Finite Group Databasebeta

← C23×C32×C4×D3 #13181711

Subgroups — C23×C32×C4×D3 #13181711

Order 1728.

9116 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C32×D3C211yesmaximal, characteristic
2C23×C33×C4C211yesmaximal, characteristic, complemented
3C23×C32×Dic3C211yesmaximal, characteristic, complemented
4C22×C32×C4×D3C2128yesmaximal, direct product factor
5C24×C32×C4—31nomaximal
6C23×C3×C4×D3C314yesmaximal, direct product factor
7C24×C33C2211yescharacteristic
8C23×C32×D3C4116yesdirect product factor
9C23×C32×D3C22114yes—
10C22×C33×C4C22114yescomplemented
11C22×C32×Dic3C22114yescomplemented
12C2×C32×C4×D3C221112yesdirect product factor
13C25×C32—31no—
14C24×C3×D3C2×C314yes—
15C23×C32×C4—24no—
16C23×C32×C4—328no—
17C23×C32×C4—31no—
18C23×C32×C4D311yescharacteristic, direct product factor
19C23×C32×C4C2×C314yescomplemented
20C23×C3×Dic3C2×C314yescomplemented
21C22×C3×C4×D3C2×C31112yesdirect product factor
22C23×C33C2×C418yescomplemented
23C23×C33C2317yes—
24C22×C32×D3C2×C41112yesdirect product factor
25C22×C32×D3C23128yes—
26C2×C33×C4C23128yescomplemented
27C2×C32×Dic3C23128yescomplemented
28C32×C4×D3C23164yesdirect product factor
29C24×C3×C4—34no—
30C23×C4×D3C3211yescharacteristic, direct product factor
31C24×C32—24no—
32C24×C32—316no—
33C24×C32—314no—
34C24×C32C2×D311yescharacteristic
35C24×C32C22×C314yes—
36C23×C3×D3C3×C4164yesdirect product factor
37C23×C3×D3C22×C3156yes—
38C22×C32×C4—256no—
39C22×C32×C4—314no—
40C22×C32×C4—3112no—
41C22×C32×C4C2×D3114yesdirect product factor
42C22×C32×C4C22×C3156yescomplemented
43C22×C3×Dic3C22×C3156yescomplemented
44C2×C3×C4×D3C22×C31448yesdirect product factor
45C22×C33C22×C4128yescomplemented
46C22×C33C2417yes—
47C2×C32×D3C22×C41112yesdirect product factor
48C2×C32×D3C2418yes—
49C33×C4C2418yescomplemented
50C32×Dic3C2418yescomplemented
51C25×C3—34no—
52C24×D3C2×C3211yescharacteristic
53C23×C3×C4—24no—
54C23×C3×C4—34no—
55C23×C3×C4—3112no—
56C23×C3×C4C3×D314yesdirect product factor
57C23×C3×C4C2×C3211yescharacteristic, complemented
58C23×Dic3C2×C3211yescharacteristic, complemented
59C22×C4×D3C2×C32128yesdirect product factor
60C23×C32—232no—
61C23×C32—228no—
62C23×C32—3112no—
63C23×C32—328no—
64C23×C32C4×D318yesdirect product factor
65C23×C32C2×C3×C4132yescomplemented
66C23×C32C22×D317yes—
67C23×C32C23×C3128yes—
68C22×C3×D3C2×C3×C41448yesdirect product factor
69C22×C3×D3C23×C31112yes—
70C2×C32×C4—2112no—
71C2×C32×C4—328no—
72C2×C32×C4—364no—
73C2×C32×C4C22×D3128yesdirect product factor
74C2×C32×C4C23×C31112yescomplemented
75C2×C3×Dic3C23×C31112yescomplemented
76C3×C4×D3C23×C31256yesdirect product factor
77C24×C4—31noSylow
78C2×C33C23×C4114yescomplemented
79C2×C33C2511yescharacteristic
80C32×D3C23×C4116yesdirect product factor
81C24×C3—24no—
82C24×C3—364no—
83C24×C3—356no—
84C24×C3C2×C3×D314yes—
85C24×C3C22×C3211yescharacteristic
86C23×D3C32×C4116yesdirect product factor
87C23×D3C22×C32114yes—
88C22×C3×C4—256no—
89C22×C3×C4—356no—
90C22×C3×C4—3448no—
91C22×C3×C4C2×C3×D3156yesdirect product factor
92C22×C3×C4C22×C32114yescomplemented
