Finite Group Databasebeta

← #27135818

Subgroups — #27135818

Order 12352.

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⋊D193C211yesmaximal, characteristic, complemented
2#20259882C212yesmaximal, complemented
3#20259880C212yesmaximal, complemented
4#20259871C214yesmaximal, complemented
5#20259868C212yesmaximal, complemented
6#20259867C212yesmaximal, complemented
7C2×C22⋊Dic193C211yesmaximal, characteristic, complemented
8C22×D4×C193C211yesmaximal, characteristic, complemented
9C23×D193C2211yescharacteristic, complemented
10C2×C22⋊D193—24no—
11C2×C22⋊D193C2212yes—
12C2×C22⋊D193C2212yescomplemented
13C2×C22⋊D193C2214yescomplemented
14C22×Dic193C2211yescharacteristic, complemented
15C22×Dic193C2212yescomplemented
16#15690534C2214yescomplemented
17C22⋊Dic193C2212yescomplemented
18C22⋊Dic193C2212yescomplemented
19C22⋊Dic193C2214yescomplemented
20#15690532C2214yescomplemented
21C24×C193C2211yescharacteristic
22C24×C193C2211yescharacteristic, complemented
23C2×D4×C193—24no—
24C2×D4×C193C2212yes—
25C2×D4×C193C2212yescomplemented
26C22×C4×C193C2211yescharacteristic, complemented
27C22×D193—22no—
28C22×D193—42no—
29C22×D193C2312yes—
30C22⋊D193—24no—
31C22⋊D193—24no—
32C22⋊D193—44no—
33C2×Dic193—22no—
34C2×Dic193—42no—
35C2×Dic193C2314yes—
36C2×Dic193C2312yes—
37C23×C193—22no—
38C23×C193—42no—
39C23×C193—42no—
40C23×C193D414yescomplemented
41C23×C193C2312yes—
42C23×C193C2311yescharacteristic
43C23×C193C2311yescharacteristic
44C23×C193C2311yescharacteristic
45D4×C193—24no—
46D4×C193—24no—
47C2×C4×C193—22no—
48C2×C4×C193C2312yes—
49C22×C193—22no—
50C22×C193—22no—
51C22×C193—22no—
52C22×C193—22no—
53C22×C193—24no—
54C22×C193—44no—
55C22×C193—44no—
56C22×C193C2×D412yes—
57C22×C193C2×D412yes—
58C22×C193C2×D412yes—
59C22×C193C2411yescharacteristic
60C2×D193—22no—
61C2×D193—24no—
62C2×D193—44no—
63C4×C193—22no—
64Dic193—22no—
65Dic193—24no—
66C2×C193—22no—
67C2×C193—22no—
68C2×C193—22no—
69C2×C193—42no—
70C2×C193C22×D411yescharacteristic
71C2×C193H(5,𝔽2)12yes—
72D193—42no—
73C193#53411yescharacteristic, complemented, Sylow
74#534—1931nomaximal, Sylow
75C22×D4—1931no—
76C22×D4D19311yescharacteristic, complemented
77#123—1932no—
78#123—1932no—
79#121—1934no—
80#120—1932no—
81#120—1932no—
82C2×C22⋊C4—1931no—
83C24—1931no—
84C24C2×D19311yescharacteristic
85C24C2×D19311yescharacteristic, complemented
86C2×D4—24no—
87C2×D4—3864no—
88C2×D4—1934no—
89C2×D4—1932no—
90C2×D4—1932no—
91C2×D4C2×D19312yes—
92C2×D4C2×D19312yescomplemented
93C22×C4—1932no—
94C22×C4—1931no—
95C22×C4C2×D19311yescharacteristic, complemented
96C4⋊C4—1934no—
97C22⋊C4—1932no—
98C22⋊C4—1932no—
99C22⋊C4—1934no—
100C22⋊C4—1934no—
101C23—22no—
102C23—42no—
103C23—42no—
104C23—7722no—
105C23—3862no—
106C23—1932no—
107C23C22⋊D19314yescomplemented
108C23C22×D19312yes—
109C23C22×D19311yescharacteristic
110C23C22×D19311yescharacteristic
111C23C22×D19311yescharacteristic
112D4—24no—
113D4—24no—
114D4—7724no—
115D4—3864no—
116D4—3864no—
117C2×C4—22no—
118C2×C4—7722no—
119C2×C4—3862no—
120C2×C4—1934no—
121C2×C4—1932no—
122C2×C4C22×D19312yes—
123C22—22no—
124C22—22no—
125C22—22no—
126C22—22no—
127C22—24no—
128C22—44no—
129C22—44no—
130C22—7724no—
131C22—3862no—
132C22—3864no—
133C22C2×C22⋊D19312yes—
134C22C2×C22⋊D19312yes—
135C22C2×C22⋊D19312yes—
136C22C23×D19311yescharacteristic
137C4—22no—
138C4—3862no—
139C4—3864no—
140C2—22no—
141C2—22no—
142C2—22no—
143C2—42no—
144C2—7722no—
145C2C22×C22⋊D19311yescharacteristic
146C2#2025994412yes—
subgroupG/H
center, FrattiniC22C23×D193
commutator, socle, cosocleC22×C193C24
FittingC22×D4×C193C2
hypercenterC22×D4D193
hypocenterC193#534
LCSG ▷ C22×C193 ▷ C193
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D193C2semidirect
2. #20259882C2semidirect
3. #20259880C2semidirect
4. #20259871C2semidirect
5. #20259868C2semidirect
6. #20259867C2semidirect
7. C2×C22⋊Dic193C2semidirect
8. C22×D4×C193C2semidirect
9. C23×D193C22semidirect
12. C2×C22⋊D193C22semidirect
13. C2×C22⋊D193C22semidirect
14. C22×Dic193C22semidirect
15. C22×Dic193C22semidirect
16. #15690534C22semidirect
17. C22⋊Dic193C22semidirect
18. C22⋊Dic193C22semidirect
19. C22⋊Dic193C22semidirect
20. #15690532C22semidirect
22. C24×C193C22semidirect
25. C2×D4×C193C22semidirect
26. C22×C4×C193C22semidirect
40. C23×C193D4semidirect
73. C193#534semidirect
76. C22×D4D193semidirect
85. C24C2×D193semidirect
92. C2×D4C2×D193semidirect
95. C22×C4C2×D193semidirect
107. C23C22⋊D193semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D1932
#202598822
#202598802
#202598712
#202598682
#202598672
C2×C22⋊Dic1932
C22×D4×C1932
#534193