Finite Group Databasebeta

← #27151875

Subgroups — #27151875

Order 12416.

450 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 121 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#20261271C211yesmaximal, characteristic, complemented
2C2×D4×Dic97C211yesmaximal, characteristic, complemented
3#20261063C211yesmaximal, characteristic, complemented
4#20261049C211yesmaximal, characteristic, complemented
5#20261033C212yesmaximal, complemented
6#20260952C212yesmaximal, complemented
7#20260869C211yesmaximal, characteristic
8#20260844C212yesmaximal, complemented
9#20260836C212yesmaximal, complemented
10#20260789C211yesmaximal, characteristic, complemented
11#450×C97C211yesmaximal, characteristic, complemented
12C23×Dic97—21no—
13C2×#13095496C2212yes—
14#15690805—21no—
15#15690801C2211yescharacteristic, complemented
16#15690799C2211yescharacteristic, complemented
17#15690794—21no—
18#15690784C2211yescharacteristic
19Q8×Dic97C2211yescharacteristic, complemented
20D4×Dic97—21no—
21D4×Dic97—21no—
22D4×Dic97C2212yes—
23D4×Dic97C2212yes—
24#15690772C2211yescharacteristic, complemented
25#15690770—21no—
26#15690768—21no—
27C2×C4⋊Dic97C2211yescharacteristic
28C2×C22⋊Dic97—21no—
29C2×C4×Dic97C2211yescharacteristic
30#15690744C2212yescomplemented
31#15690741C2212yescomplemented
32#15690739C2212yescomplemented
33#15690734C2212yescomplemented
34#15690732C2212yescomplemented
35#15690731C2212yes—
36C22⋊#11715895C2211yescharacteristic
37#15690720C2212yescomplemented
38C22×D4×C97C2211yescharacteristic
39C2×SD16×C97C2212yes—
40C22⋊Q8×C97C2211yescharacteristic
41#102×C97C2212yes—
42C22⋊C8×C97C2211yescharacteristic
43C2×D388C2311yescharacteristic
44C2×C22⋊D97—21no—
45C22×Dic97—21no—
46C22×Dic97—22no—
47C22×Dic97—22no—
48C22×Dic97—44no—
49C22×Dic97D411yescharacteristic
50C22×Dic97D411yescharacteristic
51C2×C4×D97—21no—
52#13095496—24no—
53C2×#11715895C2312yes—
54C4⋊Dic97—21no—
55C4⋊Dic97—21no—
56C4⋊Dic97—21no—
57C4⋊Dic97C2312yes—
58C4⋊Dic97C2311yescharacteristic
59#13095490—21no—
60C22⋊Dic97—22no—
61C22⋊Dic97—22no—
62#13095488—21no—
63#13095488—21no—
64#13095488—21no—
65#13095488—21no—
66C4×Dic97—21no—
67C4×Dic97D411yescharacteristic
68C4×Dic97D411yescharacteristic
69C4×Dic97C2311yescharacteristic
70C4×Dic97C2311yescharacteristic
71C24×C97—21no—
72C2×Q8×C97C2311yescharacteristic
73C2×D4×C97—24no—
74C2×D4×C97—21no—
75C2×D4×C97C2312yes—
76C22×C4×C97C2311yescharacteristic
77SD16×C97—24no—
78C2×C8×C97C2312yes—
79C4⋊C4×C97—21no—
80C4⋊C4×C97C2311yescharacteristic
81C22⋊C4×C97—21no—
82C23×C97—24no—
83C23×C97—22no—
84C23×C97—42no—
85C23×C97—42no—
86C23×C97C2×D411yescharacteristic
87C22×D97—21no—
88Q8×C97—22no—
89D4×C97—24no—
90D4×C97—22no—
91D4×C97#4114yes—
92C2×C4×C97—22no—
93C2×C4×C97—21no—
94C2×C4×C97—21no—
95C2×C4×C97C2×D411yescharacteristic
96C2×C4×C97C2411yescharacteristic
97C22⋊D97—42no—
98C2×Dic97—22no—
99C2×Dic97—21no—
100C2×Dic97—22no—
101C2×Dic97—41no—
102C2×Dic97—44no—
103C2×Dic97—44no—
104C2×Dic97—42no—
105C2×Dic97C2×D411yescharacteristic
106C2×Dic97C2×D411yescharacteristic
107C2×Dic97C2×D411yescharacteristic
108C2×Dic97C2×D411yescharacteristic
109D388—22no—
110C4×D97—42no—
111C8×C97—22no—
112#11715895—22no—
113C22×C97—21no—
114C22×C97—21no—
115C22×C97—24no—
116C22×C97—48no—
117C22×C97—42no—
118C22×C97—42no—
119C22×C97AGL(1,ℤ/8)12yes—
120C22×C97C22×D411yescharacteristic
121C2×D97—41no—
122C2×D97—41no—
123C2×D97—41no—
124C4×C97—21no—
125C4×C97—41no—
126C4×C97—41no—
127C4×C97C2×#4112yes—
128Dic97—21no—
129Dic97—21no—
130Dic97—21no—
131Dic97—21no—
132Dic97—42no—
133Dic97—41no—
134C2×C97—22no—
135C2×C97—44no—
136C2×C97#52511yescharacteristic
