Finite Group Databasebeta

← C3×C4×Dic32 #13158172

Subgroups — C3×C4×Dic32 #13158172

Order 1728.

1410 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
1C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C32×C4×Dic3C212yesmaximal
3C2×C3×Dic32C214yesmaximal
4C3×C42×Dic3—32nomaximal
5C4×Dic32C311yesmaximal, characteristic, direct product factor
6C22×C33×C4C2211yescharacteristic
7C22×C3×C3⋊Dic3C2212yes—
8C22×C32×Dic3C2214yes—
9C3×C4×C3⋊Dic3C414yescomplemented
10C32×C4×Dic3C418yescomplemented
11C3×Dic32C4116yesdirect product factor
12C2×C32×C42—32no—
13C2×C4×C3⋊Dic3C2×C311yescharacteristic
14C2×C3×C4×Dic3—22no—
15C2×C3×C4×Dic3—32no—
16C2×C3×C4×Dic3—38no—
17C2×C3×C4×Dic3—61no—
18C2×C3×C4×Dic3D312yes—
19C2×C3×C4×Dic3C2×C312yes—
20C2×Dic32C2×C314yes—
21C23×C33C2311yescharacteristic
22C2×C33×C4C2×C412yes—
23C2×C33×C4C2×C414yes—
24C2×C3×C3⋊Dic3C2×C414yes—
25C2×C3×C3⋊Dic3C2×C418yes—
26C2×C32×Dic3C2×C4116yes—
27C2×C32×Dic3C2×C418yes—
28C3×C43—91no—
29C42×Dic3—32no—
30C22×C32×C4—21no—
31C22×C32×C4—22no—
32C22×C32×C4—34no—
33C22×C32×C4—41no—
34C22×C32×C4C2×D312yes—
35C22×C32×C4C22×C311yescharacteristic
36C22×C3⋊Dic3C22×C312yes—
37C22×C3×Dic3—24no—
38C22×C3×Dic3—34no—
39C22×C3×Dic3—62no—
40C22×C3×Dic3C2×D314yes—
41C22×C3×Dic3C22×C314yes—
42C32×C42—38no—
43C4×C3⋊Dic3C3×C414yescomplemented
44C3×C4×Dic3—28no—
45C3×C4×Dic3—38no—
46C3×C4×Dic3—332no—
47C3×C4×Dic3—64no—
48C3×C4×Dic3Dic318yesdirect product factor
49C3×C4×Dic3C3×C418yescomplemented
50Dic32C3×C4116yesdirect product factor
51C22×C33C22×C414yes—
52C22×C33C22×C411yescharacteristic
53C22×C33C22×C412yes—
54C33×C4C4214yescomplemented
55C3×C3⋊Dic3C4218yescomplemented
56C32×Dic3C42116yescomplemented
57C2×C3×C42—32no—
58C2×C3×C42—32no—
59C2×C3×C42—62no—
60C2×C3×C42—94no—
61C2×C3×C42—91no—
62C2×C4×Dic3—32no—
63C2×C4×Dic3—38no—
64C2×C4×Dic3—61no—
65C2×C4×Dic3C3×D312yes—
66C23×C32—21no—
67C23×C32—22no—
68C23×C32—41no—
69C23×C32C22×D312yes—
70C23×C32C23×C311yescharacteristic
71C2×C32×C4—24no—
72C2×C32×C4—22no—
73C2×C32×C4—24no—
74C2×C32×C4—24no—
75C2×C32×C4—24no—
76C2×C32×C4—316no—
77C2×C32×C4—38no—
78C2×C32×C4—44no—
79C2×C32×C4—42no—
80C2×C32×C4C4×D314yes—
81C2×C32×C4C4×D314yes—
82C2×C32×C4C2×Dic314yes—
83C2×C32×C4C2×C3×C412yes—
84C2×C32×C4C2×C3×C414yes—
85C2×C3⋊Dic3C2×C3×C418yes—
86C2×C3⋊Dic3C2×C3×C414yes—
87C2×C3×Dic3—216no—
88C2×C3×Dic3—28no—
89C2×C3×Dic3—316no—
90C2×C3×Dic3—38no—
91C2×C3×Dic3—64no—
92C2×C3×Dic3—68no—
93C2×C3×Dic3C4×D3116yes—
94C2×C3×Dic3C2×Dic318yes—
95C2×C3×Dic3C2×C3×C4116yes—
