Finite Group Databasebeta

← C22×C3×Dic32 #13175071

Subgroups — C22×C3×Dic32 #13175071

Order 1728.

2834 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic
2C23×C32×Dic3C212yesmaximal
3C2×C3×Dic32C2112yesmaximal, direct product factor
4C22×C3×C4×Dic3—32nomaximal
5C22×Dic32C311yesmaximal, characteristic, direct product factor
6C24×C33C2211yescharacteristic
7C22×C3×C3⋊Dic3C418yescomplemented
8C22×C3×C3⋊Dic3C2216yes—
9C22×C32×Dic3C4116yescomplemented
10C22×C32×Dic3C22112yes—
11C3×Dic32C22116yesdirect product factor
12C23×C32×C4—32no—
13C23×C3⋊Dic3C2×C311yescharacteristic
14C23×C3×Dic3—22no—
15C23×C3×Dic3—32no—
16C23×C3×Dic3—61no—
17C23×C3×Dic3D312yes—
18C23×C3×Dic3C2×C312yes—
19C2×C3×C4×Dic3—324no—
20C2×Dic32C2×C3112yesdirect product factor
21C23×C33C2×C414yes—
22C23×C33C2×C418yes—
23C23×C33C2313yes—
24C2×C3×C3⋊Dic3C2×C4124yescomplemented
25C2×C3×C3⋊Dic3C2314yes—
26C2×C32×Dic3C2×C4148yescomplemented
27C2×C32×Dic3C2318yes—
28C22×C3×C42—91no—
29C22×C4×Dic3—32no—
30C24×C32—21no—
31C24×C32—22no—
32C24×C32—41no—
33C24×C32C2×D312yes—
34C24×C32C22×C311yescharacteristic
35C22×C32×C4—316no—
36C22×C32×C4—312no—
37C22×C3⋊Dic3C3×C418yescomplemented
38C22×C3⋊Dic3C22×C316yes—
39C22×C3×Dic3—216no—
40C22×C3×Dic3—212no—
41C22×C3×Dic3—316no—
42C22×C3×Dic3—312no—
43C22×C3×Dic3—68no—
44C22×C3×Dic3—66no—
45C22×C3×Dic3Dic3116yesdirect product factor
46C22×C3×Dic3C3×C4116yescomplemented
47C22×C3×Dic3C2×D3112yes—
48C22×C3×Dic3C22×C3112yes—
49C3×C4×Dic3—332no—
50Dic32C22×C3116yesdirect product factor
51C22×C33C42116yescomplemented
52C22×C33C22×C416yes—
53C22×C33C22×C4112yes—
54C22×C33C2411yescharacteristic
55C3×C3⋊Dic3C22×C418yescomplemented
56C32×Dic3C22×C4116yescomplemented
57C23×C3×C4—32no—
58C23×C3×C4—32no—
59C23×C3×C4—62no—
60C23×C3×C4—91no—
61C23×Dic3—32no—
62C23×Dic3—61no—
63C23×Dic3C3×D312yes—
64C2×C3×C42—912no—
65C2×C4×Dic3—324no—
66C23×C32—28no—
67C23×C32—24no—
68C23×C32—28no—
69C23×C32—28no—
70C23×C32—28no—
71C23×C32—23no—
72C23×C32—26no—
73C23×C32—48no—
74C23×C32—44no—
75C23×C32—43no—
76C23×C32C4×D318yes—
77C23×C32C4×D318yes—
78C23×C32C2×Dic318yes—
79C23×C32C2×C3×C414yes—
80C23×C32C2×C3×C418yes—
81C23×C32C22×D316yes—
82C23×C32C23×C313yes—
83C2×C32×C4—348no—
84C2×C32×C4—38no—
85C2×C3⋊Dic3C2×C3×C4124yescomplemented
86C2×C3⋊Dic3C23×C314yes—
87C2×C3×Dic3—248no—
88C2×C3×Dic3—28no—
89C2×C3×Dic3—348no—
90C2×C3×Dic3—38no—
91C2×C3×Dic3—624no—
92C2×C3×Dic3—64no—
93C2×C3×Dic3C2×Dic3148yesdirect product factor
94C2×C3×Dic3C2×C3×C4148yescomplemented
95C2×C3×Dic3C22×D318yes—
96C2×C3×Dic3C23×C318yes—
97C22×C42—91noSylow
98C2×C33C2×C42112yescomplemented
99C2×C33C23×C411yescharacteristic
100C2×C33C23×C412yes—
101C24×C3—22no—
102C24×C3—21no—
103C24×C3—41no—
104C24×C3D3211yescharacteristic
105C24×C3C2×C3×D312yes—
