no—
350C2×C4C23×D9712yes—
351C2×C4C23×D9714yes—
352C2×C4C23×D9712yes—
353C2×C4C23×D9711yescharacteristic
354C2×C4C23×D9711yescharacteristic
355C22—21no—
356C22—24no—
357C22—24no—
358C22—24no—
359C22—22no—
360C22—972no—
361C22—3884no—
362C22—1941no—
363C22—1942no—
364C22—1942no—
365C22C24×D9711yescharacteristic
366C4—21no—
367C4—22no—
368C4—24no—
369C4—22no—
370C4—972no—
371C4—1941no—
372C4—1944no—
373C4—1942no—
374C4—1944no—
375C4—1942no—
376C4#41×D9712yes—
377C2—41no—
378C2—44no—
379C2—3881no—
380C2—1942no—
381C2C2×#41×D9711yescharacteristic
382C2#2026157012yes—
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#540×C97C2
hypercenter#540D97
hypocenterC97#3455
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #20261352C2semidirect
2. #20261350C2semidirect
3. #20261348C2semidirect
4. #20261347C2semidirect
5. #20261339C2semidirect
6. #20261328C2semidirect
7. #20261317C2semidirect
8. #20261291C2semidirect
9. #20261283C2semidirect
10. #20261266C2semidirect
11. #20261262C2semidirect
12. #20261245C2semidirect
13. C4×#13095510C2semidirect
14. #117×D97C2semidirect
15. #540×C97C2semidirect
18. #15690810C22semidirect
19. #15690809C22semidirect
22. #15690803C22semidirect
23. #15690803C22semidirect
24. #15690802C22semidirect
25. #15690801C22semidirect
26. #15690800C22semidirect
28. #15690797C22semidirect
33. #15690791C22semidirect
36. #15690789C22semidirect
38. #15690787C22semidirect
43. C4×Dic194C22semidirect
44. #15690775C22semidirect
45. #15690775C22semidirect
47. D4×Dic97C22semidirect
48. D4×Dic97C22semidirect
58. #15690766C22semidirect
61. C2×C4×Dic97C22semidirect
64. #127×C97C22semidirect
65. #125×C97C22semidirect
66. #123×C97C22semidirect
69. C4×D4×C97C22semidirect
192. C97#3455semidirect
198. #540D97semidirect
212. #127C2×D97semidirect
220. #125C2×D97semidirect
228. #123C2×D97semidirect
249. C4×D4C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613522
#202613502
#202613482
#202613472
#202613392
#202613282
#202613172
#202612912
#202612832
#202612662
#202612622
#202612452
C4×#130955102
#117×D972
#540×C972
#345597