Finite Group Databasebeta

← C23×C4⋊(C3⋊D9) #13164973

Subgroups — C23×C4⋊(C3⋊D9) #13164973

Order 1728.

6810 conjugacy classes of subgroups; only the 1487 normal subgroups are shown.

normal subgroups only

This group has 6810 conjugacy classes of subgroups — too many to draw. Only its 1487 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⋊D9C212yesmaximal, complemented
2C23×C3×C4×C9C211yesmaximal, characteristic, complemented
3C22×C4⋊(C3⋊D9)C2128yesmaximal, direct product factor
4C23×C4⋊(C3⋊D3)—31nomaximal
5C23×D36—33nomaximal
6C24×C3×C9C2211yescharacteristic
7C23×C3⋊D9—216no—
8C23×C3⋊D9C22128yescomplemented
9C22×C3×C4×C9C22114yescomplemented
10C2×C4⋊(C3⋊D9)C221112yesdirect product factor
11C24×C3⋊D3—32no—
12C23×C32×C4D311yescharacteristic
13C22×C4⋊(C3⋊D3)—328no—
14C24×D9—36no—
15C23×C4×C9D313yescomplemented
16C22×D36—384no—
17C23×C3×C9D418yescomplemented
18C23×C3×C9C2317yes—
19C22×C3⋊D9—2112no—
20C22×C3⋊D9C23156yescomplemented
21C2×C3×C4×C9C23128yescomplemented
22C4⋊(C3⋊D9)C23164yesdirect product factor
23C23×D12—93no—
24C23×D12—91no—
25C24×C32C2×D311yescharacteristic
26C23×C3⋊D3—328no—
27C23×C3⋊D3—616no—
28C22×C32×C4C2×D3114yes—
29C2×C4⋊(C3⋊D3)—3112no—
30C24×C9C2×D313yes—
31C23×D9—384no—
32C23×D9—648no—
33C22×C4×C9C2×D3142yescomplemented
34C2×D36—3336no—
35C22×C3×C9C2×D4128yescomplemented
36C22×C3×C9C2417yes—
37C2×C3⋊D9—2112no—
38C2×C3⋊D9C24116yescomplemented
39C3×C4×C9C2418yescomplemented
40C24×D3—96no—
41C24×D3—92no—
42C23×C3×C4D913yescomplemented
43C23×C3×C4C3⋊D311yescharacteristic
44C22×D12—984no—
45C22×D12—928no—
46C23×C32D1218yes—
47C23×C32C22×D317yes—
48C22×C3⋊D3—356no—
49C22×C3⋊D3—6112no—
50C2×C32×C4C22×D3128yes—
51C4⋊(C3⋊D3)—364no—
52C23×C9D12124yescomplemented
53C23×C9C22×D3121yes—
54C22×D9—3168no—
55C22×D9—6336no—
56C2×C4×C9C22×D3184yescomplemented
57D36—3192no—
58C23×D4—271noSylow
59C2×C3×C9C22×D4114yescomplemented
60C2×C3×C9C2511yescharacteristic
61C3⋊D9—216no—
62C24×C3C2×D913yes—
63C24×C3C2×C3⋊D311yescharacteristic
64C23×D3—984no—
65C23×D3—928no—
66C23×D3—1848no—
67C23×D3—1816no—
68C22×C3×C4C2×D9142yescomplemented
69C22×C3×C4C2×C3⋊D3114yes—
70C2×D12—9336no—
71C2×D12—9112no—
72C22×C32C2×D12128yes—
73C22×C32C23×D317yes—
74C2×C3⋊D3—316no—
75C2×C3⋊D3—6112no—
76C32×C4C23×D318yes—
77C22×C9C2×D12184yescomplemented
78C22×C9C23×D3121yes—
79C2×D9—348no—
80C2×D9—6336no—
81C4×C9C23×D3124yescomplemented
82C25—272no—
83C22×D4—2728no—
84C23×C4C3⋊D911yescharacteristic, complemented
85C3×C9C23×D411yescharacteristic, complemented, Sylow
86C23×C3D36124yescomplemented
87C23×C3C22×D9121yes—
88C23×C3C4⋊(C3⋊D3)18yes—
89C23×C3C22×C3⋊D317yes—
90C22×D3—956no—
91C22×D3—9168no—
92C22×D3—18336no—