106C22×C3×C4—316no—
107C22×C3×C4—316no—
108C22×C3×C4—312no—
109C22×C3×C4—312no—
110C22×C3×C4—616no—
111C22×C3×C4—612no—
112C22×C3×C4—98no—
113C22×C3×C4—96no—
114C22×Dic3—316no—
115C22×Dic3—312no—
116C22×Dic3—68no—
117C22×Dic3—66no—
118C22×Dic3C3×Dic3116yesdirect product factor
119C22×Dic3C2×C3×D3112yes—
120C3×C42—916no—
121C4×Dic3—332no—
122C22×C32—21no—
123C22×C32—22no—
124C22×C32—216no—
125C22×C32—232no—
126C22×C32—212no—
127C22×C32—26no—
128C22×C32—212no—
129C22×C32—212no—
130C22×C32—212no—
131C22×C32—416no—
132C22×C32—412no—
133C22×C32—46no—
134C22×C32—41no—
135C22×C32C4×Dic3132yescomplemented
136C22×C32C3×C42116yescomplemented
137C22×C32C2×C4×D3112yes—
138C22×C32C2×C4×D3112yes—
139C22×C32C22×Dic3112yes—
140C22×C32C22×C3×C416yes—
141C22×C32C22×C3×C4112yes—
142C22×C32C23×D312yes—
143C22×C32C24×C311yescharacteristic
144C32×C4—316no—
145C3⋊Dic3C22×C3×C418yescomplemented
146C3×Dic3—216no—
147C3×Dic3—316no—
148C3×Dic3—68no—
149C3×Dic3C22×Dic3116yesdirect product factor
150C3×Dic3C22×C3×C4116yescomplemented
151C23×C4—32no—
152C23×C4—91no—
153C2×C42—912no—
154C33C22×C4211yescharacteristic, complemented, Sylow
155C23×C3—28no—
156C23×C3—28no—
157C23×C3—28no—
158C23×C3—28no—
159C23×C3—24no—
160C23×C3—26no—
161C23×C3—23no—
162C23×C3—48no—
163C23×C3—44no—
164C23×C3—43no—
165C23×C3D3×Dic318yes—
166C23×C3#62414yes—
167C23×C3C3×C4×D318yes—
168C23×C3C3×C4×D318yes—
169C23×C3C2×C3×Dic318yes—
170C23×C3C2×D3213yes—
171C23×C3C22×C3×D316yes—
172C2×C3×C4—38no—
173C2×C3×C4—38no—
174C2×C3×C4—348no—
175C2×C3×C4—348no—
176C2×C3×C4—68no—
177C2×C3×C4—648no—
178C2×C3×C4—94no—
179C2×C3×C4—924no—
180C2×Dic3—38no—
181C2×Dic3—348no—
182C2×Dic3—624no—
183C2×Dic3—64no—
184C2×Dic3C2×C3×Dic3148yesdirect product factor
185C2×Dic3C22×C3×D318yes—
186C2×C32—224no—
187C2×C32—22no—
188C2×C32—21no—
189C2×C32—22no—
190C2×C32—22no—
191C2×C32—22no—
192C2×C32—212no—
193C2×C32—412no—
194C2×C32—42no—
195C2×C32—41no—
196C2×C32C2×C4×Dic3124yescomplemented
197C2×C32C2×C3×C42112yescomplemented
198C2×C32C22×C4×D312yes—
199C2×C32C22×C4×D312yes—
200C2×C32C23×Dic312yes—
201C2×C32C23×C3×C411yescharacteristic
202C2×C32C23×C3×C412yes—
203C24C3×D3211yescharacteristic
204C22×C4—316no—
205C22×C4—312no—
206C22×C4—98no—
207C22×C4—96no—
208C42—916no—
209C22×C3—21no—
210C22×C3—22no—
211C22×C3—212no—
212C22×C3—212no—
213C22×C3—212no—
214C22×C3—26no—
215C22×C3—212no—
216C22×C3—232no—
217C22×C3—216no—
218C22×C3—41no—
219C22×C3—412no—
220C22×C3—46no—
221C22×C3—416no—
222C22×C3Dic32116yesdirect product factor
223C22×C3C3×C4×Dic3132yescomplemented
224C22×C3C2×D3×Dic3112yes—
225C22×C3C2×#62416yes—
226C22×C3C2×C3×C4×D3112yes—
227C22×C3C2×C3×C4×D3112yes—
228C22×C3C22×C3×Dic3112yes—
229C22×C3C22×D3211yescharacteristic
230C22×C3C23×C3×D312yes—
