Finite Group Databasebeta

← D3×ΓL(1,𝔽8)×C11⋊D11 #44479118

Subgroups — D3×ΓL(1,𝔽8)×C11⋊D11 #44479118

Order 30492.

616 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 138 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
1ΓL(1,𝔽8)×C3⋊(C11⋊D11)C211yesmaximal, characteristic, complemented
2D3×C112×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C3×ΓL(1,𝔽8)×C11⋊D11C211yesmaximal, characteristic, complemented
4D3×C7×C11⋊D11C311yesmaximal, characteristic, complemented
5C2×ΓL(1,𝔽8)×C11⋊D11—31nomaximal
6C3×C112×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
7C7×C3⋊(C11⋊D11)C2×C311yescharacteristic, complemented
8D3×C7×C112C2×C311yescharacteristic, complemented
9C3×C7×C11⋊D11C2×C311yescharacteristic, complemented
10C2×C112×ΓL(1,𝔽8)—31no—
11ΓL(1,𝔽8)×C11⋊D11—21no—
12ΓL(1,𝔽8)×C11⋊D11—31no—
13ΓL(1,𝔽8)×C11⋊D11D311yescharacteristic, direct product factor
14C3×D3×C11⋊D11—71nomaximal, Hall
15C2×C7×C11⋊D11—31noHall
16D3×ΓL(1,𝔽8)×D11—1112nomaximal
17C112×ΓL(1,𝔽8)—21no—
18C112×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
19C3×C7×C112C22×C311yescharacteristic, complemented
20C3×C3⋊(C11⋊D11)—71no—
21C3×D3×C112—71no—
22C32×C11⋊D11—71no—
23C2×C7×C112—31no—
24C7×C11⋊D11—31no—
25C7×C11⋊D11C3×D311yescharacteristic, complemented
26C2×C3×C11⋊D11—211no—
27D3×C11⋊D11ΓL(1,𝔽8)11yescharacteristic, direct product factor
28ΓL(1,𝔽8)×D33—1112no—
29D3×C11×ΓL(1,𝔽8)D11112yescomplemented
30C3×ΓL(1,𝔽8)×D11—1112no—
31C32×C112—71noHall
32D3×C7×D11—1112no—
33C2×ΓL(1,𝔽8)×D11—3312no—
34C7×C112C2×C3×D311yescharacteristic, complemented, Hall
35C2×C3×C112—211no—
36C3⋊(C11⋊D11)C2×ΓL(1,𝔽8)11yescharacteristic, complemented
37D3×C112C2×ΓL(1,𝔽8)11yescharacteristic, complemented
38C3×C11⋊D11—71no—
39C3×C11⋊D11—141no—
40C3×C11⋊D11—211no—
41C3×C11⋊D11C2×ΓL(1,𝔽8)11yescharacteristic, complemented
42C3×C11×ΓL(1,𝔽8)C2×D11112yescomplemented
43C2×C11⋊D11—31noHall
44C7×D33—1112no—
45D3×C7×C11C3×D11112yescomplemented
46C3×C7×D11—1112no—
47C2×C11×ΓL(1,𝔽8)—312no—
48ΓL(1,𝔽8)×D11—1112no—
49ΓL(1,𝔽8)×D11—2212no—
50ΓL(1,𝔽8)×D11—3312no—
51C3×D3×D11—7712no—
52C3×C112—71no—
53C3×C112—141no—
54C3×C112C22×ΓL(1,𝔽8)11yescharacteristic, complemented
55C2×C7×D11—3312no—
56C2×D3×ΓL(1,𝔽8)—1211noHall
57C2×C112—31no—
58C11⋊D11—31no—
59C11⋊D11D3×ΓL(1,𝔽8)11yescharacteristic, direct product factor
60C3×C7×C11C2×C3×D11112yescomplemented
61C11×ΓL(1,𝔽8)—212no—
62C11×ΓL(1,𝔽8)D3×D11112yescomplemented
63C3×D33—7712no—
64C3×D3×C11—712no—
65C32×D11—7712no—
66C2×C7×C11—312no—
67C7×D11—1112no—
68C7×D11—3312no—
69C2×C3×D11—23112no—
70D3×D11—1112no—
71C2×C3×ΓL(1,𝔽8)—1211no—
72D3×ΓL(1,𝔽8)—1211no—
73D3×ΓL(1,𝔽8)C11⋊D1111yescharacteristic, direct product factor
74C112C2×D3×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
75C32×C11—712no—
76C2×D3×C7—1211no—
77C22×ΓL(1,𝔽8)—3631no—
