Finite Group Databasebeta

← #38754537

Subgroups — #38754537

Order 25216.

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

normal subgroups only

This group has 462 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#27161786C211yesmaximal, characteristic, complemented
2D4×D788C211yesmaximal, characteristic
3#27161426C211yesmaximal, characteristic, complemented
4#27161398C212yesmaximal, complemented
5#27161370C211yesmaximal, characteristic, complemented
6#27161308C212yesmaximal, complemented
7#27161280C211yesmaximal, characteristic, complemented
8#27161232C212yesmaximal, complemented
9C4×#15691347C211yesmaximal, characteristic, complemented
10C2×#20263248C212yesmaximal, complemented
11C8×D4×C197C211yesmaximal, characteristic, complemented
12C22×D788C2212yes—
13C2×D4×D197—21no—
14C2×#15691347C2211yescharacteristic
15C2×#15691347C2212yescomplemented
16#20263326C2211yescharacteristic, complemented
17#20263323C2211yescharacteristic
18#20263317—21no—
19#20263305C2212yescomplemented
20#20263303—22no—
21#20263301—21no—
22#20263300C2212yes—
23#20263296—22no—
24C4×Dic394C2211yescharacteristic, complemented
25C4×D788C2211yescharacteristic
26D4×Dic197—21no—
27C2×C4⋊Dic197C2212yescomplemented
28#20263258C2212yescomplemented
29#20263258C2211yescharacteristic, complemented
30#20263251C2211yescharacteristic, complemented
31#20263248C2212yes—
32#20263248C2211yescharacteristic
33#20263248C2212yescomplemented
34#20263248C2212yescomplemented
35#20263248C2212yescomplemented
36C22×C8×C197C2212yescomplemented
37C4×D4×C197C2211yescharacteristic
38C4⋊C8×C197C2211yescharacteristic, complemented
39C22⋊C8×C197C2212yescomplemented
40C4×C8×C197C2211yescharacteristic, complemented
41C23×D197—22no—
42C2×Dic394—21no—
43C2×Dic394C2311yescharacteristic
44C2×D788—24no—
45C2×D788—22no—
46C2×D788—21no—
47C2×D788—21no—
48C2×D788—42no—
49C2×D788C2312yes—
50C2×D788C2311yescharacteristic
51C2×C22⋊D197—22no—
52D4×D197—44no—
53C22×Dic197—22no—
54C2×C4×D197—21no—
55#15691347—22no—
56#15691347—42no—
57C4⋊Dic197—22no—
58C4⋊Dic197—21no—
59C4⋊Dic197C2311yescharacteristic
60C4⋊Dic197C2312yes—
61C4⋊Dic197C2311yescharacteristic
62#15691337—21no—
63#15691337—22no—
64C22⋊Dic197—22no—
65#15691335—21no—
66#15691335—22no—
67C4×Dic197—21no—
68C2×D4×C197D411yescharacteristic
69C22×C4×C197C2312yes—
70C2×C8×C197—42no—
71C2×C8×C197C2311yescharacteristic
72C2×C8×C197C2311yescharacteristic
73C2×C8×C197C2312yes—
74C4⋊C4×C197D411yescharacteristic
75C22⋊C4×C197D412yes—
76C42×C197C2311yescharacteristic
77C22×D197—21no—
78C22×D197—22no—
79C22×D197—44no—
80C22×D197—44no—
81C22×D197—82no—
82Dic394—21no—
83Dic394—21no—
84Dic394—81no—
85D788—22no—
86D788—21no—
87D788—22no—
88D788—44no—
89D788—42no—
90D788—81no—
91D788#4112yes—
92C22⋊D197—44no—
93C2×Dic197—21no—
94C2×Dic197—21no—
95C2×Dic197—21no—
96C2×Dic197—22no—
97C2×Dic197—82no—
98C4×D197—42no—
99C23×C197C2×D412yes—
100D4×C197SD1614yescomplemented
101C2×C4×C197—22no—
102C2×C4×C197—22no—
103C2×C4×C197C2×D411yescharacteristic
104C2×C4×C197C2×D412yes—
105C2×C4×C197C2×D411yescharacteristic
106C2×C4×C197C2411yescharacteristic
107C8×C197—21no—
108C8×C197—21no—
109C8×C197—22no—
110C22×C197—22no—
111C22×C197—22no—
112C22×C197C2×SD1614yes—
113C22×C197C22×D411yescharacteristic
114C2×D197—22no—
115C2×D197—41no—
116C2×D197—41no—
117C2×D197—42no—
118C2×D197—42no—
119C2×D197—42no—
120C2×D197—88no—
121C4×C197—22no—
122C4×C197—21no—
123C4×C197C2×SD1612yes—
124C4×C197C2×#4111yescharacteristic
125C4×C197H(5,𝔽2)11yescharacteristic
126Dic197—41no—
127Dic197—41no—
128Dic197—41no—
129Dic197—42no—
130C2×C197—24no—
131C2×C197#54611yescharacteristic
132C2×C197C22×SD1611yescharacteristic
133C2×C197#57611yescharacteristic
134D197—42no—
135D197—82no—
136C197#329511yescharacteristic, complemented, Sylow
137#3295—1971nomaximal, Sylow
138#554—1971no—
139D42—1971no—
140#486—1971no—
141#481—1972no—
142#476—1971no—
143#465—1972no—
