Finite Group Databasebeta

← D4×D796 #27164654

Subgroups — D4×D796 #27164654

Order 12736.

428 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 123 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×D796C212yesmaximal, complemented
2C2×D4×D199C212yesmaximal, complemented
3#20263952C211yesmaximal, characteristic, complemented
4#20263930C212yesmaximal, complemented
5#20263929C212yesmaximal, complemented
6#20263925C214yesmaximal, complemented
7C4×D796C211yesmaximal, characteristic, complemented
8C4×D4×C199C211yesmaximal, characteristic, complemented
9C23×D199C2214yescomplemented
10C2×D796—28no—
11C2×D796C2211yescharacteristic
12C2×D796C2212yescomplemented
13C2×D796C2212yescomplemented
14C2×D796C2214yescomplemented
15C2×D796C2212yescomplemented
16C2×C22⋊D199C2214yescomplemented
17D4×D199—28no—
18C2×C4×D199C2212yescomplemented
19C4⋊Dic199C2211yescharacteristic, complemented
20(C2×D199)⋊C4C2212yescomplemented
21(C2×D199)⋊C4C2214yescomplemented
22C2×D4×C199C2211yescharacteristic
23C22×C4×C199C2212yescomplemented
24C4⋊C4×C199C2211yescharacteristic, complemented
25C22⋊C4×C199C2212yescomplemented
26C42×C199C2211yescharacteristic, complemented
27C22×D199—28no—
28C22×D199—28no—
29C22×D199—44no—
30C22×D199C2314yes—
31C22×D199C2312yes—
32D796—28no—
33D796—24no—
34D796—24no—
35D796—42no—
36D796D414yesdirect product factor
37C22⋊D199—28no—
38C2×Dic199C2312yes—
39C4×D199—24no—
40C23×C199C2312yes—
41D4×C199D414yescomplemented
42C2×C4×C199—24no—
43C2×C4×C199C2311yescharacteristic
44C2×C4×C199C2312yes—
45C2×C4×C199C2311yescharacteristic
46C2×C4×C199C2311yescharacteristic
47C22×C199—22no—
48C22×C199—22no—
49C22×C199C2×D414yescomplemented
50C22×C199C2411yescharacteristic
51C2×D199—22no—
52C2×D199—22no—
53C2×D199—24no—
54C2×D199—24no—
55C2×D199—24no—
56C2×D199—416no—
57C2×D199C2×D414yescomplemented
58C4×C199—22no—
59C4×C199—21no—
60C4×C199C2×D412yescomplemented
61C4×C199C2×D412yescomplemented
62Dic199—22no—
63C2×C199—24no—
64C2×C199C22×D411yescharacteristic
65C2×C199C22×D411yescharacteristic
66C2×C199H(5,𝔽2)11yescharacteristic
67D199—24no—
68D199—44no—
69C199D4211yescharacteristic, complemented, Sylow
70D42—1991nomaximal, Sylow
71C22×D4—1992no—
72C22×D4—1992no—
73#127—1991no—
74#121—1992no—
75#121—1992no—
76#120—1994no—
77C4×D4—1991no—
78C4×D4D19911yescharacteristic, complemented
79C24—1994no—
80C2×D4—3988no—
81C2×D4—3988no—
82C2×D4—1994no—
83C2×D4—1992no—
84C2×D4—1994no—
85C2×D4—1992no—
86C2×D4—1992no—
87C2×D4—1991no—
88C2×D4C2×D19911yescharacteristic
89C22×C4—1992no—
90C22×C4C2×D19912yescomplemented
91C4⋊C4—1991no—
92C4⋊C4C2×D19911yescharacteristic, complemented
93C22⋊C4—1992no—
94C22⋊C4—1994no—
95C22⋊C4C2×D19912yescomplemented
96C42C2×D19911yescharacteristic, complemented
97C23—7964no—
98C23—3988no—
99C23—3988no—
100C23—1994no—
101C23—1992no—
102C23C22×D19912yes—
103D4—7962no—
104D4—3988no—
105D4—3988no—
106D4—3984no—
107D4—3984no—
108D4—1994no—
109D4D79614yesdirect product factor
110C2×C4—24no—
111C2×C4—3984no—
112C2×C4—1992no—
113C2×C4C22×D19911yescharacteristic
114C2×C4C22×D19912yes—
115C2×C4C22×D19911yescharacteristic
116C2×C4C22×D19911yescharacteristic
117C22—22no—
118C22—22no—
119C22—79616no—
120C22—3984no—
121C22—3984no—
122C22—3984no—
123C22—3982no—
124C22—3982no—
125C22—1994no—
126C22C2×D79614yescomplemented
127C22C23×D19911yescharacteristic
128C4—22no—
129C4—21no—
130C4—3982no—
131C4D4×D19912yescomplemented
132C4C2×D79612yescomplemented
133C2—24no—
134C2—7964no—
135C2—3984no—
136C2C2×D4×D19911yescharacteristic
137C2C22×D79611yescharacteristic
138C2#2026400111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D199
commutator, socle, cosocleC22×C199C24
FittingC4×D4×C199C2
hypercenterC4×D4D199
hypocenterC199D42
LCSG ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×D796C2semidirect
2. C2×D4×D199C2semidirect
3. #20263952C2semidirect
4. #20263930C2semidirect
5. #20263929C2semidirect
6. #20263925C2semidirect
7. C4×D796C2semidirect
8. C4×D4×C199C2semidirect
9. C23×D199C22semidirect
12. C2×D796C22semidirect
13. C2×D796C22semidirect
14. C2×D796C22semidirect
15. C2×D796C22semidirect
16. C2×C22⋊D199C22semidirect
18. C2×C4×D199C22semidirect
19. C4⋊Dic199C22semidirect
20. (C2×D199)⋊C4C22semidirect
21. (C2×D199)⋊C4C22semidirect
23. C22×C4×C199C22semidirect
24. C4⋊C4×C199C22semidirect
25. C22⋊C4×C199C22semidirect
26. C42×C199C22semidirect
36. D796D4direct (N × H)
41. D4×C199D4semidirect
49. C22×C199C2×D4semidirect
57. C2×D199C2×D4semidirect
60. C4×C199C2×D4semidirect
61. C4×C199C2×D4semidirect
69. C199D42semidirect
78. C4×D4D199semidirect
90. C22×C4C2×D199semidirect
92. C4⋊C4C2×D199semidirect
95. C22⋊C4C2×D199semidirect
96. C42C2×D199semidirect
109. D4D796direct (N × H)
126. C22C2×D796semidirect
131. C4D4×D199semidirect
132. C4C2×D796semidirect

Maximal subgroups

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

subgroupindex
C22×D7962
C2×D4×D1992
#202639522
#202639302
#202639292
#202639252
C4×D7962
C4×D4×C1992
D42199