Finite Group Databasebeta

← C22×D42 #62695

Subgroups — C22×D42 #62695

Order 256.

10019 conjugacy classes of subgroups; only the 3479 normal subgroups are shown.

normal subgroups only

This group has 10019 conjugacy classes of subgroups — too many to draw. Only its 3479 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
1C24×D4C214yesmaximal, complemented
2C2×D42C2148yesmaximal, direct product factor
3C22×#127C211yesmaximal, characteristic, complemented
4C22×#121C214yesmaximal, complemented
5C22×#120C214yesmaximal, complemented
6C22×C4×D4C212yesmaximal, complemented
7C26C2214yescomplemented
8C23×D4—264no—
9C23×D4C2212yes—
10C23×D4C2218yescomplemented
11C23×D4C2214yescomplemented
12C23×D4C2212yescomplemented
13C23×D4C22196yescomplemented
14C24×C4C2214yescomplemented
15D42C221256yesdirect product factor
16C2×#127C22124yescomplemented
17C2×#121C22196yescomplemented
18C2×#120C22196yescomplemented
19C2×C4×D4C22148yescomplemented
20C22×C4⋊C4C2212yescomplemented
21C22×C22⋊C4C2214yescomplemented
22C22×C22⋊C4C2214yescomplemented
23C22×C42C2211yescharacteristic, complemented
24C25—264no—
25C25—416no—
26C25C2314yes—
27C25C2314yes—
28C25C23148yescomplemented
29C22×D4—264no—
30C22×D4—2768no—
31C22×D4—232no—
32C22×D4—48no—
33C22×D4D4132yesdirect product factor
34C22×D4C23124yes—
35C22×D4C23196yescomplemented
36C22×D4C23148yescomplemented
37C22×D4C23124yescomplemented
38C22×D4C231256yescomplemented
39C23×C4—232no—
40C23×C4C2311yescharacteristic
41C23×C4C2314yes—
42C23×C4C2312yes—
43C23×C4C23148yescomplemented
44#127C23164yescomplemented
45#121C231256yescomplemented
46#120C231256yescomplemented
47C4×D4C231128yescomplemented
48C2×C4⋊C4C23124yescomplemented
49C2×C22⋊C4C23148yescomplemented
50C2×C22⋊C4C23148yescomplemented
51C2×C42C23112yescomplemented
52C24—216no—
53C24—216no—
54C24—232no—
55C24—2384no—
56C24—216no—
57C24—496no—
58C24—4256no—
59C24C2×D4132yescomplemented
60C24C24124yes—
61C24C24124yes—
62C24C2411yescharacteristic
63C24C24164yescomplemented
64C2×D4—2384no—
65C2×D4—21024no—
66C2×D4—2192no—
67C2×D4—448no—
68C2×D4C2×D41192yesdirect product factor
69C2×D4C24132yes—
70C2×D4C241128yescomplemented
71C2×D4C24164yescomplemented
72C2×D4C24132yescomplemented
73C22×C4—216no—
74C22×C4—24no—
75C22×C4—2192no—
76C22×C4C2×D4116yescomplemented
77C22×C4C2416yes—
78C22×C4C24124yes—
79C22×C4C24112yes—
80C22×C4C24164yescomplemented
81C4⋊C4C24132yescomplemented
82C22⋊C4C24164yescomplemented
83C22⋊C4C24164yescomplemented
84C42C24116yescomplemented
85C23—2128no—
86C23—248no—
87C23—248no—
88C23—296no—
89C23—2256no—
90C23—248no—
91C23—464no—
92C23—4768no—
93C23—464no—
94C23C22×D418yes—
95C23C22×D4196yescomplemented
96C23H(5,𝔽2)14yes—
97C23C25116yes—
98C23C25116yes—
99C23C2513yes—
100D4—2256no—
101D4—2128no—
102D4—432no—
103D4C22×D41128yesdirect product factor
104C2×C4—248no—
105C2×C4—212no—
106C2×C4—2128no—
107C2×C4C22×D4148yescomplemented
108C2×C4C2514yes—
109C2×C4C25116yes—
110C2×C4C2518yes—
111C22—216no—
112C22—216no—
113C22—2192no—
114C22—232no—
115C22—216no—
116C22—4256no—
117C22—496no—
118C22D42116yesdirect product factor
119C22C23×D4112yes—
120C22C23×D4132yescomplemented
121C22C2×H(5,𝔽2)16yes—
122C22C2611yescharacteristic
123C4—216no—
124C4—24no—
125C4C23×D4116yescomplemented
126C2—232no—
127C2—416no—
128C2C2×D42112yesdirect product factor
129C2C24×D412yes—
130C2C22×H(5,𝔽2)11yescharacteristic
subgroupG/H
center, socleC24C24
commutator, Frattini, cosocleC22C26
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C24×D4C2semidirect
2. C2×D42C2direct (N × H)
3. C22×#127C2semidirect
4. C22×#121C2semidirect
5. C22×#120C2semidirect
6. C22×C4×D4C2semidirect
7. C26C22semidirect
10. C23×D4C22semidirect
11. C23×D4C22semidirect
12. C23×D4C22semidirect
13. C23×D4C22semidirect
14. C24×C4C22semidirect
15. D42C22direct (N × H)
16. C2×#127C22semidirect
17. C2×#121C22semidirect
18. C2×#120C22semidirect
19. C2×C4×D4C22semidirect
20. C22×C4⋊C4C22semidirect
21. C22×C22⋊C4C22semidirect
22. C22×C22⋊C4C22semidirect
23. C22×C42C22semidirect
28. C25C23semidirect
33. C22×D4D4direct (N × H)
35. C22×D4C23semidirect
36. C22×D4C23semidirect
37. C22×D4C23semidirect
38. C22×D4C23semidirect
43. C23×C4C23semidirect
44. #127C23semidirect
45. #121C23semidirect
46. #120C23semidirect
47. C4×D4C23semidirect
48. C2×C4⋊C4C23semidirect
49. C2×C22⋊C4C23semidirect
50. C2×C22⋊C4C23semidirect
51. C2×C42C23semidirect
59. C24C2×D4semidirect
63. C24C24semidirect
68. C2×D4C2×D4direct (N × H)
70. C2×D4C24semidirect
71. C2×D4C24semidirect
72. C2×D4C24semidirect
76. C22×C4C2×D4semidirect
80. C22×C4C24semidirect
81. C4⋊C4C24semidirect
82. C22⋊C4C24semidirect
83. C22⋊C4C24semidirect
84. C42C24semidirect
95. C23C22×D4semidirect
103. D4C22×D4direct (N × H)
107. C2×C4C22×D4semidirect
118. C22D42direct (N × H)
120. C22C23×D4semidirect
125. C4C23×D4semidirect
128. C2C2×D42direct (N × H)

Maximal subgroups

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

subgroupindex
C24×D42
C2×D422
C22×#1272
C22×#1212
C22×#1202
C22×C4×D42