Finite Group Databasebeta

← C23×C4×C3⋊(C3⋊D3) #13181751

Subgroups — C23×C4×C3⋊(C3⋊D3) #13181751

Order 1728.

19068 conjugacy classes of subgroups; only the 3867 normal subgroups are shown.

normal subgroups only

This group has 19068 conjugacy classes of subgroups — too many to draw. Only its 3867 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
1C24×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2C23×C33×C4C211yesmaximal, characteristic, complemented
3C23×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4C22×C4×C3⋊(C3⋊D3)C2128yesmaximal, direct product factor
5C23×C4×C3⋊D3—313nomaximal
6C24×C33C2211yescharacteristic
7C23×C3⋊(C3⋊D3)C4116yesdirect product factor
8C23×C3⋊(C3⋊D3)C22114yes—
9C22×C33×C4C22114yescomplemented
10C22×C3⋊(C3⋊Dic3)C22114yescomplemented
11C2×C4×C3⋊(C3⋊D3)C221112yesdirect product factor
12C24×C3⋊D3—313no—
13C23×C32×C4D3113yescomplemented
14C23×C3⋊Dic3—313no—
15C22×C4×C3⋊D3—3364no—
16C23×C33C2×C418yescomplemented
17C23×C33C2317yes—
18C22×C3⋊(C3⋊D3)C2×C41112yesdirect product factor
19C22×C3⋊(C3⋊D3)C23128yes—
20C2×C33×C4C23128yescomplemented
21C2×C3⋊(C3⋊Dic3)C23128yescomplemented
22C4×C3⋊(C3⋊D3)C23164yesdirect product factor
23C23×C4×D3—913no—
24C24×C32C2×D3113yes—
25C23×C3⋊D3—3208no—
26C23×C3⋊D3—3182no—
27C22×C32×C4C2×D31182yescomplemented
28C22×C3⋊Dic3—3182no—
29C2×C4×C3⋊D3—31456no—
30C22×C33C22×C4128yescomplemented
31C22×C33C2417yes—
32C2×C3⋊(C3⋊D3)C22×C41112yesdirect product factor
33C2×C3⋊(C3⋊D3)C2418yes—
34C33×C4C2418yescomplemented
35C3⋊(C3⋊Dic3)C2418yescomplemented
36C24×D3—913no—
37C23×C3×C4C3⋊D3113yescomplemented
38C23×Dic3—913no—
39C22×C4×D3—9364no—
40C23×C32C4×D31104yescomplemented
41C23×C32C22×D3191yes—
42C22×C3⋊D3—31456no—
43C22×C3⋊D3—3364no—
44C2×C32×C4C22×D31364yescomplemented
45C2×C3⋊Dic3—3364no—
46C4×C3⋊D3—3832no—
47C24×C4—271noSylow
48C2×C33C23×C4114yescomplemented
49C2×C33C2511yescharacteristic
50C3⋊(C3⋊D3)C23×C4116yesdirect product factor
51C24×C3C2×C3⋊D3113yes—
52C23×D3—9208no—
53C23×D3—9182no—
54C22×C3×C4C2×C3⋊D31182yescomplemented
55C22×Dic3—9182no—
56C2×C4×D3—91456no—
57C22×C32C2×C4×D31364yescomplemented
58C22×C32C23×D3191yes—
59C2×C3⋊D3—31456no—
60C2×C3⋊D3—3104no—
61C32×C4C23×D31104yescomplemented
62C3⋊Dic3—3104no—
63C25—271no—
64C23×C4—271no—
65C23×C4—2728no—
66C23×C4C3⋊(C3⋊D3)11yescharacteristic, direct product factor
67C33C24×C411yescharacteristic, complemented, Sylow
68C23×C3C4×C3⋊D31104yescomplemented
69C23×C3C22×C3⋊D3191yes—
70C22×D3—9364no—
71C22×D3—91456no—
72C2×C3×C4C22×C3⋊D31364yescomplemented
73C2×Dic3—9364no—
74C4×D3—9832no—
75C2×C32C22×C4×D31182yescomplemented
76C2×C32C24×D3113yes—
