Finite Group Databasebeta

← ΓL(1,𝔽8)×D33⋊D11 #44479117

Subgroups — ΓL(1,𝔽8)×D33⋊D11 #44479117

Order 30492.

296 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C11×ΓL(1,𝔽8)×D33C212yesmaximal, complemented
2C3×ΓL(1,𝔽8)×C11⋊D11C211yesmaximal, characteristic, complemented
3C7×D33⋊D11C311yesmaximal, characteristic, complemented
4ΓL(1,𝔽8)×D112—31nomaximal
5C3×C112×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
6C7×C11×D33C2×C312yescomplemented
7C3×C7×C11⋊D11C2×C311yescharacteristic, complemented
8ΓL(1,𝔽8)×C11⋊D11—21no—
9ΓL(1,𝔽8)×C11⋊D11D311yescharacteristic, complemented
10C11×ΓL(1,𝔽8)×D11—32no—
11C3×D33⋊D11—71nomaximal, Hall
12C7×D112—31noHall
13D3×ΓL(1,𝔽8)×D11—112nomaximal
14C112×ΓL(1,𝔽8)—21no—
15C112×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
16C3×C7×C112C22×C311yescharacteristic, complemented
17C3×C11×D33—72no—
18C32×C11⋊D11—71no—
19C7×C11⋊D11C3×D311yescharacteristic, complemented
20C7×C11×D11—32no—
21D33⋊D11ΓL(1,𝔽8)11yescharacteristic, direct product factor
22C3×D112—211no—
23ΓL(1,𝔽8)×D33D1112yescomplemented
24D3×C11×ΓL(1,𝔽8)—112no—
25C3×ΓL(1,𝔽8)×D11—112no—
26C3×ΓL(1,𝔽8)×D11—225no—
27C32×C112—71noHall
28D3×C7×D11—112no—
29C2×ΓL(1,𝔽8)×D11—332no—
30C7×C112C2×C3×D311yescharacteristic, complemented, Hall
31C11×D33C2×ΓL(1,𝔽8)12yescomplemented
32C3×C11⋊D11—71no—
33C3×C11⋊D11—141no—
34C3×C11⋊D11C2×ΓL(1,𝔽8)11yescharacteristic, complemented
35C3×C11×D11—212no—
36C3×C11×ΓL(1,𝔽8)—25no—
37C3×C11×ΓL(1,𝔽8)C2×D1112yescomplemented
38D112—31noHall
39C7×D33C3×D1112yescomplemented
40D3×C7×C11—112no—
41C3×C7×D11—112no—
42C3×C7×D11—225no—
43C2×C11×ΓL(1,𝔽8)—332no—
44ΓL(1,𝔽8)×D11—32no—
45ΓL(1,𝔽8)×D11—112no—
46ΓL(1,𝔽8)×D11—222no—
47ΓL(1,𝔽8)×D11—225no—
48ΓL(1,𝔽8)×D11—2210no—
49C3×D3×D11—772no—
50C3×C112—71no—
51C3×C112—141no—
52C3×C112C22×ΓL(1,𝔽8)11yescharacteristic, complemented
53C2×C7×D11—332no—
54C2×D3×ΓL(1,𝔽8)—1211noHall
55C11⋊D11D3×ΓL(1,𝔽8)11yescharacteristic, complemented
56C11×D11—32no—
57C3×C7×C11—25no—
58C3×C7×C11C2×C3×D1112yescomplemented
59C11×ΓL(1,𝔽8)—210no—
60C11×ΓL(1,𝔽8)—22no—
61C11×ΓL(1,𝔽8)—25no—
62C11×ΓL(1,𝔽8)D3×D1112yescomplemented
63C3×D33—72no—
64C3×D3×C11—772no—
65C32×D11—772no—
66C32×D11—1545no—
67C2×C7×C11—332no—
68C7×D11—32no—
69C7×D11—112no—
70C7×D11—225no—
71C2×C3×D11—2312no—
72D3×D11—112no—
73C2×C3×ΓL(1,𝔽8)—1211no—
74D3×ΓL(1,𝔽8)—112no—
75C112C2×D3×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
76C32×C11—72no—
77C32×C11—145no—
78C2×D3×C7—1211no—
79C22×ΓL(1,𝔽8)—3631no—
80C7×C11—25no—
81C7×C11C3×D3×D1112yescomplemented
82C2×C3×C11—2312no—
83D33ΓL(1,𝔽8)×D1112yescomplemented
84C3×D11—112no—
85C3×D11—212no—
86C3×D11—225no—
87C3×D11—772no—
88C3×D11—1545no—
89C3×D11—1542no—
90C3×D11—15410no—
91D3×C11—112no—
92C3×ΓL(1,𝔽8)D11211yescharacteristic, complemented, Hall
93C2×D11—332no—
94C2×C3×C7—1211no—
95D3×C7—112no—
96C2×ΓL(1,𝔽8)—332no—
97C2×ΓL(1,𝔽8)—1211no—
98C2×ΓL(1,𝔽8)—2421no—
99C2×C3×D3—8471noHall
100C3×C11—25no—
101C3×C11—72no—
102C3×C11—145no—
103C3×C11—1410no—
104C3×C11—142no—
105C3×C11C2×ΓL(1,𝔽8)×D1112yescomplemented
106C22×C7—3631noHall
107C2×C11—332no—
108D11—32no—
109D11—112no—
110D11—225no—
111C3×C7C3×D11211yescharacteristic, complemented
112ΓL(1,𝔽8)—21no—
113ΓL(1,𝔽8)D33⋊D1111yescharacteristic, direct product factor
114C2×C32—8471no—
115C3×D3—772no—
116C2×C7—332no—
117C2×C7—1211no—
118C22×C3—25411no—
119C2×D3—1211no—
120C11—25no—
121C11D3×ΓL(1,𝔽8)×D1112yescomplemented
122C32—71noSylow
123C7C3×D33⋊D1111yescharacteristic, complemented, Sylow
124C2×C3—8471no—
125C2×C3—1211no—
126C2×C3—16941no—
127C2×C3—2312no—
128D3—112no—
129C22—3631noSylow
130C3—71no—
131C3—141no—
132C3ΓL(1,𝔽8)×D11211yescharacteristic, complemented
133C2—332no—
134C2—1211no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C7×C112C22×C3
LCSG ▷ C3×C7×C112
UCSC1

