Finite Group Databasebeta

← #38752504

Subgroups — #38752504

Order 25216.

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#27161634C211yesmaximal, characteristic, complemented
2C4×D4×D197C211yesmaximal, characteristic
3#27161382C212yesmaximal, complemented
4#27161381C212yesmaximal, complemented
5#27161370C211yesmaximal, characteristic, complemented
6#27161369C211yesmaximal, characteristic, complemented
7#27161356C211yesmaximal, characteristic, complemented
8#27161120C212yesmaximal, complemented
9#27161101C211yesmaximal, characteristic, complemented
10#27161082C212yesmaximal
11#474×C197C211yesmaximal, characteristic, complemented
12C2×D4×D197C2211yescharacteristic
13C22×C4×D197—21no—
14C2×#15691341C2212yes—
15#20263326C2211yescharacteristic, complemented
16#20263317C2212yescomplemented
17#20263306—21no—
18#20263302—21no—
19Q8×Dic197—21no—
20#20263291—21no—
21C4×D788C2211yescharacteristic
22D4×Dic197C2211yescharacteristic
23#20263288C2211yescharacteristic
24C4×C22⋊D197—21no—
25#20263286—21no—
26#20263284C2212yes—
27C4⋊C4×D197C2211yescharacteristic
28C22⋊C4×D197—21no—
29C42×D197C2211yescharacteristic
30#20263259C2212yescomplemented
31#20263258C2212yescomplemented
32#20263257C2212yescomplemented
33C4⋊#13096802C2211yescharacteristic, complemented
34#20263255C2212yescomplemented
35#20263248C2212yes—
36#20263247C2212yes—
37#20263246C2212yes—
38C4⋊Q8×C197C2211yescharacteristic, complemented
39C4×D4×C197C2211yescharacteristic
40#107×C197C2212yescomplemented
41C4⋊C8×C197C2211yescharacteristic, complemented
42#102×C197C2212yes—
43C23×D197—21no—
44C2×Dic394—21no—
45C2×D788C2311yescharacteristic
46C2×C22⋊D197—21no—
47D4×D197—22no—
48D4×D197Q814yescomplemented
49C22×Dic197—21no—
50C2×C4×D197—21no—
51C2×C4×D197—21no—
52C2×C4×D197—21no—
53C2×C4×D197—44no—
54C2×C4×D197D411yescharacteristic
55C2×C4×D197D411yescharacteristic
56C2×C4×D197C2311yescharacteristic
57#15691341—24no—
58C2×#13096802C2312yes—
59C4⋊Dic197—22no—
60C4⋊Dic197—21no—
61C4⋊Dic197C2312yes—
62C4⋊Dic197C2311yescharacteristic
63#15691337—21no—
64#15691337—22no—
65C22⋊Dic197—21no—
66#15691335—22no—
67#15691335—21no—
68#15691335—21no—
69C4×Dic197—22no—
70C4×Dic197D411yescharacteristic
71C4×Dic197D411yescharacteristic
72C4×Dic197C2311yescharacteristic
73C2×Q8×C197—21no—
74C2×D4×C197C2311yescharacteristic
75C22×C4×C197—21no—
76C2×C8×C197C2312yes—
77C4⋊C4×C197—21no—
78C4⋊C4×C197C2312yes—
79C4⋊C4×C197C2311yescharacteristic
80C22⋊C4×C197—21no—
81C42×C197C2311yescharacteristic
82C22×D197—21no—
83C22×D197—24no—
84C22×D197—42no—
85C22×D197—42no—
86C22×D197C2×D411yescharacteristic
87Dic394—42no—
88D788—21no—
89D788C2×Q812yes—
90C22⋊D197—24no—
91C2×Dic197—22no—
92C2×Dic197—21no—
93C2×Dic197—42no—
94C2×Dic197C2×D411yescharacteristic
95C2×Dic197C2×D411yescharacteristic
96C2×Dic197C2×D411yescharacteristic
97C4×D197—22no—
98C4×D197—41no—
99C4×D197—42no—
100C4×D197—42no—
101C4×D197—42no—
102C4×D197C2×Q812yes—
103C4×D197#4112yes—
104#13096802—22no—
105C23×C197—21no—
106Q8×C197—42no—
107D4×C197—21no—
108D4×C197C2×Q812yes—
109C2×C4×C197—21no—
110C2×C4×C197—22no—
111C2×C4×C197—42no—
112C2×C4×C197C2×D411yescharacteristic
113C2×C4×C197C2×D411yescharacteristic
114C2×C4×C197C2411yescharacteristic
115C8×C197—22no—
116C22×C197—22no—
117C22×C197—41no—
118C22×C197—41no—
119C22×C197C22×D411yescharacteristic
120C2×D197—21no—
121C2×D197—21no—
122C2×D197—22no—
123C2×D197—48no—
124C2×D197—41no—
125C2×D197—41no—
126C2×D197C22⋊Q812yes—
127C4×C197—21no—
128C4×C197—41no—
129C4×C197—42no—
130C4×C197AGL(1,ℤ/8)12yes—
131C4×C197C22×Q811yescharacteristic
132C4×C197C2×#4111yescharacteristic
133Dic197—21no—
134Dic197—21no—
135Dic197—41no—
136Dic197—42no—
137Dic197C22⋊Q812yes—
138C2×C197—42no—
139C2×C197C2×C22⋊Q811yescharacteristic
140C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
141C2×C197#57511yescharacteristic
