Finite Group Databasebeta

← D3×(C11⋊D11)⋊ΓL(1,𝔽8) #44479113

Subgroups — D3×(C11⋊D11)⋊ΓL(1,𝔽8) #44479113

Order 30492.

168 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 27 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
1D3×C7⋊(C112⋊C3)C211yesmaximal, characteristic, complemented
2C3⋊((C11⋊D11)⋊ΓL(1,𝔽8))C211yesmaximal, characteristic, complemented
3C3×(C11⋊D11)⋊ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
4D3×C7×C11⋊D11C311yesmaximal, characteristic, complemented
5C2×(C11⋊D11)⋊ΓL(1,𝔽8)—31nomaximal
6C3×C7⋊(C112⋊C3)C2211yescharacteristic, complemented, Hall
7C7×C3⋊(C11⋊D11)C2×C311yescharacteristic, complemented
8D3×C7×C112C2×C311yescharacteristic, complemented
9C3×C7×C11⋊D11C2×C311yescharacteristic, complemented
10C2×C7⋊(C112⋊C3)—31no—
11(C11⋊D11)⋊ΓL(1,𝔽8)—21no—
12(C11⋊D11)⋊ΓL(1,𝔽8)—31no—
13(C11⋊D11)⋊ΓL(1,𝔽8)D311yescharacteristic, direct product factor
14D3×(C11⋊D11)⋊C3—71nomaximal, Hall
15C2×C7×C11⋊D11—31noHall
16C7⋊(C112⋊C3)—21no—
17C7⋊(C112⋊C3)C2×D311yescharacteristic, complemented
18C3×C7×C112C22×C311yescharacteristic, complemented
19D3×C112⋊C3—71no—
20C3⋊((C11⋊D11)⋊C3)—71no—
21C3×(C11⋊D11)⋊C3—71no—
22C2×C7×C112—31no—
23C7×C11⋊D11—31no—
24C7×C11⋊D11C3×D311yescharacteristic, complemented
25D3×C11⋊D11ΓL(1,𝔽8)11yescharacteristic, complemented
26C2×(C11⋊D11)⋊C3—211no—
27C3×C112⋊C3—71noHall
28D3×C7×D11—334no—
29C7×C112C2×C3×D311yescharacteristic, complemented, Hall
30C3⋊(C11⋊D11)C2×ΓL(1,𝔽8)11yescharacteristic, complemented
31D3×C112C2×ΓL(1,𝔽8)11yescharacteristic, complemented
32C3×C11⋊D11C2×ΓL(1,𝔽8)11yescharacteristic, complemented
33C2×C112⋊C3—211no—
34(C11⋊D11)⋊C3—71no—
35(C11⋊D11)⋊C3—141no—
36(C11⋊D11)⋊C3—211no—
37C2×C11⋊D11—31noHall
38C7×D33—334no—
39D3×C7×C11—34no—
40C3×C7×D11—334no—
41C3×C112C22×ΓL(1,𝔽8)11yescharacteristic, complemented
42C112⋊C3—71no—
43C112⋊C3—141no—
44C2×C7×D11—994no—
45C2×D3×ΓL(1,𝔽8)—1211nomaximal, Hall
46C2×C112—31no—
47C11⋊D11—31no—
48C11⋊D11D3×ΓL(1,𝔽8)11yescharacteristic, complemented
49C3×C7×C11—34no—
50C2×C7×C11—94no—
51C7×D11—334no—
52C7×D11—994no—
53D3×D11—334no—
54C2×C3×ΓL(1,𝔽8)—1211no—
55D3×ΓL(1,𝔽8)—1211no—
56D3×ΓL(1,𝔽8)—2421no—
57C112C2×D3×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
