Finite Group Databasebeta

← #27153546

Subgroups — #27153546

Order 12416.

494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 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#20261388C211yesmaximal, characteristic, complemented
2D4×D388C211yesmaximal, characteristic, complemented
3#20261083C211yesmaximal, characteristic, complemented
4#20261050C212yesmaximal, complemented
5#20260972C212yesmaximal, complemented
6#20260931C211yesmaximal, characteristic, complemented
7#20260870C212yesmaximal, complemented
8C4×#13095496C211yesmaximal, characteristic, complemented
9#20260800C211yesmaximal, characteristic, complemented
10#102×D97C212yesmaximal, complemented
11#474×C97C211yesmaximal, characteristic, complemented
12C2×#13095512—21no—
13C22×D388—21no—
14C2×D4×D97C2212yes—
15C2×D776C2212yescomplemented
16C2×#13095496C2211yescharacteristic
17C2×C8×D97C2212yescomplemented
18#15690807C2211yescharacteristic
19#15690795—22no—
20#15690786—21no—
21#15690785C2212yes—
22#15690785C2212yescomplemented
23#15690784—21no—
24#15690780—22no—
25C4×D388C2211yescharacteristic
26C4×D388C2211yescharacteristic, complemented
27C4⋊C4×D97C2212yescomplemented
28#15690743C2211yescharacteristic, complemented
29#15690732C2212yescomplemented
30#15690731C2211yescharacteristic
31#15690730C2212yescomplemented
32#15690730C2211yescharacteristic, complemented
33#15690730C2212yescomplemented
34#15690722C2212yescomplemented
35C4×#11715895C2211yescharacteristic, complemented
36C4⋊Q8×C97C2211yescharacteristic, complemented
37C4×D4×C97C2211yescharacteristic
38#107×C97C2212yescomplemented
39C4⋊C8×C97C2211yescharacteristic, complemented
40#102×C97C2212yes—
41C23×D97—22no—
42#13095512—44no—
43C2×D388—22no—
44C2×D388—22no—
45C2×D388—21no—
46C2×D388—21no—
47C2×D388—44no—
48C2×D388D411yescharacteristic
49C2×D388C2311yescharacteristic
50C2×D388C2312yes—
51C2×C22⋊D97—22no—
52D4×D97—24no—
53D4×D97—42no—
54C2×C4×D97—22no—
55C2×C4×D97—21no—
56C2×C4×D97C2312yes—
57D776—42no—
58#13095496—22no—
59C2×#11715895C2311yescharacteristic
60C2×#11715895C2311yescharacteristic
61C8×D97—42no—
62C4⋊Dic97D411yescharacteristic
63C4⋊Dic97C2311yescharacteristic
64#13095490—22no—
65#13095488—21no—
66#13095488—22no—
67#13095488—22no—
68#13095488D412yes—
69C2×Q8×C97—21no—
70C2×D4×C97C2311yescharacteristic
71C22×C4×C97—21no—
72C2×C8×C97C2312yes—
73C4⋊C4×C97—21no—
74C4⋊C4×C97C2312yes—
75C4⋊C4×C97C2311yescharacteristic
76C22⋊C4×C97—21no—
77C42×C97C2311yescharacteristic
78C23×C97—21no—
79C22×D97—21no—
80C22×D97—22no—
81C22×D97—44no—
82C22×D97—44no—
83C22×D97—82no—
84C22×D97C2×D412yes—
85Q8×C97—42no—
86D4×C97—21no—
87D4×C97#4112yescomplemented
88C2×C4×C97—21no—
89C2×C4×C97—22no—
90C2×C4×C97—42no—
91C2×C4×C97C2×D411yescharacteristic
92C2×C4×C97C2×D411yescharacteristic
93C2×C4×C97C2411yescharacteristic
94C22⋊D97—44no—
95C2×Dic97—21no—
96C2×Dic97C2×D412yes—
97D388—21no—
98D388—22no—
99D388—21no—
100D388—22no—
101D388—44no—
102D388—44no—
103D388—42no—
104D388—81no—
105D388D814yescomplemented
106C4×D97—22no—
107C4×D97—22no—
108C4×D97—44no—
109C4×D97—42no—
110C8×C97—22no—
111#11715895—21no—
112#11715895—21no—
113C22×C97—22no—
114C22×C97—41no—
115C22×C97—41no—
116C22×C97C22×D411yescharacteristic
117C2×D97—22no—
118C2×D97—22no—
119C2×D97—42no—
120C2×D97—41no—
121C2×D97—41no—
122C2×D97—42no—
123C2×D97—42no—
124C2×D97—41no—
125C2×D97—88no—
126C2×D97C2×D814yes—
127C4×C97—21no—
128C4×C97—42no—
129C4×C97—41no—
130C4×C97C2×D812yes—
131C4×C97C2×#4111yescharacteristic
132C4×C97H(5,𝔽2)11yescharacteristic
133Dic97—22no—
134Dic97—41no—
135C2×C97—42no—
136C2×C97#54611yescharacteristic
137C2×C97C22×D811yescharacteristic
138C2×C97#57711yescharacteristic
139D97—24no—
140D97—81no—
141D97—82no—
142#3294—971nomaximal, Sylow
143C97#329411yescharacteristic, complemented, Sylow
144#547—971no—
145D42—971no—
146#486—971no—
