Finite Group Databasebeta

← #27153549

Subgroups — #27153549

Order 12416.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 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#20261389C211yesmaximal, characteristic, complemented
2#20261081C211yesmaximal, characteristic, complemented
3#20261046C212yesmaximal, complemented
4#20261015C211yesmaximal, characteristic, complemented
5#20260969C212yesmaximal, complemented
6#20260927C211yesmaximal, characteristic, complemented
7#20260869C212yesmaximal, complemented
8C4×#13095496C211yesmaximal, characteristic, complemented
9D4×#11715895C211yesmaximal, characteristic, complemented
10C2×#15690731C212yesmaximal, complemented
11D42×C97C211yesmaximal, characteristic, complemented
12C2×#13095510—21no—
13C2×#13095496C2211yescharacteristic
14C2×#13095496C2212yescomplemented
15C22×#11715895C2212yescomplemented
16#15690810C2211yescharacteristic, complemented
17#15690794—22no—
18#15690790—21no—
19#15690784C2212yescomplemented
20C4×D388C2211yescharacteristic, complemented
21C2×C4⋊Dic97C2212yescomplemented
22#15690743C2212yescomplemented
23#15690743C2211yescharacteristic, complemented
24C4⋊#11715895C2211yescharacteristic, complemented
25#15690731C2212yes—
26#15690731C2211yescharacteristic
27#15690731C2212yescomplemented
28#15690731C2212yescomplemented
29#15690731C2212yescomplemented
30#15690730C2211yescharacteristic, complemented
31C22⋊#11715895C2212yescomplemented
32C4×#11715895C2211yescharacteristic, complemented
33C22×D4×C97—21no—
34C22×D4×C97C2212yes—
35#127×C97C2211yescharacteristic
36#121×C97—21no—
37#121×C97C2212yes—
38#120×C97—22no—
39C4×D4×C97C2211yescharacteristic
40C4×D4×C97C2211yescharacteristic
41#13095510—44no—
42C2×Dic194—21no—
43C2×D388C2311yescharacteristic
44C2×C22⋊D97—22no—
45C22×Dic97—22no—
46C2×C4×D97—21no—
47#13095496—22no—
48#13095496—42no—
49C2×#11715895—42no—
50C2×#11715895C2311yescharacteristic
51C2×#11715895C2312yes—
52C2×#11715895C2311yescharacteristic
53C4⋊Dic97—22no—
54C4⋊Dic97—21no—
55C4⋊Dic97C2311yescharacteristic
56C4⋊Dic97C2312yes—
57#13095488—21no—
58#13095488—22no—
59C24×C97—22no—
60C2×D4×C97—24no—
61C2×D4×C97—22no—
62C2×D4×C97—22no—
63C2×D4×C97—21no—
64C2×D4×C97—21no—
65C2×D4×C97—42no—
66C2×D4×C97—44no—
67C2×D4×C97D411yescharacteristic
68C2×D4×C97C2312yes—
69C2×D4×C97C2311yescharacteristic
70C22×C4×C97—21no—
71C22×C4×C97C2312yes—
72C4⋊C4×C97D411yescharacteristic
73C4⋊C4×C97C2311yescharacteristic
74C22⋊C4×C97—21no—
75C22⋊C4×C97—22no—
76C22⋊C4×C97D412yes—
77C42×C97C2311yescharacteristic
78C23×C97—21no—
79C23×C97—22no—
80C23×C97—44no—
81C23×C97—44no—
82C23×C97—82no—
83C23×C97C2×D412yes—
84C22×D97—21no—
85D4×C97—22no—
86D4×C97—21no—
87D4×C97—22no—
88D4×C97—44no—
89D4×C97—44no—
90D4×C97—42no—
91D4×C97—81no—
92D4×C97D814yescomplemented
93D4×C97#4112yescomplemented
94C2×C4×C97—22no—
95C2×C4×C97—22no—
96C2×C4×C97—21no—
97C2×C4×C97—42no—
98C2×C4×C97C2×D411yescharacteristic
99C2×C4×C97C2×D412yes—
100C2×C4×C97C2×D411yescharacteristic
101C2×C4×C97C2411yescharacteristic
102C22⋊D97—44no—
103C2×Dic97—21no—
104C2×Dic97—22no—
105C2×Dic97—44no—
106D388—21no—
107D388—21no—
108C4×D97—42no—
109Dic194—42no—
110#11715895—21no—
111#11715895—21no—
112#11715895—22no—
113C22×C97—22no—
114C22×C97—22no—
115C22×C97—22no—
116C22×C97—41no—
117C22×C97—41no—
118C22×C97—42no—
119C22×C97—42no—
120C22×C97—42no—
121C22×C97—88no—
122C22×C97C2×D814yes—
123C22×C97C22×D411yescharacteristic
124C2×D97—41no—
125C2×D97—41no—
126C2×D97—41no—
127C4×C97—22no—
128C4×C97—21no—
129C4×C97—41no—
130C4×C97C2×D812yes—
131C4×C97C2×#4111yescharacteristic
132C4×C97H(5,𝔽2)11yescharacteristic
133Dic97—42no—
134Dic97—41no—
135C2×C97—24no—
136C2×C97—42no—
137C2×C97—82no—
138C2×C97#54611yescharacteristic
139C2×C97C22×D811yescharacteristic
140C2×C97#57711yescharacteristic
141D97—81no—
142#3294—971nomaximal, Sylow
