Finite Group Databasebeta

← C22×C3×C4×C3⋊Dic3 #13179533

Subgroups — C22×C3×C4×C3⋊Dic3 #13179533

Order 1728.

3932 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C33×C4C211yesmaximal, characteristic
2C23×C3×C3⋊Dic3C212yesmaximal
3C2×C3×C4×C3⋊Dic3C2112yesmaximal, direct product factor
4C22×C4×C3⋊Dic3C311yesmaximal, characteristic, direct product factor
5C22×C3×C4×Dic3—34nomaximal
6C24×C33C2211yescharacteristic
7C22×C33×C4C418yescomplemented
8C22×C33×C4C2216yes—
9C22×C3×C3⋊Dic3C4116yesdirect product factor
10C22×C3×C3⋊Dic3C22112yes—
11C3×C4×C3⋊Dic3C22116yesdirect product factor
12C23×C32×C4—24no—
13C23×C32×C4D314yes—
14C23×C32×C4C2×C311yescharacteristic
15C23×C3⋊Dic3C2×C312yes—
16C23×C3×Dic3—38no—
17C2×C4×C3⋊Dic3C2×C3112yesdirect product factor
18C2×C3×C4×Dic3—348no—
19C23×C33C2×C418yes—
20C23×C33C2×C414yes—
21C23×C33C2313yes—
22C2×C33×C4C2×C4124yescomplemented
23C2×C33×C4C2314yes—
24C2×C3×C3⋊Dic3C2×C4148yesdirect product factor
25C2×C3×C3⋊Dic3C2318yes—
26C22×C3×C42—91no—
27C22×C4×Dic3—34no—
28C24×C32—24no—
29C24×C32C2×D314yes—
30C24×C32C22×C311yescharacteristic
31C22×C32×C4—232no—
32C22×C32×C4—224no—
33C22×C32×C4Dic3132yescomplemented
34C22×C32×C4C3×C418yescomplemented
35C22×C32×C4C2×D3124yes—
36C22×C32×C4C22×C316yes—
37C22×C3⋊Dic3C3×C4116yesdirect product factor
38C22×C3⋊Dic3C22×C3112yes—
39C22×C3×Dic3—364no—
40C22×C3×Dic3—348no—
41C4×C3⋊Dic3C22×C3116yesdirect product factor
42C3×C4×Dic3—364no—
43C22×C33C42116yescomplemented
44C22×C33C22×C4112yes—
45C22×C33C22×C416yes—
46C22×C33C2411yescharacteristic
47C33×C4C22×C418yescomplemented
48C3×C3⋊Dic3C22×C4116yesdirect product factor
49C23×C3×C4—24no—
50C23×C3×C4—92no—
51C23×C3×C4C3×D314yes—
52C23×C3×C4C3⋊D311yescharacteristic
53C23×Dic3—38no—
54C2×C3×C42—912no—
55C2×C4×Dic3—348no—
56C23×C32—232no—
57C23×C32—216no—
58C23×C32—212no—
59C23×C32C4×D3132yes—
60C23×C32C2×Dic3116yes—
61C23×C32C2×C3×C418yes—
62C23×C32C2×C3×C414yes—
63C23×C32C22×D3112yes—
64C23×C32C23×C313yes—
65C2×C32×C4—296no—
66C2×C32×C4—216no—
67C2×C32×C4C2×Dic3196yescomplemented
68C2×C32×C4C2×C3×C4124yescomplemented
69C2×C32×C4C22×D3116yes—
70C2×C32×C4C23×C314yes—
71C2×C3⋊Dic3C2×C3×C4148yesdirect product factor
72C2×C3⋊Dic3C23×C318yes—
73C2×C3×Dic3—3192no—
74C2×C3×Dic3—332no—
75C22×C42—91noSylow
76C2×C33C2×C42112yescomplemented
77C2×C33C23×C412yes—
78C2×C33C23×C411yescharacteristic
79C24×C3—24no—
80C24×C3C2×C3×D314yes—
81C24×C3C2×C3⋊D311yescharacteristic
82C22×C3×C4—232no—
83C22×C3×C4—224no—
84C22×C3×C4—916no—
85C22×C3×C4—912no—
86C22×C3×C4C3×Dic3132yescomplemented
87C22×C3×C4C3⋊Dic318yesdirect product factor
88C22×C3×C4C2×C3×D3124yes—
89C22×C3×C4C2×C3⋊D316yes—
90C22×Dic3—364no—
91C22×Dic3—348no—
92C3×C42—916no—
93C4×Dic3—364no—
94C22×C32—24no—
95C22×C32—264no—
96C22×C32—248no—
97C22×C32—224no—
98C22×C32C4×Dic3164yescomplemented
99C22×C32C3×C42116yescomplemented
100C22×C32C2×C4×D3148yes—
101C22×C32C22×Dic3124yes—
