Finite Group Databasebeta

← #38754398

Subgroups — #38754398

Order 25216.

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

normal subgroups only

This group has 526 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
1C2×#20263345C211yesmaximal, characteristic, complemented
2C2×D8×D197C211yesmaximal, characteristic, complemented
3#27161724C212yesmaximal, complemented
4#27161463C211yesmaximal, characteristic, complemented
5#27161321C211yesmaximal, characteristic, complemented
6#27161316C211yesmaximal, characteristic, complemented
7#27161294C212yesmaximal, complemented
8#27161231C212yesmaximal, complemented
9#27161225C212yesmaximal, complemented
10C8×D788C211yesmaximal, characteristic, complemented
11C4×D8×C197C211yesmaximal, characteristic, complemented
12C2×#15691357C2212yescomplemented
13C2×D4×D197C2212yes—
14#20263345—24no—
15C2×Dic788C2211yescharacteristic, complemented
16C2×#15691347C2212yescomplemented
17C2×#15691346C2212yescomplemented
18C2×D1576C2211yescharacteristic
19C2×#15691343C2212yes—
20D8×D197—24no—
21C2×C8×D197C2211yescharacteristic
22C2×C8×D197C2211yescharacteristic, complemented
23#20263314C2212yescomplemented
24#20263310—22no—
25#20263306C2212yes—
26#20263301C2212yescomplemented
27#20263300—22no—
28#20263296—22no—
29C4×D788C2211yescharacteristic
30C2×#15691335—22no—
31C4⋊#13096802C2211yescharacteristic, complemented
32#20263249C2212yescomplemented
33#20263246C2212yescomplemented
34#20263238C2211yescharacteristic
35#20263238C2211yescharacteristic, complemented
36C2×D8×C197C2211yescharacteristic
37C4×D4×C197C2212yescomplemented
38#107×C197C2211yescharacteristic, complemented
39#102×C197C2212yescomplemented
40C4×C8×C197C2211yescharacteristic, complemented
41C23×D197—22no—
42#15691357—24no—
43#15691357—42no—
44C2×Dic394C2312yes—
45C2×D788—22no—
46C2×D788D411yescharacteristic
47C2×D788C2312yes—
48C2×C22⋊D197—22no—
49C2×C22⋊D197—22no—
50D4×D197—24no—
51D4×D197—42no—
52C22×Dic197—22no—
53C2×C4×D197C2311yescharacteristic
54C2×C4×D197C2311yescharacteristic
55Dic788—22no—
56#15691347—42no—
57#15691346—24no—
58D1576—22no—
59#15691343—24no—
60C2×#13096802C2311yescharacteristic
61C2×#13096802C2311yescharacteristic
62C8×D197—22no—
63C8×D197—22no—
64C4⋊Dic197—22no—
65C4⋊Dic197D411yescharacteristic
66#15691335—22no—
67#15691335—22no—
68#15691335—22no—
69#15691335—22no—
70#15691335D411yescharacteristic
71#15691335D411yescharacteristic
72C2×D4×C197C2312yes—
73C22×C4×C197—22no—
74D8×C197D414yescomplemented
75C2×C8×C197C2311yescharacteristic
76C2×C8×C197C2311yescharacteristic
77C4⋊C4×C197C2312yes—
78C22⋊C4×C197—22no—
79C42×C197C2311yescharacteristic
80C22×D197—22no—
81C22×D197—44no—
82C22×D197—44no—
83C22×D197—82no—
84C22×D197C2×D411yescharacteristic
85C22×D197C2×D411yescharacteristic
86Dic394—22no—
87Dic394—22no—
88D788—22no—
89D788—22no—
90D788—22no—
91D788—82no—
92C22⋊D197—44no—
93C22⋊D197—44no—
94C2×Dic197—22no—
95C2×Dic197—44no—
96C2×Dic197C2×D411yescharacteristic
97C2×Dic197C2×D411yescharacteristic
98C4×D197—21no—
99C4×D197—21no—
100C4×D197—21no—
101C4×D197—21no—
102#13096802—21no—
103#13096802—21no—
104C23×C197—22no—
105D4×C197—22no—
106D4×C197C2×D414yes—
107C2×C4×C197—22no—
108C2×C4×C197—82no—
109C2×C4×C197C2×D411yescharacteristic
110C2×C4×C197C2×D411yescharacteristic
111C2×C4×C197C2411yescharacteristic
112C8×C197—21no—
113C8×C197C2×D412yes—
114C22×C197—24no—
115C22×C197—42no—
116C22×C197—42no—
117C22×C197C22×D411yescharacteristic
118C2×D197—21no—
119C2×D197—21no—
120C2×D197—21no—
121C2×D197—21no—
122C2×D197—21no—
123C2×D197—42no—
124C2×D197—42no—
125C2×D197—42no—
126C2×D197—88no—
127C2×D197#13512yescomplemented
128C4×C197—21no—
129C4×C197—21no—
130C4×C197—42no—
131C4×C197C22×D411yescharacteristic
132C4×C197H(5,𝔽2)11yescharacteristic
133Dic197—21no—
134Dic197—21no—
135Dic197—42no—
136C2×C197—44no—
137C2×C197D4211yescharacteristic
138C2×C197C2×#13511yescharacteristic
139C2×C197#57611yescharacteristic
140D197—22no—
141D197—41no—
142D197—82no—
143C197#328011yescharacteristic, complemented, Sylow
