Finite Group Databasebeta

← C23×C3×C4⋊(C3⋊D3) #13181732

Subgroups — C23×C3×C4⋊(C3⋊D3) #13181732

Order 1728.

9116 conjugacy classes of subgroups; only the 2030 normal subgroups are shown.

normal subgroups only

This group has 9116 conjugacy classes of subgroups — too many to draw. Only its 2030 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⋊D3C212yesmaximal, complemented
2C23×C33×C4C211yesmaximal, characteristic, complemented
3C22×C3×C4⋊(C3⋊D3)C2128yesmaximal, direct product factor
4C23×C4⋊(C3⋊D3)C311yesmaximal, characteristic, direct product factor
5C23×C3×D12—34nomaximal
6C24×C33C2211yescharacteristic
7C23×C3×C3⋊D3—216no—
8C23×C3×C3⋊D3C22128yescomplemented
9C22×C33×C4C22114yescomplemented
10C2×C3×C4⋊(C3⋊D3)C221112yesdirect product factor
11C24×C3⋊D3C2×C312yescomplemented
12C24×C3×D3—38no—
13C23×C32×C4—24no—
14C23×C32×C4D314yescomplemented
15C23×C32×C4C2×C311yescharacteristic, complemented
16C22×C4⋊(C3⋊D3)C2×C3128yesdirect product factor
17C22×C3×D12—3112no—
18C23×C33D418yescomplemented
19C23×C33C2317yes—
20C22×C3×C3⋊D3—2112no—
21C22×C3×C3⋊D3C23156yescomplemented
22C2×C33×C4C23128yescomplemented
23C3×C4⋊(C3⋊D3)C23164yesdirect product factor
24C23×C3×D4—91no—
25C23×D12—34no—
26C24×C32—24no—
27C24×C32C2×D314yes—
28C24×C32C22×C311yescharacteristic
29C23×C3⋊D3—216no—
30C23×C3⋊D3C22×C3128yescomplemented
31C23×C3×D3—3112no—
32C23×C3×D3—664no—
33C22×C32×C4—256no—
34C22×C32×C4C2×D3156yescomplemented
35C22×C32×C4C22×C3114yescomplemented
36C2×C4⋊(C3⋊D3)C22×C31112yesdirect product factor
37C2×C3×D12—3448no—
38C22×C33C2×D4128yescomplemented
39C22×C33C2417yes—
40C2×C3×C3⋊D3—2112no—
41C2×C3×C3⋊D3C24116yescomplemented
42C33×C4C2418yescomplemented
43C25×C3—92no—
44C24×D3—38no—
45C22×C3×D4—928no—
46C23×C3×C4—24no—
47C23×C3×C4C3×D314yescomplemented
48C23×C3×C4C3⋊D311yescharacteristic, complemented
49C22×D12—3112no—
50C23×C32—232no—
51C23×C32—228no—
52C23×C32D12132yescomplemented
53C23×C32C3×D418yescomplemented
54C23×C32C22×D3128yes—
55C23×C32C23×C317yes—
56C22×C3⋊D3—2112no—
57C22×C3⋊D3C23×C3156yescomplemented
58C22×C3×D3—3224no—
59C22×C3×D3—6448no—
60C2×C32×C4—2112no—
61C2×C32×C4C22×D31112yescomplemented
62C2×C32×C4C23×C3128yescomplemented
63C4⋊(C3⋊D3)C23×C3164yesdirect product factor
64C3×D12—3256no—
65C23×D4—91noSylow
66C2×C33C22×D4114yescomplemented
67C2×C33C2511yescharacteristic
68C3×C3⋊D3—216no—
69C24×C3—24no—
70C24×C3—928no—
71C24×C3—1816no—
72C24×C3C2×C3×D314yes—
73C24×C3C2×C3⋊D311yescharacteristic
74C23×D3—3112no—
75C23×D3—664no—
76C2×C3×D4—9112no—
77C22×C3×C4—256no—
78C22×C3×C4C2×C3×D3156yescomplemented