102C22×C32C22×C3×C4112yes—
103C22×C32C22×C3×C416yes—
104C22×C32C23×D314yes—
105C22×C32C24×C311yescharacteristic
106C32×C4—232no—
107C32×C4C22×Dic3132yescomplemented
108C32×C4C22×C3×C418yescomplemented
109C3⋊Dic3C22×C3×C4116yesdirect product factor
110C3×Dic3—364no—
111C23×C4—92no—
112C23×C4C3×C3⋊D311yescharacteristic
113C2×C42—912no—
114C33C22×C4211yescharacteristic, complemented, Sylow
115C23×C3—232no—
116C23×C3—216no—
117C23×C3—212no—
118C23×C3C3×C4×D3132yes—
119C23×C3C2×C3×Dic3116yes—
120C23×C3C4×C3⋊D318yes—
121C23×C3C2×C3⋊Dic314yes—
122C23×C3C22×C3×D3112yes—
123C23×C3C22×C3⋊D313yes—
124C2×C3×C4—296no—
125C2×C3×C4—216no—
126C2×C3×C4—98no—
127C2×C3×C4—948no—
128C2×C3×C4C2×C3×Dic3196yescomplemented
129C2×C3×C4C2×C3⋊Dic3124yesdirect product factor
130C2×C3×C4C22×C3×D3116yes—
131C2×C3×C4C22×C3⋊D314yes—
132C2×Dic3—332no—
133C2×Dic3—3192no—
134C2×C32—248no—
135C2×C32—28no—
136C2×C32—24no—
137C2×C32C2×C4×Dic3148yescomplemented
138C2×C32C2×C3×C42112yescomplemented
139C2×C32C22×C4×D318yes—
140C2×C32C23×Dic314yes—
141C2×C32C23×C3×C412yes—
142C2×C32C23×C3×C411yescharacteristic
143C24C2×C3×C3⋊D311yescharacteristic
144C22×C4—916no—
145C22×C4—912no—
146C22×C4C3×C3⋊Dic318yesdirect product factor
147C22×C4C2×C3×C3⋊D316yes—
148C42—916no—
149C22×C3—264no—
150C22×C3—24no—
151C22×C3—248no—
152C22×C3—224no—
153C22×C3C3×C4×Dic3164yescomplemented
154C22×C3C4×C3⋊Dic3116yesdirect product factor
155C22×C3C2×C3×C4×D3148yes—
156C22×C3C22×C3×Dic3124yes—
157C22×C3C2×C4×C3⋊D3112yes—
158C22×C3C22×C3⋊Dic316yes—
159C22×C3C23×C3×D314yes—
160C22×C3C23×C3⋊D311yescharacteristic
161C3×C4—232no—
162C3×C4—916no—
163C3×C4C22×C3×Dic3132yescomplemented
164C3×C4C22×C3⋊Dic318yesdirect product factor
165Dic3—364no—
166C32—24no—
167C32C22×C4×Dic314yescomplemented
168C32C22×C3×C4211yescharacteristic, complemented
169C23C3×C4×C3⋊D318yes—
170C23C2×C3×C3⋊Dic314yes—
171C23C22×C3×C3⋊D313yes—
172C2×C4—98no—
173C2×C4—948no—
174C2×C4C2×C3×C3⋊Dic3124yesdirect product factor
175C2×C4C22×C3×C3⋊D314yes—
176C2×C3—248no—
177C2×C3—28no—
178C2×C3—24no—
179C2×C3C2×C3×C4×Dic3148yescomplemented
180C2×C3C2×C4×C3⋊Dic3112yesdirect product factor
181C2×C3C22×C3×C4×D318yes—
182C2×C3C23×C3×Dic314yes—
183C2×C3C22×C4×C3⋊D312yes—
184C2×C3C23×C3⋊Dic311yescharacteristic
185C22C3×C4×C3⋊Dic3116yesdirect product factor
186C22C2×C3×C4×C3⋊D3112yes—
187C22C22×C3×C3⋊Dic316yes—
188C22C23×C3×C3⋊D311yescharacteristic
189C4—916no—
190C4C22×C3×C3⋊Dic318yesdirect product factor
191C3—24no—
192C3C22×C3×C4×Dic314yescomplemented
193C3C22×C4×C3⋊Dic311yescharacteristic, direct product factor
194C2C2×C3×C4×C3⋊Dic3112yesdirect product factor
195C2C22×C3×C4×C3⋊D312yes—
196C2C23×C3×C3⋊Dic311yescharacteristic
subgroupG/H
center, hypercenterC23×C3×C4C3⋊D3
commutator, hypocenterC32C22×C3×C42
FrattiniC22C23×C3×C3⋊D3
FittingC23×C33×C4C2
socle, prime radicalC24×C33C22
cosocleC22×C32C24×C3
LCSG ▷ C32
UCSC1 ◁ C23×C3×C4

