Finite Group Databasebeta

← C22×C11×H(3,𝔽4) #14639881

Subgroups — C22×C11×H(3,𝔽4) #14639881

Order 2816.

10246 conjugacy classes of subgroups; only the 5774 normal subgroups are shown.

normal subgroups only

This group has 10246 conjugacy classes of subgroups — too many to draw. Only its 5774 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
1C2×C11×H(3,𝔽4)C2148yesmaximal, direct product factor
2C22×C11×#124C219yesmaximal, complemented
3C22×C11×#120C216yesmaximal, complemented
4C26×C11C2212yescomplemented
5C23×Q8×C11C2213yescomplemented
6C23×D4×C11C2219yes—
7C11×H(3,𝔽4)C221256yesdirect product factor
8C2×C11×#124C221216yescomplemented
9C2×C11×#120C221144yescomplemented
10C22×C11×C22⋊C4C22118yescomplemented
11C22×C42×C11C2213yescomplemented
12C25×C11—424no—
13C25×C11C2316yes—
14C25×C11C23124yescomplemented
15C22×Q8×C11—412no—
16C22×Q8×C11C23136yescomplemented
17C22×D4×C11—436no—
18C22×D4×C11C231108yes—
19C23×C4×C11C2319yes—
20C11×#124C231576yescomplemented
21C11×#120C231384yescomplemented
22C2×C11×C22⋊C4C231216yescomplemented
23C2×C42×C11C23136yescomplemented
24C22×H(3,𝔽4)C1111yesmaximal, characteristic, direct product factor, Sylow
25C24×C11—272no—
26C24×C11—4144no—
27C24×C11—4128no—
28C24×C11C24136yes—
29C24×C11C2411yescharacteristic
30C24×C11C24132yescomplemented
31C2×Q8×C11—472no—
32C2×Q8×C11C24148yescomplemented
33C2×D4×C11—4216no—
34C2×D4×C11C241144yes—
35C22×C4×C11—236no—
36C22×C4×C11C24154yes—
37C11×C22⋊C4C241288yescomplemented
38C42×C11C24148yescomplemented
39C2×H(3,𝔽4)C2×C11148yesdirect product factor
40C22×#124C2×C1119yescomplemented
41C22×#120C2×C1116yescomplemented
42C23×C11—2216no—
43C23×C11—496no—
44C23×C11—4384no—
45C23×C11—496no—
46C23×C11H(5,𝔽2)112yes—
47C23×C11C25124yes—
48C23×C11C2513yes—
49Q8×C11—448no—
50D4×C11—4144no—
51C2×C4×C11—2108no—
52C2×C4×C11C25136yes—
53C26C22×C1112yescomplemented
54C23×Q8C22×C1113yescomplemented
55C23×D4C22×C1119yes—
56H(3,𝔽4)C22×C111256yesdirect product factor
57C2×#124C22×C111216yescomplemented
58C2×#120C22×C111144yescomplemented
59C22×C22⋊C4C22×C11118yescomplemented
60C22×C42C22×C1113yescomplemented
61C22×C11—272no—
62C22×C11—4128no—
63C22×C11—4144no—
64C22×C11H(3,𝔽4)116yesdirect product factor
65C22×C11C2×H(5,𝔽2)118yes—
66C22×C11C2611yescharacteristic
67C4×C11—236no—
68C25—424no—
69C25C23×C1116yes—
70C25C23×C11124yescomplemented
71C22×Q8—412no—
72C22×Q8C23×C11136yescomplemented
73C22×D4—436no—
74C22×D4C23×C111108yes—
75C23×C4C23×C1119yes—
76#124C23×C111576yescomplemented
77#120C23×C111384yescomplemented
78C2×C22⋊C4C23×C111216yescomplemented
79C2×C42C23×C11136yescomplemented
80C2×C11—424no—
81C2×C11C2×H(3,𝔽4)112yesdirect product factor
82C2×C11C22×H(5,𝔽2)13yes—
83C24—272no—
84C24—4144no—
85C24—4128no—
86C24C24×C11136yes—
87C24C24×C1111yescharacteristic
88C24C24×C11132yescomplemented
89C2×Q8—472no—
90C2×Q8C24×C11148yescomplemented
91C2×D4—4216no—
92C2×D4C24×C111144yes—
93C22×C4—236no—
94C22×C4C24×C11154yes—
95C22⋊C4C24×C111288yescomplemented
96C42C24×C11148yescomplemented
97C11C22×H(3,𝔽4)11yescharacteristic, direct product factor, Sylow
98C23—2216no—
99C23—496no—
100C23—4384no—
101C23—496no—
102C23C11×H(5,𝔽2)112yes—
103C23C25×C11124yes—
104C23C25×C1113yes—
105Q8—448no—
106D4—4144no—
107C2×C4—2108no—
108C2×C4C25×C11136yes—
109C22—272no—
110C22—4128no—
111C22—4144no—
112C22C11×H(3,𝔽4)116yesdirect product factor
113C22C2×C11×H(5,𝔽2)118yes—
114C22C26×C1111yescharacteristic
115C4—236no—
116C2—424no—
117C2C2×C11×H(3,𝔽4)112yesdirect product factor
118C2C22×C11×H(5,𝔽2)13yes—
subgroupG/H
center, socleC24×C11C24
commutator, Frattini, cosocleC22C26×C11
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C11 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C11×H(3,𝔽4)C2direct (N × H)
2. C22×C11×#124C2semidirect
3. C22×C11×#120C2semidirect
4. C26×C11C22semidirect
5. C23×Q8×C11C22semidirect
7. C11×H(3,𝔽4)C22direct (N × H)
8. C2×C11×#124C22semidirect
9. C2×C11×#120C22semidirect
10. C22×C11×C22⋊C4C22semidirect
11. C22×C42×C11C22semidirect
14. C25×C11C23semidirect
16. C22×Q8×C11C23semidirect
20. C11×#124C23semidirect
21. C11×#120C23semidirect
22. C2×C11×C22⋊C4C23semidirect
23. C2×C42×C11C23semidirect
24. C22×H(3,𝔽4)C11direct (N × H)
30. C24×C11C24semidirect
32. C2×Q8×C11C24semidirect
37. C11×C22⋊C4C24semidirect
38. C42×C11C24semidirect
39. C2×H(3,𝔽4)C2×C11direct (N × H)
40. C22×#124C2×C11semidirect
41. C22×#120C2×C11semidirect
53. C26C22×C11semidirect
54. C23×Q8C22×C11semidirect
56. H(3,𝔽4)C22×C11direct (N × H)
57. C2×#124C22×C11semidirect
58. C2×#120C22×C11semidirect
59. C22×C22⋊C4C22×C11semidirect
60. C22×C42C22×C11semidirect
64. C22×C11H(3,𝔽4)direct (N × H)
70. C25C23×C11semidirect
72. C22×Q8C23×C11semidirect
76. #124C23×C11semidirect
77. #120C23×C11semidirect
78. C2×C22⋊C4C23×C11semidirect
79. C2×C42C23×C11semidirect
81. C2×C11C2×H(3,𝔽4)direct (N × H)
88. C24C24×C11semidirect
90. C2×Q8C24×C11semidirect
95. C22⋊C4C24×C11semidirect
96. C42C24×C11semidirect
97. C11C22×H(3,𝔽4)direct (N × H)
112. C22C11×H(3,𝔽4)direct (N × H)
117. C2C2×C11×H(3,𝔽4)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C11×H(3,𝔽4)2
C22×C11×#1242
C22×C11×#1202
C22×H(3,𝔽4)11