Finite Group Databasebeta

← #27151528

Subgroups — #27151528

Order 12416.

440 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 131 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#20261290C211yesmaximal, characteristic, complemented
2C4×D4×D97C211yesmaximal, characteristic
3#20261038C212yesmaximal, complemented
4#20261037C212yesmaximal, complemented
5#20261026C211yesmaximal, characteristic, complemented
6#20261025C211yesmaximal, characteristic, complemented
7#20261012C211yesmaximal, characteristic, complemented
8#20260776C212yesmaximal, complemented
9#20260757C211yesmaximal, characteristic, complemented
10#20260738C212yesmaximal
11#474×C97C211yesmaximal, characteristic, complemented
12C2×D4×D97C2211yescharacteristic
13C22×C4×D97—21no—
14C2×#13095494C2212yes—
15#15690810C2211yescharacteristic, complemented
16#15690801C2212yescomplemented
17#15690790—21no—
18#15690786—21no—
19Q8×Dic97—21no—
20#15690775—21no—
21C4×D388C2211yescharacteristic
22D4×Dic97C2211yescharacteristic
23#15690772C2211yescharacteristic
24C4×C22⋊D97—21no—
25#15690770—21no—
26#15690768C2212yes—
27C4⋊C4×D97C2211yescharacteristic
28C22⋊C4×D97—21no—
29C42×D97C2211yescharacteristic
30#15690743C2212yescomplemented
31#15690742C2212yescomplemented
32#15690741C2212yescomplemented
33C4⋊#11715895C2211yescharacteristic, complemented
34#15690739C2212yescomplemented
35#15690732C2212yes—
36#15690731C2212yes—
37#15690730C2212yes—
38C4⋊Q8×C97C2211yescharacteristic, complemented
39C4×D4×C97C2211yescharacteristic
40#107×C97C2212yescomplemented
41C4⋊C8×C97C2211yescharacteristic, complemented
42#102×C97C2212yes—
43C23×D97—21no—
44C2×Dic194—21no—
45C2×D388C2311yescharacteristic
46C2×C22⋊D97—21no—
47D4×D97—22no—
48D4×D97Q814yescomplemented
49C22×Dic97—21no—
50C2×C4×D97—21no—
51C2×C4×D97—21no—
52C2×C4×D97—21no—
53C2×C4×D97—44no—
54C2×C4×D97D411yescharacteristic
55C2×C4×D97D411yescharacteristic
56C2×C4×D97C2311yescharacteristic
57#13095494—24no—
58C2×#11715895C2312yes—
59C4⋊Dic97—21no—
60C4⋊Dic97—22no—
61C4⋊Dic97C2311yescharacteristic
62C4⋊Dic97C2312yes—
63#13095490—21no—
64#13095490—22no—
65C22⋊Dic97—21no—
66#13095488—22no—
67#13095488—21no—
68#13095488—21no—
69C4×Dic97—22no—
70C4×Dic97D411yescharacteristic
71C4×Dic97D411yescharacteristic
72C4×Dic97C2311yescharacteristic
73C2×Q8×C97—21no—
74C2×D4×C97C2311yescharacteristic
75C22×C4×C97—21no—
76C2×C8×C97C2312yes—
77C4⋊C4×C97—21no—
78C4⋊C4×C97C2311yescharacteristic
79C4⋊C4×C97C2312yes—
80C22⋊C4×C97—21no—
81C42×C97C2311yescharacteristic
82C23×C97—21no—
83C22×D97—21no—
84C22×D97—24no—
85C22×D97—42no—
86C22×D97—42no—
87C22×D97C2×D411yescharacteristic
88Q8×C97—42no—
89D4×C97—21no—
90D4×C97C2×Q812yes—
91C2×C4×C97—21no—
92C2×C4×C97—22no—
93C2×C4×C97—42no—
94C2×C4×C97C2×D411yescharacteristic
95C2×C4×C97C2×D411yescharacteristic
96C2×C4×C97C2411yescharacteristic
97C22⋊D97—24no—
98C2×Dic97—22no—
99C2×Dic97—21no—
100C2×Dic97—42no—
101C2×Dic97C2×D411yescharacteristic
102C2×Dic97C2×D411yescharacteristic
103C2×Dic97C2×D411yescharacteristic
104D388—21no—
105D388C2×Q812yes—
106C4×D97—22no—
107C4×D97—41no—
108C4×D97—42no—
109C4×D97—42no—
110C4×D97—42no—
111C4×D97C2×Q812yes—
112C4×D97#4112yes—
113Dic194—42no—
114C8×C97—22no—
115#11715895—22no—
116C22×C97—22no—
117C22×C97—41no—
118C22×C97—41no—
119C22×C97C22×D411yescharacteristic
120C2×D97—21no—
121C2×D97—21no—
122C2×D97—22no—
123C2×D97—48no—
124C2×D97—41no—
125C2×D97—41no—
126C2×D97C22⋊Q812yes—
127C4×C97—21no—
128C4×C97—41no—
129C4×C97—42no—
130C4×C97AGL(1,ℤ/8)12yes—
131C4×C97C22×Q811yescharacteristic
132C4×C97C2×#4111yescharacteristic
133Dic97—21no—
134Dic97—21no—
135Dic97—41no—
136Dic97—42no—
137Dic97C22⋊Q812yes—
138C2×C97—42no—
139C2×C97C2×C22⋊Q811yescharacteristic
140C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