142D197—22no—
143D197—42no—
144C197#307711yescharacteristic, complemented, Sylow
145#3077—1971nomaximal, Sylow
146#533—1971no—
147C2×C4×D4—1971no—
148#478—1972no—
149#478—1972no—
150#476—1971no—
151#476—1971no—
152#474—1971no—
153#474D19711yescharacteristic, complemented
154#428—1972no—
155#423—1971no—
156#418—1972no—
157C22×D4—1971no—
158C23×C4—3941no—
159C2×M16—1972no—
160C4⋊Q8—1971no—
161C4⋊Q8C2×D19711yescharacteristic, complemented
162#125—1972no—
163C22⋊Q8—3941no—
164C22⋊Q8—3941no—
165C4×Q8—3941no—
166C4×Q8—3941no—
167C4×D4—3941no—
168C4×D4—3941no—
169C4×D4—1971no—
170C4×D4—1971no—
171C4×D4—1971no—
172C4×D4C2×D19711yescharacteristic
173#117—1972no—
174C2×C4⋊C4—1971no—
175C2×C22⋊C4—3941no—
176C2×C42—1971no—
177#107—1972no—
178#107C2×D19712yescomplemented
179#106—1972no—
180#106—1972no—
181C4⋊C8—1972no—
182C4⋊C8—1971no—
183C4⋊C8C2×D19711yescharacteristic, complemented
184#102—1972no—
185#102—1972no—
186#102—1972no—
187#102C2×D19712yes—
188C24—3941no—
189C2×Q8—21no—
190C2×Q8—3941no—
191C2×D4—3942no—
192C2×D4—3941no—
193C2×D4—1974no—
194C2×D4—1971no—
195C2×D4C22×D19711yescharacteristic
196C22×C4—21no—
197C22×C4—7884no—
198C22×C4—3941no—
199C22×C4—3941no—
200C22×C4—3941no—
201C22×C4—3941no—
202C22×C4—1971no—
203C22×C4—1971no—
204C22×C4—1971no—
205M16—3944no—
206C2×C8—1972no—
207C2×C8C22×D19712yes—
208C4⋊C4—21no—
209C4⋊C4—3941no—
210C4⋊C4—3942no—
211C4⋊C4—3942no—
212C4⋊C4—3941no—
213C4⋊C4—1972no—
214C4⋊C4—1971no—
215C4⋊C4C22×D19712yes—
216C4⋊C4C22×D19711yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—3942no—
219C22⋊C4—3941no—
220C22⋊C4—3941no—
221C22⋊C4—3941no—
222C42—3942no—
223C42—1971no—
224C42—1971no—
225C42—1971no—
226C42C22×D19711yescharacteristic
227C23—21no—
228C23—7882no—
229C23—7882no—
230C23—3944no—
231C23—3941no—
232C23—1971no—
233Q8—42no—
234Q8—7882no—
235D4—21no—
236D4—3944no—
237D4—3941no—
238D4—1972no—
239D4Q8×D19712yes—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—7881no—
244C2×C4—7882no—
245C2×C4—7882no—
246C2×C4—7882no—
247C2×C4—7882no—
248C2×C4—3942no—
249C2×C4—3941no—
250C2×C4—3942no—
251C2×C4—1971no—
252C2×C4—1971no—
253C2×C4—1971no—
254C2×C4—1972no—
255C2×C4—1972no—
256C2×C4D4×D19711yescharacteristic
257C2×C4D4×D19711yescharacteristic
258C2×C4C23×D19711yescharacteristic
259C8—22no—
260C8—3942no—
261C22—22no—
262C22—41no—
263C22—41no—
264C22—7888no—
265C22—7881no—
266C22—7881no—
267C22—3942no—
268C22—3941no—
269C22—3941no—
270C22—1972no—
271C22C2×D4×D19711yescharacteristic
272C4—21no—
273C4—41no—
274C4—42no—
275C4—7881no—
276C4—7882no—
277C4—3941no—
278C4—3941no—
279C4—1972no—
280C4#2026335012yes—
281C4C2×Q8×D19711yescharacteristic
282C4#41×D19711yescharacteristic
283C2—42no—
284C2—7882no—
285C2—3942no—
286C2C22⋊Q8×D19711yescharacteristic
287C2C2×#2026335011yescharacteristic
288C2#2716187811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#474×C197C2
socleC22×C197C22×D4
prime radicalC2×D4×D197C22
hypercenter#474D197
hypocenterC197#3077
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #27161634C2semidirect
3. #27161382C2semidirect
4. #27161381C2semidirect
5. #27161370C2semidirect
6. #27161369C2semidirect
7. #27161356C2semidirect
8. #27161120C2semidirect
9. #27161101C2semidirect
11. #474×C197C2semidirect
15. #20263326C22semidirect
16. #20263317C22semidirect
30. #20263259C22semidirect
31. #20263258C22semidirect
32. #20263257C22semidirect
33. C4⋊#13096802C22semidirect
34. #20263255C22semidirect
38. C4⋊Q8×C197C22semidirect
40. #107×C197C22semidirect
41. C4⋊C8×C197C22semidirect
48. D4×D197Q8semidirect
144. C197#3077semidirect
153. #474D197semidirect
161. C4⋊Q8C2×D197semidirect
178. #107C2×D197semidirect
183. C4⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616342
C4×D4×D1972
#271613822
#271613812
#271613702
#271613692
#271613562
#271611202
#271611012
#271610822
#474×C1972
#3077197