93C22×D3—18112no—
94C2×C3×C4C22×D9184yescomplemented
95C2×C3×C4C22×C3⋊D3128yes—
96D12—964no—
97D12—9192no—
98C2×C32C22×D12114yes—
99C2×C32C24×D311yescharacteristic
100C3⋊D3—616no—
101C2×C9C22×D12142yescomplemented
102C2×C9C24×D313yes—
103D9—648no—
104C24—2728no—
105C24—5416no—
106C24C2×C3⋊D911yescharacteristic
107C2×D4—27112no—
108C22×C4C2×C3⋊D9114yescomplemented
109C22×C3C2×D36184yescomplemented
110C22×C3C23×D9121yes—
111C22×C3C2×C4⋊(C3⋊D3)128yes—
112C22×C3C23×C3⋊D317yes—
113C2×D3—916no—
114C2×D3—948no—
115C2×D3—18112no—
116C2×D3—18336no—
117C3×C4C23×D9124yescomplemented
118C3×C4C23×C3⋊D318yes—
119C32C23×D1211yescharacteristic
120C9C23×D1213yescomplemented
121C23—2756no—
122C23—54112no—
123C23C4⋊(C3⋊D9)18yesdirect product factor
124C23C22×C3⋊D917yes—
125D4—2764no—
126C2×C4C22×C3⋊D9128yescomplemented
127C2×C3C22×D36142yescomplemented
128C2×C3C24×D913yes—
129C2×C3C22×C4⋊(C3⋊D3)114yes—
130C2×C3C24×C3⋊D311yescharacteristic
131D3—1848no—
132D3—1816no—
133C22—2716no—
134C22—54112no—
135C22C2×C4⋊(C3⋊D9)128yesdirect product factor
136C22C23×C3⋊D917yes—
137C4C23×C3⋊D918yescomplemented
138C3C23×D3613yescomplemented
139C3C23×C4⋊(C3⋊D3)11yescharacteristic
140C2—5416no—
141C2C22×C4⋊(C3⋊D9)114yesdirect product factor
142C2C24×C3⋊D911yescharacteristic
subgroupG/H
centerC24C2×C3⋊D9
commutator, cosocleC2×C3×C9C25
FrattiniC2×C3C24×C3⋊D3
FittingC23×C3×C4×C9C2
socleC24×C32C2×D3
hypercenterC23×C4C3⋊D9
hypocenterC3×C9C23×D4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C24×C3⋊D9C2semidirect
2. C23×C3×C4×C9C2semidirect
3. C22×C4⋊(C3⋊D9)C2direct (N × H)
8. C23×C3⋊D9C22semidirect
9. C22×C3×C4×C9C22semidirect
10. C2×C4⋊(C3⋊D9)C22direct (N × H)
15. C23×C4×C9D3semidirect
17. C23×C3×C9D4semidirect
20. C22×C3⋊D9C23semidirect
21. C2×C3×C4×C9C23semidirect
22. C4⋊(C3⋊D9)C23direct (N × H)
33. C22×C4×C9C2×D3semidirect
35. C22×C3×C9C2×D4semidirect
38. C2×C3⋊D9C24semidirect
39. C3×C4×C9C24semidirect
42. C23×C3×C4D9semidirect
52. C23×C9D12semidirect
56. C2×C4×C9C22×D3semidirect
59. C2×C3×C9C22×D4semidirect
68. C22×C3×C4C2×D9semidirect
77. C22×C9C2×D12semidirect
81. C4×C9C23×D3semidirect
84. C23×C4C3⋊D9semidirect
85. C3×C9C23×D4semidirect
86. C23×C3D36semidirect
94. C2×C3×C4C22×D9semidirect
101. C2×C9C22×D12semidirect
108. C22×C4C2×C3⋊D9semidirect
109. C22×C3C2×D36semidirect
117. C3×C4C23×D9semidirect
120. C9C23×D12semidirect
123. C23C4⋊(C3⋊D9)direct (N × H)
126. C2×C4C22×C3⋊D9semidirect
127. C2×C3C22×D36semidirect
135. C22C2×C4⋊(C3⋊D9)direct (N × H)
137. C4C23×C3⋊D9semidirect
138. C3C23×D36semidirect
141. C2C22×C4⋊(C3⋊D9)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⋊D92
C23×C3×C4×C92
C22×C4⋊(C3⋊D9)2
C23×C4⋊(C3⋊D3)3
C23×D363