Finite Group Databasebeta

← C23×C3×C4×C3⋊D3 #13181731

Subgroups — C23×C3×C4×C3⋊D3 #13181731

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×C3×C3⋊D3C211yesmaximal, characteristic
2C23×C33×C4C211yesmaximal, characteristic, complemented
3C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C22×C3×C4×C3⋊D3C2128yesmaximal, direct product factor
5C23×C4×C3⋊D3C311yesmaximal, characteristic, direct product factor
6C23×C3×C4×D3—34nomaximal
7C24×C33C2211yescharacteristic
8C23×C3×C3⋊D3C4116yesdirect product factor
9C23×C3×C3⋊D3C22114yes—
10C22×C33×C4C22114yescomplemented
11C22×C3×C3⋊Dic3C22114yescomplemented
12C2×C3×C4×C3⋊D3C221112yesdirect product factor
13C24×C3⋊D3C2×C311yescharacteristic
14C24×C3×D3—34no—
15C23×C32×C4—24no—
16C23×C32×C4D314yescomplemented
17C23×C32×C4C2×C311yescharacteristic, complemented
18C23×C3⋊Dic3C2×C311yescharacteristic, complemented
19C22×C4×C3⋊D3C2×C3128yesdirect product factor
20C23×C3×Dic3—34no—
21C22×C3×C4×D3—3112no—
22C23×C33C2×C418yescomplemented
23C23×C33C2317yes—
24C22×C3×C3⋊D3C2×C41112yesdirect product factor
25C22×C3×C3⋊D3C23128yes—
26C2×C33×C4C23128yescomplemented
27C2×C3×C3⋊Dic3C23128yescomplemented
28C3×C4×C3⋊D3C23164yesdirect product factor
29C24×C3×C4—91no—
30C23×C4×D3—34no—
31C24×C32—24no—
32C24×C32C2×D314yes—
33C24×C32C22×C311yescharacteristic
34C23×C3⋊D3C3×C4116yesdirect product factor
35C23×C3⋊D3C22×C3114yes—
36C23×C3×D3—364no—
37C23×C3×D3—356no—
38C22×C32×C4—256no—
39C22×C32×C4C2×D3156yescomplemented
40C22×C32×C4C22×C3114yescomplemented
41C22×C3⋊Dic3C22×C3114yescomplemented
42C2×C4×C3⋊D3C22×C31112yesdirect product factor
43C22×C3×Dic3—356no—
44C2×C3×C4×D3—3448no—
45C22×C33C22×C4128yescomplemented
46C22×C33C2417yes—
47C2×C3×C3⋊D3C22×C41112yesdirect product factor
48C2×C3×C3⋊D3C2418yes—
49C33×C4C2418yescomplemented
50C3×C3⋊Dic3C2418yescomplemented
51C25×C3—91no—
52C24×D3—34no—
53C23×C3×C4—24no—
54C23×C3×C4—91no—
55C23×C3×C4—928no—
56C23×C3×C4C3×D314yescomplemented
57C23×C3×C4C3⋊D311yescharacteristic, direct product factor
58C23×Dic3—34no—
59C22×C4×D3—3112no—
60C23×C32—232no—
61C23×C32—228no—
62C23×C32C4×D3132yescomplemented
63C23×C32C2×C3×C418yescomplemented
64C23×C32C22×D3128yes—
65C23×C32C23×C317yes—
66C22×C3⋊D3C2×C3×C41112yesdirect product factor
67C22×C3⋊D3C23×C3128yes—
68C22×C3×D3—3448no—
69C22×C3×D3—3112no—
70C2×C32×C4—2112no—
71C2×C32×C4C22×D31112yescomplemented
72C2×C32×C4C23×C3128yescomplemented
73C2×C3⋊Dic3C23×C3128yescomplemented
74C4×C3⋊D3C23×C3164yesdirect product factor
75C2×C3×Dic3—3112no—
76C3×C4×D3—3256no—
77C24×C4—91noSylow
78C2×C33C23×C4114yescomplemented
79C2×C33C2511yescharacteristic
80C3×C3⋊D3C23×C4116yesdirect product factor
81C24×C3—24no—
82C24×C3—916no—
83C24×C3—914no—
84C24×C3C2×C3×D314yes—
85C24×C3C2×C3⋊D311yescharacteristic
86C23×D3—364no—
87C23×D3—356no—
88C22×C3×C4—256no—
89C22×C3×C4—914no—
90C22×C3×C4—9112no—
91C22×C3×C4C2×C3×D3156yescomplemented