Complements

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

NH ≅ G/Ntype
3. C2×C3×C4×C3⋊Dic3C2direct (N × H)
4. C22×C4×C3⋊Dic3C3direct (N × H)
7. C22×C33×C4C4semidirect
9. C22×C3×C3⋊Dic3C4direct (N × H)
11. C3×C4×C3⋊Dic3C22direct (N × H)
17. C2×C4×C3⋊Dic3C2×C3direct (N × H)
22. C2×C33×C4C2×C4semidirect
24. C2×C3×C3⋊Dic3C2×C4direct (N × H)
33. C22×C32×C4Dic3semidirect
34. C22×C32×C4C3×C4semidirect
37. C22×C3⋊Dic3C3×C4direct (N × H)
41. C4×C3⋊Dic3C22×C3direct (N × H)
43. C22×C33C42semidirect
47. C33×C4C22×C4semidirect
48. C3×C3⋊Dic3C22×C4direct (N × H)
67. C2×C32×C4C2×Dic3semidirect
68. C2×C32×C4C2×C3×C4semidirect
71. C2×C3⋊Dic3C2×C3×C4direct (N × H)
76. C2×C33C2×C42semidirect
86. C22×C3×C4C3×Dic3semidirect
87. C22×C3×C4C3⋊Dic3direct (N × H)
98. C22×C32C4×Dic3semidirect
99. C22×C32C3×C42semidirect
107. C32×C4C22×Dic3semidirect
108. C32×C4C22×C3×C4semidirect
109. C3⋊Dic3C22×C3×C4direct (N × H)
114. C33C22×C42semidirect
128. C2×C3×C4C2×C3×Dic3semidirect
129. C2×C3×C4C2×C3⋊Dic3direct (N × H)
137. C2×C32C2×C4×Dic3semidirect
138. C2×C32C2×C3×C42semidirect
146. C22×C4C3×C3⋊Dic3direct (N × H)
153. C22×C3C3×C4×Dic3semidirect
154. C22×C3C4×C3⋊Dic3direct (N × H)
163. C3×C4C22×C3×Dic3semidirect
164. C3×C4C22×C3⋊Dic3direct (N × H)
167. C32C22×C4×Dic3semidirect
168. C32C22×C3×C42semidirect
174. C2×C4C2×C3×C3⋊Dic3direct (N × H)
179. C2×C3C2×C3×C4×Dic3semidirect
180. C2×C3C2×C4×C3⋊Dic3direct (N × H)
185. C22C3×C4×C3⋊Dic3direct (N × H)
190. C4C22×C3×C3⋊Dic3direct (N × H)
192. C3C22×C3×C4×Dic3semidirect
193. C3C22×C4×C3⋊Dic3direct (N × H)
194. C2C2×C3×C4×C3⋊Dic3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C33×C42
C23×C3×C3⋊Dic32
C2×C3×C4×C3⋊Dic32
C22×C4×C3⋊Dic33
C22×C3×C4×Dic33