Finite Group Databasebeta

← C3×C4×D3×Dic3 #11723200

Subgroups — C3×C4×D3×Dic3 #11723200

Order 864.

562 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×D3C211yesmaximal, characteristic
2C2×C3×D3×Dic3C212yesmaximal
3C3×C4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C32×C4×Dic3C211yesmaximal, characteristic, complemented
5C3×Dic32C212yesmaximal, complemented
6C2×C3×C4×Dic3—31nomaximal
7C3×C42×D3—31nomaximal
8C4×D3×Dic3C311yesmaximal, characteristic, direct product factor
9C22×C32×D3C2211yescharacteristic
10C2×C33×C4C2211yescharacteristic
11C2×C3×C3⋊Dic3C2212yes—
12C2×C32×Dic3C2212yes—
13C2×C32×Dic3C2211yescharacteristic
14C32×C4×D3C414yescomplemented
15C3×D3×Dic3C418yesdirect product factor
16C22×C32×C4—31no—
17C22×C3×Dic3—32no—
18C2×C3×C4×D3—21no—
19C2×C3×C4×D3—32no—
20C2×C3×C4×D3D311yescharacteristic
21C2×C3×C4×D3C2×C311yescharacteristic
22C2×D3×Dic3C2×C312yes—
23C32×C42—31no—
24C4×C3⋊Dic3C2×C311yescharacteristic, complemented
25C3×C4×Dic3—21no—
26C3×C4×Dic3—31no—
27C3×C4×Dic3—31no—
28C3×C4×Dic3—32no—
29C3×C4×Dic3—32no—
30C3×C4×Dic3—61no—
31C3×C4×Dic3D311yescharacteristic, direct product factor
32C3×C4×Dic3C2×C311yescharacteristic, complemented
33Dic32C2×C312yescomplemented
34C22×C33C2311yescharacteristic
35C2×C32×D3C2×C414yes—
36C2×C32×D3C2×C412yes—
37C33×C4C2×C412yescomplemented
38C3×C3⋊Dic3C2×C414yescomplemented
39C32×Dic3C2×C414yescomplemented
40C32×Dic3C2×C412yescomplemented
41C2×C3×C42—91no—
42C2×C4×Dic3—31no—
43C42×D3—31no—
44C23×C32—31no—
45C22×C3×D3—21no—
46C22×C3×D3C2×D311yescharacteristic
47C22×C3×D3C22×C311yescharacteristic
48C2×C32×C4—21no—
49C2×C32×C4—21no—
50C2×C32×C4—21no—
51C2×C32×C4—31no—
52C2×C32×C4—32no—
53C2×C32×C4—34no—
54C2×C32×C4—41no—
55C2×C32×C4C2×D311yescharacteristic
56C2×C32×C4C2×D311yescharacteristic
57C2×C32×C4C22×C311yescharacteristic
58C2×C3⋊Dic3C22×C312yes—
59C2×C3×Dic3—22no—
60C2×C3×Dic3—21no—
61C2×C3×Dic3—32no—
62C2×C3×Dic3—32no—
63C2×C3×Dic3—38no—
64C2×C3×Dic3—62no—
65C2×C3×Dic3C2×D312yes—
66C2×C3×Dic3C2×D311yescharacteristic
67C2×C3×Dic3C22×C311yescharacteristic
68C2×C3×Dic3C22×C312yes—
69C3×C4×D3—24no—
70C3×C4×D3—38no—
71C3×C4×D3Dic314yesdirect product factor
72C3×C4×D3C3×C414yescomplemented
73D3×Dic3C3×C418yesdirect product factor
74C2×C33C22×C412yes—
75C2×C33C22×C411yescharacteristic
76C32×D3C4214yescomplemented
77C22×C3×C4—31no—
78C22×C3×C4—31no—
79C22×C3×C4—61no—
80C22×C3×C4—92no—
81C22×Dic3—32no—
82C2×C4×D3—32no—
83C2×C4×D3C3×D311yescharacteristic
84C3×C42—31no—
85C3×C42—31no—
86C3×C42—61no—
87C3×C42—92no—
88C3×C42—91no—
89C4×Dic3—31no—
90C4×Dic3—31no—
91C4×Dic3—32no—
92C4×Dic3—32no—
93C4×Dic3—61no—
94C4×Dic3C3×D311yescharacteristic, direct product factor
95C22×C32—21no—
96C22×C32—21no—
97C22×C32—21no—
98C22×C32—34no—
