Finite Group Databasebeta

← #27153504

Subgroups — #27153504

Order 12416.

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

normal subgroups only

This group has 464 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#20261543C212yesmaximal, complemented
2#20261539C212yesmaximal, complemented
3#20261125C211yesmaximal, characteristic, complemented
4#20261003C212yesmaximal, complemented
5#20261001C212yesmaximal, complemented
6#20260912C211yesmaximal, characteristic, complemented
7#20260904C212yesmaximal, complemented
8#20260854C212yesmaximal, complemented
9#454×C97C211yesmaximal, characteristic, complemented
10#15690856C2212yescomplemented
11#15690855C2211yescharacteristic
12#15690841—22no—
13#15690841C2212yescomplemented
14#15690838C2212yes—
15#15690837—22no—
16#15690835—22no—
17#15690835C2212yescomplemented
18#15690834—22no—
19#15690831—22no—
20#15690830—22no—
21#15690830C2212yes—
22C2×#13095498—22no—
23C2×#13095498C2212yescomplemented
24C2×#13095496C2212yescomplemented
25D8×D97—22no—
26#15690818C2212yescomplemented
27#15690817C2212yescomplemented
28#15690807C2211yescharacteristic, complemented
29#15690745C2212yescomplemented
30#15690738C2212yescomplemented
31#15690727C2211yescharacteristic, complemented
32#15690726C2212yescomplemented
33#15690721C2211yescharacteristic, complemented
342−1+4×C97C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C97C2212yescomplemented
36#124×C97C2211yescharacteristic, complemented
37#104×C97C2212yescomplemented
38#101×C97C2211yescharacteristic, complemented
39#13095512—22no—
40#13095511—22no—
41#13095511C2312yes—
42#13095510—22no—
43C2×D388—22no—
44C2×D388—41no—
45C2×D388C2312yes—
46C2×C22⋊D97—41no—
47D4×D97—22no—
48D4×D97—22no—
49D4×D97—42no—
50#13095501—22no—
51#13095501D412yescomplemented
52#13095500—22no—
53#13095499D412yescomplemented
54#13095498—22no—
55#13095498—22no—
56#13095498—22no—
57#13095498—22no—
58#13095498D412yescomplemented
59D776—22no—
60#13095496—22no—
61#13095496—22no—
62#13095496—22no—
63#13095496D412yescomplemented
64#13095495—22no—
65#13095495C2312yes—
66#13095494—22no—
67C2×#11715895—22no—
68C2×#11715895C2312yes—
69C8×D97—22no—
70#41×C97—22no—
71#41×C97—41no—
72#41×C97C2312yes—
73C2×Q8×C97—22no—
74C2×Q8×C97C2311yescharacteristic
75C2×D4×C97C2311yescharacteristic
76SD16×C97—22no—
77D8×C97—22no—
78M16×C97C2312yes—
79C22⋊C4×C97—41no—
80C42×C97C2311yescharacteristic
81C23×C97—21no—
82C22×D97—22no—
83C22×D97—41no—
84C22×D97—42no—
85Q8×C97—22no—
86Q8×C97—41no—
87Q8×C97C2×D412yes—
88D4×C97—22no—
89D4×C97—22no—
90D4×C97—41no—
91D4×C97C2×D412yes—
92C2×C4×C97—22no—
93C2×C4×C97—21no—
94C2×C4×C97—21no—
95C2×C4×C97—42no—
96C2×C4×C97C2411yescharacteristic
97C22⋊D97—22no—
98C22⋊D97—42no—
99C22⋊D97—42no—
100C2×Dic97—41no—
101D388—22no—
102D388—22no—
103D388—42no—
104D388—82no—
105D388C2×D412yes—
106C4×D97—22no—
107C4×D97—42no—
108Dic194C2×D412yes—
109C8×C97—22no—
110#11715895—22no—
111#11715895C2×D412yes—
112#11715895C2×D412yes—
113C22×C97—22no—
114C22×C97—41no—
115C22×C97—81no—
116C22×C97H(5,𝔽2)11yescharacteristic
117C2×D97—22no—
118C2×D97—42no—
119C2×D97—82no—
120C2×D97—161no—
121C2×D97—162no—
122C4×C97—22no—
123C4×C97—41no—
124C4×C97—41no—
125C4×C97C22×D412yes—
126Dic97—22no—
127C2×C97—21no—
128C2×C97—42no—
129C2×C97—81no—
130C2×C97D4211yescharacteristic
131D97—42no—
132D97—82no—
133#3288—971nomaximal, Sylow
134C97#328811yescharacteristic, complemented, Sylow
135#577—972no—
136#576—972no—
137#496—971no—
138#472—972no—
139#471—972no—
140#455—971no—
141#454D9711yescharacteristic, complemented