92C22×C3×C4C2×C3⋊D3114yesdirect product factor
93C22×Dic3—356no—
94C2×C4×D3—3448no—
95C22×C32—228no—
96C22×C32—2112no—
97C22×C32C2×C4×D31112yescomplemented
98C22×C32C22×C3×C4128yescomplemented
99C22×C32C23×D3128yes—
100C22×C32C24×C317yes—
101C2×C3⋊D3C22×C3×C41112yesdirect product factor
102C2×C3⋊D3C24×C318yes—
103C2×C3×D3—3448no—
104C2×C3×D3—332no—
105C32×C4—232no—
106C32×C4C23×D3132yescomplemented
107C32×C4C24×C318yescomplemented
108C3⋊Dic3C24×C318yescomplemented
109C3×Dic3—332no—
110C25—91no—
111C23×C4—91no—
112C23×C4—928no—
113C23×C4C3×C3⋊D311yescharacteristic, direct product factor
114C33C24×C411yescharacteristic, complemented, Sylow
115C23×C3—232no—
116C23×C3—228no—
117C23×C3—928no—
118C23×C3—9112no—
119C23×C3C3×C4×D3132yescomplemented
120C23×C3C4×C3⋊D318yesdirect product factor
121C23×C3C22×C3×D3128yes—
122C23×C3C22×C3⋊D317yes—
123C22×D3—3112no—
124C22×D3—3448no—
125C2×C3×C4—2112no—
126C2×C3×C4—928no—
127C2×C3×C4—964no—
128C2×C3×C4C22×C3×D31112yescomplemented
129C2×C3×C4C22×C3⋊D3128yesdirect product factor
130C2×Dic3—3112no—
131C4×D3—3256no—
132C2×C32—256no—
133C2×C32—24no—
134C2×C32C22×C4×D3156yescomplemented
135C2×C32C23×C3×C4114yescomplemented
136C2×C32C24×D314yes—
137C2×C32C25×C311yescharacteristic
138C3⋊D3C23×C3×C4116yesdirect product factor
139C3×D3—364no—
140C24—916no—
141C24—914no—
142C24C2×C3×C3⋊D311yescharacteristic
143C22×C4—914no—
144C22×C4—9112no—
145C22×C4C2×C3×C3⋊D3114yesdirect product factor
146C22×C3—228no—
147C22×C3—2112no—
148C22×C3—9112no—
149C22×C3—98no—
150C22×C3C2×C3×C4×D31112yescomplemented
151C22×C3C2×C4×C3⋊D3128yesdirect product factor
152C22×C3C23×C3×D3128yes—
153C22×C3C23×C3⋊D317yes—
154C2×D3—3448no—
155C2×D3—332no—
156C3×C4—232no—
157C3×C4—98no—
158C3×C4C23×C3×D3132yescomplemented
159C3×C4C23×C3⋊D318yesdirect product factor
160Dic3—332no—
161C32—24no—
162C32C23×C4×D314yescomplemented
163C32C24×C3×C411yescharacteristic, complemented
164C23—928no—
165C23—9112no—
166C23C3×C4×C3⋊D318yesdirect product factor
167C23C22×C3×C3⋊D317yes—
168C2×C4—928no—
169C2×C4—964no—
170C2×C4C22×C3×C3⋊D3128yesdirect product factor
171C2×C3—256no—
172C2×C3—24no—
173C2×C3—916no—
174C2×C3C22×C3×C4×D3156yescomplemented
175C2×C3C22×C4×C3⋊D3114yesdirect product factor
176C2×C3C24×C3×D314yes—
177C2×C3C24×C3⋊D311yescharacteristic
178D3—364no—
179C22—9112no—
180C22—98no—
181C22C2×C3×C4×C3⋊D3128yesdirect product factor
182C22C23×C3×C3⋊D317yes—
183C4—98no—
184C4C23×C3×C3⋊D318yesdirect product factor
185C3—24no—
186C3C23×C3×C4×D314yescomplemented
187C3C23×C4×C3⋊D311yescharacteristic, direct product factor
188C2—916no—
189C2C22×C3×C4×C3⋊D3114yesdirect product factor
190C2C24×C3×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC23×C3×C4C3⋊D3
commutator, hypocenterC32C24×C3×C4
FrattiniC2C24×C3×C3⋊D3
FittingC23×C33×C4C2
socleC24×C33C22
prime radicalC24×C3×C3⋊D3C2
cosocleC2×C32C25×C3
LCSG ▷ C32
UCSC1 ◁ C23×C3×C4

