Finite Group Databasebeta

← C23×C9×D12 #13164897

Subgroups — C23×C9×D12 #13164897

Order 1728.

4322 conjugacy classes of subgroups; only the 1629 normal subgroups are shown.

normal subgroups only

This group has 4322 conjugacy classes of subgroups — too many to draw. Only its 1629 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×D3×C9C212yesmaximal, complemented
2C23×C3×C4×C9C211yesmaximal, characteristic, complemented
3C22×C9×D12C2128yesmaximal, direct product factor
4C23×C3×D12C311yesmaximal, characteristic
5C23×D4×C9—31nomaximal
6C24×C3×C9C2211yescharacteristic
7C23×D3×C9—216no—
8C23×D3×C9C22128yescomplemented
9C22×C3×C4×C9C22114yescomplemented
10C2×C9×D12C221112yesdirect product factor
11C24×C3×D3C2×C312yes—
12C23×C32×C4C2×C311yescharacteristic
13C22×C3×D12C2×C3128yes—
14C25×C9—32no—
15C22×D4×C9—328no—
16C23×C4×C9—21no—
17C23×C4×C9D311yescharacteristic, complemented
18C23×C3×C9D418yescomplemented
19C23×C3×C9C2317yes—
20C22×D3×C9—2112no—
21C22×D3×C9C23156yescomplemented
22C2×C3×C4×C9C23128yescomplemented
23C9×D12C23164yesdirect product factor
24C23×C3×D4—31no—
25C23×D12C911yescharacteristic, direct product factor
26C24×C32C22×C311yescharacteristic
27C23×C3×D3—216no—
28C23×C3×D3C22×C3128yes—
29C22×C32×C4C22×C3114yes—
30C2×C3×D12C22×C31112yes—
31C24×C9—21no—
32C24×C9—328no—
33C24×C9—616no—
34C24×C9C2×D311yescharacteristic
35C2×D4×C9—3112no—
36C22×C4×C9—214no—
37C22×C4×C9C2×D3114yescomplemented
38C22×C3×C9C2×D4128yescomplemented
39C22×C3×C9C2417yes—
40C2×D3×C9—2112no—
41C2×D3×C9C24116yescomplemented
42C3×C4×C9C2418yescomplemented
43C25×C3—32no—
44C24×D3C2×C912yescomplemented
45C22×C3×D4—328no—
46C23×C3×C4—21no—
47C23×C3×C4C2×C911yescharacteristic, complemented
48C23×C3×C4C3×D311yescharacteristic
49C22×D12C2×C9128yesdirect product factor
50C23×C32C3×D418yes—
51C23×C32C23×C317yes—
52C22×C3×D3—2112no—
53C22×C3×D3C23×C3156yes—
54C2×C32×C4C23×C3128yes—
55C3×D12C23×C3164yes—
56C23×C9—28no—
57C23×C9—27no—
58C23×C9—356no—
59C23×C9—6112no—
60C23×C9D1218yesdirect product factor
61C23×C9C22×D317yes—
62D4×C9—364no—
63C2×C4×C9—228no—
64C2×C4×C9C22×D3128yescomplemented
65C23×D4—31noSylow
66C2×C3×C9C22×D4114yescomplemented
67C2×C3×C9C2511yescharacteristic
68D3×C9—216no—
69C24×C3—21no—
70C24×C3—328no—
71C24×C3—616no—
72C24×C3C22×C911yescharacteristic
73C24×C3C2×C3×D311yescharacteristic
74C23×D3—216no—
75C23×D3C22×C9128yescomplemented
76C2×C3×D4—3112no—
77C22×C3×C4—214no—
78C22×C3×C4C22×C9114yescomplemented
79C22×C3×C4C2×C3×D3114yes—
80C2×D12C22×C91112yesdirect product factor
81C22×C32C2×C3×D4128yes—
82C22×C32C24×C317yes—
83C2×C3×D3—2112no—
84C2×C3×D3C24×C3116yes—
85C32×C4C24×C318yes—
86C22×C9—27no—
87C22×C9—228no—
88C22×C9—316no—
89C22×C9—6112no—
90C22×C9C2×D12128yesdirect product factor
91C22×C9C23×D317yes—
92C4×C9—28no—
93C4×C9C23×D318yescomplemented
94C25—32no—
95C22×D4—328no—
96C23×C4D3×C911yescharacteristic, complemented
97C3×C9C23×D411yescharacteristic, complemented, Sylow
98C23×C3—28no—
99C23×C3—27no—
100C23×C3—356no—
101C23×C3—6112no—
102C23×C3D4×C918yescomplemented
103C23×C3C23×C917yes—
104C23×C3C3×D1218yes—
105C23×C3C22×C3×D317yes—
106C22×D3—2112no—
107C22×D3C23×C9156yescomplemented
108C3×D4—364no—
109C2×C3×C4—228no—
110C2×C3×C4C23×C9128yescomplemented
111C2×C3×C4C22×C3×D3128yes—
112D12C23×C9164yesdirect product factor
113C2×C32C22×C3×D4114yes—
114C2×C32C25×C311yescharacteristic
115C3×D3—216no—
116C2×C9—214no—
117C2×C9—21no—
118C2×C9—616no—
119C2×C9C22×D12114yesdirect product factor
120C2×C9C24×D311yescharacteristic
121C24—328no—
122C24—616no—
123C24C2×D3×C911yescharacteristic
124C2×D4—3112no—
125C22×C4C2×D3×C9114yescomplemented
126C22×C3—27no—
127C22×C3—228no—
128C22×C3—316no—
129C22×C3—6112no—
130C22×C3C2×D4×C9128yescomplemented
131C22×C3C24×C917yes—
132C22×C3C2×C3×D12128yes—
133C22×C3C23×C3×D317yes—
134C2×D3—2112no—
135C2×D3C24×C9116yescomplemented
136C3×C4—28no—
137C3×C4C24×C918yescomplemented
138C3×C4C23×C3×D318yes—
139C32C23×C3×D411yescharacteristic
140C9—21no—
141C9C23×D1211yescharacteristic, direct product factor
142C23—356no—
143C23—6112no—
144C23C9×D1218yesdirect product factor
145C23C22×D3×C917yes—
146D4—364no—
147C2×C4C22×D3×C9128yescomplemented
148C2×C3—21no—
149C2×C3—214no—
150C2×C3—616no—
151C2×C3C22×D4×C9114yescomplemented
152C2×C3C25×C911yescharacteristic
153C2×C3C22×C3×D12114yes—
154C2×C3C24×C3×D311yescharacteristic
155D3—216no—
156C22—316no—
157C22—6112no—
158C22C2×C9×D12128yesdirect product factor
159C22C23×D3×C917yes—
160C4C23×D3×C918yescomplemented
161C3—21no—
162C3C23×D4×C911yescharacteristic, complemented
163C3C23×C3×D1211yescharacteristic
164C2—616no—
165C2C22×C9×D12114yesdirect product factor
166C2C24×D3×C911yescharacteristic
subgroupG/H
centerC24×C9C2×D3
commutatorC2×C3C25×C9
FrattiniC2×C3C24×C3×D3
FittingC23×C3×C4×C9C2
socleC24×C32C22×C3
prime radicalC23×C3×D12C3
cosocleC2×C32C25×C3
hypercenterC23×C4×C9D3
hypocenterC3C23×D4×C9
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C24×C9 ◁ C23×C4×C9

