Finite Group Databasebeta

← C22×H(3,𝔽4)×C83 #35290512

Subgroups — C22×H(3,𝔽4)×C83 #35290512

Order 21248.

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×H(3,𝔽4)×C83C2148yesmaximal, direct product factor
2C22×#124×C83C219yesmaximal, complemented
3C22×#120×C83C216yesmaximal, complemented
4C26×C83C2212yescomplemented
5C23×Q8×C83C2213yescomplemented
6C23×D4×C83C2219yes—
7H(3,𝔽4)×C83C221256yesdirect product factor
8C2×#124×C83C221216yescomplemented
9C2×#120×C83C221144yescomplemented
10C22×C22⋊C4×C83C22118yescomplemented
11C22×C42×C83C2213yescomplemented
12C25×C83—424no—
13C25×C83C2316yes—
14C25×C83C23124yescomplemented
15C22×Q8×C83—412no—
16C22×Q8×C83C23136yescomplemented
17C22×D4×C83—436no—
18C22×D4×C83C231108yes—
19C23×C4×C83C2319yes—
20#124×C83C231576yescomplemented
21#120×C83C231384yescomplemented
22C2×C22⋊C4×C83C231216yescomplemented
23C2×C42×C83C23136yescomplemented
24C24×C83—272no—
25C24×C83—4144no—
26C24×C83—4128no—
27C24×C83C24136yes—
28C24×C83C2411yescharacteristic
29C24×C83C24132yescomplemented
30C2×Q8×C83—472no—
31C2×Q8×C83C24148yescomplemented
32C2×D4×C83—4216no—
33C2×D4×C83C241144yes—
34C22×C4×C83—236no—
35C22×C4×C83C24154yes—
36C22⋊C4×C83C241288yescomplemented
37C42×C83C24148yescomplemented
38C23×C83—2216no—
39C23×C83—496no—
40C23×C83—4384no—
41C23×C83—496no—
42C23×C83H(5,𝔽2)112yes—
43C23×C83C25124yes—
44C23×C83C2513yes—
45Q8×C83—448no—
46D4×C83—4144no—
47C2×C4×C83—2108no—
48C2×C4×C83C25136yes—
49C22×C83—272no—
50C22×C83—4128no—
51C22×C83—4144no—
52C22×C83H(3,𝔽4)116yesdirect product factor
53C22×C83C2×H(5,𝔽2)118yes—
54C22×C83C2611yescharacteristic
55C4×C83—236no—
56C22×H(3,𝔽4)C8311yesmaximal, characteristic, direct product factor, Sylow
57C2×C83—424no—
58C2×C83C2×H(3,𝔽4)112yesdirect product factor
59C2×C83C22×H(5,𝔽2)13yes—
60C2×H(3,𝔽4)C2×C83148yesdirect product factor
61C22×#124C2×C8319yescomplemented
62C22×#120C2×C8316yescomplemented
63C83C22×H(3,𝔽4)11yescharacteristic, direct product factor, Sylow
64C26C22×C8312yescomplemented
65C23×Q8C22×C8313yescomplemented
66C23×D4C22×C8319yes—
67H(3,𝔽4)C22×C831256yesdirect product factor
68C2×#124C22×C831216yescomplemented
69C2×#120C22×C831144yescomplemented
70C22×C22⋊C4C22×C83118yescomplemented
71C22×C42C22×C8313yescomplemented
72C25—424no—
73C25C23×C8316yes—
74C25C23×C83124yescomplemented
75C22×Q8—412no—
76C22×Q8C23×C83136yescomplemented
77C22×D4—436no—
78C22×D4C23×C831108yes—
79C23×C4C23×C8319yes—
80#124C23×C831576yescomplemented
81#120C23×C831384yescomplemented
82C2×C22⋊C4C23×C831216yescomplemented
83C2×C42C23×C83136yescomplemented
84C24—272no—
85C24—4144no—
86C24—4128no—
87C24C24×C83136yes—
88C24C24×C8311yescharacteristic
89C24C24×C83132yescomplemented
90C2×Q8—472no—
91C2×Q8C24×C83148yescomplemented
92C2×D4—4216no—
93C2×D4C24×C831144yes—
94C22×C4—236no—
95C22×C4C24×C83154yes—
96C22⋊C4C24×C831288yescomplemented
97C42C24×C83148yescomplemented
98C23—2216no—
99C23—496no—
100C23—4384no—
101C23—496no—
102C23H(5,𝔽2)×C83112yes—
103C23C25×C83124yes—
104C23C25×C8313yes—
105Q8—448no—
106D4—4144no—
107C2×C4—2108no—
108C2×C4C25×C83136yes—
109C22—272no—
110C22—4128no—
111C22—4144no—
112C22H(3,𝔽4)×C83116yesdirect product factor
113C22C2×H(5,𝔽2)×C83118yes—
114C22C26×C8311yescharacteristic
115C4—236no—
116C2—424no—
117C2C2×H(3,𝔽4)×C83112yesdirect product factor
118C2C22×H(5,𝔽2)×C8313yes—
subgroupG/H
center, socleC24×C83C24
commutator, Frattini, cosocleC22C26×C83
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×H(3,𝔽4)×C83C2direct (N × H)
2. C22×#124×C83C2semidirect
3. C22×#120×C83C2semidirect
4. C26×C83C22semidirect
5. C23×Q8×C83C22semidirect
7. H(3,𝔽4)×C83C22direct (N × H)
8. C2×#124×C83C22semidirect
9. C2×#120×C83C22semidirect
10. C22×C22⋊C4×C83C22semidirect
11. C22×C42×C83C22semidirect
14. C25×C83C23semidirect
16. C22×Q8×C83C23semidirect
20. #124×C83C23semidirect
21. #120×C83C23semidirect
22. C2×C22⋊C4×C83C23semidirect
23. C2×C42×C83C23semidirect
29. C24×C83C24semidirect
31. C2×Q8×C83C24semidirect
36. C22⋊C4×C83C24semidirect
37. C42×C83C24semidirect
52. C22×C83H(3,𝔽4)direct (N × H)
56. C22×H(3,𝔽4)C83direct (N × H)
58. C2×C83C2×H(3,𝔽4)direct (N × H)
60. C2×H(3,𝔽4)C2×C83direct (N × H)
61. C22×#124C2×C83semidirect
62. C22×#120C2×C83semidirect
63. C83C22×H(3,𝔽4)direct (N × H)
64. C26C22×C83semidirect
65. C23×Q8C22×C83semidirect
67. H(3,𝔽4)C22×C83direct (N × H)
68. C2×#124C22×C83semidirect
69. C2×#120C22×C83semidirect
70. C22×C22⋊C4C22×C83semidirect
71. C22×C42C22×C83semidirect
74. C25C23×C83semidirect
76. C22×Q8C23×C83semidirect
80. #124C23×C83semidirect
81. #120C23×C83semidirect
82. C2×C22⋊C4C23×C83semidirect
83. C2×C42C23×C83semidirect
89. C24C24×C83semidirect
91. C2×Q8C24×C83semidirect
96. C22⋊C4C24×C83semidirect
97. C42C24×C83semidirect
112. C22H(3,𝔽4)×C83direct (N × H)
117. C2C2×H(3,𝔽4)×C83direct (N × H)

Maximal subgroups

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

subgroupindex
C2×H(3,𝔽4)×C832
C22×#124×C832
C22×#120×C832
C22×H(3,𝔽4)83