Finite Group Databasebeta

← #27151871

Subgroups — #27151871

Order 12416.

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

normal subgroups only

This group has 450 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#20261266C211yesmaximal, characteristic, complemented
2C2×#15690772C211yesmaximal, characteristic, complemented
3#20261065C211yesmaximal, characteristic, complemented
4#20261038C212yesmaximal, complemented
5#20260949C212yesmaximal, complemented
6#20260888C211yesmaximal, characteristic, complemented
7#20260868C211yesmaximal, characteristic
8#20260844C212yesmaximal, complemented
9#20260834C212yesmaximal, complemented
10#20260789C211yesmaximal, characteristic, complemented
11#480×C97C211yesmaximal, characteristic, complemented
12C22×D388C2211yescharacteristic
13C22×C4×D97—21no—
14C2×#13095500C2212yes—
15C2×#13095496C2212yes—
16#15690803—21no—
17#15690801C2211yescharacteristic, complemented
18#15690797C2211yescharacteristic, complemented
19#15690791—21no—
20#15690789C2211yescharacteristic
21#15690775C2211yescharacteristic, complemented
22D4×Dic97—21no—
23D4×Dic97C2211yescharacteristic, complemented
24#15690772—21no—
25#15690772—21no—
26#15690772C2212yes—
27#15690772C2212yes—
28#15690766—21no—
29C2×#13095488—21no—
30C2×C4×Dic97C2211yescharacteristic
31#15690741C2212yescomplemented
32#15690739C2212yescomplemented
33#15690733C2212yescomplemented
34#15690732C2212yes—
35#15690730C2212yes—
36C22⋊#11715895C2211yescharacteristic
37#15690720C2212yescomplemented
38#121×C97C2211yescharacteristic
39C2×C4⋊C4×C97C2211yescharacteristic
40#107×C97C2212yescomplemented
41#102×C97C2212yescomplemented
42C22⋊C8×C97C2211yescharacteristic
43C23×D97—21no—
44C2×Dic194C2311yescharacteristic
45C2×D388—21no—
46C2×D388—24no—
47C2×D388C2312yes—
48C22×Dic97—21no—
49C22×Dic97D411yescharacteristic
50C22×Dic97D411yescharacteristic
51C2×C4×D97—22no—
52C2×C4×D97—22no—
53C2×C4×D97—44no—
54#13095500—24no—
55#13095496—24no—
56C2×#11715895C2312yes—
57C4⋊Dic97—21no—
58C4⋊Dic97C2311yescharacteristic
59#13095490—21no—
60#13095490—21no—
61#13095490—21no—
62C22⋊Dic97—21no—
63C22⋊Dic97—21no—
64C22⋊Dic97—21no—
65C22⋊Dic97—21no—
66#13095488—22no—
67#13095488—22no—
68C4×Dic97—21no—
69C4×Dic97D411yescharacteristic
70C4×Dic97D411yescharacteristic
71C4×Dic97C2311yescharacteristic
72C4×Dic97C2311yescharacteristic
73C2×D4×C97—21no—
74C2×D4×C97C2311yescharacteristic
75C22×C4×C97—21no—
76C22×C4×C97C2311yescharacteristic
77C2×C8×C97C2312yes—
78C4⋊C4×C97—21no—
79C4⋊C4×C97C2312yes—
80C4⋊C4×C97C2311yescharacteristic
81C22⋊C4×C97—21no—
82C23×C97—21no—
83C23×C97C2×D411yescharacteristic
84C22×D97—24no—
85C22×D97—22no—
86C22×D97—42no—
87C22×D97—42no—
88D4×C97—22no—
89D4×C97—42no—
90C2×C4×C97—22no—
91C2×C4×C97—21no—
92C2×C4×C97—22no—
93C2×C4×C97—42no—
94C2×C4×C97C2×D411yescharacteristic
95C2×C4×C97C2411yescharacteristic
96C2×Dic97—22no—
97C2×Dic97—21no—
98C2×Dic97—21no—
99C2×Dic97—41no—
100C2×Dic97—42no—
101C2×Dic97C2×D411yescharacteristic
102C2×Dic97C2×D411yescharacteristic
103C2×Dic97C2×D411yescharacteristic
104C2×Dic97C2×D411yescharacteristic
105D388—24no—
106D388—22no—
107D388#4114yes—
108C4×D97—44no—
109C4×D97—44no—
110Dic194—22no—
111C8×C97—22no—
112#11715895—22no—
113C22×C97—21no—
114C22×C97—21no—
115C22×C97—41no—
116C22×C97—41no—
117C22×C97—41no—
118C22×C97AGL(1,ℤ/8)12yes—
119C22×C97C22×D411yescharacteristic
120C2×D97—24no—
121C2×D97—48no—
122C2×D97—42no—
123C2×D97—42no—
124C4×C97—21no—
125C4×C97—42no—
126C4×C97—41no—
127C4×C97C2×#4112yes—
128Dic97—21no—
129Dic97—21no—
130Dic97—21no—
131Dic97—21no—
132Dic97—41no—
133Dic97—41no—
134C2×C97—22no—
135C2×C97—81no—