143C97#329411yescharacteristic, complemented, Sylow
144#547—971no—
145D42D9711yescharacteristic, complemented
146#486—971no—
147#480—972no—
148#474—971no—
149#466—972no—
150#459—971no—
151#447—972no—
152C4×D8—971no—
153C8×D4—971no—
154C2×#102—972no—
155C2×#41—1941no—
156C22×D4—21no—
157C22×D4C2×D9712yes—
158C2×D8—972no—
159C2×D8—971no—
160C22×C8—972no—
161C4⋊Q8—971no—
162#127C2×D9711yescharacteristic
163#123—1942no—
164C22⋊Q8—1941no—
165#121—21no—
166#121—972no—
167#121C2×D9712yes—
168#120—22no—
169C4×D4—971no—
170C4×D4C2×D9711yescharacteristic
171C4×D4C2×D9711yescharacteristic
172C2×C4⋊C4—972no—
173#107—971no—
174#107—972no—
175C4⋊C8—971no—
176#102—971no—
177#102—972no—
178#102—972no—
179#102—972no—
180#102—972no—
181#102—971no—
182C22⋊C8—972no—
183C4×C8—971no—
184C24—22no—
185#41—3884no—
186C2×Q8—1941no—
187C2×D4—24no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—42no—
193C2×D4—44no—
194C2×D4—971no—
195C2×D4—1942no—
196C2×D4C22⋊D9711yescharacteristic
197C2×D4C22×D9712yes—
198C2×D4C22×D9711yescharacteristic
199C22×C4—21no—
200C22×C4—1942no—
201C22×C4—1941no—
202C22×C4C22×D9712yes—
203D8—3882no—
204D8—1942no—
205C2×C8—971no—
206C2×C8—971no—
207C2×C8—972no—
208C2×C8—3882no—
209C4⋊C4—971no—
210C4⋊C4—972no—
211C4⋊C4—1942no—
212C4⋊C4—1941no—
213C4⋊C4C22⋊D9711yescharacteristic
214C4⋊C4C22×D9711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—1941no—
218C22⋊C4—1942no—
219C22⋊C4C22⋊D9712yes—
220C42C22×D9711yescharacteristic
221C23—21no—
222C23—22no—
223C23—44no—
224C23—44no—
225C23—82no—
226C23—1941no—
227C23C2×C22⋊D9712yes—
228Q8—3882no—
229D4—22no—
230D4—21no—
231D4—22no—
232D4—44no—
233D4—44no—
234D4—42no—
235D4—81no—
236D4—3884no—
237D4—1941no—
238D4—1941no—
239D4#1309549614yescomplemented
240D4#1309551012yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—42no—
245C2×C4—3884no—
246C2×C4—3882no—
247C2×C4—1941no—
248C2×C4—1942no—
249C2×C4C2×C22⋊D9711yescharacteristic
250C2×C4C2×C22⋊D9712yes—
251C2×C4C2×C22⋊D9711yescharacteristic
252C2×C4C23×D9711yescharacteristic
253C8—1941no—
254C8—1941no—
255C8—1942no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—88no—
265C22—3881no—
266C22—3881no—
267C22—3881no—
268C22C2×#1309549614yes—
269C22C22×C22⋊D9711yescharacteristic
270C4—22no—
271C4—21no—
272C4—41no—
273C4—3881no—
274C4—3882no—
275C4C2×#1309549612yes—
276C4C2×#1309551011yescharacteristic
277C4#1569085511yescharacteristic
278C2—24no—
279C2—42no—
280C2—82no—
281C2—7761no—
282C2#2026137911yescharacteristic
283C2C22×#1309549611yescharacteristic
284C2#2026154611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingD42×C97C2
socleC22×C97C22×D4
hypercenterD42D97
hypocenterC97#3294
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #20261389C2semidirect
2. #20261081C2semidirect
3. #20261046C2semidirect
4. #20261015C2semidirect
5. #20260969C2semidirect
6. #20260927C2semidirect
7. #20260869C2semidirect
8. C4×#13095496C2semidirect
9. D4×#11715895C2semidirect
10. C2×#15690731C2semidirect
11. D42×C97C2semidirect
14. C2×#13095496C22semidirect
15. C22×#11715895C22semidirect
16. #15690810C22semidirect
19. #15690784C22semidirect
20. C4×D388C22semidirect
21. C2×C4⋊Dic97C22semidirect
22. #15690743C22semidirect
23. #15690743C22semidirect
24. C4⋊#11715895C22semidirect
27. #15690731C22semidirect
28. #15690731C22semidirect
29. #15690731C22semidirect
30. #15690730C22semidirect
31. C22⋊#11715895C22semidirect
32. C4×#11715895C22semidirect
92. D4×C97D8semidirect
93. D4×C97#41semidirect
143. C97#3294semidirect
145. D42D97semidirect
239. D4#13095496semidirect
240. D4#13095510semidirect

Maximal subgroups

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

subgroupindex
#202613892
#202610812
#202610462
#202610152
#202609692
#202609272
#202608692
C4×#130954962
D4×#117158952
C2×#156907312
D42×C972
#329497