Finite Group Databasebeta

← #38752655

Subgroups — #38752655

Order 25216.

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

normal subgroups only

This group has 410 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
1C22⋊Q8×D197C211yesmaximal, characteristic
2C2×#20263284C211yesmaximal, characteristic
3#27161423C212yesmaximal, complemented
4#27161419C212yesmaximal, complemented
5#27161415C211yesmaximal, characteristic, complemented
6#27161412C211yesmaximal, characteristic, complemented
7#27161409C211yesmaximal, characteristic, complemented
8#27161118C212yesmaximal, complemented
9#27161086C212yesmaximal, complemented
10#27161041C211yesmaximal, characteristic
11#483×C197C211yesmaximal, characteristic, complemented
12C2×Q8×D197C2211yescharacteristic
13C22×C4×D197C2211yescharacteristic
14C2×#15691341C2212yes—
15#20263306C2211yescharacteristic
16#20263305C2211yescharacteristic
17#20263304—21no—
18#20263303—21no—
19#20263302C2211yescharacteristic
20#20263284—21no—
21#20263284—21no—
22#20263284C2212yes—
23#20263284C2212yes—
24C2×C4⋊Dic197C2211yescharacteristic
25C4⋊C4×D197—21no—
26C4⋊C4×D197C2211yescharacteristic
27C2×#15691335—21no—
28C22⋊C4×D197—21no—
29C2×C4×Dic197—21no—
30#20263260C2212yescomplemented
31#20263259C2212yescomplemented
32#20263257C2212yescomplemented
33#20263251C2212yescomplemented
34#20263250C2212yescomplemented
35#20263249C2212yescomplemented
36C22⋊#13096802C2211yescharacteristic
37#20263238C2212yes—
38C22⋊Q8×C197C2211yescharacteristic
39C2×C4⋊C4×C197C2211yescharacteristic
40#106×C197C2212yescomplemented
41#103×C197C2212yescomplemented
42C22⋊C8×C197C2211yescharacteristic
43C23×D197D411yescharacteristic
44C2×Dic394—21no—
45C2×Dic394C2311yescharacteristic
46Q8×D197—24no—
47C22×Dic197—21no—
48C22×Dic197D411yescharacteristic
49C2×C4×D197—24no—
50C2×C4×D197—21no—
51C2×C4×D197—21no—
52C2×C4×D197D411yescharacteristic
53C2×C4×D197D411yescharacteristic
54C2×C4×D197C2311yescharacteristic
55C2×C4×D197C2311yescharacteristic
56#15691341—24no—
57C2×#13096802C2312yes—
58C4⋊Dic197—21no—
59C4⋊Dic197—21no—
60C4⋊Dic197C2312yes—
61C4⋊Dic197C2311yescharacteristic
62#15691337—21no—
63#15691337—21no—
64#15691337—21no—
65C22⋊Dic197—21no—
66#15691335—21no—
67#15691335—21no—
68#15691335—22no—
69#15691335—22no—
70C4×Dic197—22no—
71C4×Dic197—22no—
72C2×Q8×C197C2311yescharacteristic
73C22×C4×C197—21no—
74C22×C4×C197C2311yescharacteristic
75C2×C8×C197C2312yes—
76C4⋊C4×C197—21no—
77C4⋊C4×C197—21no—
78C4⋊C4×C197C2312yes—
79C4⋊C4×C197C2311yescharacteristic
80C22⋊C4×C197—21no—
81C22×D197—22no—
82C22×D197—22no—
83C22×D197—41no—
84C22×D197C2×D411yescharacteristic
85C22×D197C2×D411yescharacteristic
86Dic394—22no—
87Dic394—42no—
88C2×Dic197—22no—
89C2×Dic197—22no—
90C2×Dic197—21no—
91C2×Dic197—21no—
92C2×Dic197—42no—
93C2×Dic197C2×D411yescharacteristic
94C2×Dic197C2×D411yescharacteristic
95C4×D197—22no—
96C4×D197—22no—
97C4×D197—22no—
98C4×D197—42no—
99C4×D197—42no—
100C4×D197#4114yes—
101#13096802—22no—
102C23×C197C2×D411yescharacteristic
103Q8×C197—22no—
104C2×C4×C197—22no—
105C2×C4×C197—21no—
106C2×C4×C197—21no—
107C2×C4×C197—22no—
108C2×C4×C197—42no—
109C2×C4×C197C2×D411yescharacteristic
110C2×C4×C197C2411yescharacteristic
111C8×C197—22no—
112C22×C197—21no—
113C22×C197—21no—
114C22×C197#13712yes—
115C22×C197C22×D411yescharacteristic
116C2×D197—21no—
117C2×D197—21no—
118C2×D197—21no—
119C2×D197—21no—
120C2×D197—21no—
121C2×D197—44no—
122C2×D197#12312yes—
123C4×C197—21no—
124C4×C197—41no—
125C4×C197—42no—
126C4×C197—41no—
127C4×C197C2×#4112yes—
128Dic197—21no—
129Dic197—41no—
130Dic197—41no—
131Dic197—42no—
132Dic197#12312yes—
133C2×C197—22no—
134C2×C197C2×#12311yescharacteristic