231C3×C4—316no—
232C3×C4—316no—
233C3×C4—616no—
234C3×C4—98no—
235Dic3—316no—
236Dic3—68no—
237Dic3C22×C3×Dic3116yesdirect product factor
238C32—22no—
239C32—21no—
240C32—41no—
241C32C22×C4×Dic312yescomplemented
242C32C22×C3×C4211yescharacteristic, complemented
243C23C3×D3×Dic318yes—
244C23C3×#62414yes—
245C23C2×C3×D3213yes—
246C2×C4—38no—
247C2×C4—348no—
248C2×C4—94no—
249C2×C4—924no—
250C2×C3—22no—
251C2×C3—22no—
252C2×C3—22no—
253C2×C3—21no—
254C2×C3—22no—
255C2×C3—224no—
256C2×C3—212no—
257C2×C3—42no—
258C2×C3—41no—
259C2×C3—412no—
260C2×C3C2×Dic32112yesdirect product factor
261C2×C3C2×C3×C4×Dic3124yescomplemented
262C2×C3C22×D3×Dic312yes—
263C2×C3C22×#62411yescharacteristic
264C2×C3C22×C3×C4×D312yes—
265C2×C3C22×C3×C4×D312yes—
266C2×C3C23×C3×Dic312yes—
267C22C3×Dic32116yesdirect product factor
268C22C2×C3×D3×Dic3112yes—
269C22C2×C3×#62416yes—
270C22C22×C3×D3211yescharacteristic
271C4—316no—
272C4—98no—
273C3—22no—
274C3—21no—
275C3—41no—
276C3C22×Dic3211yescharacteristic, direct product factor
277C3C22×C3×C4×Dic312yescomplemented
278C2C2×C3×Dic32112yesdirect product factor
279C2C22×C3×D3×Dic312yes—
280C2C22×C3×#62411yescharacteristic
subgroupG/H
center, hypercenterC24×C3D32
commutator, hypocenterC32C22×C3×C42
FrattiniC22C22×C3×D32
Fitting, socle, prime radicalC24×C33C22
cosocleC22×C32C24×C3
LCSG ▷ C32
UCSC1 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
3. C2×C3×Dic32C2direct (N × H)
5. C22×Dic32C3direct (N × H)
7. C22×C3×C3⋊Dic3C4semidirect
9. C22×C32×Dic3C4semidirect
11. C3×Dic32C22direct (N × H)
20. C2×Dic32C2×C3direct (N × H)
24. C2×C3×C3⋊Dic3C2×C4semidirect
26. C2×C32×Dic3C2×C4semidirect
37. C22×C3⋊Dic3C3×C4semidirect
45. C22×C3×Dic3Dic3direct (N × H)
46. C22×C3×Dic3C3×C4semidirect
50. Dic32C22×C3direct (N × H)
51. C22×C33C42semidirect
55. C3×C3⋊Dic3C22×C4semidirect
56. C32×Dic3C22×C4semidirect
85. C2×C3⋊Dic3C2×C3×C4semidirect
93. C2×C3×Dic3C2×Dic3direct (N × H)
94. C2×C3×Dic3C2×C3×C4semidirect
98. C2×C33C2×C42semidirect
118. C22×Dic3C3×Dic3direct (N × H)
135. C22×C32C4×Dic3semidirect
136. C22×C32C3×C42semidirect
145. C3⋊Dic3C22×C3×C4semidirect
149. C3×Dic3C22×Dic3direct (N × H)
150. C3×Dic3C22×C3×C4semidirect
154. C33C22×C42semidirect
184. C2×Dic3C2×C3×Dic3direct (N × H)
196. C2×C32C2×C4×Dic3semidirect
197. C2×C32C2×C3×C42semidirect
222. C22×C3Dic32direct (N × H)
223. C22×C3C3×C4×Dic3semidirect
237. Dic3C22×C3×Dic3direct (N × H)
241. C32C22×C4×Dic3semidirect
242. C32C22×C3×C42semidirect
260. C2×C3C2×Dic32direct (N × H)
261. C2×C3C2×C3×C4×Dic3semidirect
267. C22C3×Dic32direct (N × H)
276. C3C22×Dic32direct (N × H)
277. C3C22×C3×C4×Dic3semidirect
278. C2C2×C3×Dic32direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C23×C32×Dic32
C2×C3×Dic322
C22×C3×C4×Dic33
C22×Dic323