93C22×Dic3C22×C32114yescomplemented
94C2×C4×D3C22×C321112yesdirect product factor
95C22×C32—228no—
96C22×C32—2112no—
97C22×C32—3112no—
98C22×C32—38no—
99C22×C32C2×C4×D3128yesdirect product factor
100C22×C32C22×C3×C41112yescomplemented
101C22×C32C23×D317yes—
102C22×C32C24×C3128yes—
103C2×C3×D3C22×C3×C41448yesdirect product factor
104C2×C3×D3C24×C3132yes—
105C32×C4—232no—
106C32×C4—38no—
107C32×C4C23×D318yesdirect product factor
108C32×C4C24×C3132yescomplemented
109C3×Dic3C24×C3132yescomplemented
110C25—31no—
111C23×C4—31no—
112C23×C4—328no—
113C23×C4C32×D311yescharacteristic, direct product factor
114C33C24×C411yescharacteristic, complemented, Sylow
115C23×C3—232no—
116C23×C3—228no—
117C23×C3—3112no—
118C23×C3—3448no—
119C23×C3C3×C4×D3132yesdirect product factor
120C23×C3C2×C32×C418yescomplemented
121C23×C3C22×C3×D3128yes—
122C23×C3C23×C3217yes—
123C22×D3C2×C32×C41112yesdirect product factor
124C22×D3C23×C32128yes—
125C2×C3×C4—2112no—
126C2×C3×C4—3112no—
127C2×C3×C4—3256no—
128C2×C3×C4C22×C3×D31112yesdirect product factor
129C2×C3×C4C23×C32128yescomplemented
130C2×Dic3C23×C32128yescomplemented
131C4×D3C23×C32164yesdirect product factor
132C2×C32—256no—
133C2×C32—24no—
134C2×C32—316no—
135C2×C32C22×C4×D3114yesdirect product factor
136C2×C32C23×C3×C4156yescomplemented
137C2×C32C24×D311yescharacteristic
138C2×C32C25×C314yes—
139C3×D3C23×C3×C4164yesdirect product factor
140C24—316no—
141C24—314no—
142C24C2×C32×D311yescharacteristic
143C22×C4—314no—
144C22×C4—3112no—
145C22×C4C2×C32×D3114yesdirect product factor
146C22×C3—228no—
147C22×C3—2112no—
148C22×C3—3448no—
149C22×C3—332no—
150C22×C3C2×C3×C4×D31112yesdirect product factor
151C22×C3C22×C32×C4128yescomplemented
152C22×C3C23×C3×D3128yes—
153C22×C3C24×C3217yes—
154C2×D3C22×C32×C41112yesdirect product factor
155C2×D3C24×C3218yes—
156C3×C4—232no—
157C3×C4—332no—
158C3×C4C23×C3×D3132yesdirect product factor
159C3×C4C24×C3218yescomplemented
160Dic3C24×C3218yescomplemented
161C32—24no—
162C32C23×C4×D311yescharacteristic, direct product factor
163C32C24×C3×C414yescomplemented
164C23—328no—
165C23—3112no—
166C23C32×C4×D318yesdirect product factor
167C23C22×C32×D317yes—
168C2×C4—328no—
169C2×C4—364no—
170C2×C4C22×C32×D3128yesdirect product factor
171C2×C3—256no—
172C2×C3—24no—
173C2×C3—364no—
174C2×C3C22×C3×C4×D3156yesdirect product factor
175C2×C3C23×C32×C4114yescomplemented
176C2×C3C24×C3×D314yes—
177C2×C3C25×C3211yescharacteristic
178D3C23×C32×C4116yesdirect product factor
179C22—3112no—
180C22—38no—
181C22C2×C32×C4×D3128yesdirect product factor
182C22C23×C32×D317yes—
183C4—38no—
184C4C23×C32×D318yesdirect product factor
185C3—24no—
186C3C23×C3×C4×D314yesdirect product factor
187C3C24×C32×C411yescharacteristic, complemented
188C2—316no—
189C2C22×C32×C4×D3114yesdirect product factor
190C2C24×C32×D311yescharacteristic
subgroupG/H
center, hypercenterC23×C32×C4D3
commutator, hypocenterC3C24×C32×C4
FrattiniC2C24×C32×D3
FittingC23×C33×C4C2
socleC24×C33C22
prime radicalC24×C32×D3C2
cosocleC2×C3C25×C32
LCSG ▷ C3
UCSC1 ◁ C23×C32×C4

