Finite Group Databasebeta

← #27153534

Subgroups — #27153534

Order 12416.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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#20261540C211yesmaximal, characteristic, complemented
2#20261539C212yesmaximal, complemented
3#20261536C211yesmaximal, characteristic, complemented
4#20261110C211yesmaximal, characteristic, complemented
5#20261006C211yesmaximal, characteristic, complemented
6#20261004C212yesmaximal, complemented
7#20260905C212yesmaximal, complemented
8#20260901C212yesmaximal, complemented
9#20260851C211yesmaximal, characteristic, complemented
10#20260850C211yesmaximal, characteristic, complemented
11#472×C97C211yesmaximal, characteristic, complemented
12#15690856C2212yescomplemented
13#15690855C2212yescomplemented
14#15690838—22no—
15#15690838C2212yescomplemented
16#15690838C2212yescomplemented
17#15690837—22no—
18#15690836C2212yescomplemented
19#15690834—22no—
20#15690834—22no—
21#15690832—21no—
22#15690832C2211yescharacteristic
23#15690831—22no—
24#15690830C2211yescharacteristic
25#15690829—21no—
26C2×D776—21no—
27C2×D776C2211yescharacteristic, complemented
28C2×#13095496C2212yescomplemented
29D8×D97—21no—
30D8×D97—22no—
31#15690818C2212yescomplemented
32#15690816C2211yescharacteristic, complemented
33#15690806C2212yescomplemented
34#15690745C2211yescharacteristic, complemented
35#15690737C2211yescharacteristic, complemented
36#15690736C2212yescomplemented
37#15690736C2211yescharacteristic, complemented
38#15690726C2212yescomplemented
39C8×Dic97C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C97C2212yescomplemented
41C2×D8×C97C2211yescharacteristic, complemented
42#131×C97C2211yescharacteristic, complemented
43#108×C97C2211yescharacteristic, complemented
44#100×C97C2212yescomplemented
45#13095512—22no—
46#13095511—22no—
47#13095511C2311yescharacteristic
48#13095511C2312yes—
49#13095510—22no—
50#13095510—22no—
51C2×D388—22no—
52C2×D388C2312yes—
53C2×C22⋊D97—42no—
54C2×C22⋊D97—42no—
55D4×D97—22no—
56D4×D97—22no—
57D4×D97—22no—
58D4×D97—42no—
59Dic388D411yescharacteristic, complemented
60#13095501D411yescharacteristic, complemented
61#13095500—22no—
62#13095500—22no—
63#13095500D412yescomplemented
64#13095499—22no—
65#13095499D411yescharacteristic, complemented
66#13095498D411yescharacteristic, complemented
67D776—22no—
68D776—22no—
69D776—21no—
70D776—21no—
71D776D411yescharacteristic, complemented
72#13095496—22no—
73#13095496—22no—
74#13095496—22no—
75#13095496D411yescharacteristic, complemented
76#13095495C2311yescharacteristic
77#13095494—21no—
78#13095494—22no—
79C2×#11715895C2311yescharacteristic
80C8×D97—21no—
81C8×D97—22no—
82C4×Dic97C2311yescharacteristic
83#41×C97C2311yescharacteristic
84C2×D4×C97C2312yes—
85SD16×C97—22no—
86D8×C97—22no—
87D8×C97—21no—
88D8×C97—21no—
89M16×C97—21no—
90M16×C97C2312yes—
91M16×C97C2311yescharacteristic
92C2×C8×C97—21no—
93C2×C8×C97C2311yescharacteristic
94C23×C97—22no—
95C22×D97—22no—
96C22×D97—42no—
97C22×D97—42no—
98Q8×C97C2×D411yescharacteristic
99D4×C97—21no—
100D4×C97—22no—
101D4×C97—22no—
102D4×C97C2×D411yescharacteristic
103C2×C4×C97—21no—
104C2×C4×C97C2411yescharacteristic
105C22⋊D97—22no—
106C22⋊D97—21no—
107C22⋊D97—42no—
108C22⋊D97—42no—
109C22⋊D97—42no—
110C22⋊D97—84no—
111C2×Dic97—21no—
112C2×Dic97—42no—
113D388—22no—
114D388—22no—
115D388—42no—
116D388C2×D412yes—
117D388C2×D411yescharacteristic
118C4×D97—21no—
119C4×D97—22no—
120C4×D97—42no—
121Dic194C2×D412yes—
122Dic194C2×D411yescharacteristic
123C8×C97—21no—
124C8×C97—22no—
125C8×C97C2×D411yescharacteristic
126C8×C97C2×D411yescharacteristic
127#11715895—21no—
128#11715895C2×D411yescharacteristic
129#11715895C2×D411yescharacteristic
130C22×C97—21no—
131C22×C97—42no—
132C22×C97—82no—
133C22×C97H(5,𝔽2)11yescharacteristic
134C2×D97—22no—
135C2×D97—21no—
136C2×D97—42no—
137C2×D97—82no—
138C2×D97—162no—
139C2×D97—162no—
140C4×C97—21no—
141C4×C97C22×D411yescharacteristic
142C4×C97C22×D411yescharacteristic
143Dic97—22no—
144Dic97—22no—
145Dic97—21no—
146C2×C97—21no—
147C2×C97—41no—
148C2×C97—82no—
149C2×C97D4211yescharacteristic
150D97—42no—
151D97—41no—
152D97—82no—
153#3292—971nomaximal, Sylow
