Finite Group Databasebeta

← C24×C3×C3⋊Dic3 #13181748

Subgroups — C24×C3×C3⋊Dic3 #13181748

Order 1728.

11164 conjugacy classes of subgroups; only the 5102 normal subgroups are shown.

normal subgroups only

This group has 11164 conjugacy classes of subgroups — too many to draw. Only its 5102 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
1C25×C33C211yesmaximal, characteristic
2C23×C3×C3⋊Dic3C2130yesmaximal, direct product factor
3C24×C3⋊Dic3C311yesmaximal, characteristic, direct product factor
4C24×C3×Dic3—34nomaximal
5C24×C33C4116yescomplemented
6C24×C33C22115yes—
7C22×C3×C3⋊Dic3C221140yesdirect product factor
8C25×C32—24no—
9C25×C32D314yes—
10C25×C32C2×C311yescharacteristic
11C23×C3⋊Dic3C2×C3130yesdirect product factor
12C23×C3×Dic3—3120no—
13C23×C33C2×C41120yescomplemented
14C23×C33C23135yes—
15C2×C3×C3⋊Dic3C231120yesdirect product factor
16C24×C3×C4—91no—
17C24×Dic3—34no—
18C24×C32—264no—
19C24×C32—260no—
20C24×C32Dic3164yescomplemented
21C24×C32C3×C4116yescomplemented
22C24×C32C2×D3160yes—
23C24×C32C22×C3115yes—
24C22×C3⋊Dic3C22×C31140yesdirect product factor
25C22×C3×Dic3—3560no—
26C22×C33C22×C41140yescomplemented
27C22×C33C24115yes—
28C3×C3⋊Dic3C24116yesdirect product factor
29C25×C3—24no—
30C25×C3C3×D314yes—
31C25×C3C3⋊D311yescharacteristic
32C23×C3×C4—930no—
33C23×Dic3—3120no—
34C23×C32—2480no—
35C23×C32—2140no—
36C23×C32C2×Dic31480yescomplemented
37C23×C32C2×C3×C41120yescomplemented
38C23×C32C22×D31140yes—
39C23×C32C23×C3135yes—
40C2×C3⋊Dic3C23×C31120yesdirect product factor
41C2×C3×Dic3—3480no—
42C24×C4—91noSylow
43C2×C33C23×C4130yescomplemented
44C2×C33C2511yescharacteristic
45C24×C3—264no—
46C24×C3—260no—
47C24×C3C3×Dic3164yescomplemented
48C24×C3C3⋊Dic3116yesdirect product factor
49C24×C3C2×C3×D3160yes—
50C24×C3C2×C3⋊D3115yes—
51C22×C3×C4—9140no—
52C22×Dic3—3560no—
53C22×C32—260no—
54C22×C32—2560no—
55C22×C32C22×Dic31560yescomplemented
56C22×C32C22×C3×C41140yescomplemented
57C22×C32C23×D3160yes—
58C22×C32C24×C3115yes—
59C3⋊Dic3C24×C3116yesdirect product factor
60C3×Dic3—364no—
61C25C3×C3⋊D311yescharacteristic
62C23×C4—930no—
63C33C24×C411yescharacteristic, complemented, Sylow
64C23×C3—2480no—
65C23×C3—2140no—
66C23×C3C2×C3×Dic31480yescomplemented
67C23×C3C2×C3⋊Dic31120yesdirect product factor
68C23×C3C22×C3×D31140yes—
69C23×C3C22×C3⋊D3135yes—
70C2×C3×C4—9120no—
71C2×Dic3—3480no—
72C2×C32—2120no—
73C2×C32—24no—
74C2×C32C23×Dic31120yescomplemented
75C2×C32C23×C3×C4130yescomplemented
76C2×C32C24×D314yes—
77C2×C32C25×C311yescharacteristic
78C24C3×C3⋊Dic3116yesdirect product factor
79C24C2×C3×C3⋊D3115yes—
80C22×C4—9140no—
81C22×C3—260no—
82C22×C3—2560no—
83C22×C3C22×C3×Dic31560yescomplemented
84C22×C3C22×C3⋊Dic31140yesdirect product factor
85C22×C3C23×C3×D3160yes—
86C22×C3C23×C3⋊D3115yes—
87C3×C4—916no—
88Dic3—364no—
89C32—24no—
90C32C24×Dic314yescomplemented
91C32C24×C3×C411yescharacteristic, complemented
92C23C2×C3×C3⋊Dic31120yesdirect product factor
93C23C22×C3×C3⋊D3135yes—
94C2×C4—9120no—
95C2×C3—2120no—
96C2×C3—24no—
97C2×C3C23×C3×Dic31120yescomplemented
98C2×C3C23×C3⋊Dic3130yesdirect product factor
99C2×C3C24×C3×D314yes—
100C2×C3C24×C3⋊D311yescharacteristic
101C22C22×C3×C3⋊Dic31140yesdirect product factor
102C22C23×C3×C3⋊D3115yes—
103C4—916no—
104C3—24no—
105C3C24×C3×Dic314yescomplemented
106C3C24×C3⋊Dic311yescharacteristic, direct product factor
107C2C23×C3×C3⋊Dic3130yesdirect product factor
108C2C24×C3×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC25×C3C3⋊D3
commutator, hypocenterC32C24×C3×C4
FrattiniC2C24×C3×C3⋊D3
Fitting, socle, prime radicalC25×C33C2
cosocleC2×C32C25×C3
LCSG ▷ C32
UCSC1 ◁ C25×C3