96C2×C3×Dic3C2×C3×C418yes—
97C43—91noSylow
98C2×C33C2×C4212yes—
99C2×C33C2×C4214yes—
100C2×C33C2×C4211yescharacteristic
101C22×C3×C4—21no—
102C22×C3×C4—22no—
103C22×C3×C4—34no—
104C22×C3×C4—34no—
105C22×C3×C4—41no—
106C22×C3×C4—64no—
107C22×C3×C4—92no—
108C22×C3×C4D3211yescharacteristic
109C22×C3×C4C2×C3×D312yes—
110C22×Dic3—34no—
111C22×Dic3—62no—
112C22×Dic3C2×C3×D314yes—
113C3×C42—38no—
114C3×C42—38no—
115C3×C42—68no—
116C3×C42—916no—
117C3×C42—94no—
118C4×Dic3—38no—
119C4×Dic3—332no—
120C4×Dic3—64no—
121C4×Dic3C3×Dic318yesdirect product factor
122C22×C32—24no—
123C22×C32—22no—
124C22×C32—21no—
125C22×C32—28no—
126C22×C32—22no—
127C22×C32—22no—
128C22×C32—22no—
129C22×C32—42no—
130C22×C32—41no—
131C22×C32—44no—
132C22×C32C2×C4×D318yes—
133C22×C32C2×C4×D312yes—
134C22×C32C2×C4×D312yes—
135C22×C32C22×Dic312yes—
136C22×C32C22×C3×C414yes—
137C22×C32C22×C3×C411yescharacteristic
138C22×C32C22×C3×C412yes—
139C32×C4—24no—
140C32×C4—28no—
141C32×C4—316no—
142C32×C4—44no—
143C32×C4C4×Dic318yescomplemented
144C32×C4C3×C4214yescomplemented
145C3⋊Dic3C3×C4218yescomplemented
146C3×Dic3—216no—
147C3×Dic3—316no—
148C3×Dic3—68no—
149C3×Dic3C4×Dic3116yesdirect product factor
150C3×Dic3C3×C42116yescomplemented
151C2×C42—32no—
152C2×C42—94no—
153C2×C42—91no—
154C33C4311yescharacteristic, complemented, Sylow
155C23×C3—22no—
156C23×C3—21no—
157C23×C3—41no—
158C23×C3C2×D3211yescharacteristic
159C23×C3C22×C3×D312yes—
160C2×C3×C4—24no—
161C2×C3×C4—24no—
162C2×C3×C4—24no—
163C2×C3×C4—24no—
164C2×C3×C4—22no—
165C2×C3×C4—316no—
166C2×C3×C4—38no—
167C2×C3×C4—316no—
168C2×C3×C4—38no—
169C2×C3×C4—44no—
170C2×C3×C4—42no—
171C2×C3×C4—68no—
172C2×C3×C4—616no—
173C2×C3×C4—98no—
174C2×C3×C4—94no—
175C2×C3×C4D3×Dic314yes—
176C2×C3×C4#62412yes—
177C2×C3×C4C3×C4×D314yes—
178C2×C3×C4C3×C4×D314yes—
179C2×C3×C4C2×C3×Dic314yes—
180C2×Dic3—316no—
181C2×Dic3—38no—
182C2×Dic3—64no—
183C2×Dic3—68no—
184C2×Dic3C3×C4×D3116yes—
185C2×Dic3C2×C3×Dic318yes—
186C2×C32—24no—
187C2×C32—22no—
188C2×C32—24no—
189C2×C32—22no—
190C2×C32—21no—
191C2×C32—24no—
192C2×C32—24no—
193C2×C32—41no—
194C2×C32—44no—
195C2×C32—42no—
196C2×C32C42×D314yes—
197C2×C32C42×D314yes—
198C2×C32C2×C4×Dic314yes—
199C2×C32C2×C4×Dic312yes—
200C2×C32C2×C3×C4211yescharacteristic
201C2×C32C2×C3×C4214yes—
202C2×C32C2×C3×C4212yes—
203C22×C4—34no—
204C22×C4—92no—
205C22×C4C3×D3211yescharacteristic
206C42—38no—
207C42—916no—
208C42—94no—
209C22×C3—24no—
210C22×C3—28no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—21no—
215C22×C3—22no—
