Finite Group Databasebeta

← C22×#576 #63084

Subgroups — C22×#576 #63084

Order 256.

5731 conjugacy classes of subgroups; only the 3143 normal subgroups are shown.

normal subgroups only

This group has 5731 conjugacy classes of subgroups — too many to draw. Only its 3143 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×H(5,𝔽2)C212yesmaximal, complemented
2C2×#576C2148yesmaximal, direct product factor
3C22×AGL(1,ℤ/8)C216yesmaximal, complemented
4C22×#135C213yesmaximal, complemented
5C23×D8C213yesmaximal, complemented
6C22×#131C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)—224no—
8C2×H(5,𝔽2)C22148yescomplemented
9C22×#41—22no—
10C22×#41C2211yescharacteristic
11C22×#41C2216yescomplemented
12C23×D4—26no—
13C23×D4C2216yescomplemented
14#576C221256yesdirect product factor
15C2×AGL(1,ℤ/8)C221144yescomplemented
16C2×#135C22172yescomplemented
17C22×Q16C2211yescharacteristic, complemented
18C22×SD16C2216yescomplemented
19C22×D8C2213yescomplemented
20C22×D8C2216yescomplemented
21C22×D8C22172yescomplemented
22C2×#131C22124yescomplemented
23C22×M16C2213yescomplemented
24C23×C8C2213yescomplemented
25C25—26no—
26H(5,𝔽2)—2192no—
27H(5,𝔽2)C231128yescomplemented
28C2×#41—248no—
29C2×#41—236no—
30C2×#41D4112yes—
31C2×#41C23112yes—
32C2×#41C23172yescomplemented
33C22×Q8D411yescharacteristic
34C22×Q8C2312yes—
35C22×D4—26no—
36C22×D4—2144no—
37C22×D4—26no—
38C22×D4—236no—
39C22×D4D413yes—
40C22×D4C2316yes—
41C22×D4C23172yescomplemented
42C23×C4—26no—
43C23×C4C2313yes—
44AGL(1,ℤ/8)C231384yescomplemented
45#135C231192yescomplemented
46C2×Q16C23112yescomplemented
47C2×SD16C23172yescomplemented
48C2×D8C23136yescomplemented
49C2×D8C23172yescomplemented
50C2×D8C231192yescomplemented
51#131C23164yescomplemented
52C2×M16C23136yescomplemented
53C22×C8C2311yescharacteristic
54C22×C8C2313yes—
55C22×C8C23136yescomplemented
56C24—26no—
57C24—272no—
58C24—824no—
59C24C2×D413yes—
60#41—2128no—
61#41—2144no—
62#41C2×D4148yes—
63#41C24116yes—
64#41C24196yescomplemented
65C2×Q8—26no—
66C2×Q8C2×D4112yes—
67C2×Q8C24112yes—
68C2×D4—272no—
69C2×D4—2384no—
70C2×D4—218no—
71C2×D4—272no—
72C2×D4—2144no—
73C2×D4C2×D4136yes—
74C2×D4C24136yes—
75C2×D4C24196yescomplemented
76C22×C4—22no—
77C22×C4—272no—
78C22×C4C2×D4118yes—
79C22×C4C2×D413yes—
80C22×C4C24118yes—
81C22×C4C2411yescharacteristic
82Q16C24116yescomplemented
83SD16C24196yescomplemented
84D8C24148yescomplemented
85D8C24196yescomplemented
86M16C24148yescomplemented
87C2×C8C2416yes—
88C2×C8C24118yes—
89C2×C8C24148yescomplemented
90C23—212no—
91C23—236no—
92C23—296no—
93C23—424no—
94C23—8144no—
95C23C22×D4118yes—
96C23C22×D411yescharacteristic
97Q8—212no—
98Q8C22×D4116yes—
99Q8C2518yes—
100D4—296no—
101D4—236no—
102D4—296no—
103D4C22×D4148yes—
104D4C25124yes—
105C2×C4—212no—
106C2×C4—296no—
107C2×C4C22×D4118yes—
108C2×C4C22×D413yes—
109C2×C4C22×D4136yes—
110C2×C4C2513yes—
111C2×C4C25112yes—
112C8C2514yes—
113C8C25112yes—
114C22—236no—
115C22—224no—
116C22—472no—
117C22—896no—
118C22#57614yesdirect product factor
119C22C23×D4112yes—
120C22C23×D413yes—
121C4—28no—
122C4C23×D4112yes—
123C4C23×D413yes—
124C4C2611yescharacteristic
125C2—212no—
126C2—424no—
127C2C2×#57616yesdirect product factor
128C2C24×D411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC4C26
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×H(5,𝔽2)C2semidirect
2. C2×#576C2direct (N × H)
3. C22×AGL(1,ℤ/8)C2semidirect
4. C22×#135C2semidirect
5. C23×D8C2semidirect
6. C22×#131C2semidirect
8. C2×H(5,𝔽2)C22semidirect
11. C22×#41C22semidirect
13. C23×D4C22semidirect
14. #576C22direct (N × H)
15. C2×AGL(1,ℤ/8)C22semidirect
16. C2×#135C22semidirect
17. C22×Q16C22semidirect
18. C22×SD16C22semidirect
19. C22×D8C22semidirect
20. C22×D8C22semidirect
21. C22×D8C22semidirect
22. C2×#131C22semidirect
23. C22×M16C22semidirect
24. C23×C8C22semidirect
27. H(5,𝔽2)C23semidirect
32. C2×#41C23semidirect
41. C22×D4C23semidirect
44. AGL(1,ℤ/8)C23semidirect
45. #135C23semidirect
46. C2×Q16C23semidirect
47. C2×SD16C23semidirect
48. C2×D8C23semidirect
49. C2×D8C23semidirect
50. C2×D8C23semidirect
51. #131C23semidirect
52. C2×M16C23semidirect
55. C22×C8C23semidirect
64. #41C24semidirect
75. C2×D4C24semidirect
82. Q16C24semidirect
83. SD16C24semidirect
84. D8C24semidirect
85. D8C24semidirect
86. M16C24semidirect
89. C2×C8C24semidirect
118. C22#576direct (N × H)
127. C2C2×#576direct (N × H)

Maximal subgroups

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

subgroupindex
C22×H(5,𝔽2)2
C2×#5762
C22×AGL(1,ℤ/8)2
C22×#1352
C23×D82
C22×#1312