136C2×C97#52511yescharacteristic
137C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C97#57511yescharacteristic
139D97—44no—
140#3114—971nomaximal, Sylow
141C97#311411yescharacteristic, complemented, Sylow
142#529—971no—
143C2×C4×D4—971no—
144#483—971no—
145#480D9711yescharacteristic, complemented
146#478—972no—
147#463—972no—
148#450—971no—
149#447—971no—
150#442—972no—
151#440—972no—
152#432—971no—
153C22×D4—971no—
154C23×C4—1941no—
155C2×SD16—972no—
156C2×D8—972no—
157#126—1941no—
158#125—971no—
159#124—971no—
160#123—1941no—
161C22⋊Q8—971no—
162#121C2×D9711yescharacteristic
163C4×Q8—971no—
164C4×D4—971no—
165C4×D4—972no—
166C4×D4—972no—
167C4×D4—1941no—
168C4×D4—1941no—
169C4×D4—1941no—
170#117—1941no—
171C2×C4⋊C4C2×D9711yescharacteristic
172C2×C22⋊C4—1941no—
173C2×C42—971no—
174#107C2×D9712yescomplemented
175#106—972no—
176C4⋊C8—972no—
177#103—972no—
178#102—972no—
179#102—972no—
180#102C2×D9712yescomplemented
181C22⋊C8—971no—
182C22⋊C8C2×D9711yescharacteristic
183#97—972no—
184C24—1941no—
185C2×Q8—971no—
186C2×D4—21no—
187C2×D4—972no—
188C2×D4—1944no—
189C2×D4—1941no—
190C2×D4C22×D9711yescharacteristic
191C22×C4—21no—
192C22×C4—971no—
193C22×C4—971no—
194C22×C4—3884no—
195C22×C4—1942no—
196C22×C4—1942no—
197C22×C4—1941no—
198C22×C4C22×D9711yescharacteristic
199SD16—1944no—
200D8—1944no—
201C2×C8—972no—
202C2×C8C22×D9712yes—
203C4⋊C4—21no—
204C4⋊C4—971no—
205C4⋊C4—1941no—
206C4⋊C4—1941no—
207C4⋊C4—1941no—
208C4⋊C4—1941no—
209C4⋊C4C22×D9711yescharacteristic
210C4⋊C4C22×D9712yes—
211C22⋊C4—21no—
212C22⋊C4—1942no—
213C22⋊C4—1942no—
214C22⋊C4—1941no—
215C22⋊C4—1941no—
216C22⋊C4—1941no—
217C22⋊C4—1941no—
218C42—971no—
219C42—971no—
220C42—971no—
221C42—971no—
222C42—1941no—
223C23—21no—
224C23—3882no—
225C23—3882no—
226C23—1944no—
227C23—1942no—
228C23D4×D9711yescharacteristic
229Q8—1942no—
230D4—22no—
231D4—42no—
232D4—974no—
233D4—1944no—
234D4—1942no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—971no—
240C2×C4—971no—
241C2×C4—971no—
242C2×C4—971no—
243C2×C4—3884no—
244C2×C4—3884no—
245C2×C4—3882no—
246C2×C4—3881no—
247C2×C4—1941no—
248C2×C4—1941no—
249C2×C4—1942no—
250C2×C4D4×D9711yescharacteristic
251C2×C4C23×D9711yescharacteristic
252C8—22no—
253C8—1942no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—3888no—
260C22—3882no—
261C22—3882no—
262C22—1944no—
263C22#1569083412yes—
264C22C2×D4×D9711yescharacteristic
265C4—21no—
266C4—42no—
267C4—41no—
268C4—3881no—
269C4—3881no—
270C4—1941no—
271C4—1941no—
272C4—1941no—
273C4—1941no—
274C4#41×D9712yes—
275C2—22no—
276C2—81no—
277C2—3884no—
278C2#2026124311yescharacteristic
279C2C2×#1569083411yescharacteristic
280C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#480×C97C2
socleC22×C97C22×D4
prime radical#20260868C2
hypercenter#480D97
hypocenterC97#3114
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #20261266C2semidirect
2. C2×#15690772C2semidirect
3. #20261065C2semidirect
4. #20261038C2semidirect
5. #20260949C2semidirect
6. #20260888C2semidirect
8. #20260844C2semidirect
9. #20260834C2semidirect
10. #20260789C2semidirect
11. #480×C97C2semidirect
17. #15690801C22semidirect
18. #15690797C22semidirect
21. #15690775C22semidirect
23. D4×Dic97C22semidirect
31. #15690741C22semidirect
32. #15690739C22semidirect
33. #15690733C22semidirect
37. #15690720C22semidirect
40. #107×C97C22semidirect
41. #102×C97C22semidirect
141. C97#3114semidirect
145. #480D97semidirect
174. #107C2×D97semidirect
180. #102C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612662
C2×#156907722
#202610652
#202610382
#202609492
#202608882
#202608682
#202608442
#202608342
#202607892
#480×C972
#311497