135C2×C197C2×#13711yescharacteristic
136C2×C197#57511yescharacteristic
137D197—22no—
138D197—41no—
139C197#309511yescharacteristic, complemented, Sylow
140#3095—1971nomaximal, Sylow
141C2×C22⋊Q8—1971no—
142C2×#117—1971no—
143#485—1972no—
144#485—1972no—
145#484—1971no—
146#484—1971no—
147#483—1971no—
148#483D19711yescharacteristic, complemented
149#428—1972no—
150#419—1972no—
151#407—1971no—
152C22×Q8—1971no—
153C23×C4—1971no—
154C2×M16—1972no—
155C22⋊Q8—3941no—
156C22⋊Q8—3941no—
157C22⋊Q8—1971no—
158C22⋊Q8—1971no—
159C22⋊Q8—1971no—
160C22⋊Q8C2×D19711yescharacteristic
161#117—3941no—
162#117—3941no—
163#117—1972no—
164#117—1972no—
165C2×C4⋊C4—3941no—
166C2×C4⋊C4—1971no—
167C2×C4⋊C4—1971no—
168C2×C4⋊C4C2×D19711yescharacteristic
169C2×C22⋊C4—3941no—
170C2×C22⋊C4—3941no—
171C2×C42—3941no—
172#107—1972no—
173#107—1972no—
174#106—1972no—
175#106C2×D19712yescomplemented
176#103—1972no—
177#103—1972no—
178#103—1972no—
179#103C2×D19712yescomplemented
180C22⋊C8—1971no—
181C22⋊C8—1972no—
182C22⋊C8C2×D19711yescharacteristic
183C24—1971no—
184C2×Q8—3941no—
185C2×Q8—3944no—
186C2×Q8—1971no—
187C2×Q8C22×D19711yescharacteristic
188C22×C4—21no—
189C22×C4—3941no—
190C22×C4—3941no—
191C22×C4—3944no—
192C22×C4—3941no—
193C22×C4—1971no—
194C22×C4—1971no—
195C22×C4—1971no—
196C22×C4—1971no—
197C22×C4—1971no—
198C22×C4C22×D19711yescharacteristic
199M16—3944no—
200C2×C8—1972no—
201C2×C8C22×D19712yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—3941no—
205C4⋊C4—3941no—
206C4⋊C4—3941no—
207C4⋊C4—3941no—
208C4⋊C4—3941no—
209C4⋊C4—1972no—
210C4⋊C4—1971no—
211C4⋊C4C22×D19712yes—
212C4⋊C4C22×D19711yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—3942no—
215C22⋊C4—3942no—
216C22⋊C4—3941no—
217C22⋊C4—3941no—
218C22⋊C4—3941no—
219C42—3942no—
220C42—3942no—
221C23—7881no—
222C23—3942no—
223C23—3942no—
224C23—1971no—
225C23—1971no—
226C23D4×D19711yescharacteristic
227Q8—22no—
228Q8—7882no—
229Q8—3942no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—7882no—
236C2×C4—7882no—
237C2×C4—7882no—
238C2×C4—3942no—
239C2×C4—3942no—
240C2×C4—3942no—
241C2×C4—3941no—
242C2×C4—3941no—
243C2×C4—3942no—
244C2×C4—3942no—
245C2×C4—1971no—
246C2×C4—1971no—
247C2×C4—1974no—
248C2×C4D4×D19711yescharacteristic
249C2×C4C23×D19711yescharacteristic
250C8—22no—
251C8—3942no—
252C22—21no—
253C22—21no—
254C22—7884no—
255C22—3941no—
256C22—3941no—
257C22—3941no—
258C22—3941no—
259C22—3941no—
260C22—1972no—
261C22#2026335512yes—
262C22C2×D4×D19711yescharacteristic
263C4—21no—
264C4—42no—
265C4—41no—
266C4—41no—
267C4—7881no—
268C4—7882no—
269C4—7881no—
270C4—3941no—
271C4—1972no—
272C4#41×D19712yes—
273C2—22no—
274C2—7881no—
275C2—3942no—
276C2#123×D19711yescharacteristic
277C2C2×#2026335511yescharacteristic
278C2#2716187811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#483×C197C2
socleC22×C197C22×D4
prime radicalC23×D197D4
hypercenter#483D197
hypocenterC197#3095
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #27161423C2semidirect
4. #27161419C2semidirect
5. #27161415C2semidirect
6. #27161412C2semidirect
7. #27161409C2semidirect
8. #27161118C2semidirect
9. #27161086C2semidirect
11. #483×C197C2semidirect
30. #20263260C22semidirect
31. #20263259C22semidirect
32. #20263257C22semidirect
33. #20263251C22semidirect
34. #20263250C22semidirect
35. #20263249C22semidirect
40. #106×C197C22semidirect
41. #103×C197C22semidirect
139. C197#3095semidirect
148. #483D197semidirect
175. #106C2×D197semidirect
179. #103C2×D197semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1972
C2×#202632842
#271614232
#271614192
#271614152
#271614122
#271614092
#271611182
#271610862
#271610412
#483×C1972
#3095197