Finite Group Databasebeta

← #27153525

Subgroups — #27153525

Order 12416.

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

normal subgroups only

This group has 454 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#20261548C211yesmaximal, characteristic, complemented
2#20261544C212yesmaximal, complemented
3#20261540C211yesmaximal, characteristic, complemented
4#20261120C211yesmaximal, characteristic, complemented
5#20261006C211yesmaximal, characteristic, complemented
6#20261001C212yesmaximal, complemented
7#20260911C212yesmaximal, complemented
8#20260910C212yesmaximal, complemented
9#20260851C211yesmaximal, characteristic, complemented
10#20260850C211yesmaximal, characteristic, complemented
11#473×C97C211yesmaximal, characteristic, complemented
12#15690858C2212yescomplemented
13#15690856C2212yes—
14#15690842—22no—
15#15690842—22no—
16#15690841C2212yescomplemented
17#15690838—22no—
18#15690838C2212yes—
19#15690838C2212yescomplemented
20#15690837—22no—
21#15690833—22no—
22#15690833—21no—
23#15690832—21no—
24#15690832C2211yescharacteristic
25#15690830C2211yescharacteristic
26Q16×D97—21no—
27C2×#13095498C2212yescomplemented
28SD16×D97—22no—
29C2×D776—21no—
30C2×D776C2211yescharacteristic
31#15690818C2212yescomplemented
32#15690816C2211yescharacteristic, complemented
33#15690799C2212yescomplemented
34#15690745C2211yescharacteristic, complemented
35#15690737C2211yescharacteristic
36#15690736C2211yescharacteristic, complemented
37#15690736C2212yescomplemented
38#15690726C2212yescomplemented
39C8×Dic97C2211yescharacteristic, complemented
40#137×C97C2212yes—
41C2×Q16×C97C2211yescharacteristic, complemented
42#131×C97C2211yescharacteristic
43#108×C97C2211yescharacteristic, complemented
44#101×C97C2212yescomplemented
45#13095512—22no—
46#13095512—22no—
47#13095512—22no—
48#13095511—22no—
49#13095511—42no—
50#13095511C2311yescharacteristic
51#13095511C2312yes—
52Q8×D97—22no—
53Q8×D97—22no—
54C2×D388—22no—
55C2×D388C2312yes—
56D4×D97—22no—
57D4×D97—42no—
58Dic388D411yescharacteristic
59#13095501—22no—
60#13095501D411yescharacteristic
61#13095500—22no—
62#13095500—22no—
63#13095500D412yes—
64#13095499D411yescharacteristic
65#13095498—22no—
66#13095498—22no—
67#13095498—22no—
68#13095498D411yescharacteristic
69D776—22no—
70D776—22no—
71D776—21no—
72D776—21no—
73D776D411yescharacteristic
74#13095496D411yescharacteristic
75#13095495C2311yescharacteristic
76#13095494—21no—
77#13095494—22no—
78C2×#11715895C2311yescharacteristic
79C8×D97—21no—
80C8×D97—22no—
81#13095488—42no—
82C4×Dic97C2311yescharacteristic
83#41×C97C2311yescharacteristic
84C2×Q8×C97C2312yes—
85Q16×C97—22no—
86Q16×C97—21no—
87Q16×C97—21no—
88SD16×C97—22no—
89M16×C97—21no—
90M16×C97C2312yes—
91M16×C97C2311yescharacteristic
92C2×C8×C97—21no—
93C2×C8×C97C2311yescharacteristic
94C22×D97—22no—
95C22×D97—42no—
96Q8×C97—22no—
97Q8×C97—22no—
98Q8×C97C2×D411yescharacteristic
99D4×C97—21no—
100D4×C97C2×D411yescharacteristic
101C2×C4×C97—21no—
102C2×C4×C97—22no—
103C2×C4×C97C2411yescharacteristic
104C22⋊D97—22no—
105C22⋊D97—21no—
106C22⋊D97—42no—
107C2×Dic97—21no—
108D388—22no—
109D388—22no—
110D388—42no—
111D388—42no—
112D388C2×D411yescharacteristic
113D388C2×D412yes—
114C4×D97—22no—
115C4×D97—21no—
116C4×D97—42no—
117C4×D97—42no—
118C4×D97—42no—
119Dic194—42no—
120Dic194C2×D411yescharacteristic
121Dic194C2×D412yes—
122C8×C97—21no—
123C8×C97—22no—
124C8×C97C2×D411yescharacteristic
125C8×C97C2×D411yescharacteristic
126#11715895—21no—
127#11715895C2×D411yescharacteristic
128#11715895C2×D411yescharacteristic
129C22×C97—21no—
130C22×C97H(5,𝔽2)11yescharacteristic
131C2×D97—21no—
132C2×D97—22no—
133C2×D97—42no—
134C2×D97—82no—
135C2×D97—162no—
136C4×C97—21no—
137C4×C97—42no—
138C4×C97C22×D411yescharacteristic
139C4×C97C22×D411yescharacteristic
140Dic97—21no—
141Dic97—22no—
142Dic97—41no—
143C2×C97—21no—
144C2×C97—41no—
145C2×C97D4211yescharacteristic
146D97—41no—