154C97#329211yescharacteristic, complemented, Sylow
155#576—972no—
156#576—971no—
157#576—971no—
158#493—971no—
159#472—971no—
160#472—972no—
161#472D9711yescharacteristic, complemented
162#453—972no—
163#453—972no—
164#443—971no—
165#443—971no—
166H(5,𝔽2)—972no—
167H(5,𝔽2)—972no—
168AGL(1,ℤ/8)—972no—
169AGL(1,ℤ/8)—972no—
170AGL(1,ℤ/8)—972no—
171AGL(1,ℤ/8)—1942no—
172AGL(1,ℤ/8)—1942no—
173AGL(1,ℤ/8)—1942no—
174AGL(1,ℤ/8)—1942no—
175AGL(1,ℤ/8)C2×D9712yescomplemented
176#135—971no—
177#135—971no—
178#135—1941no—
179#135—1941no—
180#135—1942no—
181C2×D8—972no—
182C2×D8—971no—
183C2×D8—1941no—
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#108C2×D9711yescharacteristic, complemented
193#104—971no—
194#104—972no—
195#104—971no—
196#100—972no—
197#100C2×D9712yescomplemented
198C4×C8—971no—
199#41—971no—
200#41—972no—
201#41—1942no—
202#41—1942no—
203#41—1942no—
204#41—1942no—
205#41C22×D9711yescharacteristic
206C2×D4—972no—
207C2×D4—3882no—
208C2×D4—3882no—
209C2×D4—3882no—
210C2×D4—1942no—
211C2×D4—1942no—
212C2×D4—1942no—
213C2×D4—1942no—
214C2×D4C22×D9712yes—
215Q16—971no—
216Q16—971no—
217SD16—22no—
218SD16—971no—
219SD16—971no—
220SD16—972no—
221SD16—1942no—
222SD16—1942no—
223SD16—1942no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—971no—
228D8—971no—
229D8—1942no—
230D8—1942no—
231D8—1942no—
232D8—1942no—
233D8—1942no—
234D8—1941no—
235D8—1941no—
236M16—21no—
237M16—971no—
238M16—1942no—
239M16—1941no—
240M16C22×D9712yes—
241M16C22×D9711yescharacteristic
242C2×C8—21no—
243C2×C8—971no—
244C2×C8—1942no—
245C2×C8—1941no—
246C2×C8C22×D9711yescharacteristic
247C42—971no—
248C23—22no—
249C23—3882no—
250C23—3882no—
251C23—1942no—
252Q8—971no—
253Q8—972no—
254Q8D4×D9711yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—7764no—
259D4—971no—
260D4—972no—
261D4—3882no—
262D4—3882no—
263D4—3882no—
264D4—3882no—
265D4—1942no—
266D4—1941no—
267D4—1942no—
268D4—1942no—
269D4D4×D9711yescharacteristic
270C2×C4—21no—
271C2×C4—3882no—
272C2×C4—3882no—
273C2×C4—1941no—
274C2×C4—1942no—
275C2×C4—1941no—
276C2×C4C23×D9711yescharacteristic
277C8—21no—
278C8—22no—
279C8—971no—
280C8—971no—
281C8—1941no—
282C8D4×D9711yescharacteristic
283C8D4×D9711yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—7762no—
288C22—15522no—
289C22—15522no—
290C22—3882no—
291C22—1942no—
292C22—1941no—
293C22#1569085611yescharacteristic
294C4—21no—
295C4—1942no—
296C4—1942no—
297C4—1941no—
298C4C2×D4×D9711yescharacteristic
299C4C2×D4×D9711yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—7762no—
304C2—3882no—
305C2—3881no—
306C2#2026136911yescharacteristic
subgroupG/H
centerC2#20261369
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#472×C97C2
socleC2×C97D42
hypercenter#472D97
hypocenterC97#3292
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #20261540C2semidirect
2. #20261539C2semidirect
3. #20261536C2semidirect
4. #20261110C2semidirect
5. #20261006C2semidirect
6. #20261004C2semidirect
7. #20260905C2semidirect
8. #20260901C2semidirect
9. #20260851C2semidirect
10. #20260850C2semidirect
11. #472×C97C2semidirect
12. #15690856C22semidirect
13. #15690855C22semidirect
15. #15690838C22semidirect
16. #15690838C22semidirect
18. #15690836C22semidirect
27. C2×D776C22semidirect
28. C2×#13095496C22semidirect
31. #15690818C22semidirect
32. #15690816C22semidirect
33. #15690806C22semidirect
34. #15690745C22semidirect
35. #15690737C22semidirect
36. #15690736C22semidirect
37. #15690736C22semidirect
38. #15690726C22semidirect
39. C8×Dic97C22semidirect
40. AGL(1,ℤ/8)×C97C22semidirect
41. C2×D8×C97C22semidirect
42. #131×C97C22semidirect
43. #108×C97C22semidirect
44. #100×C97C22semidirect
59. Dic388D4semidirect
60. #13095501D4semidirect
63. #13095500D4semidirect
65. #13095499D4semidirect
66. #13095498D4semidirect
71. D776D4semidirect
75. #13095496D4semidirect
154. C97#3292semidirect
161. #472D97semidirect
175. AGL(1,ℤ/8)C2×D97semidirect
186. C2×D8C2×D97semidirect
189. #131C2×D97semidirect
192. #108C2×D97semidirect
197. #100C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615402
#202615392
#202615362
#202611102
#202610062
#202610042
#202609052
#202609012
#202608512
#202608502
#472×C972
#329297