144#3280—1971nomaximal, Sylow
145C2×#135—1971no—
146C22×D8—1971no—
147#546—1972no—
148#495—1971no—
149#467—1971no—
150#466—1971no—
151#463—1972no—
152#450—1972no—
153#449—1972no—
154C4×D8D19711yescharacteristic, complemented
155C8×D4—1971no—
156C2×#41—1972no—
157C22×D4—1972no—
158#135—3944no—
159C2×Q16—1971no—
160C2×SD16—1972no—
161C2×SD16—1972no—
162C2×D8—3944no—
163C2×D8—1972no—
164C2×D8—1971no—
165C2×D8C2×D19711yescharacteristic
166C22×C8—1971no—
167C22×C8—1971no—
168#124—1972no—
169#123—3942no—
170C22⋊Q8—1972no—
171#121—3942no—
172#121—1972no—
173#120—3942no—
174C4×D4—1971no—
175C4×D4C2×D19712yescomplemented
176C2×C22⋊C4—3942no—
177#107C2×D19711yescharacteristic, complemented
178C4⋊C8—1971no—
179#103—1972no—
180#102—1972no—
181#102C2×D19712yescomplemented
182C22⋊C8—1971no—
183C22⋊C8—1971no—
184C4×C8C2×D19711yescharacteristic, complemented
185C24—3942no—
186#41—7882no—
187#41—3944no—
188C2×Q8—1972no—
189C2×D4—7882no—
190C2×D4—3942no—
191C2×D4—3942no—
192C2×D4—3944no—
193C2×D4—3942no—
194C2×D4—1971no—
195C2×D4—1972no—
196C2×D4C22×D19712yes—
197C22×C4—22no—
198C22×C4—3942no—
199C22×C4—1971no—
200C22×C4—1971no—
201Q16—3942no—
202SD16—7882no—
203SD16—3944no—
204D8—3944no—
205D8—3942no—
206D8D78814yescomplemented
207C2×C8—3942no—
208C2×C8—3942no—
209C2×C8—1971no—
210C2×C8—1971no—
211C2×C8C22×D19711yescharacteristic
212C2×C8C22×D19711yescharacteristic
213C4⋊C4—3942no—
214C4⋊C4—1971no—
215C4⋊C4C22×D19712yes—
216C22⋊C4—22no—
217C22⋊C4—3942no—
218C22⋊C4—3942no—
219C22⋊C4—3942no—
220C22⋊C4—3942no—
221C22⋊C4—1971no—
222C22⋊C4—1971no—
223C42C22×D19711yescharacteristic
224C23—22no—
225C23—7884no—
226C23—7884no—
227C23—15762no—
228C23—3942no—
229C23—1971no—
230C23—1971no—
231Q8—3942no—
232Q8—3942no—
233D4—22no—
234D4—7884no—
235D4—7884no—
236D4—15762no—
237D4—3942no—
238D4—3942no—
239D4—3942no—
240D4C2×D78814yes—
241C2×C4—22no—
242C2×C4—82no—
243C2×C4—7884no—
244C2×C4—3942no—
245C2×C4—3941no—
246C2×C4—3941no—
247C2×C4—3941no—
248C2×C4—3941no—
249C2×C4—1971no—
250C2×C4—1971no—
251C2×C4D4×D19711yescharacteristic
252C2×C4D4×D19711yescharacteristic
253C2×C4C23×D19711yescharacteristic
254C8—21no—
255C8—3941no—
256C8—3941no—
257C8C2×D78812yes—
258C22—24no—
259C22—42no—
260C22—42no—
261C22—7882no—
262C22—7882no—
263C22—7882no—
264C22—15768no—
265C22—3941no—
266C22—3941no—
267C22—3941no—
268C22—3941no—
269C22—3941no—
270C22—1972no—
271C22C2×D4×D19711yescharacteristic
272C4—21no—
273C4—21no—
274C4—42no—
275C4—7882no—
276C4—3941no—
277C4—3941no—
278C4C22×D78811yescharacteristic
279C4#2026337211yescharacteristic
280C2—44no—
281C2—7881no—
282C2—15762no—
283C2—3942no—
284C2D4×D78811yescharacteristic
285C2#135×D19711yescharacteristic
286C2#2716188011yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC4×D8×C197C2
socleC22×C197C22×D4
hypercenterC4×D8D197
hypocenterC197#3280
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#20263345C2semidirect
2. C2×D8×D197C2semidirect
3. #27161724C2semidirect
4. #27161463C2semidirect
5. #27161321C2semidirect
6. #27161316C2semidirect
7. #27161294C2semidirect
8. #27161231C2semidirect
9. #27161225C2semidirect
10. C8×D788C2semidirect
11. C4×D8×C197C2semidirect
12. C2×#15691357C22semidirect
15. C2×Dic788C22semidirect
16. C2×#15691347C22semidirect
17. C2×#15691346C22semidirect
22. C2×C8×D197C22semidirect
23. #20263314C22semidirect
26. #20263301C22semidirect
31. C4⋊#13096802C22semidirect
32. #20263249C22semidirect
33. #20263246C22semidirect
35. #20263238C22semidirect
37. C4×D4×C197C22semidirect
38. #107×C197C22semidirect
39. #102×C197C22semidirect
40. C4×C8×C197C22semidirect
74. D8×C197D4semidirect
127. C2×D197#135semidirect
143. C197#3280semidirect
154. C4×D8D197semidirect
175. C4×D4C2×D197semidirect
177. #107C2×D197semidirect
181. #102C2×D197semidirect
184. C4×C8C2×D197semidirect
206. D8D788semidirect

Maximal subgroups

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

subgroupindex
C2×#202633452
C2×D8×D1972
#271617242
#271614632
#271613212
#271613162
#271612942
#271612312
#271612252
C8×D7882
C4×D8×C1972
#3280197