144#460—1971no—
145#450—1972no—
146C4×SD16—1971no—
147C8×D4D19711yescharacteristic, complemented
148C2×#102—1972no—
149C22×D4—3941no—
150C22×D4—1972no—
151C2×SD16—1972no—
152C2×SD16—1971no—
153C22×C8C2×D19712yescomplemented
154C4⋊Q8—1971no—
155#127—1971no—
156#125—3941no—
157C22⋊Q8—3942no—
158C22⋊Q8—1972no—
159#121—3941no—
160#121—1972no—
161#120—3942no—
162C4×Q8—1971no—
163C4×D4—3941no—
164C4×D4—1971no—
165C4×D4C2×D19711yescharacteristic
166C2×C4⋊C4—1972no—
167#106—1971no—
168#106—1972no—
169C4⋊C8C2×D19711yescharacteristic, complemented
170#103—1971no—
171#102—1971no—
172#102—1972no—
173#102—1972no—
174#102—1972no—
175#102—1972no—
176C22⋊C8C2×D19712yescomplemented
177C4×C8C2×D19711yescharacteristic, complemented
178C24—3942no—
179C2×Q8—3941no—
180C2×Q8—1971no—
181C2×D4—7882no—
182C2×D4—7884no—
183C2×D4—3944no—
184C2×D4—3942no—
185C2×D4—3942no—
186C2×D4—3941no—
187C2×D4—3941no—
188C2×D4—1972no—
189C2×D4—1971no—
190C2×D4D78811yescharacteristic
191C22×C4—3941no—
192C22×C4—3942no—
193C22×C4C22×D19712yes—
194SD16—7882no—
195SD16—3942no—
196C2×C8—42no—
197C2×C8C22×D19711yescharacteristic
198C2×C8C22×D19711yescharacteristic
199C2×C8C22×D19712yes—
200C4⋊C4—3941no—
201C4⋊C4—3942no—
202C4⋊C4—3942no—
203C4⋊C4—3941no—
204C4⋊C4—1971no—
205C4⋊C4—1972no—
206C4⋊C4—1971no—
207C4⋊C4D78811yescharacteristic
208C22⋊C4—3942no—
209C22⋊C4—3941no—
210C22⋊C4—3942no—
211C22⋊C4D78812yes—
212C42—3941no—
213C42C22×D19711yescharacteristic
214C23—7884no—
215C23—7884no—
216C23—15762no—
217C23—3942no—
218C23—3941no—
219C23C2×D78812yes—
220Q8—15761no—
221Q8—3941no—
222Q8—3941no—
223D4—7884no—
224D4—7884no—
225D4—7882no—
226D4—15761no—
227D4—3942no—
228D4—3942no—
229D4—3941no—
230D4—1972no—
231D4#1569134714yescomplemented
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—7882no—
235C2×C4—15762no—
236C2×C4—3941no—
237C2×C4—3941no—
238C2×C4—3941no—
239C2×C4—3942no—
240C2×C4C2×D78811yescharacteristic
241C2×C4C2×D78812yes—
242C2×C4C2×D78811yescharacteristic
243C2×C4C23×D19711yescharacteristic
244C8—21no—
245C8—21no—
246C8—22no—
247C22—22no—
248C22—22no—
249C22—7881no—
250C22—7881no—
251C22—7882no—
252C22—7882no—
253C22—7882no—
254C22—15768no—
255C22—3942no—
256C22C2×#1569134714yes—
257C22C22×D78811yescharacteristic
258C4—21no—
259C4—22no—
260C4—7881no—
261C4—7882no—
262C4—7881no—
263C4—7881no—
264C4C2×#1569134712yes—
265C4#41×D19711yescharacteristic
266C4#2026337111yescharacteristic
267C2—24no—
268C2—7882no—
269C2—15762no—
270C2#2716172411yescharacteristic
271C2C22×#1569134711yescharacteristic
272C2#2716188411yescharacteristic
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC8×D4×C197C2
socleC22×C197C22×D4
prime radicalD4×D788C2
hypercenterC8×D4D197
hypocenterC197#3295
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. #27161786C2semidirect
3. #27161426C2semidirect
4. #27161398C2semidirect
5. #27161370C2semidirect
6. #27161308C2semidirect
7. #27161280C2semidirect
8. #27161232C2semidirect
9. C4×#15691347C2semidirect
10. C2×#20263248C2semidirect
11. C8×D4×C197C2semidirect
15. C2×#15691347C22semidirect
16. #20263326C22semidirect
19. #20263305C22semidirect
24. C4×Dic394C22semidirect
27. C2×C4⋊Dic197C22semidirect
28. #20263258C22semidirect
29. #20263258C22semidirect
30. #20263251C22semidirect
33. #20263248C22semidirect
34. #20263248C22semidirect
35. #20263248C22semidirect
36. C22×C8×C197C22semidirect
38. C4⋊C8×C197C22semidirect
39. C22⋊C8×C197C22semidirect
40. C4×C8×C197C22semidirect
100. D4×C197SD16semidirect
136. C197#3295semidirect
147. C8×D4D197semidirect
153. C22×C8C2×D197semidirect
169. C4⋊C8C2×D197semidirect
176. C22⋊C8C2×D197semidirect
177. C4×C8C2×D197semidirect
231. D4#15691347semidirect

Maximal subgroups

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

subgroupindex
#271617862
D4×D7882
#271614262
#271613982
#271613702
#271613082
#271612802
#271612322
C4×#156913472
C2×#202632482
C8×D4×C1972
#3295197