Finite Group Databasebeta

← C2×C3⋊(C3⋊(C3⋊S4)) #13049871

Subgroups — C2×C3⋊(C3⋊(C3⋊S4)) #13049871

Order 1296.

2484 conjugacy classes of subgroups; only the 483 normal subgroups are shown.

normal subgroups only

This group has 2484 conjugacy classes of subgroups — too many to draw. Only its 483 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×C33×A4C211yesmaximal, characteristic, complemented
2C3⋊(C3⋊(C3⋊S4))C212yesmaximal, direct product factor
3C2×C3⋊(C3⋊S4)—339nomaximal
4C2×#6312—31nomaximal
5C2×C3⋊(C3⋊(C3⋊D3))—41nomaximal
6C33×A4C2211yescharacteristic, complemented
7C23×C33D311yescharacteristic, complemented
8C22×C3⋊(C3⋊D3)—31no—
9C2×C32×A4D3139yescomplemented
10C3⋊(C3⋊S4)—378no—
11#6312—32no—
12#6312—32no—
13C2×C3⋊(C3⋊Dic3)—31no—
14C2×C34—41no—
15C3⋊(C3⋊(C3⋊D3))—42no—
16C2×C3⋊S4—9117no—
17C2×#638—913no—
18C22×C33—31no—
19C22×C33—31no—
20C22×C33C2×D311yescharacteristic, complemented
21C2×C3⋊(C3⋊D3)—32no—
22C2×C3⋊(C3⋊D3)—61no—
23C2×C3⋊(C3⋊D3)—61no—
24C2×C3⋊(C3⋊D3)—1239no—
25C32×A4C2×D3139yescomplemented
26C3⋊(C3⋊Dic3)—32no—
27C34—41noSylow
28C23×C32C3⋊D3113yescomplemented
29C22×C3⋊D3—913no—
30C2×C3×A4C3⋊D31117yescomplemented
31C3⋊S4—9234no—
32#638—926no—
33#638—926no—
34C2×C3⋊Dic3—913no—
35C2×C33—31no—
36C2×C33—31no—
37C2×C33—439no—
38C2×C33S411yescharacteristic, complemented
39C3⋊(C3⋊D3)—62no—
40C3⋊(C3⋊D3)—1278no—
41C2×S4—2727no—
42C2×#67—2713no—
43C22×C32—313no—
44C22×C32—313no—
45C22×C32C2×C3⋊D3113yescomplemented
46C2×C3⋊D3—926no—
47C2×C3⋊D3—1813no—
48C2×C3⋊D3—1813no—
49C2×C3⋊D3—36117no—
50C3×A4C2×C3⋊D31117yescomplemented
51C3⋊Dic3—926no—
52C33—439no—
53C33C2×S411yescharacteristic, complemented
54C23×C3C3⋊(C3⋊D3)113yescomplemented
55C22×D3—2713no—
56C2×A4C3⋊(C3⋊D3)127yescomplemented
57S4—2754no—
58#67—2726no—
59#67—2726no—
60C2×Dic3—2713no—
61C2×C32—313no—
62C2×C32—313no—
63C2×C32—4117no—
64C2×C32C3⋊S4113yescomplemented
65C3⋊D3—1826no—
66C3⋊D3—36234no—
67C2×D4—811noSylow
68C22×C3—313no—
69C22×C3—313no—
70C22×C3C2×C3⋊(C3⋊D3)113yescomplemented
71C2×D3—2726no—
72C2×D3—5413no—
73C2×D3—5413no—
74C2×D3—10827no—
75A4C2×C3⋊(C3⋊D3)127yescomplemented
76Dic3—2726no—
77C32—4117no—
78C32C2×C3⋊S4113yescomplemented
79C23—811no—
80C23C3⋊(C3⋊(C3⋊D3))11yescharacteristic, complemented
81D4—812no—
82D4—812no—
83C2×C4—811no—
84C2×C3—313no—
85C2×C3—313no—
86C2×C3—427no—
87C2×C3C3⋊(C3⋊S4)113yescomplemented
88D3—5426no—
89D3—10854no—
90C22—31no—
91C22—31no—
92C22—812no—
93C22—1621no—
94C22—1621no—
95C22C2×C3⋊(C3⋊(C3⋊D3))11yescharacteristic, complemented
96C4—812no—
97C3—427no—
98C3C2×C3⋊(C3⋊S4)113yescomplemented
99C2—31no—
100C2—31no—
101C2—1622no—
102C2C3⋊(C3⋊(C3⋊S4))11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2C3⋊(C3⋊(C3⋊S4))
commutator, cosocle, hypocenterC33×A4C22
FrattiniC1G
Fitting, socleC23×C33D3
LCSG ▷ C33×A4
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C33×A4C2semidirect
2. C3⋊(C3⋊(C3⋊S4))C2direct (N × H)
6. C33×A4C22semidirect
7. C23×C33D3semidirect
9. C2×C32×A4D3semidirect
20. C22×C33C2×D3semidirect
25. C32×A4C2×D3semidirect
28. C23×C32C3⋊D3semidirect
30. C2×C3×A4C3⋊D3semidirect
38. C2×C33S4semidirect
45. C22×C32C2×C3⋊D3semidirect
50. C3×A4C2×C3⋊D3semidirect
53. C33C2×S4semidirect
54. C23×C3C3⋊(C3⋊D3)semidirect
56. C2×A4C3⋊(C3⋊D3)semidirect
64. C2×C32C3⋊S4semidirect
70. C22×C3C2×C3⋊(C3⋊D3)semidirect
75. A4C2×C3⋊(C3⋊D3)semidirect
78. C32C2×C3⋊S4semidirect
80. C23C3⋊(C3⋊(C3⋊D3))semidirect
87. C2×C3C3⋊(C3⋊S4)semidirect
95. C22C2×C3⋊(C3⋊(C3⋊D3))semidirect
98. C3C2×C3⋊(C3⋊S4)semidirect
102. C2C3⋊(C3⋊(C3⋊S4))direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×A42
C3⋊(C3⋊(C3⋊S4))2
C2×C3⋊(C3⋊S4)3
C2×#63123
C2×C3⋊(C3⋊(C3⋊D3))4