Finite Group Databasebeta

← #27153533

Subgroups — #27153533

Order 12416.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#20261538C212yesmaximal, complemented
2#20261537C211yesmaximal, characteristic, complemented
3#20261110C211yesmaximal, characteristic, complemented
4#20261005C211yesmaximal, characteristic, complemented
5#20261003C212yesmaximal, complemented
6#20261002C211yesmaximal, characteristic, complemented
7#20260903C212yesmaximal, complemented
8#20260901C212yesmaximal, complemented
9#20260852C211yesmaximal, characteristic, complemented
10#20260850C211yesmaximal, characteristic, complemented
11#576×C97C211yesmaximal, characteristic, complemented
12#15690856C2212yescomplemented
13#15690837—22no—
14#15690836—22no—
15#15690836C2212yescomplemented
16#15690835—22no—
17#15690835C2212yescomplemented
18#15690835C2212yescomplemented
19#15690833—21no—
20#15690830—22no—
21#15690830C2211yescharacteristic
22C2×D776C2211yescharacteristic, complemented
23C2×#13095496—22no—
24C2×#13095496C2212yescomplemented
25D8×D97—21no—
26#15690817C2212yescomplemented
27#15690816C2211yescharacteristic, complemented
28#15690806C2212yescomplemented
29#15690746C2211yescharacteristic, complemented
30#15690745C2211yescharacteristic, complemented
31#15690737C2212yescomplemented
32#15690737C2211yescharacteristic, complemented
33#15690736C2211yescharacteristic, complemented
34#15690727C2212yescomplemented
35#15690726C2212yescomplemented
36C8×Dic97C2211yescharacteristic, complemented
37H(5,𝔽2)×C97C2212yescomplemented
38AGL(1,ℤ/8)×C97—22no—
39AGL(1,ℤ/8)×C97C2212yescomplemented
40#135×C97—21no—
41#135×C97C2211yescharacteristic
42C2×D8×C97—21no—
43C2×D8×C97C2211yescharacteristic, complemented
44#131×C97C2211yescharacteristic, complemented
45#13095512—22no—
46#13095511—22no—
47#13095511C2311yescharacteristic
48C2×D388—22no—
49C2×D388C2312yes—
50C2×C22⋊D97—42no—
51D4×D97—22no—
52D4×D97—42no—
53#13095501D411yescharacteristic, complemented
54#13095499—22no—
55#13095499D411yescharacteristic, complemented
56#13095498—22no—
57#13095498—22no—
58#13095498D411yescharacteristic, complemented
59D776—21no—
60D776—21no—
61#13095496—22no—
62#13095496—22no—
63#13095496—22no—
64#13095496—22no—
65#13095496—22no—
66#13095496D411yescharacteristic, complemented
67#13095495—22no—
68#13095495C2311yescharacteristic
69#13095495C2312yes—
70#13095494—21no—
71C2×#11715895—22no—
72C2×#11715895C2311yescharacteristic
73C8×D97—21no—
74C4×Dic97C2311yescharacteristic
75#41×C97—22no—
76#41×C97—22no—
77#41×C97C2312yes—
78#41×C97C2311yescharacteristic
79C2×D4×C97—22no—
80C2×D4×C97—22no—
81C2×D4×C97—42no—
82C2×D4×C97C2312yes—
83Q16×C97D411yescharacteristic, complemented
84SD16×C97—22no—
85SD16×C97D412yescomplemented
86D8×C97—22no—
87D8×C97—21no—
88D8×C97—21no—
89D8×C97D411yescharacteristic, complemented
90M16×C97—21no—
91M16×C97C2311yescharacteristic
92C2×C8×C97—21no—
93C2×C8×C97C2311yescharacteristic
94C23×C97—22no—
95C23×C97—42no—
96C22×D97—22no—
97C22×D97—42no—
98Q8×C97C2×D411yescharacteristic
99Q8×C97C2×D412yes—
100D4×C97—21no—
101D4×C97—22no—
102D4×C97—22no—
103D4×C97—22no—
104D4×C97—42no—
105D4×C97—42no—
106D4×C97C2×D411yescharacteristic
107D4×C97C2×D412yes—
108C2×C4×C97—21no—
109C2×C4×C97—22no—
110C2×C4×C97—42no—
111C2×C4×C97C2411yescharacteristic
112C22⋊D97—21no—
113C22⋊D97—42no—
114C22⋊D97—84no—
115C2×Dic97—21no—
116D388—22no—
117D388—22no—
118D388—42no—
119D388C2×D411yescharacteristic
120C4×D97—21no—
121C4×D97—42no—
122Dic194C2×D411yescharacteristic
123C8×C97—21no—
124C8×C97C2×D411yescharacteristic
125C8×C97C2×D411yescharacteristic
126#11715895—21no—
127#11715895—22no—
128#11715895C2×D411yescharacteristic
129#11715895C2×D411yescharacteristic
130C22×C97—22no—
131C22×C97—21no—
132C22×C97—42no—
133C22×C97—82no—
134C22×C97—162no—
135C22×C97H(5,𝔽2)11yescharacteristic
136C2×D97—21no—
137C2×D97—42no—
138C2×D97—82no—
139C2×D97—162no—
140C4×C97—22no—
141C4×C97—21no—
142C4×C97C22×D411yescharacteristic
143C4×C97C22×D411yescharacteristic
144Dic97—22no—
145Dic97—21no—
146C2×C97—21no—
147C2×C97—42no—
148C2×C97—41no—
149C2×C97—82no—
150C2×C97D4211yescharacteristic
151D97—41no—
152D97—82no—
153#3292—971nomaximal, Sylow
154C97#329211yescharacteristic, complemented, Sylow
155#576—972no—
156#576—971no—