79C22×C3×C4C2×C3⋊D3114yescomplemented
80C2×D12—3448no—
81C22×C32—228no—
82C22×C32—2112no—
83C22×C32C2×D121112yescomplemented
84C22×C32C2×C3×D4128yescomplemented
85C22×C32C23×D3128yes—
86C22×C32C24×C317yes—
87C2×C3⋊D3—2112no—
88C2×C3⋊D3C24×C3116yescomplemented
89C2×C3×D3—364no—
90C2×C3×D3—6448no—
91C32×C4—232no—
92C32×C4C23×D3132yescomplemented
93C32×C4C24×C318yescomplemented
94C25—92no—
95C22×D4—928no—
96C23×C4C3×C3⋊D311yescharacteristic, complemented
97C33C23×D411yescharacteristic, complemented, Sylow
98C23×C3—232no—
99C23×C3—228no—
100C23×C3—956no—
101C23×C3—18112no—
102C23×C3C3×D12132yescomplemented
103C23×C3C4⋊(C3⋊D3)18yesdirect product factor
104C23×C3C22×C3×D3128yes—
105C23×C3C22×C3⋊D317yes—
106C22×D3—3224no—
107C22×D3—6448no—
108C3×D4—964no—
109C2×C3×C4—2112no—
110C2×C3×C4C22×C3×D31112yescomplemented
111C2×C3×C4C22×C3⋊D3128yescomplemented
112D12—3256no—
113C2×C32—256no—
114C2×C32—24no—
115C2×C32C22×D12156yescomplemented
116C2×C32C22×C3×D4114yescomplemented
117C2×C32C24×D314yes—
118C2×C32C25×C311yescharacteristic
119C3⋊D3—216no—
120C3×D3—664no—
121C24—928no—
122C24—1816no—
123C24C2×C3×C3⋊D311yescharacteristic
124C2×D4—9112no—
125C22×C4C2×C3×C3⋊D3114yescomplemented
126C22×C3—228no—
127C22×C3—2112no—
128C22×C3—916no—
129C22×C3—18112no—
130C22×C3C2×C3×D121112yescomplemented
131C22×C3C2×C4⋊(C3⋊D3)128yesdirect product factor
132C22×C3C23×C3×D3128yes—
133C22×C3C23×C3⋊D317yes—
134C2×D3—364no—
135C2×D3—6448no—
136C3×C4—232no—
137C3×C4C23×C3×D3132yescomplemented
138C3×C4C23×C3⋊D318yescomplemented
139C32—24no—
140C32C23×D1214yescomplemented
141C32C23×C3×D411yescharacteristic, complemented
142C23—956no—
143C23—18112no—
144C23C3×C4⋊(C3⋊D3)18yesdirect product factor
145C23C22×C3×C3⋊D317yes—
146D4—964no—
147C2×C4C22×C3×C3⋊D3128yescomplemented
148C2×C3—256no—
149C2×C3—24no—
150C2×C3—1816no—
151C2×C3C22×C3×D12156yescomplemented
152C2×C3C22×C4⋊(C3⋊D3)114yesdirect product factor
153C2×C3C24×C3×D314yes—
154C2×C3C24×C3⋊D311yescharacteristic
155D3—664no—
156C22—916no—
157C22—18112no—
158C22C2×C3×C4⋊(C3⋊D3)128yesdirect product factor
159C22C23×C3×C3⋊D317yes—
160C4C23×C3×C3⋊D318yescomplemented
161C3—24no—
162C3C23×C3×D1214yescomplemented
163C3C23×C4⋊(C3⋊D3)11yescharacteristic, direct product factor
164C2—1816no—
165C2C22×C3×C4⋊(C3⋊D3)114yesdirect product factor
166C2C24×C3×C3⋊D311yescharacteristic
subgroupG/H
centerC24×C3C2×C3⋊D3
commutator, cosocleC2×C32C25×C3
FrattiniC2C24×C3×C3⋊D3
FittingC23×C33×C4C2
socleC24×C33C22
hypercenterC23×C3×C4C3⋊D3
hypocenterC32C23×C3×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C24×C3 ◁ C23×C3×C4