77C3⋊D3—3208no—
78C24—2716no—
79C24—2714no—
80C24C2×C3⋊(C3⋊D3)11yescharacteristic
81C22×C4—2714no—
82C22×C4—27112no—
83C22×C4C2×C3⋊(C3⋊D3)114yesdirect product factor
84C22×C3C2×C4×C3⋊D31364yescomplemented
85C22×C3C23×C3⋊D3191yes—
86C2×D3—91456no—
87C2×D3—9104no—
88C3×C4C23×C3⋊D31104yescomplemented
89Dic3—9104no—
90C32C23×C4×D3113yescomplemented
91C23—2728no—
92C23—27112no—
93C23C4×C3⋊(C3⋊D3)18yesdirect product factor
94C23C22×C3⋊(C3⋊D3)17yes—
95C2×C4—2728no—
96C2×C4—2764no—
97C2×C4C22×C3⋊(C3⋊D3)128yesdirect product factor
98C2×C3C22×C4×C3⋊D31182yescomplemented
99C2×C3C24×C3⋊D3113yes—
100D3—9208no—
101C22—27112no—
102C22—278no—
103C22C2×C4×C3⋊(C3⋊D3)128yesdirect product factor
104C22C23×C3⋊(C3⋊D3)17yes—
105C4—278no—
106C4C23×C3⋊(C3⋊D3)18yesdirect product factor
107C3C23×C4×C3⋊D3113yescomplemented
108C2—2716no—
109C2C22×C4×C3⋊(C3⋊D3)114yesdirect product factor
110C2C24×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
center, hypercenterC23×C4C3⋊(C3⋊D3)
commutator, hypocenterC33C24×C4
FrattiniC2C24×C3⋊(C3⋊D3)
FittingC23×C33×C4C2
socleC24×C33C22
prime radicalC24×C3⋊(C3⋊D3)C2
cosocleC2×C33C25
LCSG ▷ C33
UCSC1 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C23×C33×C4C2semidirect
3. C23×C3⋊(C3⋊Dic3)C2semidirect
4. C22×C4×C3⋊(C3⋊D3)C2direct (N × H)
7. C23×C3⋊(C3⋊D3)C4direct (N × H)
9. C22×C33×C4C22semidirect
10. C22×C3⋊(C3⋊Dic3)C22semidirect
11. C2×C4×C3⋊(C3⋊D3)C22direct (N × H)
13. C23×C32×C4D3semidirect
16. C23×C33C2×C4semidirect
18. C22×C3⋊(C3⋊D3)C2×C4direct (N × H)
20. C2×C33×C4C23semidirect
21. C2×C3⋊(C3⋊Dic3)C23semidirect
22. C4×C3⋊(C3⋊D3)C23direct (N × H)
27. C22×C32×C4C2×D3semidirect
30. C22×C33C22×C4semidirect
32. C2×C3⋊(C3⋊D3)C22×C4direct (N × H)
34. C33×C4C24semidirect
35. C3⋊(C3⋊Dic3)C24semidirect
37. C23×C3×C4C3⋊D3semidirect
40. C23×C32C4×D3semidirect
44. C2×C32×C4C22×D3semidirect
48. C2×C33C23×C4semidirect
50. C3⋊(C3⋊D3)C23×C4direct (N × H)
54. C22×C3×C4C2×C3⋊D3semidirect
57. C22×C32C2×C4×D3semidirect
61. C32×C4C23×D3semidirect
66. C23×C4C3⋊(C3⋊D3)direct (N × H)
67. C33C24×C4semidirect
68. C23×C3C4×C3⋊D3semidirect
72. C2×C3×C4C22×C3⋊D3semidirect
75. C2×C32C22×C4×D3semidirect
83. C22×C4C2×C3⋊(C3⋊D3)direct (N × H)
84. C22×C3C2×C4×C3⋊D3semidirect
88. C3×C4C23×C3⋊D3semidirect
90. C32C23×C4×D3semidirect
93. C23C4×C3⋊(C3⋊D3)direct (N × H)
97. C2×C4C22×C3⋊(C3⋊D3)direct (N × H)
98. C2×C3C22×C4×C3⋊D3semidirect
103. C22C2×C4×C3⋊(C3⋊D3)direct (N × H)
106. C4C23×C3⋊(C3⋊D3)direct (N × H)
107. C3C23×C4×C3⋊D3semidirect
109. C2C22×C4×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
C24×C3⋊(C3⋊D3)2
C23×C33×C42
C23×C3⋊(C3⋊Dic3)2
C22×C4×C3⋊(C3⋊D3)2
C23×C4×C3⋊D33