Finite Group Databasebeta

← #27161817

Subgroups — #27161817

Order 12608.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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#20263322C211yesmaximal, characteristic, complemented
2#20263321C211yesmaximal, characteristic, complemented
3#20263319C211yesmaximal, characteristic, complemented
4#20263311C211yesmaximal, characteristic, complemented
5#20263310C211yesmaximal, characteristic, complemented
6#20263309C211yesmaximal, characteristic, complemented
7#20263300C211yesmaximal, characteristic, complemented
8#20263299C211yesmaximal, characteristic, complemented
9#20263298C211yesmaximal, characteristic, complemented
10#20263298C211yesmaximal, characteristic, complemented
11#20263297C211yesmaximal, characteristic, complemented
12#20263296C211yesmaximal, characteristic, complemented
13#20263294C211yesmaximal, characteristic, complemented
14#20263294C211yesmaximal, characteristic, complemented
15#121×C197C211yesmaximal, characteristic, complemented
16C23×D197C2211yescharacteristic
17C2×D788C2211yescharacteristic
18C2×D788C2211yescharacteristic
19C2×C22⋊D197C2211yescharacteristic
20C2×C22⋊D197C2211yescharacteristic
21C2×C22⋊D197C2211yescharacteristic
22C2×C22⋊D197C2211yescharacteristic
23C2×C22⋊D197C2211yescharacteristic
24C2×C22⋊D197C2211yescharacteristic, complemented
25C2×C22⋊D197C2211yescharacteristic, complemented
26C22×Dic197C2211yescharacteristic, complemented
27C2×C4×D197C2211yescharacteristic, complemented
28C4⋊Dic197C2211yescharacteristic, complemented
29C4⋊Dic197C2211yescharacteristic, complemented
30#15691337C2211yescharacteristic, complemented
31#15691337C2211yescharacteristic, complemented
32#15691337C2211yescharacteristic, complemented
33C22⋊Dic197C2211yescharacteristic
34C22⋊Dic197C2211yescharacteristic
35C22⋊Dic197C2211yescharacteristic, complemented
36#15691335C2211yescharacteristic
37#15691335C2211yescharacteristic
38#15691335C2211yescharacteristic
39#15691335C2211yescharacteristic
40#15691335C2211yescharacteristic
41#15691335C2211yescharacteristic
42#15691335C2211yescharacteristic, complemented
43C4×Dic197C2211yescharacteristic, complemented
44C2×D4×C197C2211yescharacteristic
45C2×D4×C197C2211yescharacteristic
46C2×D4×C197C2211yescharacteristic, complemented
47C22×C4×C197C2211yescharacteristic, complemented
48C4⋊C4×C197C2211yescharacteristic, complemented
49C22⋊C4×C197C2211yescharacteristic
50C22⋊C4×C197C2211yescharacteristic, complemented
51C22×D197—41no—
52C22×D197—41no—
53C22×D197—41no—
54C22×D197C2311yescharacteristic
55C22×D197C2311yescharacteristic
56C22×D197C2311yescharacteristic
57D788—41no—
58D788—41no—
59C22⋊D197—41no—
60C22⋊D197—41no—
61C22⋊D197—41no—
62C22⋊D197—41no—
63C22⋊D197—41no—
64C22⋊D197—41no—
65C22⋊D197—41no—
66C2×Dic197—41no—
67C2×Dic197C2311yescharacteristic
68C2×Dic197C2311yescharacteristic
69C2×Dic197C2311yescharacteristic
70C2×Dic197C2311yescharacteristic
71C2×Dic197C2311yescharacteristic
72C4×D197—41no—
73C23×C197C2311yescharacteristic
74C23×C197C2311yescharacteristic
75C23×C197C2311yescharacteristic
76D4×C197—41no—
77D4×C197—41no—
78D4×C197—41no—
79C2×C4×C197—41no—
80C2×C4×C197C2311yescharacteristic
81C2×C4×C197C2311yescharacteristic
82C2×C4×C197C2311yescharacteristic
83C2×C4×C197C2311yescharacteristic
84C22×C197—21no—
85C22×C197—21no—
86C22×C197—21no—
87C22×C197—21no—
88C22×C197—21no—
89C22×C197—21no—
90C22×C197—21no—
91C22×C197—21no—
92C22×C197—21no—
93C22×C197C2411yescharacteristic
94C2×D197—21no—
95C2×D197—21no—
96C2×D197—21no—
97C2×D197—21no—
98C2×D197—21no—
99C2×D197—21no—
100C2×D197—21no—
101C2×D197—21no—
102C2×D197—21no—
103C2×D197—44no—
104C4×C197—21no—
105C4×C197—21no—
106C4×C197—21no—
107C4×C197—21no—
108Dic197—21no—
109Dic197—21no—
110Dic197—21no—
111Dic197—21no—
112Dic197—21no—
113C2×C197—41no—
114C2×C197—41no—
115C2×C197—41no—