157#576D9711yescharacteristic, complemented
158#493—971no—
159#472—972no—
160#472—971no—
161#472—971no—
162#453—972no—
163#453—972no—
164#443—971no—
165#443—971no—
166H(5,𝔽2)—972no—
167H(5,𝔽2)C2×D9712yescomplemented
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—972no—
170AGL(1,ℤ/8)—972no—
171AGL(1,ℤ/8)—972no—
172AGL(1,ℤ/8)—1942no—
173AGL(1,ℤ/8)—1942no—
174AGL(1,ℤ/8)—1942no—
175AGL(1,ℤ/8)C2×D9712yescomplemented
176#135—21no—
177#135—971no—
178#135—1942no—
179#135—1941no—
180#135C2×D9711yescharacteristic
181C2×D8—21no—
182C2×D8—972no—
183C2×D8—971no—
184C2×D8—1942no—
185C2×D8—1941no—
186C2×D8C2×D9711yescharacteristic, complemented
187#131—972no—
188#131—971no—
189#131C2×D9711yescharacteristic, complemented
190#127—972no—
191#108—971no—
192#108—971no—
193#104—971no—
194#104—972no—
195#104—971no—
196#100—972no—
197#100—972no—
198C4×C8—971no—
199#41—22no—
200#41—22no—
201#41—971no—
202#41—1942no—
203#41—1942no—
204#41C22×D9711yescharacteristic
205#41C22×D9712yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—972no—
210C2×D4—3882no—
211C2×D4—3882no—
212C2×D4—1942no—
213C2×D4—1942no—
214C2×D4C22×D9712yes—
215Q16—971no—
216Q16C22⋊D9711yescharacteristic, complemented
217SD16—22no—
218SD16—971no—
219SD16—971no—
220SD16—1942no—
221SD16—1942no—
222SD16—1942no—
223SD16C22⋊D9712yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—971no—
228D8—1942no—
229D8—1942no—
230D8—1942no—
231D8—1942no—
232D8—1942no—
233D8—1941no—
234D8—1941no—
235D8C22⋊D9711yescharacteristic, complemented
236M16—21no—
237M16—972no—
238M16—971no—
239M16—1942no—
240M16—1941no—
241M16C22×D9711yescharacteristic
242C2×C8—21no—
243C2×C8—971no—
244C2×C8—1942no—
245C2×C8—1941no—
246C2×C8C22×D9711yescharacteristic
247C42—971no—
248C23—22no—
249C23—42no—
250C23—3882no—
251C23—1942no—
252Q8—971no—
253Q8D4×D9711yescharacteristic
254Q8C2×C22⋊D9712yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—7764no—
262D4—971no—
263D4—3882no—
264D4—3882no—
265D4—1941no—
266D4—1942no—
267D4—1942no—
268D4D4×D9711yescharacteristic
269D4C2×C22⋊D9712yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—3882no—
274C2×C4—1941no—
275C2×C4—1941no—
276C2×C4C23×D9711yescharacteristic
277C8—21no—
278C8—971no—
279C8—971no—
280C8—1942no—
281C8—1941no—
282C8C2×C22⋊D9711yescharacteristic
283C8C2×C22⋊D9711yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—7762no—
289C22—162no—
290C22—15522no—
291C22—3882no—
292C22—1941no—
293C22#1569085511yescharacteristic
294C4—21no—
295C4—22no—
296C4—1942no—
297C4—1941no—
298C4C2×D4×D9711yescharacteristic
299C4C22×C22⋊D9711yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—7762no—
305C2—3881no—
306C2D4×C22⋊D9711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#576×C97C2
socleC2×C97D42
hypercenter#576D97
hypocenterC97#3292
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #20261538C2semidirect
2. #20261537C2semidirect
3. #20261110C2semidirect
4. #20261005C2semidirect
5. #20261003C2semidirect
6. #20261002C2semidirect
7. #20260903C2semidirect
8. #20260901C2semidirect
9. #20260852C2semidirect
10. #20260850C2semidirect
11. #576×C97C2semidirect
12. #15690856C22semidirect
15. #15690836C22semidirect
17. #15690835C22semidirect
18. #15690835C22semidirect
22. C2×D776C22semidirect
24. C2×#13095496C22semidirect
26. #15690817C22semidirect
27. #15690816C22semidirect
28. #15690806C22semidirect
29. #15690746C22semidirect
30. #15690745C22semidirect
31. #15690737C22semidirect
32. #15690737C22semidirect
33. #15690736C22semidirect
34. #15690727C22semidirect
35. #15690726C22semidirect
36. C8×Dic97C22semidirect
37. H(5,𝔽2)×C97C22semidirect
39. AGL(1,ℤ/8)×C97C22semidirect
43. C2×D8×C97C22semidirect
44. #131×C97C22semidirect
53. #13095501D4semidirect
55. #13095499D4semidirect
58. #13095498D4semidirect
66. #13095496D4semidirect
83. Q16×C97D4semidirect
85. SD16×C97D4semidirect
89. D8×C97D4semidirect
154. C97#3292semidirect
157. #576D97semidirect
167. H(5,𝔽2)C2×D97semidirect
175. AGL(1,ℤ/8)C2×D97semidirect
186. C2×D8C2×D97semidirect
189. #131C2×D97semidirect
216. Q16C22⋊D97semidirect
223. SD16C22⋊D97semidirect
235. D8C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202615382
#202615372
#202611102
#202610052
#202610032
#202610022
#202609032
#202609012
#202608522
#202608502
#576×C972
#329297