78C7×C11C3×D3×D11112yescomplemented
79C2×C3×C11—2112no—
80D33—1112no—
81C3×D11—1112no—
82C3×D11—7712no—
83C3×D11—15412no—
84C3×D11—23112no—
85D3×C11ΓL(1,𝔽8)×D11112yescomplemented
86C3×ΓL(1,𝔽8)C2×C11⋊D1111yescharacteristic, complemented, Hall
87C2×D11—3312no—
88C2×C3×C7—1211no—
89D3×C7—1211no—
90D3×C7C3×C11⋊D1111yescharacteristic, complemented
91C2×ΓL(1,𝔽8)—31no—
92C2×ΓL(1,𝔽8)—3631no—
93C2×ΓL(1,𝔽8)—1211no—
94C2×ΓL(1,𝔽8)—2421no—
95C2×C3×D3—8471noHall
96C3×C11—712no—
97C3×C11—1412no—
98C3×C11C2×ΓL(1,𝔽8)×D11112yescomplemented
99C22×C7—3631noHall
100C2×C11—312no—
101D11—1112no—
102D11—3312no—
103C3×C7C2×C3×C11⋊D1111yescharacteristic, complemented
104ΓL(1,𝔽8)—21no—
105ΓL(1,𝔽8)D3×C11⋊D1111yescharacteristic, direct product factor
106C2×C32—8471no—
107C3×D3—71no—
108C3×D3—8471no—
109C2×C7—31no—
110C2×C7—3631no—
111C2×C7—1211no—
112C22×C3—25411no—
113C2×D3—1211no—
114C11D3×ΓL(1,𝔽8)×D11112yescomplemented
115C32—71noSylow
116C7C3×D3×C11⋊D1111yescharacteristic, complemented, Sylow
117C2×C3—211no—
118C2×C3—8471no—
119C2×C3—1211no—
120C2×C3—16941no—
121C2×C3—25411no—
122D3—1211no—
123D3ΓL(1,𝔽8)×C11⋊D1111yescharacteristic, direct product factor
124C22—3631noSylow
125C3—71no—
126C3—141no—
127C3C2×ΓL(1,𝔽8)×C11⋊D1111yescharacteristic, complemented
128C2—31no—
129C2—3631no—
130C2—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. ΓL(1,𝔽8)×C3⋊(C11⋊D11)C2semidirect
2. D3×C112×ΓL(1,𝔽8)C2semidirect
3. C3×ΓL(1,𝔽8)×C11⋊D11C2semidirect
4. D3×C7×C11⋊D11C3semidirect
6. C3×C112×ΓL(1,𝔽8)C22semidirect
7. C7×C3⋊(C11⋊D11)C2×C3semidirect
8. D3×C7×C112C2×C3semidirect
9. C3×C7×C11⋊D11C2×C3semidirect
13. ΓL(1,𝔽8)×C11⋊D11D3direct (N × H)
18. C112×ΓL(1,𝔽8)C2×D3semidirect
19. C3×C7×C112C22×C3semidirect
25. C7×C11⋊D11C3×D3semidirect
27. D3×C11⋊D11ΓL(1,𝔽8)direct (N × H)
29. D3×C11×ΓL(1,𝔽8)D11semidirect
34. C7×C112C2×C3×D3semidirect
36. C3⋊(C11⋊D11)C2×ΓL(1,𝔽8)semidirect
37. D3×C112C2×ΓL(1,𝔽8)semidirect
41. C3×C11⋊D11C2×ΓL(1,𝔽8)semidirect
42. C3×C11×ΓL(1,𝔽8)C2×D11semidirect
45. D3×C7×C11C3×D11semidirect
54. C3×C112C22×ΓL(1,𝔽8)semidirect
59. C11⋊D11D3×ΓL(1,𝔽8)direct (N × H)
60. C3×C7×C11C2×C3×D11semidirect
62. C11×ΓL(1,𝔽8)D3×D11semidirect
73. D3×ΓL(1,𝔽8)C11⋊D11direct (N × H)
74. C112C2×D3×ΓL(1,𝔽8)semidirect
78. C7×C11C3×D3×D11semidirect
85. D3×C11ΓL(1,𝔽8)×D11semidirect
86. C3×ΓL(1,𝔽8)C2×C11⋊D11semidirect
90. D3×C7C3×C11⋊D11semidirect
98. C3×C11C2×ΓL(1,𝔽8)×D11semidirect
103. C3×C7C2×C3×C11⋊D11semidirect
105. ΓL(1,𝔽8)D3×C11⋊D11direct (N × H)
114. C11D3×ΓL(1,𝔽8)×D11semidirect
116. C7C3×D3×C11⋊D11semidirect
123. D3ΓL(1,𝔽8)×C11⋊D11direct (N × H)
127. C3C2×ΓL(1,𝔽8)×C11⋊D11semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×C3⋊(C11⋊D11)2
D3×C112×ΓL(1,𝔽8)2
C3×ΓL(1,𝔽8)×C11⋊D112
D3×C7×C11⋊D113
C2×ΓL(1,𝔽8)×C11⋊D113
C3×D3×C11⋊D117
D3×ΓL(1,𝔽8)×D1111