137C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C97#57511yescharacteristic
139D97—81no—
140#3114—971nomaximal, Sylow
141C97#311411yescharacteristic, complemented, Sylow
142#529—971no—
143C2×C4×D4—971no—
144#483—971no—
145#480—971no—
146#478—972no—
147#463—972no—
148#450D9711yescharacteristic, complemented
149#447—971no—
150#442—972no—
151#440—972no—
152#432—971no—
153C22×D4C2×D9711yescharacteristic
154C23×C4—1941no—
155C2×SD16C2×D9712yes—
156C2×D8—972no—
157#126—1941no—
158#125—971no—
159#124—971no—
160#123—1941no—
161C22⋊Q8C2×D9711yescharacteristic
162#121—971no—
163C4×Q8—971no—
164C4×D4—972no—
165C4×D4—972no—
166C4×D4—971no—
167C4×D4—1941no—
168C4×D4—1941no—
169C4×D4—1941no—
170#117—1941no—
171C2×C4⋊C4—971no—
172C2×C22⋊C4—1941no—
173C2×C42—971no—
174#107—972no—
175#106—972no—
176C4⋊C8—972no—
177#103—972no—
178#102—972no—
179#102—972no—
180#102C2×D9712yes—
181C22⋊C8—971no—
182C22⋊C8C2×D9711yescharacteristic
183#97—972no—
184C24—21no—
185C2×Q8C22×D9711yescharacteristic
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—971no—
189C2×D4—1941no—
190C2×D4C22×D9712yes—
191C22×C4—971no—
192C22×C4—971no—
193C22×C4—3884no—
194C22×C4—1941no—
195C22×C4—1942no—
196C22×C4—1942no—
197C22×C4—1941no—
198C22×C4C22×D9711yescharacteristic
199SD16—24no—
200D8—1944no—
201C2×C8—972no—
202C2×C8C22×D9712yes—
203C4⋊C4—21no—
204C4⋊C4—971no—
205C4⋊C4—972no—
206C4⋊C4—1941no—
207C4⋊C4—1941no—
208C4⋊C4—1941no—
209C4⋊C4—1941no—
210C4⋊C4C22×D9711yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—1941no—
213C22⋊C4—1941no—
214C22⋊C4—1941no—
215C22⋊C4—1942no—
216C22⋊C4—1942no—
217C22⋊C4—1941no—
218C42—971no—
219C42—971no—
220C42—971no—
221C42—971no—
222C42—1941no—
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—1941no—
228C23D4×D9711yescharacteristic
229Q8—22no—
230D4—24no—
231D4—22no—
232D4—3882no—
233D4—1942no—
234D4#1309551014yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—971no—
239C2×C4—971no—
240C2×C4—971no—
241C2×C4—971no—
242C2×C4—3882no—
243C2×C4—3882no—
244C2×C4—3884no—
245C2×C4—3884no—
246C2×C4—3881no—
247C2×C4—1941no—
248C2×C4—1942no—
249C2×C4—1942no—
250C2×C4D4×D9711yescharacteristic
251C2×C4C23×D9711yescharacteristic
252C8—22no—
253C8—1942no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—3881no—
261C22—3881no—
262C22—3881no—
263C22#1569083712yes—
264C22C2×D4×D9711yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—3882no—
269C4—3881no—
270C4—1941no—
271C4—1941no—
272C4—1941no—
273C4—1941no—
274C4C2×#1309551012yes—
275C2—22no—
276C2—44no—
277C2—7761no—
278C2#2026124111yescharacteristic
279C2C2×#1569083711yescharacteristic
280C2#2026153111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#450×C97C2
socleC22×C97C22×D4
prime radical#20260869C2
hypercenter#450D97
hypocenterC97#3114
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #20261271C2semidirect
2. C2×D4×Dic97C2semidirect
3. #20261063C2semidirect
4. #20261049C2semidirect
5. #20261033C2semidirect
6. #20260952C2semidirect
8. #20260844C2semidirect
9. #20260836C2semidirect
10. #20260789C2semidirect
11. #450×C97C2semidirect
15. #15690801C22semidirect
16. #15690799C22semidirect
19. Q8×Dic97C22semidirect
24. #15690772C22semidirect
30. #15690744C22semidirect
31. #15690741C22semidirect
32. #15690739C22semidirect
33. #15690734C22semidirect
34. #15690732C22semidirect
37. #15690720C22semidirect
141. C97#3114semidirect
148. #450D97semidirect

Maximal subgroups

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

subgroupindex
#202612712
C2×D4×Dic972
#202610632
#202610492
#202610332
#202609522
#202608692
#202608442
#202608362
#202607892
#450×C972
#311497