Complements

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

NH ≅ G/Ntype
2. C23×C33×C4C2semidirect
3. C23×C32×Dic3C2semidirect
4. C22×C32×C4×D3C2direct (N × H)
6. C23×C3×C4×D3C3direct (N × H)
8. C23×C32×D3C4direct (N × H)
10. C22×C33×C4C22semidirect
11. C22×C32×Dic3C22semidirect
12. C2×C32×C4×D3C22direct (N × H)
18. C23×C32×C4D3direct (N × H)
19. C23×C32×C4C2×C3semidirect
20. C23×C3×Dic3C2×C3semidirect
21. C22×C3×C4×D3C2×C3direct (N × H)
22. C23×C33C2×C4semidirect
24. C22×C32×D3C2×C4direct (N × H)
26. C2×C33×C4C23semidirect
27. C2×C32×Dic3C23semidirect
28. C32×C4×D3C23direct (N × H)
30. C23×C4×D3C32direct (N × H)
36. C23×C3×D3C3×C4direct (N × H)
41. C22×C32×C4C2×D3direct (N × H)
42. C22×C32×C4C22×C3semidirect
43. C22×C3×Dic3C22×C3semidirect
44. C2×C3×C4×D3C22×C3direct (N × H)
45. C22×C33C22×C4semidirect
47. C2×C32×D3C22×C4direct (N × H)
49. C33×C4C24semidirect
50. C32×Dic3C24semidirect
56. C23×C3×C4C3×D3direct (N × H)
57. C23×C3×C4C2×C32semidirect
58. C23×Dic3C2×C32semidirect
59. C22×C4×D3C2×C32direct (N × H)
64. C23×C32C4×D3direct (N × H)
65. C23×C32C2×C3×C4semidirect
68. C22×C3×D3C2×C3×C4direct (N × H)
73. C2×C32×C4C22×D3direct (N × H)
74. C2×C32×C4C23×C3semidirect
75. C2×C3×Dic3C23×C3semidirect
76. C3×C4×D3C23×C3direct (N × H)
78. C2×C33C23×C4semidirect
80. C32×D3C23×C4direct (N × H)
86. C23×D3C32×C4direct (N × H)
91. C22×C3×C4C2×C3×D3direct (N × H)
92. C22×C3×C4C22×C32semidirect
93. C22×Dic3C22×C32semidirect
94. C2×C4×D3C22×C32direct (N × H)
99. C22×C32C2×C4×D3direct (N × H)
100. C22×C32C22×C3×C4semidirect
103. C2×C3×D3C22×C3×C4direct (N × H)
107. C32×C4C23×D3direct (N × H)
108. C32×C4C24×C3semidirect
109. C3×Dic3C24×C3semidirect
113. C23×C4C32×D3direct (N × H)
114. C33C24×C4semidirect
119. C23×C3C3×C4×D3direct (N × H)
120. C23×C3C2×C32×C4semidirect
123. C22×D3C2×C32×C4direct (N × H)
128. C2×C3×C4C22×C3×D3direct (N × H)
129. C2×C3×C4C23×C32semidirect
130. C2×Dic3C23×C32semidirect
131. C4×D3C23×C32direct (N × H)
135. C2×C32C22×C4×D3direct (N × H)
136. C2×C32C23×C3×C4semidirect
139. C3×D3C23×C3×C4direct (N × H)
145. C22×C4C2×C32×D3direct (N × H)
150. C22×C3C2×C3×C4×D3direct (N × H)
151. C22×C3C22×C32×C4semidirect
154. C2×D3C22×C32×C4direct (N × H)
158. C3×C4C23×C3×D3direct (N × H)
159. C3×C4C24×C32semidirect
160. Dic3C24×C32semidirect
162. C32C23×C4×D3direct (N × H)
163. C32C24×C3×C4semidirect
166. C23C32×C4×D3direct (N × H)
170. C2×C4C22×C32×D3direct (N × H)
174. C2×C3C22×C3×C4×D3direct (N × H)
175. C2×C3C23×C32×C4semidirect
178. D3C23×C32×C4direct (N × H)
181. C22C2×C32×C4×D3direct (N × H)
184. C4C23×C32×D3direct (N × H)
186. C3C23×C3×C4×D3direct (N × H)
187. C3C24×C32×C4semidirect
189. C2C22×C32×C4×D3direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C32×D32
C23×C33×C42
C23×C32×Dic32
C22×C32×C4×D32
C24×C32×C43
C23×C3×C4×D33