147D97—42no—
148D97—82no—
149#3290—971nomaximal, Sylow
150C97#329011yescharacteristic, complemented, Sylow
151#578—971no—
152#577—972no—
153#576—971no—
154#495—971no—
155#473D9711yescharacteristic, complemented
156#472—971no—
157#471—972no—
158#455—972no—
159#454—972no—
160#443—971no—
161#443—971no—
1622−1+4—972no—
163H(5,𝔽2)—972no—
164#137—972no—
165#137—1942no—
166#137—1942no—
167#137C2×D9712yes—
168AGL(1,ℤ/8)—972no—
169AGL(1,ℤ/8)—972no—
170AGL(1,ℤ/8)—1942no—
171AGL(1,ℤ/8)—1942no—
172#135—971no—
173#135—971no—
174#135—1941no—
175#135—1941no—
176#135—1942no—
177C2×Q16—1941no—
178C2×Q16C2×D9711yescharacteristic, complemented
179C2×SD16—972no—
180C2×SD16—1942no—
181C2×D8—971no—
182C2×D8—1941no—
183#131—972no—
184#131—971no—
185#131C2×D9711yescharacteristic
186#124—972no—
187#108—971no—
188#108C2×D9711yescharacteristic, complemented
189#104—971no—
190#104—971no—
191#104—972no—
192#101C2×D9712yescomplemented
193#100—972no—
194C4×C8—971no—
195#41—971no—
196#41—972no—
197#41—3882no—
198#41—1942no—
199#41—1942no—
200#41—1942no—
201#41—1942no—
202#41C22×D9711yescharacteristic
203C2×Q8—1942no—
204C2×Q8—1942no—
205C2×Q8C22×D9712yes—
206C2×D4—972no—
207C2×D4—3882no—
208C2×D4—1942no—
209C2×D4—1942no—
210Q16—22no—
211Q16—21no—
212Q16—21no—
213Q16—971no—
214Q16—971no—
215Q16—1942no—
216SD16—22no—
217SD16—971no—
218SD16—971no—
219SD16—972no—
220SD16—1942no—
221SD16—1942no—
222SD16—1942no—
223SD16—1942no—
224SD16—1942no—
225D8—971no—
226D8—971no—
227D8—1942no—
228D8—1942no—
229D8—1941no—
230D8—1941no—
231M16—21no—
232M16—971no—
233M16—1942no—
234M16—1941no—
235M16C22×D9712yes—
236M16C22×D9711yescharacteristic
237C2×C8—21no—
238C2×C8—971no—
239C2×C8—1942no—
240C2×C8—1941no—
241C2×C8C22×D9711yescharacteristic
242C22⋊C4—3882no—
243C42—971no—
244C23—3882no—
245C23—1942no—
246Q8—22no—
247Q8—22no—
248Q8—971no—
249Q8—972no—
250Q8—3882no—
251Q8D4×D9711yescharacteristic
252D4—21no—
253D4—971no—
254D4—972no—
255D4—3882no—
256D4—3882no—
257D4—3882no—
258D4—1942no—
259D4—1941no—
260D4—1942no—
261D4—1942no—
262D4D4×D9711yescharacteristic
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—3882no—
266C2×C4—3882no—
267C2×C4—3882no—
268C2×C4—1941no—
269C2×C4—1942no—
270C2×C4—1941no—
271C2×C4C23×D9711yescharacteristic
272C8—21no—
273C8—22no—
274C8—971no—
275C8—971no—
276C8—1941no—
277C8D4×D9711yescharacteristic
278C8D4×D9711yescharacteristic
279C22—21no—
280C22—7762no—
281C22—15522no—
282C22—3882no—
283C22—1941no—
284C22—1942no—
285C22#1569085611yescharacteristic
286C4—21no—
287C4—42no—
288C4—3881no—
289C4—1942no—
290C4—1941no—
291C4C2×D4×D9711yescharacteristic
292C4C2×D4×D9711yescharacteristic
293C2—21no—
294C2—41no—
295C2—7762no—
296C2—3882no—
297C2—3881no—
298C2#2026136911yescharacteristic
subgroupG/H
centerC2#20261369
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#473×C97C2
socleC2×C97D42
hypercenter#473D97
hypocenterC97#3290
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #20261548C2semidirect
2. #20261544C2semidirect
3. #20261540C2semidirect
4. #20261120C2semidirect
5. #20261006C2semidirect
6. #20261001C2semidirect
7. #20260911C2semidirect
8. #20260910C2semidirect
9. #20260851C2semidirect
10. #20260850C2semidirect
11. #473×C97C2semidirect
12. #15690858C22semidirect
16. #15690841C22semidirect
19. #15690838C22semidirect
27. C2×#13095498C22semidirect
31. #15690818C22semidirect
32. #15690816C22semidirect
33. #15690799C22semidirect
34. #15690745C22semidirect
36. #15690736C22semidirect
37. #15690736C22semidirect
38. #15690726C22semidirect
39. C8×Dic97C22semidirect
41. C2×Q16×C97C22semidirect
43. #108×C97C22semidirect
44. #101×C97C22semidirect
150. C97#3290semidirect
155. #473D97semidirect
178. C2×Q16C2×D97semidirect
188. #108C2×D97semidirect
192. #101C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615482
#202615442
#202615402
#202611202
#202610062
#202610012
#202609112
#202609102
#202608512
#202608502
#473×C972
#329097