Finite Group Databasebeta

← #27153543

Subgroups — #27153543

Order 12416.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D388C211yesmaximal, characteristic, complemented
2#20261083C211yesmaximal, characteristic, complemented
3#20261048C212yesmaximal, complemented
4#20261012C211yesmaximal, characteristic, complemented
5#20260969C212yesmaximal, complemented
6#20260930C211yesmaximal, characteristic, complemented
7#20260868C212yesmaximal, complemented
8C4×#13095496C211yesmaximal, characteristic, complemented
9D4×#11715895C211yesmaximal, characteristic, complemented
10C2×#15690730C212yesmaximal, complemented
11#547×C97C211yesmaximal, characteristic, complemented
12C22×D388C2212yes—
13C2×D4×D97—21no—
14C2×#13095496C2211yescharacteristic
15C2×#13095496C2212yescomplemented
16C22×#11715895C2212yescomplemented
17#15690807C2211yescharacteristic
18#15690785—21no—
19#15690784C2212yes—
20#15690780—22no—
21C4×D388C2211yescharacteristic
22#15690743C2212yescomplemented
23#15690743C2211yescharacteristic, complemented
24C4⋊#11715895C2211yescharacteristic, complemented
25#15690731C2211yescharacteristic, complemented
26#15690730C2212yes—
27#15690730C2211yescharacteristic
28#15690730C2212yescomplemented
29#15690730C2212yescomplemented
30#15690730C2212yescomplemented
31C22⋊#11715895C2212yescomplemented
32C4×#11715895C2211yescharacteristic, complemented
33C2×#41×C97—21no—
34C4⋊Q8×C97C2211yescharacteristic, complemented
35#123×C97—22no—
36C22⋊Q8×C97—21no—
37#121×C97C2212yescomplemented
38C4×D4×C97C2211yescharacteristic
39C4×D4×C97C2211yescharacteristic, complemented
40C2×C4⋊C4×C97C2212yescomplemented
41C23×D97—22no—
42C2×D388—24no—
43C2×D388—22no—
44C2×D388—21no—
45C2×D388—21no—
46C2×D388—42no—
47C2×D388C2312yes—
48C2×D388C2311yescharacteristic
49C2×C22⋊D97—22no—
50D4×D97—44no—
51C2×C4×D97—21no—
52#13095496—22no—
53#13095496—42no—
54C2×#11715895—42no—
55C2×#11715895C2311yescharacteristic
56C2×#11715895C2312yes—
57C2×#11715895C2311yescharacteristic
58C4⋊Dic97C2311yescharacteristic
59#13095488—21no—
60#13095488—22no—
61#41×C97—44no—
62C2×Q8×C97—21no—
63C2×D4×C97—22no—
64C2×D4×C97D411yescharacteristic
65C2×D4×C97C2311yescharacteristic
66C22×C4×C97—22no—
67C22×C4×C97—21no—
68C22×C4×C97C2312yes—
69C4⋊C4×C97—22no—
70C4⋊C4×C97—21no—
71C4⋊C4×C97D411yescharacteristic
72C4⋊C4×C97C2311yescharacteristic
73C4⋊C4×C97C2312yes—
74C22⋊C4×C97—21no—
75C22⋊C4×C97—22no—
76C22⋊C4×C97D412yes—
77C42×C97C2311yescharacteristic
78C23×C97—21no—
79C23×C97C2×D412yes—
80C22×D97—21no—
81C22×D97—22no—
82C22×D97—44no—
83C22×D97—44no—
84C22×D97—82no—
85Q8×C97—42no—
86D4×C97—21no—
87D4×C97—21no—
88D4×C97—44no—
89D4×C97D814yescomplemented
90C2×C4×C97—22no—
91C2×C4×C97—22no—
92C2×C4×C97—21no—
93C2×C4×C97—22no—
94C2×C4×C97—44no—
95C2×C4×C97—42no—
96C2×C4×C97C2×D411yescharacteristic
97C2×C4×C97C2×D412yes—
98C2×C4×C97C2×D411yescharacteristic
99C2×C4×C97C2411yescharacteristic
100C22⋊D97—44no—
101C2×Dic97—21no—
102D388—22no—
103D388—21no—
104D388—22no—
105D388—44no—
106D388—42no—
107D388—81no—
108D388#4112yescomplemented
109C4×D97—42no—
110#11715895—21no—
111#11715895—21no—
112#11715895—22no—
113C22×C97—22no—
114C22×C97—22no—
115C22×C97—41no—
116C22×C97—41no—
117C22×C97—41no—
118C22×C97C2×D814yes—
119C22×C97C22×D411yescharacteristic
120C2×D97—22no—
121C2×D97—41no—
122C2×D97—41no—
123C2×D97—42no—
124C2×D97—42no—
125C2×D97—42no—
126C2×D97—88no—
127C4×C97—22no—
128C4×C97—21no—
129C4×C97—41no—
130C4×C97—42no—
131C4×C97C2×D812yes—
132C4×C97C2×#4111yescharacteristic
133C4×C97H(5,𝔽2)11yescharacteristic
134Dic97—41no—
135C2×C97—24no—
136C2×C97—81no—
137C2×C97#54611yescharacteristic
138C2×C97C22×D811yescharacteristic
139C2×C97#57711yescharacteristic
140D97—42no—
141D97—82no—
142#3294—971nomaximal, Sylow