99C22×C32—32no—
100C22×C32—41no—
101C22×C32C22×D311yescharacteristic
102C22×C32C22×D311yescharacteristic
103C22×C32C23×C311yescharacteristic
104C2×C3×D3—24no—
105C2×C3×D3—22no—
106C2×C3×D3C4×D314yes—
107C2×C3×D3C2×Dic312yes—
108C2×C3×D3C2×C3×C414yes—
109C2×C3×D3C2×C3×C412yes—
110C32×C4—22no—
111C32×C4—22no—
112C32×C4—22no—
113C32×C4—34no—
114C32×C4—32no—
115C32×C4—42no—
116C32×C4C4×D312yescomplemented
117C32×C4C2×Dic312yescomplemented
118C32×C4C2×C3×C412yescomplemented
119C3⋊Dic3C2×C3×C414yescomplemented
120C3×Dic3—24no—
121C3×Dic3—22no—
122C3×Dic3—34no—
123C3×Dic3—34no—
124C3×Dic3—64no—
125C3×Dic3C4×D314yesdirect product factor
126C3×Dic3C2×Dic312yescomplemented
127C3×Dic3C2×C3×C414yescomplemented
128C3×Dic3C2×C3×C412yescomplemented
129C2×C42—91noSylow
130C33C2×C4211yescharacteristic, complemented, Sylow
131C23×C3—31no—
132C23×C3—31no—
133C23×C3—61no—
134C22×D3C2×C3×D311yescharacteristic
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—32no—
139C2×C3×C4—31no—
140C2×C3×C4—31no—
141C2×C3×C4—32no—
142C2×C3×C4—34no—
143C2×C3×C4—34no—
144C2×C3×C4—41no—
145C2×C3×C4—64no—
146C2×C3×C4—62no—
147C2×C3×C4—61no—
148C2×C3×C4—98no—
149C2×C3×C4—92no—
150C2×C3×C4D3211yescharacteristic
151C2×C3×C4C2×C3×D311yescharacteristic
152C2×C3×C4C2×C3×D311yescharacteristic
153C2×Dic3—32no—
154C2×Dic3—32no—
155C2×Dic3—38no—
156C2×Dic3—62no—
157C2×Dic3C2×C3×D312yes—
158C2×Dic3C2×C3×D311yescharacteristic
159C4×D3—38no—
160C4×D3C3×Dic314yesdirect product factor
161C2×C32—22no—
162C2×C32—21no—
163C2×C32—22no—
164C2×C32—21no—
165C2×C32—22no—
166C2×C32—21no—
167C2×C32—34no—
168C2×C32—41no—
169C2×C32—42no—
170C2×C32C2×C4×D312yes—
171C2×C32C2×C4×D312yes—
172C2×C32C2×C4×D311yescharacteristic
173C2×C32C22×Dic311yescharacteristic
174C2×C32C22×C3×C412yes—
175C2×C32C22×C3×C411yescharacteristic
176C3×D3—24no—
177C3×D3C4×Dic314yesdirect product factor
178C3×D3C3×C4214yescomplemented
179C22×C4—31no—
180C22×C4—92no—
181C42—31no—
182C42—92no—
183C42—91no—
184C22×C3—21no—
185C22×C3—21no—
186C22×C3—21no—
187C22×C3—34no—
188C22×C3—32no—
189C22×C3—34no—
190C22×C3—32no—
191C22×C3—41no—
192C22×C3—62no—
193C22×C3—64no—
194C22×C3C2×D3211yescharacteristic
195C22×C3C22×C3×D311yescharacteristic
196C22×C3C22×C3×D311yescharacteristic
197C2×D3C3×C4×D314yes—
198C2×D3C2×C3×Dic312yes—
199C3×C4—22no—
200C3×C4—22no—
201C3×C4—22no—
202C3×C4—34no—
203C3×C4—32no—
204C3×C4—34no—
205C3×C4—32no—
206C3×C4—42no—
207C3×C4—62no—
208C3×C4—64no—
209C3×C4—94no—
210C3×C4D3×Dic312yesdirect product factor
211C3×C4C3×C4×D312yescomplemented
212C3×C4C2×C3×Dic312yescomplemented
213Dic3—34no—
214Dic3—34no—
215Dic3—64no—
216Dic3C3×C4×D314yesdirect product factor
217Dic3C2×C3×Dic312yescomplemented
218C32—21no—
219C32—21no—
220C32—21no—
221C32—41no—
