Finite Group Databasebeta

← #38754526

Subgroups — #38754526

Order 25216.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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#27161733C211yesmaximal, characteristic, complemented
2D4×D788C211yesmaximal, characteristic, complemented
3#27161426C211yesmaximal, characteristic, complemented
4#27161393C212yesmaximal, complemented
5#27161360C211yesmaximal, characteristic, complemented
6#27161315C212yesmaximal, complemented
7#27161275C211yesmaximal, characteristic, complemented
8#27161215C212yesmaximal, complemented
9C4×D1576C211yesmaximal, characteristic, complemented
10C2×#20263248C212yesmaximal, complemented
11C8×D4×C197C211yesmaximal, characteristic, complemented
12C2×#15691357—21no—
13C22×D788C2212yes—
14C2×D4×D197—21no—
15C2×D1576C2211yescharacteristic
16C2×D1576C2212yescomplemented
17#20263326C2211yescharacteristic, complemented
18#20263323C2211yescharacteristic
19#20263310—22no—
20#20263306—21no—
21#20263301—21no—
22#20263300C2212yes—
23#20263300C2212yescomplemented
24#20263296—22no—
25C4×D788C2211yescharacteristic
26C4×D788C2211yescharacteristic, complemented
27C2×C4⋊Dic197C2212yescomplemented
28#20263260C2212yescomplemented
29#20263260C2211yescharacteristic, complemented
30#20263248C2212yes—
31#20263248C2211yescharacteristic
32#20263248C2212yescomplemented
33#20263248C2212yescomplemented
34#20263248C2211yescharacteristic, complemented
35#20263248C2212yescomplemented
36C22×C8×C197C2212yescomplemented
37C4×D4×C197C2211yescharacteristic
38C4⋊C8×C197C2211yescharacteristic, complemented
39C22⋊C8×C197C2212yescomplemented
40C4×C8×C197C2211yescharacteristic, complemented
41C23×D197—22no—
42#15691357—44no—
43C2×Dic394—21no—
44C2×D788—24no—
45C2×D788—22no—
46C2×D788—21no—
47C2×D788—21no—
48C2×D788—42no—
49C2×D788C2312yes—
50C2×D788C2311yescharacteristic
51C2×D788C2311yescharacteristic
52C2×C22⋊D197—22no—
53C2×C22⋊D197—22no—
54D4×D197—44no—
55C22×Dic197—22no—
56C2×C4×D197—21no—
57C2×C4×D197—21no—
58D1576—22no—
59D1576—42no—
60C4⋊Dic197—22no—
61C4⋊Dic197—21no—
62C4⋊Dic197C2311yescharacteristic
63C4⋊Dic197C2312yes—
64C4⋊Dic197C2311yescharacteristic
65#15691335—21no—
66#15691335—21no—
67#15691335—22no—
68#15691335—22no—
69C2×D4×C197D411yescharacteristic
70C22×C4×C197C2312yes—
71C2×C8×C197—42no—
72C2×C8×C197C2311yescharacteristic
73C2×C8×C197C2311yescharacteristic
74C2×C8×C197C2312yes—
75C4⋊C4×C197D411yescharacteristic
76C22⋊C4×C197D412yes—
77C42×C197C2311yescharacteristic
78C22×D197—21no—
79C22×D197—21no—
80C22×D197—22no—
81C22×D197—44no—
82C22×D197—44no—
83C22×D197—82no—
84Dic394—42no—
85D788—21no—
86D788—22no—
87D788—21no—
88D788—21no—
89D788—22no—
90D788—44no—
91D788—42no—
92D788—81no—
93D788#4112yescomplemented
94C22⋊D197—44no—
95C22⋊D197—44no—
96C2×Dic197—21no—
97C2×Dic197—21no—
98C2×Dic197—22no—
99C2×Dic197—44no—
100C4×D197—42no—
101C4×D197—42no—
102C23×C197C2×D412yes—
103D4×C197D814yescomplemented
104C2×C4×C197—22no—
105C2×C4×C197—22no—
106C2×C4×C197C2×D411yescharacteristic
107C2×C4×C197C2×D412yes—
108C2×C4×C197C2×D411yescharacteristic
109C2×C4×C197C2411yescharacteristic
110C8×C197—21no—
111C8×C197—21no—
112C8×C197—22no—
113C22×C197—22no—
114C22×C197—22no—
115C22×C197C2×D814yes—
116C22×C197C22×D411yescharacteristic
117C2×D197—22no—
118C2×D197—41no—
119C2×D197—41no—
120C2×D197—42no—
121C2×D197—41no—
122C2×D197—41no—
123C2×D197—42no—
124C2×D197—42no—
125C2×D197—41no—
126C2×D197—88no—
127C4×C197—22no—
128C4×C197—21no—
129C4×C197C2×D812yes—
130C4×C197C2×#4111yescharacteristic
131C4×C197H(5,𝔽2)11yescharacteristic
132Dic197—41no—
133Dic197—41no—
134Dic197—42no—
135C2×C197—24no—
136C2×C197#54611yescharacteristic
137C2×C197C22×D811yescharacteristic
138C2×C197#57711yescharacteristic
139D197—42no—
140D197—81no—
141D197—82no—
142C197#329411yescharacteristic, complemented, Sylow
143#3294—1971nomaximal, Sylow
144#547—1971no—
145D42—1971no—
146#486—1971no—
147#480—1972no—
148#474—1971no—
149#466—1972no—