142#453—972no—
143#444—972no—
1442−1+4C2×D9711yescharacteristic, complemented
145H(5,𝔽2)—972no—
146H(5,𝔽2)—971no—
147#137—972no—
148#137—1942no—
149AGL(1,ℤ/8)—972no—
150AGL(1,ℤ/8)—972no—
151AGL(1,ℤ/8)—1942no—
152AGL(1,ℤ/8)—1942no—
153AGL(1,ℤ/8)—1942no—
154AGL(1,ℤ/8)C2×D9712yescomplemented
155#135—972no—
156#135—1942no—
157#135—1942no—
158C2×SD16—972no—
159C2×SD16—1942no—
160C2×D8—972no—
161C2×D8—1942no—
162#131—972no—
163#131—972no—
164#127—971no—
165#124C2×D9711yescharacteristic, complemented
166#108—972no—
167#104—972no—
168#104C2×D9712yescomplemented
169#101C2×D9711yescharacteristic, complemented
170#100—971no—
171#100—972no—
172#97—971no—
173#41—22no—
174#41—41no—
175#41—972no—
176#41—1942no—
177#41—1942no—
178#41—1942no—
179#41C22×D9712yes—
180C2×Q8—22no—
181C2×Q8C22×D9711yescharacteristic
182C2×D4—972no—
183C2×D4—3881no—
184C2×D4—3882no—
185C2×D4—3881no—
186C2×D4—1942no—
187C2×D4—1942no—
188C2×D4—1942no—
189C2×D4C22×D9711yescharacteristic
190Q16—972no—
191Q16—1942no—
192SD16—22no—
193SD16—972no—
194SD16—972no—
195SD16—1942no—
196SD16—1942no—
197SD16—1942no—
198SD16—1942no—
199SD16—1942no—
200D8—22no—
201D8—972no—
202D8—1942no—
203D8—1942no—
204D8—1942no—
205D8—1942no—
206M16—972no—
207M16—1942no—
208M16—1942no—
209M16C22×D9712yes—
210C2×C8—972no—
211C2×C8—1942no—
212C2×C8—1942no—
213C22⋊C4—41no—
214C42C22×D9711yescharacteristic
215C23—21no—
216C23—3882no—
217C23—3881no—
218C23—1942no—
219Q8—22no—
220Q8—41no—
221Q8—972no—
222Q8D4×D9712yes—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—7762no—
227D4—972no—
228D4—3882no—
229D4—3882no—
230D4—3882no—
231D4—1942no—
232D4—1942no—
233D4—1942no—
234D4D4×D9712yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—3882no—
240C2×C4—3881no—
241C2×C4—1942no—
242C2×C4C23×D9711yescharacteristic
243C8—22no—
244C8—972no—
245C8—972no—
246C8—1942no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—7762no—
251C22—15522no—
252C22—15521no—
253C22—3882no—
254C22—1942no—
255C22#1569085511yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—1942no—
260C4C2×D4×D9712yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—7762no—
265C2—3882no—
266C2#2026136711yescharacteristic
subgroupG/H
centerC2#20261367
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#454×C97C2
socleC2×C97D42
hypercenter#454D97
hypocenterC97#3288
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #20261543C2semidirect
2. #20261539C2semidirect
3. #20261125C2semidirect
4. #20261003C2semidirect
5. #20261001C2semidirect
6. #20260912C2semidirect
7. #20260904C2semidirect
8. #20260854C2semidirect
9. #454×C97C2semidirect
10. #15690856C22semidirect
13. #15690841C22semidirect
17. #15690835C22semidirect
23. C2×#13095498C22semidirect
24. C2×#13095496C22semidirect
26. #15690818C22semidirect
27. #15690817C22semidirect
28. #15690807C22semidirect
29. #15690745C22semidirect
30. #15690738C22semidirect
31. #15690727C22semidirect
32. #15690726C22semidirect
33. #15690721C22semidirect
34. 2−1+4×C97C22semidirect
35. AGL(1,ℤ/8)×C97C22semidirect
36. #124×C97C22semidirect
37. #104×C97C22semidirect
38. #101×C97C22semidirect
51. #13095501D4semidirect
53. #13095499D4semidirect
58. #13095498D4semidirect
63. #13095496D4semidirect
134. C97#3288semidirect
141. #454D97semidirect
144. 2−1+4C2×D97semidirect
154. AGL(1,ℤ/8)C2×D97semidirect
165. #124C2×D97semidirect
168. #104C2×D97semidirect
169. #101C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615432
#202615392
#202611252
#202610032
#202610012
#202609122
#202609042
#202608542
#454×C972
#328897