222C32C42×D311yescharacteristic, complemented
223C32C2×C4×Dic311yescharacteristic, complemented
224C32C2×C3×C4211yescharacteristic, complemented
225C23—31no—
226C2×C4—32no—
227C2×C4—31no—
228C2×C4—34no—
229C2×C4—98no—
230C2×C4—92no—
231C2×C4C3×D3211yescharacteristic
232C2×C3—22no—
233C2×C3—22no—
234C2×C3—22no—
235C2×C3—21no—
236C2×C3—21no—
237C2×C3—21no—
238C2×C3—34no—
239C2×C3—34no—
240C2×C3—42no—
241C2×C3—41no—
242C2×C3—64no—
243C2×C3C4×D3212yes—
244C2×C3C2×D3×Dic311yescharacteristic
245C2×C3C2×C3×C4×D312yes—
246C2×C3C2×C3×C4×D312yes—
247C2×C3C2×C3×C4×D311yescharacteristic
248C2×C3C22×C3×Dic311yescharacteristic
249D3C3×C4×Dic314yesdirect product factor
250C22—34no—
251C22—32no—
252C22C2×C3×D3211yescharacteristic
253C4—34no—
254C4—32no—
255C4—94no—
256C4C3×D3×Dic312yesdirect product factor
257C3—21no—
258C3—21no—
259C3—21no—
260C3—41no—
261C3C4×D3×Dic311yescharacteristic, direct product factor
262C3C3×C42×D311yescharacteristic, complemented
263C3C2×C3×C4×Dic311yescharacteristic, complemented
264C2—34no—
265C2C3×C4×D3212yes—
266C2C2×C3×D3×Dic311yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4D32
commutator, hypocenterC32C2×C3×C42
FrattiniC22C2×C3×D32
FittingC2×C33×C4C22
socleC22×C33C23
prime radicalC22×C32×D3C22
cosocleC22×C32C23×C3
LCSG ▷ C32
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×C4×C3⋊Dic3C2semidirect
4. C32×C4×Dic3C2semidirect
5. C3×Dic32C2semidirect
8. C4×D3×Dic3C3direct (N × H)
14. C32×C4×D3C4semidirect
15. C3×D3×Dic3C4direct (N × H)
24. C4×C3⋊Dic3C2×C3semidirect
31. C3×C4×Dic3D3direct (N × H)
32. C3×C4×Dic3C2×C3semidirect
33. Dic32C2×C3semidirect
37. C33×C4C2×C4semidirect
38. C3×C3⋊Dic3C2×C4semidirect
39. C32×Dic3C2×C4semidirect
40. C32×Dic3C2×C4semidirect
71. C3×C4×D3Dic3direct (N × H)
72. C3×C4×D3C3×C4semidirect
73. D3×Dic3C3×C4direct (N × H)
76. C32×D3C42semidirect
94. C4×Dic3C3×D3direct (N × H)
116. C32×C4C4×D3semidirect
117. C32×C4C2×Dic3semidirect
118. C32×C4C2×C3×C4semidirect
119. C3⋊Dic3C2×C3×C4semidirect
125. C3×Dic3C4×D3direct (N × H)
126. C3×Dic3C2×Dic3semidirect
127. C3×Dic3C2×C3×C4semidirect
128. C3×Dic3C2×C3×C4semidirect
130. C33C2×C42semidirect
160. C4×D3C3×Dic3direct (N × H)
177. C3×D3C4×Dic3direct (N × H)
178. C3×D3C3×C42semidirect
210. C3×C4D3×Dic3direct (N × H)
211. C3×C4C3×C4×D3semidirect
212. C3×C4C2×C3×Dic3semidirect
216. Dic3C3×C4×D3direct (N × H)
217. Dic3C2×C3×Dic3semidirect
222. C32C42×D3semidirect
223. C32C2×C4×Dic3semidirect
224. C32C2×C3×C42semidirect
249. D3C3×C4×Dic3direct (N × H)
256. C4C3×D3×Dic3direct (N × H)
261. C3C4×D3×Dic3direct (N × H)
262. C3C3×C42×D3semidirect
263. C3C2×C3×C4×Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×D32
C2×C3×D3×Dic32
C3×C4×C3⋊Dic32
C32×C4×Dic32
C3×Dic322
C2×C3×C4×Dic33
C3×C42×D33
C4×D3×Dic33