150#459—1971no—
151#447—1972no—
152C4×D8—1971no—
153C8×D4D19711yescharacteristic, complemented
154C2×#102—1972no—
155C2×#41—3941no—
156C22×D4—3941no—
157C22×D4—1972no—
158C2×D8—1972no—
159C2×D8—1971no—
160C22×C8C2×D19712yescomplemented
161C4⋊Q8—1971no—
162#127—1971no—
163#123—3942no—
164C22⋊Q8—3941no—
165#121—3941no—
166#121—1972no—
167#121—1972no—
168#120—3942no—
169C4×D4—1971no—
170C4×D4—1971no—
171C4×D4C2×D19711yescharacteristic
172C2×C4⋊C4—1972no—
173#107—1971no—
174#107—1972no—
175C4⋊C8C2×D19711yescharacteristic, complemented
176#102—1971no—
177#102—1972no—
178#102—1972no—
179#102—1972no—
180#102—1971no—
181#102—1972no—
182C22⋊C8C2×D19712yescomplemented
183C4×C8C2×D19711yescharacteristic, complemented
184C24—3942no—
185#41—7884no—
186C2×Q8—3941no—
187C2×D4—7882no—
188C2×D4—7884no—
189C2×D4—3944no—
190C2×D4—3942no—
191C2×D4—3942no—
192C2×D4—3942no—
193C2×D4—3941no—
194C2×D4—3941no—
195C2×D4—1972no—
196C2×D4—1971no—
197C2×D4—1971no—
198C2×D4D78811yescharacteristic
199C22×C4—3942no—
200C22×C4—3941no—
201C22×C4—3941no—
202C22×C4C22×D19712yes—
203D8—7882no—
204D8—3942no—
205C2×C8—42no—
206C2×C8C22×D19711yescharacteristic
207C2×C8C22×D19711yescharacteristic
208C2×C8C22×D19712yes—
209C4⋊C4—3942no—
210C4⋊C4—3941no—
211C4⋊C4—1971no—
212C4⋊C4—1972no—
213C4⋊C4—1971no—
214C4⋊C4D78811yescharacteristic
215C22⋊C4—3941no—
216C22⋊C4—3941no—
217C22⋊C4—3942no—
218C22⋊C4—3942no—
219C22⋊C4D78812yes—
220C42C22×D19711yescharacteristic
221C23—7884no—
222C23—7884no—
223C23—15762no—
224C23—3941no—
225C23—3942no—
226C23—3941no—
227C23C2×D78812yes—
228Q8—7882no—
229D4—7884no—
230D4—7884no—
231D4—7884no—
232D4—7882no—
233D4—15761no—
234D4—3941no—
235D4—3942no—
236D4—3941no—
237D4—3942no—
238D4—3941no—
239D4—1972no—
240D4D157614yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—7884no—
244C2×C4—7882no—
245C2×C4—7882no—
246C2×C4—3941no—
247C2×C4—3941no—
248C2×C4—3942no—
249C2×C4C2×D78811yescharacteristic
250C2×C4C2×D78812yes—
251C2×C4C2×D78811yescharacteristic
252C2×C4C23×D19711yescharacteristic
253C8—21no—
254C8—21no—
255C8—22no—
256C22—22no—
257C22—22no—
258C22—7881no—
259C22—7881no—
260C22—7882no—
261C22—7881no—
262C22—7882no—
263C22—7881no—
264C22—7881no—
265C22—7882no—
266C22—15768no—
267C22—3942no—
268C22C2×D157614yes—
269C22C22×D78811yescharacteristic
270C4—21no—
271C4—22no—
272C4—7881no—
273C4—7881no—
274C4—7882no—
275C4C2×D157612yes—
276C4#41×D19711yescharacteristic
277C4#2026337111yescharacteristic
278C2—24no—
279C2—7882no—
280C2—15762no—
281C2—15761no—
282C2#2716172411yescharacteristic
283C2C22×D157611yescharacteristic
284C2#2716189111yescharacteristic
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC8×D4×C197C2
socleC22×C197C22×D4
hypercenterC8×D4D197
hypocenterC197#3294
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #27161733C2semidirect
2. D4×D788C2semidirect
3. #27161426C2semidirect
4. #27161393C2semidirect
5. #27161360C2semidirect
6. #27161315C2semidirect
7. #27161275C2semidirect
8. #27161215C2semidirect
9. C4×D1576C2semidirect
10. C2×#20263248C2semidirect
11. C8×D4×C197C2semidirect
16. C2×D1576C22semidirect
17. #20263326C22semidirect
23. #20263300C22semidirect
26. C4×D788C22semidirect
27. C2×C4⋊Dic197C22semidirect
28. #20263260C22semidirect
29. #20263260C22semidirect
32. #20263248C22semidirect
33. #20263248C22semidirect
34. #20263248C22semidirect
35. #20263248C22semidirect
36. C22×C8×C197C22semidirect
38. C4⋊C8×C197C22semidirect
39. C22⋊C8×C197C22semidirect
40. C4×C8×C197C22semidirect
93. D788#41semidirect
103. D4×C197D8semidirect
142. C197#3294semidirect
153. C8×D4D197semidirect
160. C22×C8C2×D197semidirect
175. C4⋊C8C2×D197semidirect
182. C22⋊C8C2×D197semidirect
183. C4×C8C2×D197semidirect
240. D4D1576semidirect

Maximal subgroups

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

subgroupindex
#271617332
D4×D7882
#271614262
#271613932
#271613602
#271613152
#271612752
#271612152
C4×D15762
C2×#202632482
C8×D4×C1972
#3294197