116C2×C197H(5,𝔽2)11yescharacteristic
117C2×C197H(5,𝔽2)11yescharacteristic
118C2×C197H(5,𝔽2)11yescharacteristic
119D197—41no—
120D197—41no—
121D197—41no—
122C197#56011yescharacteristic, complemented, Sylow
123#560—1971nomaximal, Sylow
124#127—1971no—
125#126—1971no—
126#126—1971no—
127#123—1971no—
128#123—1971no—
129#123—1971no—
130#121—1971no—
131#121—1971no—
132#121—1971no—
133#121—1971no—
134#121—1971no—
135#121D19711yescharacteristic, complemented
136#120—1971no—
137#120—1971no—
138#120—1971no—
139C24—1971no—
140C2×D4—1971no—
141C2×D4—1971no—
142C2×D4—1971no—
143C2×D4—1971no—
144C2×D4—1971no—
145C2×D4—1971no—
146C2×D4—1971no—
147C2×D4—1971no—
148C2×D4—1971no—
149C2×D4C2×D19711yescharacteristic
150C2×D4C2×D19711yescharacteristic
151C2×D4C2×D19711yescharacteristic, complemented
152C22×C4—1971no—
153C22×C4—1971no—
154C22×C4C2×D19711yescharacteristic, complemented
155C4⋊C4—1971no—
156C4⋊C4—1971no—
157C4⋊C4—1971no—
158C4⋊C4—1971no—
159C4⋊C4—1971no—
160C4⋊C4C2×D19711yescharacteristic, complemented
161C22⋊C4—1971no—
162C22⋊C4—1971no—
163C22⋊C4—1971no—
164C22⋊C4—1971no—
165C22⋊C4—1971no—
166C22⋊C4—1971no—
167C22⋊C4—1971no—
168C22⋊C4—1971no—
169C22⋊C4—1971no—
170C22⋊C4—1971no—
171C22⋊C4C2×D19711yescharacteristic
172C22⋊C4C2×D19711yescharacteristic, complemented
173C42—1971no—
174C23—7881no—
175C23—7881no—
176C23—7881no—
177C23—1971no—
178C23—1971no—
179C23—1971no—
180C23C22×D19711yescharacteristic
181C23C22×D19711yescharacteristic
182C23C22×D19711yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—7881no—
187D4—7881no—
188D4—7881no—
189D4—7881no—
190D4—7881no—
191D4—7881no—
192D4—7881no—
193D4—7881no—
194D4—7881no—
195C2×C4—41no—
196C2×C4—7881no—
197C2×C4—7881no—
198C2×C4—1971no—
199C2×C4—1971no—
200C2×C4—1971no—
201C2×C4—1971no—
202C2×C4—1971no—
203C2×C4C22×D19711yescharacteristic
204C2×C4C22×D19711yescharacteristic
205C2×C4C22×D19711yescharacteristic
206C2×C4C22×D19711yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—7884no—
217C22—3941no—
218C22—3941no—
219C22—3941no—
220C22—3941no—
221C22—3941no—
222C22—3941no—
223C22—3941no—
224C22—3941no—
225C22—3941no—
226C22C23×D19711yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—3941no—
232C4—3941no—
233C4—3941no—
234C4—3941no—
235C4—3941no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—7881no—
240C2—7881no—
241C2—7881no—
242C2#2026337111yescharacteristic
243C2#2026337111yescharacteristic
244C2#2026337211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D197
commutator, socle, cosocleC22×C197C24
Fitting#121×C197C2
hypercenter#121D197
hypocenterC197#560
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #20263322C2semidirect
2. #20263321C2semidirect
3. #20263319C2semidirect
4. #20263311C2semidirect
5. #20263310C2semidirect
6. #20263309C2semidirect
7. #20263300C2semidirect
8. #20263299C2semidirect
9. #20263298C2semidirect
10. #20263298C2semidirect
11. #20263297C2semidirect
12. #20263296C2semidirect
13. #20263294C2semidirect
14. #20263294C2semidirect
15. #121×C197C2semidirect
24. C2×C22⋊D197C22semidirect
25. C2×C22⋊D197C22semidirect
26. C22×Dic197C22semidirect
27. C2×C4×D197C22semidirect
28. C4⋊Dic197C22semidirect
29. C4⋊Dic197C22semidirect
30. #15691337C22semidirect
31. #15691337C22semidirect
32. #15691337C22semidirect
35. C22⋊Dic197C22semidirect
42. #15691335C22semidirect
43. C4×Dic197C22semidirect
46. C2×D4×C197C22semidirect
47. C22×C4×C197C22semidirect
48. C4⋊C4×C197C22semidirect
50. C22⋊C4×C197C22semidirect
122. C197#560semidirect
135. #121D197semidirect
151. C2×D4C2×D197semidirect
154. C22×C4C2×D197semidirect
160. C4⋊C4C2×D197semidirect
172. C22⋊C4C2×D197semidirect

Maximal subgroups

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

subgroupindex
#202633222
#202633212
#202633192
#202633112
#202633102
#202633092
#202633002
#202632992
#202632982
#202632972
#202632962
#202632942
#121×C1972
#560197