216C22×C3—44no—
217C22×C3—42no—
218C22×C3—41no—
219C22×C3C4×D3214yes—
220C22×C3C2×D3×Dic312yes—
221C22×C3C2×#62411yescharacteristic
222C22×C3C2×C3×C4×D318yes—
223C22×C3C2×C3×C4×D312yes—
224C22×C3C2×C3×C4×D312yes—
225C22×C3C22×C3×Dic312yes—
226C3×C4—28no—
227C3×C4—24no—
228C3×C4—316no—
229C3×C4—316no—
230C3×C4—44no—
231C3×C4—616no—
232C3×C4—98no—
233C3×C4Dic3214yesdirect product factor
234C3×C4C3×C4×Dic318yescomplemented
235Dic3—316no—
236Dic3—68no—
237Dic3C3×C4×Dic3116yesdirect product factor
238C32—21no—
239C32—22no—
240C32—41no—
241C32C42×Dic312yescomplemented
242C32C3×C4311yescharacteristic, complemented
243C23C2×C3×D3211yescharacteristic
244C2×C4—316no—
245C2×C4—38no—
246C2×C4—98no—
247C2×C4—94no—
248C2×C4C3×D3×Dic314yes—
249C2×C4C3×#62412yes—
250C2×C3—22no—
251C2×C3—21no—
252C2×C3—24no—
253C2×C3—24no—
254C2×C3—24no—
255C2×C3—22no—
256C2×C3—24no—
257C2×C3—41no—
258C2×C3—44no—
259C2×C3—42no—
260C2×C3C4×D3×Dic314yes—
261C2×C3C4×#62412yes—
262C2×C3C2×Dic3211yescharacteristic
263C2×C3C3×C42×D314yes—
264C2×C3C3×C42×D314yes—
265C2×C3C2×C3×C4×Dic312yes—
266C2×C3C2×C3×C4×Dic314yes—
267C22C3×C4×D3214yes—
268C22C2×C3×D3×Dic312yes—
269C22C2×C3×#62411yescharacteristic
270C4—316no—
271C4—98no—
272C4C3×Dic3214yesdirect product factor
273C3—22no—
274C3—21no—
275C3—41no—
276C3C4×Dic3211yescharacteristic, direct product factor
277C3C3×C42×Dic312yescomplemented
278C2C3×C4×D3×Dic314yes—
279C2C3×C4×#62412yes—
280C2C2×C3×Dic3211yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4D32
commutator, hypocenterC32C3×C43
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
LCSG ▷ C32
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
5. C4×Dic32C3direct (N × H)
9. C3×C4×C3⋊Dic3C4semidirect
10. C32×C4×Dic3C4semidirect
11. C3×Dic32C4direct (N × H)
43. C4×C3⋊Dic3C3×C4semidirect
48. C3×C4×Dic3Dic3direct (N × H)
49. C3×C4×Dic3C3×C4semidirect
50. Dic32C3×C4direct (N × H)
54. C33×C4C42semidirect
55. C3×C3⋊Dic3C42semidirect
56. C32×Dic3C42semidirect
121. C4×Dic3C3×Dic3direct (N × H)
143. C32×C4C4×Dic3semidirect
144. C32×C4C3×C42semidirect
145. C3⋊Dic3C3×C42semidirect
149. C3×Dic3C4×Dic3direct (N × H)
150. C3×Dic3C3×C42semidirect
154. C33C43semidirect
233. C3×C4Dic32direct (N × H)
234. C3×C4C3×C4×Dic3semidirect
237. Dic3C3×C4×Dic3direct (N × H)
241. C32C42×Dic3semidirect
242. C32C3×C43semidirect
272. C4C3×Dic32direct (N × H)
276. C3C4×Dic32direct (N × H)
277. C3C3×C42×Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊Dic32
C2×C32×C4×Dic32
C2×C3×Dic322
C3×C42×Dic33
C4×Dic323