Finite Group Databasebeta

← C2×A4×C3⋊(C3⋊D3) #13049874

Subgroups — C2×A4×C3⋊(C3⋊D3) #13049874

Order 1296.

1404 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 1404 conjugacy classes of subgroups — too many to draw. Only its 177 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
2A4×C3⋊(C3⋊D3)C212yesmaximal, direct product factor
3C23×C3⋊(C3⋊D3)C311yesmaximal, characteristic, complemented
4C2×A4×C3⋊D3—313nomaximal
5C2×C3×C3⋊(C3⋊D3)—41nomaximal
6C33×A4C2211yescharacteristic, complemented
7C23×C33C2×C311yescharacteristic, complemented
8C22×C3⋊(C3⋊D3)—31no—
9C22×C3⋊(C3⋊D3)—31no—
10C22×C3⋊(C3⋊D3)—32no—
11C22×C3⋊(C3⋊D3)C2×C312yescomplemented
12C2×C32×A4—213no—
13C2×C32×A4D3113yescomplemented
14A4×C3⋊D3—326no—
15C2×C34—41no—
16C3×C3⋊(C3⋊D3)—42no—
17C23×C3⋊D3—313no—
18C2×D3×A4—913no—
19C22×C33—31no—
20C22×C33—31no—
21C22×C33C22×C311yescharacteristic, complemented
22C2×C3⋊(C3⋊D3)—32no—
23C2×C3⋊(C3⋊D3)—31no—
24C2×C3⋊(C3⋊D3)—32no—
25C2×C3⋊(C3⋊D3)—32no—
26C2×C3⋊(C3⋊D3)—32no—
27C2×C3⋊(C3⋊D3)A411yescharacteristic, direct product factor
28C2×C3×C3⋊D3—1213no—
29C32×A4—213no—
30C32×A4C2×D3113yescomplemented
31C34—41noSylow
32C23×C32C3×D3113yescomplemented
33C22×C3⋊D3—326no—
34C22×C3⋊D3—913no—
35C22×C3⋊D3—926no—
36C22×C3⋊D3—913no—
37C2×C3×A4—252no—
38C2×C3×A4C3⋊D3113yescomplemented
39D3×A4—926no—
40C2×C33—31no—
41C2×C33—31no—
42C2×C33—413no—
43C2×C33—813no—
44C2×C33C2×A411yescharacteristic, complemented
45C3⋊(C3⋊D3)—32no—
46C3⋊(C3⋊D3)C2×A412yesdirect product factor
47C3×C3⋊D3—1226no—
48C23×D3—913no—
49C22×A4—271no—
50C22×C32—313no—
51C22×C32—313no—
52C22×C32C2×C3×D3113yescomplemented
53C2×C3⋊D3—313no—
54C2×C3⋊D3—926no—
55C2×C3⋊D3—926no—
56C2×C3⋊D3—926no—
57C2×C3⋊D3—913no—
58C2×C3⋊D3—926no—
59C2×C3×D3—3613no—
60C3×A4—252no—
61C3×A4C2×C3⋊D3113yescomplemented
62C33—413no—
63C33—813no—
64C33C22×A411yescharacteristic, complemented
65C23×C3C3×C3⋊D3113yescomplemented
66C22×D3—926no—
67C22×D3—2713no—
68C22×D3—2726no—
69C22×D3—2713no—
70C2×A4—213no—
71C2×A4—272no—
72C2×A4C3⋊(C3⋊D3)11yescharacteristic, direct product factor
73C2×C32—313no—
74C2×C32—313no—
75C2×C32—413no—
76C2×C32—852no—
77C2×C32D3×A4113yescomplemented
78C3⋊D3—326no—
79C3⋊D3—926no—
80C3×D3—3626no—
81C24—271noSylow
82C22×C3—313no—
83C22×C3—313no—
84C22×C3—1081no—
85C22×C3C2×C3×C3⋊D3113yescomplemented
86C2×D3—913no—
87C2×D3—2726no—
88C2×D3—2726no—
89C2×D3—2726no—
90C2×D3—2713no—
91C2×D3—2726no—
92A4—213no—
93A4C2×C3⋊(C3⋊D3)11yescharacteristic, direct product factor
94C32—413no—
95C32—852no—
96C32C2×D3×A4113yescomplemented
97C23—272no—
98C23—811no—
99C23—812no—
100C23—811no—
101C23C3×C3⋊(C3⋊D3)11yescharacteristic, complemented
102C2×C3—313no—
103C2×C3—313no—
104C2×C3—41no—
105C2×C3—813no—
106C2×C3—1082no—
107C2×C3A4×C3⋊D3113yescomplemented
108D3—926no—
109D3—2726no—
110C22—31no—
111C22—31no—
112C22—271no—
113C22—812no—
114C22—812no—
115C22—812no—
116C22—812no—
117C22—811no—
118C22C2×C3×C3⋊(C3⋊D3)11yescharacteristic, complemented
119C3—41no—
120C3—813no—
121C3C2×A4×C3⋊D3113yescomplemented
122C2—31no—
123C2—31no—
124C2—272no—
125C2—812no—
126C2A4×C3⋊(C3⋊D3)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2A4×C3⋊(C3⋊D3)
commutator, cosocle, hypocenterC22×C33C22×C3
FrattiniC1G
Fitting, socleC23×C33C2×C3
LCSG ▷ C22×C33
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C33×A4C2semidirect
2. A4×C3⋊(C3⋊D3)C2direct (N × H)
3. C23×C3⋊(C3⋊D3)C3semidirect
6. C33×A4C22semidirect
7. C23×C33C2×C3semidirect
11. C22×C3⋊(C3⋊D3)C2×C3semidirect
13. C2×C32×A4D3semidirect
21. C22×C33C22×C3semidirect
27. C2×C3⋊(C3⋊D3)A4direct (N × H)
30. C32×A4C2×D3semidirect
32. C23×C32C3×D3semidirect
38. C2×C3×A4C3⋊D3semidirect
44. C2×C33C2×A4semidirect
46. C3⋊(C3⋊D3)C2×A4direct (N × H)
52. C22×C32C2×C3×D3semidirect
61. C3×A4C2×C3⋊D3semidirect
64. C33C22×A4semidirect
65. C23×C3C3×C3⋊D3semidirect
72. C2×A4C3⋊(C3⋊D3)direct (N × H)
77. C2×C32D3×A4semidirect
85. C22×C3C2×C3×C3⋊D3semidirect
93. A4C2×C3⋊(C3⋊D3)direct (N × H)
96. C32C2×D3×A4semidirect
101. C23C3×C3⋊(C3⋊D3)semidirect
107. C2×C3A4×C3⋊D3semidirect
118. C22C2×C3×C3⋊(C3⋊D3)semidirect
121. C3C2×A4×C3⋊D3semidirect
126. C2A4×C3⋊(C3⋊D3)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
A4×C3⋊(C3⋊D3)2
C23×C3⋊(C3⋊D3)3
C2×A4×C3⋊D33
C2×C3×C3⋊(C3⋊D3)4