143C97#329411yescharacteristic, complemented, Sylow
144#547D9711yescharacteristic, complemented
145D42—971no—
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—21no—
156C22×D4—972no—
157C22×D4—1941no—
158C2×D8—972no—
159C2×D8—971no—
160C22×C8—972no—
161C4⋊Q8C2×D9711yescharacteristic, complemented
162#127—971no—
163#123—22no—
164C22⋊Q8—21no—
165#121—972no—
166#121—1941no—
167#121C2×D9712yescomplemented
168#120—1942no—
169C4×D4—971no—
170C4×D4C2×D9711yescharacteristic
171C4×D4C2×D9711yescharacteristic, complemented
172C2×C4⋊C4C2×D9712yescomplemented
173#107—971no—
174#107—972no—
175C4⋊C8—971no—
176#102—971no—
177#102—971no—
178#102—972no—
179#102—972no—
180#102—972no—
181#102—972no—
182C22⋊C8—972no—
183C4×C8—971no—
184C24—1942no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—971no—
189C2×D4—972no—
190C2×D4—3882no—
191C2×D4—3884no—
192C2×D4—1944no—
193C2×D4—1942no—
194C2×D4—1941no—
195C2×D4—1942no—
196C2×D4—1941no—
197C2×D4C22⋊D9711yescharacteristic
198C2×D4C22×D9711yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—1941no—
202C22×C4C22×D9712yes—
203D8—3882no—
204D8—1942no—
205C2×C8—971no—
206C2×C8—971no—
207C2×C8—972no—
208C2×C8—3882no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—971no—
212C4⋊C4C22⋊D9711yescharacteristic
213C4⋊C4C22×D9711yescharacteristic
214C4⋊C4C22×D9712yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—1941no—
218C22⋊C4—1942no—
219C22⋊C4C22⋊D9712yes—
220C42C22×D9711yescharacteristic
221C23—21no—
222C23—7762no—
223C23—3884no—
224C23—3884no—
225C23—1942no—
226C23—1941no—
227C23C2×C22⋊D9712yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—7761no—
233D4—972no—
234D4—3884no—
235D4—3884no—
236D4—3882no—
237D4—1942no—
238D4—1942no—
239D4—1941no—
240D4#1309549614yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—3882no—
248C2×C4—1941no—
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—41no—
259C22—41no—
260C22—41no—
261C22—7768no—
262C22—3882no—
263C22—3882no—
264C22—3882no—
265C22—3881no—
266C22—3881no—
267C22—1942no—
268C22C2×#1309549614yes—
269C22C22×C22⋊D9711yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—42no—
274C4—3881no—
275C4C2×#1309549612yes—
276C4#41×D9711yescharacteristic
277C4#1569085611yescharacteristic
278C2—24no—
279C2—81no—
280C2—7762no—
281C2—3882no—
282C2#2026137311yescharacteristic
283C2C22×#1309549611yescharacteristic
284C2#2026154311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#547×C97C2
socleC22×C97C22×D4
hypercenter#547D97
hypocenterC97#3294
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D388C2semidirect
2. #20261083C2semidirect
3. #20261048C2semidirect
4. #20261012C2semidirect
5. #20260969C2semidirect
6. #20260930C2semidirect
7. #20260868C2semidirect
8. C4×#13095496C2semidirect
9. D4×#11715895C2semidirect
10. C2×#15690730C2semidirect
11. #547×C97C2semidirect
15. C2×#13095496C22semidirect
16. C22×#11715895C22semidirect
22. #15690743C22semidirect
23. #15690743C22semidirect
24. C4⋊#11715895C22semidirect
25. #15690731C22semidirect
28. #15690730C22semidirect
29. #15690730C22semidirect
30. #15690730C22semidirect
31. C22⋊#11715895C22semidirect
32. C4×#11715895C22semidirect
34. C4⋊Q8×C97C22semidirect
37. #121×C97C22semidirect
39. C4×D4×C97C22semidirect
40. C2×C4⋊C4×C97C22semidirect
89. D4×C97D8semidirect
108. D388#41semidirect
143. C97#3294semidirect
144. #547D97semidirect
161. C4⋊Q8C2×D97semidirect
167. #121C2×D97semidirect
171. C4×D4C2×D97semidirect
172. C2×C4⋊C4C2×D97semidirect
240. D4#13095496semidirect

Maximal subgroups

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

subgroupindex
D4×D3882
#202610832
#202610482
#202610122
#202609692
#202609302
#202608682
C4×#130954962
D4×#117158952
C2×#156907302
#547×C972
#329497