Complements

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

NH ≅ G/Ntype
2. C23×C3×C3⋊Dic3C2direct (N × H)
3. C24×C3⋊Dic3C3direct (N × H)
5. C24×C33C4semidirect
7. C22×C3×C3⋊Dic3C22direct (N × H)
11. C23×C3⋊Dic3C2×C3direct (N × H)
13. C23×C33C2×C4semidirect
15. C2×C3×C3⋊Dic3C23direct (N × H)
20. C24×C32Dic3semidirect
21. C24×C32C3×C4semidirect
24. C22×C3⋊Dic3C22×C3direct (N × H)
26. C22×C33C22×C4semidirect
28. C3×C3⋊Dic3C24direct (N × H)
36. C23×C32C2×Dic3semidirect
37. C23×C32C2×C3×C4semidirect
40. C2×C3⋊Dic3C23×C3direct (N × H)
43. C2×C33C23×C4semidirect
47. C24×C3C3×Dic3semidirect
48. C24×C3C3⋊Dic3direct (N × H)
55. C22×C32C22×Dic3semidirect
56. C22×C32C22×C3×C4semidirect
59. C3⋊Dic3C24×C3direct (N × H)
63. C33C24×C4semidirect
66. C23×C3C2×C3×Dic3semidirect
67. C23×C3C2×C3⋊Dic3direct (N × H)
74. C2×C32C23×Dic3semidirect
75. C2×C32C23×C3×C4semidirect
78. C24C3×C3⋊Dic3direct (N × H)
83. C22×C3C22×C3×Dic3semidirect
84. C22×C3C22×C3⋊Dic3direct (N × H)
90. C32C24×Dic3semidirect
91. C32C24×C3×C4semidirect
92. C23C2×C3×C3⋊Dic3direct (N × H)
97. C2×C3C23×C3×Dic3semidirect
98. C2×C3C23×C3⋊Dic3direct (N × H)
101. C22C22×C3×C3⋊Dic3direct (N × H)
105. C3C24×C3×Dic3semidirect
106. C3C24×C3⋊Dic3direct (N × H)
107. C2C23×C3×C3⋊Dic3direct (N × H)

Maximal subgroups

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

subgroupindex
C25×C332
C23×C3×C3⋊Dic32
C24×C3⋊Dic33
C24×C3×Dic33