Complements

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

NH ≅ G/Ntype
1. C24×D3×C9C2semidirect
2. C23×C3×C4×C9C2semidirect
3. C22×C9×D12C2direct (N × H)
8. C23×D3×C9C22semidirect
9. C22×C3×C4×C9C22semidirect
10. C2×C9×D12C22direct (N × H)
17. C23×C4×C9D3semidirect
18. C23×C3×C9D4semidirect
21. C22×D3×C9C23semidirect
22. C2×C3×C4×C9C23semidirect
23. C9×D12C23direct (N × H)
25. C23×D12C9direct (N × H)
37. C22×C4×C9C2×D3semidirect
38. C22×C3×C9C2×D4semidirect
41. C2×D3×C9C24semidirect
42. C3×C4×C9C24semidirect
44. C24×D3C2×C9semidirect
47. C23×C3×C4C2×C9semidirect
49. C22×D12C2×C9direct (N × H)
60. C23×C9D12direct (N × H)
64. C2×C4×C9C22×D3semidirect
66. C2×C3×C9C22×D4semidirect
75. C23×D3C22×C9semidirect
78. C22×C3×C4C22×C9semidirect
80. C2×D12C22×C9direct (N × H)
90. C22×C9C2×D12direct (N × H)
93. C4×C9C23×D3semidirect
96. C23×C4D3×C9semidirect
97. C3×C9C23×D4semidirect
102. C23×C3D4×C9semidirect
107. C22×D3C23×C9semidirect
110. C2×C3×C4C23×C9semidirect
112. D12C23×C9direct (N × H)
119. C2×C9C22×D12direct (N × H)
125. C22×C4C2×D3×C9semidirect
130. C22×C3C2×D4×C9semidirect
135. C2×D3C24×C9semidirect
137. C3×C4C24×C9semidirect
141. C9C23×D12direct (N × H)
144. C23C9×D12direct (N × H)
147. C2×C4C22×D3×C9semidirect
151. C2×C3C22×D4×C9semidirect
158. C22C2×C9×D12direct (N × H)
160. C4C23×D3×C9semidirect
162. C3C23×D4×C9semidirect
165. C2C22×C9×D12direct (N × H)

Maximal subgroups

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

subgroupindex
C24×D3×C92
C23×C3×C4×C92
C22×C9×D122
C23×C3×D123
C23×D4×C93