Complements

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

NH ≅ G/Ntype
1. C24×C3×C3⋊D3C2semidirect
2. C23×C33×C4C2semidirect
3. C22×C3×C4⋊(C3⋊D3)C2direct (N × H)
4. C23×C4⋊(C3⋊D3)C3direct (N × H)
8. C23×C3×C3⋊D3C22semidirect
9. C22×C33×C4C22semidirect
10. C2×C3×C4⋊(C3⋊D3)C22direct (N × H)
11. C24×C3⋊D3C2×C3semidirect
14. C23×C32×C4D3semidirect
15. C23×C32×C4C2×C3semidirect
16. C22×C4⋊(C3⋊D3)C2×C3direct (N × H)
18. C23×C33D4semidirect
21. C22×C3×C3⋊D3C23semidirect
22. C2×C33×C4C23semidirect
23. C3×C4⋊(C3⋊D3)C23direct (N × H)
30. C23×C3⋊D3C22×C3semidirect
34. C22×C32×C4C2×D3semidirect
35. C22×C32×C4C22×C3semidirect
36. C2×C4⋊(C3⋊D3)C22×C3direct (N × H)
38. C22×C33C2×D4semidirect
41. C2×C3×C3⋊D3C24semidirect
42. C33×C4C24semidirect
47. C23×C3×C4C3×D3semidirect
48. C23×C3×C4C3⋊D3semidirect
52. C23×C32D12semidirect
53. C23×C32C3×D4semidirect
57. C22×C3⋊D3C23×C3semidirect
61. C2×C32×C4C22×D3semidirect
62. C2×C32×C4C23×C3semidirect
63. C4⋊(C3⋊D3)C23×C3direct (N × H)
66. C2×C33C22×D4semidirect
78. C22×C3×C4C2×C3×D3semidirect
79. C22×C3×C4C2×C3⋊D3semidirect
83. C22×C32C2×D12semidirect
84. C22×C32C2×C3×D4semidirect
88. C2×C3⋊D3C24×C3semidirect
92. C32×C4C23×D3semidirect
93. C32×C4C24×C3semidirect
96. C23×C4C3×C3⋊D3semidirect
97. C33C23×D4semidirect
102. C23×C3C3×D12semidirect
103. C23×C3C4⋊(C3⋊D3)direct (N × H)
110. C2×C3×C4C22×C3×D3semidirect
111. C2×C3×C4C22×C3⋊D3semidirect
115. C2×C32C22×D12semidirect
116. C2×C32C22×C3×D4semidirect
125. C22×C4C2×C3×C3⋊D3semidirect
130. C22×C3C2×C3×D12semidirect
131. C22×C3C2×C4⋊(C3⋊D3)direct (N × H)
137. C3×C4C23×C3×D3semidirect
138. C3×C4C23×C3⋊D3semidirect
140. C32C23×D12semidirect
141. C32C23×C3×D4semidirect
144. C23C3×C4⋊(C3⋊D3)direct (N × H)
147. C2×C4C22×C3×C3⋊D3semidirect
151. C2×C3C22×C3×D12semidirect
152. C2×C3C22×C4⋊(C3⋊D3)direct (N × H)
158. C22C2×C3×C4⋊(C3⋊D3)direct (N × H)
160. C4C23×C3×C3⋊D3semidirect
162. C3C23×C3×D12semidirect
163. C3C23×C4⋊(C3⋊D3)direct (N × H)
165. C2C22×C3×C4⋊(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⋊D32
C23×C33×C42
C22×C3×C4⋊(C3⋊D3)2
C23×C4⋊(C3⋊D3)3
C23×C3×D123