Complements

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

NH ≅ G/Ntype
2. C23×C33×C4C2semidirect
3. C23×C3×C3⋊Dic3C2semidirect
4. C22×C3×C4×C3⋊D3C2direct (N × H)
5. C23×C4×C3⋊D3C3direct (N × H)
8. C23×C3×C3⋊D3C4direct (N × H)
10. C22×C33×C4C22semidirect
11. C22×C3×C3⋊Dic3C22semidirect
12. C2×C3×C4×C3⋊D3C22direct (N × H)
16. C23×C32×C4D3semidirect
17. C23×C32×C4C2×C3semidirect
18. C23×C3⋊Dic3C2×C3semidirect
19. C22×C4×C3⋊D3C2×C3direct (N × H)
22. C23×C33C2×C4semidirect
24. C22×C3×C3⋊D3C2×C4direct (N × H)
26. C2×C33×C4C23semidirect
27. C2×C3×C3⋊Dic3C23semidirect
28. C3×C4×C3⋊D3C23direct (N × H)
34. C23×C3⋊D3C3×C4direct (N × H)
39. C22×C32×C4C2×D3semidirect
40. C22×C32×C4C22×C3semidirect
41. C22×C3⋊Dic3C22×C3semidirect
42. C2×C4×C3⋊D3C22×C3direct (N × H)
45. C22×C33C22×C4semidirect
47. C2×C3×C3⋊D3C22×C4direct (N × H)
49. C33×C4C24semidirect
50. C3×C3⋊Dic3C24semidirect
56. C23×C3×C4C3×D3semidirect
57. C23×C3×C4C3⋊D3direct (N × H)
62. C23×C32C4×D3semidirect
63. C23×C32C2×C3×C4semidirect
66. C22×C3⋊D3C2×C3×C4direct (N × H)
71. C2×C32×C4C22×D3semidirect
72. C2×C32×C4C23×C3semidirect
73. C2×C3⋊Dic3C23×C3semidirect
74. C4×C3⋊D3C23×C3direct (N × H)
78. C2×C33C23×C4semidirect
80. C3×C3⋊D3C23×C4direct (N × H)
91. C22×C3×C4C2×C3×D3semidirect
92. C22×C3×C4C2×C3⋊D3direct (N × H)
97. C22×C32C2×C4×D3semidirect
98. C22×C32C22×C3×C4semidirect
101. C2×C3⋊D3C22×C3×C4direct (N × H)
106. C32×C4C23×D3semidirect
107. C32×C4C24×C3semidirect
108. C3⋊Dic3C24×C3semidirect
113. C23×C4C3×C3⋊D3direct (N × H)
114. C33C24×C4semidirect
119. C23×C3C3×C4×D3semidirect
120. C23×C3C4×C3⋊D3direct (N × H)
128. C2×C3×C4C22×C3×D3semidirect
129. C2×C3×C4C22×C3⋊D3direct (N × H)
134. C2×C32C22×C4×D3semidirect
135. C2×C32C23×C3×C4semidirect
138. C3⋊D3C23×C3×C4direct (N × H)
145. C22×C4C2×C3×C3⋊D3direct (N × H)
150. C22×C3C2×C3×C4×D3semidirect
151. C22×C3C2×C4×C3⋊D3direct (N × H)
158. C3×C4C23×C3×D3semidirect
159. C3×C4C23×C3⋊D3direct (N × H)
162. C32C23×C4×D3semidirect
163. C32C24×C3×C4semidirect
166. C23C3×C4×C3⋊D3direct (N × H)
170. C2×C4C22×C3×C3⋊D3direct (N × H)
174. C2×C3C22×C3×C4×D3semidirect
175. C2×C3C22×C4×C3⋊D3direct (N × H)
181. C22C2×C3×C4×C3⋊D3direct (N × H)
184. C4C23×C3×C3⋊D3direct (N × H)
186. C3C23×C3×C4×D3semidirect
187. C3C23×C4×C3⋊D3direct (N × H)
189. C2C22×C3×C4×C3⋊D3direct (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
C23×C3×C3⋊Dic32
C22×C3×C4×C3⋊D32
C23×C4×C3⋊D33
C23×C3×C4×D33