141C2×C97#57511yescharacteristic
142D97—22no—
143D97—42no—
144#3077—971nomaximal, Sylow
145C97#307711yescharacteristic, complemented, Sylow
146#533—971no—
147C2×C4×D4—971no—
148#478—972no—
149#478—972no—
150#476—971no—
151#476—971no—
152#474—971no—
153#474D9711yescharacteristic, complemented
154#428—972no—
155#423—971no—
156#418—972no—
157C22×D4—971no—
158C23×C4—1941no—
159C2×M16—972no—
160C4⋊Q8—971no—
161C4⋊Q8C2×D9711yescharacteristic, complemented
162#125—972no—
163C22⋊Q8—1941no—
164C22⋊Q8—1941no—
165C4×Q8—1941no—
166C4×Q8—1941no—
167C4×D4—971no—
168C4×D4—971no—
169C4×D4—971no—
170C4×D4—1941no—
171C4×D4—1941no—
172C4×D4C2×D9711yescharacteristic
173#117—972no—
174C2×C4⋊C4—971no—
175C2×C22⋊C4—1941no—
176C2×C42—971no—
177#107—972no—
178#107C2×D9712yescomplemented
179#106—972no—
180#106—972no—
181C4⋊C8—972no—
182C4⋊C8—971no—
183C4⋊C8C2×D9711yescharacteristic, complemented
184#102—972no—
185#102—972no—
186#102—972no—
187#102C2×D9712yes—
188C24—1941no—
189C2×Q8—21no—
190C2×Q8—1941no—
191C2×D4—974no—
192C2×D4—971no—
193C2×D4—1942no—
194C2×D4—1941no—
195C2×D4C22×D9711yescharacteristic
196C22×C4—21no—
197C22×C4—971no—
198C22×C4—971no—
199C22×C4—971no—
200C22×C4—3884no—
201C22×C4—1941no—
202C22×C4—1941no—
203C22×C4—1941no—
204C22×C4—1941no—
205M16—1944no—
206C2×C8—972no—
207C2×C8C22×D9712yes—
208C4⋊C4—21no—
209C4⋊C4—972no—
210C4⋊C4—971no—
211C4⋊C4—1941no—
212C4⋊C4—1942no—
213C4⋊C4—1942no—
214C4⋊C4—1941no—
215C4⋊C4C22×D9712yes—
216C4⋊C4C22×D9711yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—1941no—
219C22⋊C4—1942no—
220C22⋊C4—1941no—
221C22⋊C4—1941no—
222C42—971no—
223C42—971no—
224C42—971no—
225C42—1942no—
226C42C22×D9711yescharacteristic
227C23—21no—
228C23—971no—
229C23—3882no—
230C23—3882no—
231C23—1944no—
232C23—1941no—
233Q8—42no—
234Q8—3882no—
235D4—21no—
236D4—972no—
237D4—1944no—
238D4—1941no—
239D4Q8×D9712yes—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—971no—
244C2×C4—972no—
245C2×C4—972no—
246C2×C4—971no—
247C2×C4—971no—
248C2×C4—3881no—
249C2×C4—3882no—
250C2×C4—3882no—
251C2×C4—3882no—
252C2×C4—3882no—
253C2×C4—1942no—
254C2×C4—1941no—
255C2×C4—1942no—
256C2×C4D4×D9711yescharacteristic
257C2×C4D4×D9711yescharacteristic
258C2×C4C23×D9711yescharacteristic
259C8—22no—
260C8—1942no—
261C22—22no—
262C22—41no—
263C22—41no—
264C22—972no—
265C22—3888no—
266C22—3881no—
267C22—3881no—
268C22—1942no—
269C22—1941no—
270C22—1941no—
271C22C2×D4×D9711yescharacteristic
272C4—21no—
273C4—41no—
274C4—42no—
275C4—972no—
276C4—3881no—
277C4—3882no—
278C4—1941no—
279C4—1941no—
280C4#1569083412yes—
281C4C2×Q8×D9711yescharacteristic
282C4#41×D9711yescharacteristic
283C2—42no—
284C2—3882no—
285C2—1942no—
286C2C22⋊Q8×D9711yescharacteristic
287C2C2×#1569083411yescharacteristic
288C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#474×C97C2
socleC22×C97C22×D4
prime radicalC2×D4×D97C22
hypercenter#474D97
hypocenterC97#3077
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. #20261290C2semidirect
3. #20261038C2semidirect
4. #20261037C2semidirect
5. #20261026C2semidirect
6. #20261025C2semidirect
7. #20261012C2semidirect
8. #20260776C2semidirect
9. #20260757C2semidirect
11. #474×C97C2semidirect
15. #15690810C22semidirect
16. #15690801C22semidirect
30. #15690743C22semidirect
31. #15690742C22semidirect
32. #15690741C22semidirect
33. C4⋊#11715895C22semidirect
34. #15690739C22semidirect
38. C4⋊Q8×C97C22semidirect
40. #107×C97C22semidirect
41. C4⋊C8×C97C22semidirect
48. D4×D97Q8semidirect
145. C97#3077semidirect
153. #474D97semidirect
161. C4⋊Q8C2×D97semidirect
178. #107C2×D97semidirect
183. C4⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612902
C4×D4×D972
#202610382
#202610372
#202610262
#202610252
#202610122
#202607762
#202607572
#202607382
#474×C972
#307797