58C2×D3×C7—1211no—
59C22×ΓL(1,𝔽8)—3631no—
60C7×C11—34no—
61D33—334no—
62C3×D11—334no—
63D3×C11—34no—
64C3×ΓL(1,𝔽8)—2421noHall
65C2×D11—994no—
66C2×C3×C7—1211no—
67D3×C7—1211no—
68D3×C7(C11⋊D11)⋊C311yescharacteristic, complemented
69C2×ΓL(1,𝔽8)—3631no—
70C2×ΓL(1,𝔽8)—1211no—
71C2×ΓL(1,𝔽8)—7261no—
72C2×ΓL(1,𝔽8)—2421no—
73C2×C3×D3—8471noHall
74C3×C11—34no—
75C22×C7—3631noHall
76C2×C11—94no—
77D11—334no—
78D11—994no—
79C3×C7C2×(C11⋊D11)⋊C311yescharacteristic, complemented
80ΓL(1,𝔽8)—4841no—
81ΓL(1,𝔽8)—2421no—
82C2×C32—8471no—
83C3×D3—8471no—
84C3×D3—16941no—
85C2×C7—31no—
86C2×C7—3631no—
87C2×C7—1211no—
88C22×C3—25411no—
89C2×D3—1211no—
90C11—34no—
91C32—16941noSylow
92C7D3×(C11⋊D11)⋊C311yescharacteristic, complemented, Sylow
93C2×C3—8471no—
94C2×C3—1211no—
95C2×C3—16941no—
96C2×C3—50821no—
97C2×C3—25411no—
98D3—1211no—
99D3(C11⋊D11)⋊ΓL(1,𝔽8)11yescharacteristic, direct product factor
100C22—3631noSylow
101C3—33881no—
102C3—16941no—
103C3C2×(C11⋊D11)⋊ΓL(1,𝔽8)11yescharacteristic, complemented
104C2—31no—
105C2—3631no—
106C2—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. D3×C7⋊(C112⋊C3)C2semidirect
2. C3⋊((C11⋊D11)⋊ΓL(1,𝔽8))C2semidirect
3. C3×(C11⋊D11)⋊ΓL(1,𝔽8)C2semidirect
4. D3×C7×C11⋊D11C3semidirect
6. C3×C7⋊(C112⋊C3)C22semidirect
7. C7×C3⋊(C11⋊D11)C2×C3semidirect
8. D3×C7×C112C2×C3semidirect
9. C3×C7×C11⋊D11C2×C3semidirect
13. (C11⋊D11)⋊ΓL(1,𝔽8)D3direct (N × H)
17. C7⋊(C112⋊C3)C2×D3semidirect
18. C3×C7×C112C22×C3semidirect
24. C7×C11⋊D11C3×D3semidirect
25. D3×C11⋊D11ΓL(1,𝔽8)semidirect
29. C7×C112C2×C3×D3semidirect
30. C3⋊(C11⋊D11)C2×ΓL(1,𝔽8)semidirect
31. D3×C112C2×ΓL(1,𝔽8)semidirect
32. C3×C11⋊D11C2×ΓL(1,𝔽8)semidirect
41. C3×C112C22×ΓL(1,𝔽8)semidirect
48. C11⋊D11D3×ΓL(1,𝔽8)semidirect
57. C112C2×D3×ΓL(1,𝔽8)semidirect
68. D3×C7(C11⋊D11)⋊C3semidirect
79. C3×C7C2×(C11⋊D11)⋊C3semidirect
92. C7D3×(C11⋊D11)⋊C3semidirect
99. D3(C11⋊D11)⋊ΓL(1,𝔽8)direct (N × H)
103. C3C2×(C11⋊D11)⋊ΓL(1,𝔽8)semidirect

Maximal subgroups

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

subgroupindex
D3×C7⋊(C112⋊C3)2
C3⋊((C11⋊D11)⋊ΓL(1,𝔽8))2
C3×(C11⋊D11)⋊ΓL(1,𝔽8)2
D3×C7×C11⋊D113
C2×(C11⋊D11)⋊ΓL(1,𝔽8)3
D3×(C11⋊D11)⋊C37
C2×D3×ΓL(1,𝔽8)121