147#480—972no—
148#474D9711yescharacteristic, complemented
149#466—972no—
150#459—971no—
151#447—972no—
152C4×D8—971no—
153C8×D4—971no—
154C2×#102—972no—
155C2×#41—1941no—
156C22×D4—972no—
157C22×D4—1941no—
158C2×D8—971no—
159C2×D8—972no—
160C22×C8—972no—
161C4⋊Q8C2×D9711yescharacteristic, complemented
162#127—971no—
163#123—1942no—
164C22⋊Q8—1941no—
165#121—972no—
166#121—972no—
167#121—1941no—
168#120—1942no—
169C4×D4—971no—
170C4×D4—971no—
171C4×D4C2×D9711yescharacteristic
172C2×C4⋊C4—972no—
173#107—971no—
174#107C2×D9712yescomplemented
175C4⋊C8C2×D9711yescharacteristic, complemented
176#102—971no—
177#102—972no—
178#102—972no—
179#102—971no—
180#102—972no—
181#102C2×D9712yes—
182C22⋊C8—972no—
183C4×C8—971no—
184C24—1942no—
185#41—3884no—
186C2×Q8—21no—
187C2×D4—971no—
188C2×D4—972no—
189C2×D4—971no—
190C2×D4—3884no—
191C2×D4—3882no—
192C2×D4—1942no—
193C2×D4—1942no—
194C2×D4—1942no—
195C2×D4—1944no—
196C2×D4—1941no—
197C2×D4—1941no—
198C2×D4C22×D9711yescharacteristic
199C22×C4—21no—
200C22×C4—972no—
201C22×C4—1942no—
202C22×C4—1941no—
203D8—3882no—
204D8—1942no—
205C2×C8—971no—
206C2×C8—971no—
207C2×C8—3882no—
208C2×C8C22×D9712yes—
209C4⋊C4—21no—
210C4⋊C4—971no—
211C4⋊C4—971no—
212C4⋊C4—1942no—
213C4⋊C4C22×D9712yes—
214C4⋊C4C22×D9711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—972no—
217C22⋊C4—1942no—
218C22⋊C4—1942no—
219C22⋊C4—1941no—
220C42C22×D9711yescharacteristic
221C23—21no—
222C23—7762no—
223C23—972no—
224C23—3884no—
225C23—3884no—
226C23—1941no—
227C23—1942no—
228Q8—42no—
229D4—21no—
230D4—7761no—
231D4—974no—
232D4—3884no—
233D4—3884no—
234D4—3884no—
235D4—3882no—
236D4—1941no—
237D4—1942no—
238D4—1941no—
239D4—1942no—
240D4#1309551212yescomplemented
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—972no—
245C2×C4—3884no—
246C2×C4—3882no—
247C2×C4—1941no—
248C2×C4—1942no—
249C2×C4—1942no—
250C2×C4D4×D9711yescharacteristic
251C2×C4D4×D9711yescharacteristic
252C2×C4C23×D9711yescharacteristic
253C8—22no—
254C8—1941no—
255C8—1941no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—7768no—
260C22—974no—
261C22—3882no—
262C22—3881no—
263C22—3881no—
264C22—3882no—
265C22—3882no—
266C22—3881no—
267C22—1942no—
268C22—1942no—
269C22C2×D4×D9711yescharacteristic
270C4—21no—
271C4—41no—
272C4—42no—
273C4—3881no—
274C4—1942no—
275C4D8×D9712yes—
276C4C2×#1309551211yescharacteristic
277C4#1569085611yescharacteristic
278C2—42no—
279C2—7762no—
280C2—7761no—
281C2—1944no—
282C2#2026137611yescharacteristic
283C2C2×D8×D9711yescharacteristic
284C2#2026154411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#474×C97C2
socleC22×C97C22×D4
hypercenter#474D97
hypocenterC97#3294
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #20261388C2semidirect
2. D4×D388C2semidirect
3. #20261083C2semidirect
4. #20261050C2semidirect
5. #20260972C2semidirect
6. #20260931C2semidirect
7. #20260870C2semidirect
8. C4×#13095496C2semidirect
9. #20260800C2semidirect
10. #102×D97C2semidirect
11. #474×C97C2semidirect
15. C2×D776C22semidirect
17. C2×C8×D97C22semidirect
22. #15690785C22semidirect
26. C4×D388C22semidirect
27. C4⋊C4×D97C22semidirect
28. #15690743C22semidirect
29. #15690732C22semidirect
31. #15690730C22semidirect
32. #15690730C22semidirect
33. #15690730C22semidirect
34. #15690722C22semidirect
35. C4×#11715895C22semidirect
36. C4⋊Q8×C97C22semidirect
38. #107×C97C22semidirect
39. C4⋊C8×C97C22semidirect
87. D4×C97#41semidirect
105. D388D8semidirect
143. C97#3294semidirect
148. #474D97semidirect
161. C4⋊Q8C2×D97semidirect
174. #107C2×D97semidirect
175. C4⋊C8C2×D97semidirect
240. D4#13095512semidirect

Maximal subgroups

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

subgroupindex
#202613882
D4×D3882
#202610832
#202610502
#202609722
#202609312
#202608702
C4×#130954962
#202608002
#102×D972
#474×C972
#329497