Finite Group Databasebeta

← #27161638

Subgroups — #27161638

Order 12608.

346 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C22×C22⋊D197C211yesmaximal, characteristic, complemented
2#20263309C212yesmaximal, complemented
3#20263307C212yesmaximal, complemented
4#20263298C214yesmaximal, complemented
5#20263295C212yesmaximal, complemented
6#20263294C212yesmaximal, complemented
7C2×C22⋊Dic197C211yesmaximal, characteristic, complemented
8C22×D4×C197C211yesmaximal, characteristic, complemented
9C23×D197C2211yescharacteristic, complemented
10C2×C22⋊D197—24no—
11C2×C22⋊D197C2212yes—
12C2×C22⋊D197C2212yescomplemented
13C2×C22⋊D197C2214yescomplemented
14C22×Dic197C2211yescharacteristic, complemented
15C22×Dic197C2212yescomplemented
16#15691337C2214yescomplemented
17C22⋊Dic197C2212yescomplemented
18C22⋊Dic197C2212yescomplemented
19C22⋊Dic197C2214yescomplemented
20#15691335C2214yescomplemented
21C24×C197C2211yescharacteristic
22C24×C197C2211yescharacteristic, complemented
23C2×D4×C197—24no—
24C2×D4×C197C2212yes—
25C2×D4×C197C2212yescomplemented
26C22×C4×C197C2211yescharacteristic, complemented
27C22×D197—22no—
28C22×D197—42no—
29C22×D197C2312yes—
30C22⋊D197—24no—
31C22⋊D197—24no—
32C22⋊D197—44no—
33C2×Dic197—22no—
34C2×Dic197—42no—
35C2×Dic197C2314yes—
36C2×Dic197C2312yes—
37C23×C197—22no—
38C23×C197—42no—
39C23×C197—42no—
40C23×C197D414yescomplemented
41C23×C197C2312yes—
42C23×C197C2311yescharacteristic
43C23×C197C2311yescharacteristic
44C23×C197C2311yescharacteristic
45D4×C197—24no—
46D4×C197—24no—
47C2×C4×C197—22no—
48C2×C4×C197C2312yes—
49C22×C197—22no—
50C22×C197—22no—
51C22×C197—22no—
52C22×C197—22no—
53C22×C197—24no—
54C22×C197—44no—
55C22×C197—44no—
56C22×C197C2×D412yes—
57C22×C197C2×D412yes—
58C22×C197C2×D412yes—
59C22×C197C2411yescharacteristic
60C2×D197—22no—
61C2×D197—24no—
62C2×D197—44no—
63C4×C197—22no—
64Dic197—22no—
65Dic197—24no—
66C2×C197—22no—
67C2×C197—22no—
68C2×C197—22no—
69C2×C197—42no—
70C2×C197C22×D411yescharacteristic
71C2×C197H(5,𝔽2)12yes—
72D197—42no—
73C197#53411yescharacteristic, complemented, Sylow
74#534—1971nomaximal, Sylow
75C22×D4—1971no—
76C22×D4D19711yescharacteristic, complemented
77#123—1972no—
78#123—1972no—
79#121—1974no—
80#120—1972no—
81#120—1972no—
82C2×C22⋊C4—1971no—
83C24—1971no—
84C24C2×D19711yescharacteristic
85C24C2×D19711yescharacteristic, complemented
86C2×D4—24no—
87C2×D4—3944no—
88C2×D4—1974no—
89C2×D4—1972no—
90C2×D4—1972no—
91C2×D4C2×D19712yes—
92C2×D4C2×D19712yescomplemented
93C22×C4—1972no—
94C22×C4—1971no—
95C22×C4C2×D19711yescharacteristic, complemented
96C4⋊C4—1974no—
97C22⋊C4—1972no—
98C22⋊C4—1972no—
99C22⋊C4—1974no—
100C22⋊C4—1974no—
101C23—22no—
102C23—42no—
103C23—42no—
104C23—7882no—
105C23—3942no—
106C23—1972no—
107C23C22⋊D19714yescomplemented
108C23C22×D19712yes—
109C23C22×D19711yescharacteristic
110C23C22×D19711yescharacteristic
111C23C22×D19711yescharacteristic
112D4—24no—
113D4—24no—
114D4—7884no—
115D4—3944no—
116D4—3944no—
117C2×C4—22no—
118C2×C4—7882no—
119C2×C4—3942no—
120C2×C4—1974no—
121C2×C4—1972no—
122C2×C4C22×D19712yes—
123C22—22no—
124C22—22no—
125C22—22no—
126C22—22no—
127C22—24no—
128C22—44no—
129C22—44no—
130C22—7884no—
131C22—3942no—
132C22—3944no—
133C22C2×C22⋊D19712yes—
134C22C2×C22⋊D19712yes—
135C22C2×C22⋊D19712yes—
136C22C23×D19711yescharacteristic
137C4—22no—
138C4—3942no—
139C4—3944no—
140C2—22no—
141C2—22no—
142C2—22no—
143C2—42no—
144C2—7882no—
145C2C22×C22⋊D19711yescharacteristic
146C2#2026337112yes—
subgroupG/H
center, FrattiniC22C23×D197
commutator, socle, cosocleC22×C197C24
FittingC22×D4×C197C2
hypercenterC22×D4D197
hypocenterC197#534
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D197C2semidirect
2. #20263309C2semidirect
3. #20263307C2semidirect
4. #20263298C2semidirect
5. #20263295C2semidirect
6. #20263294C2semidirect
7. C2×C22⋊Dic197C2semidirect
8. C22×D4×C197C2semidirect
9. C23×D197C22semidirect
12. C2×C22⋊D197C22semidirect
13. C2×C22⋊D197C22semidirect
14. C22×Dic197C22semidirect
15. C22×Dic197C22semidirect
16. #15691337C22semidirect
17. C22⋊Dic197C22semidirect
18. C22⋊Dic197C22semidirect
19. C22⋊Dic197C22semidirect
20. #15691335C22semidirect
22. C24×C197C22semidirect
25. C2×D4×C197C22semidirect
26. C22×C4×C197C22semidirect
40. C23×C197D4semidirect
73. C197#534semidirect
76. C22×D4D197semidirect
85. C24C2×D197semidirect
92. C2×D4C2×D197semidirect
95. C22×C4C2×D197semidirect
107. C23C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D1972
#202633092
#202633072
#202632982
#202632952
#202632942
C2×C22⋊Dic1972
C22×D4×C1972
#534197