Complements

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

NH ≅ G/Ntype
1. C11×ΓL(1,𝔽8)×D33C2semidirect
2. C3×ΓL(1,𝔽8)×C11⋊D11C2semidirect
3. C7×D33⋊D11C3semidirect
5. C3×C112×ΓL(1,𝔽8)C22semidirect
6. C7×C11×D33C2×C3semidirect
7. C3×C7×C11⋊D11C2×C3semidirect
9. ΓL(1,𝔽8)×C11⋊D11D3semidirect
15. C112×ΓL(1,𝔽8)C2×D3semidirect
16. C3×C7×C112C22×C3semidirect
19. C7×C11⋊D11C3×D3semidirect
21. D33⋊D11ΓL(1,𝔽8)direct (N × H)
23. ΓL(1,𝔽8)×D33D11semidirect
30. C7×C112C2×C3×D3semidirect
31. C11×D33C2×ΓL(1,𝔽8)semidirect
34. C3×C11⋊D11C2×ΓL(1,𝔽8)semidirect
37. C3×C11×ΓL(1,𝔽8)C2×D11semidirect
39. C7×D33C3×D11semidirect
52. C3×C112C22×ΓL(1,𝔽8)semidirect
55. C11⋊D11D3×ΓL(1,𝔽8)semidirect
58. C3×C7×C11C2×C3×D11semidirect
62. C11×ΓL(1,𝔽8)D3×D11semidirect
75. C112C2×D3×ΓL(1,𝔽8)semidirect
81. C7×C11C3×D3×D11semidirect
83. D33ΓL(1,𝔽8)×D11semidirect
92. C3×ΓL(1,𝔽8)D112semidirect
105. C3×C11C2×ΓL(1,𝔽8)×D11semidirect
111. C3×C7C3×D112semidirect
113. ΓL(1,𝔽8)D33⋊D11direct (N × H)
121. C11D3×ΓL(1,𝔽8)×D11semidirect
123. C7C3×D33⋊D11semidirect
132. C3ΓL(1,𝔽8)×D112semidirect

Maximal subgroups

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

subgroupindex
C11×ΓL(1,𝔽8)×D332
C3×ΓL(1,𝔽8)×C11⋊D112
C7×D33⋊D113
ΓL(1,𝔽8)×D1123
C3×D33⋊D117
D3×ΓL(1,𝔽8)×D1111