Finite Group Databasebeta

← C3×#10591222 #13158070

Subgroups — C3×#10591222 #13158070

Order 1728.

1090 conjugacy classes of subgroups.

The subgroup lattice has 276 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
3C3×#63579C214yesmaximal
4C3×#4932—32nomaximal
5#10591222C311yesmaximal, characteristic, direct product factor
6C22×C33×C4C2211yescharacteristic
7C22×C3×C3⋊Dic3C2212yes—
8C22×C32×Dic3C2214yes—
9C3×C4×C3⋊Dic3C414yescomplemented
10C32×C4×Dic3C418yescomplemented
11C2×C32×C42—32no—
12C2×C4×C3⋊Dic3C2×C311yescharacteristic
13C2×C3×C4×Dic3—22no—
14C2×C3×C4×Dic3—32no—
15C2×C3×C4×Dic3—61no—
16C2×C3×C4×Dic3D312yes—
17C2×C3×C4×Dic3C2×C312yes—
18C3×C3⋊#95—38no—
19#63579C2×C314yes—
20C23×C33C2311yescharacteristic
21C2×C33×C4C2×C412yes—
22C2×C33×C4C2×C414yes—
23C2×C3×C3⋊Dic3—24no—
24C2×C3×C3⋊Dic3C2×C414yes—
25C2×C32×Dic3—28no—
26C2×C32×Dic3C2×C418yes—
27C3×#376—91no—
28#4932—32no—
29C22×C32×C4—21no—
30C22×C32×C4—22no—
31C22×C32×C4—34no—
32C22×C32×C4—41no—
33C22×C32×C4C2×D312yes—
34C22×C32×C4C22×C311yescharacteristic
35C22×C3⋊Dic3C22×C312yes—
36C22×C3×Dic3—24no—
37C22×C3×Dic3—34no—
38C22×C3×Dic3—62no—
39C22×C3×Dic3C2×D314yes—
40C22×C3×Dic3C22×C314yes—
41C32×C42—38no—
42C4×C3⋊Dic3C3×C414yescomplemented
43C3×C4×Dic3—28no—
44C3×C4×Dic3—38no—
45C3×C4×Dic3—64no—
46C3×C4×Dic3Dic318yescomplemented
47C3×C4×Dic3C3×C418yescomplemented
48C22×C33C22×C411yescharacteristic
49C22×C33C22×C412yes—
50C22×C33#4114yes—
51C33×C4C4214yes—
52C3×C3⋊Dic3—24no—
53C32×Dic3—28no—
54C2×C3×C42—32no—
55C2×C3×C42—32no—
56C2×C3×C42—62no—
57C2×C3×C42—91no—
58C2×C4×Dic3—32no—
59C2×C4×Dic3—61no—
60C2×C4×Dic3C3×D312yes—
61C3×#95—94no—
62C3⋊#95—38no—
63C23×C32—21no—
64C23×C32—22no—
65C23×C32—41no—
66C23×C32C22×D312yes—
67C23×C32C23×C311yescharacteristic
68C2×C32×C4—24no—
69C2×C32×C4—22no—
70C2×C32×C4—24no—
71C2×C32×C4—24no—
72C2×C32×C4—24no—
73C2×C32×C4—38no—
74C2×C32×C4—44no—
75C2×C32×C4—42no—
76C2×C32×C4—68no—
77C2×C32×C4C4×D314yes—
78C2×C32×C4C4×D314yes—
79C2×C32×C4C2×Dic314yes—
80C2×C32×C4C2×C3×C412yes—
81C2×C32×C4C2×C3×C414yes—
82C2×C3⋊Dic3—24no—
83C2×C3⋊Dic3C2×C3×C414yes—
84C2×C3×Dic3—216no—
85C2×C3×Dic3—28no—
86C2×C3×Dic3—28no—
87C2×C3×Dic3—28no—
88C2×C3×Dic3—38no—
89C2×C3×Dic3—64no—
90C2×C3×Dic3—68no—
91C2×C3×Dic3—68no—
92C2×C3×Dic3C2×Dic318yes—
93C2×C3×Dic3C2×C3×C418yes—
94#376—91noSylow
95C2×C33C2×C4211yescharacteristic
96C2×C33#11712yes—
97C2×C33#11714yes—
98C22×C3×C4—21no—
99C22×C3×C4—22no—
100C22×C3×C4—34no—
101C22×C3×C4—34no—
102C22×C3×C4—41no—
103C22×C3×C4—64no—
104C22×C3×C4—92no—
105C22×C3×C4D3211yescharacteristic
106C22×C3×C4C2×C3×D312yes—
107C22×Dic3—34no—
108C22×Dic3—62no—
109C22×Dic3C2×C3×D314yes—
110C3×C42—38no—
111C3×C42—38no—
112C3×C42—68no—
113C3×C42—94no—
114C4×Dic3—38no—
115C4×Dic3—64no—
116C4×Dic3C3×Dic318yescomplemented
117C22×C32—24no—
118C22×C32—22no—
119C22×C32—21no—
120C22×C32—22no—
121C22×C32—22no—
122C22×C32—28no—
123C22×C32—22no—
124C22×C32—42no—
125C22×C32—41no—
126C22×C32—44no—
127C22×C32C2×C4×D312yes—
128C22×C32C2×C4×D312yes—
129C22×C32#23518yes—
130C22×C32C22×Dic312yes—
131C22×C32C22×C3×C411yescharacteristic
132C22×C32C22×C3×C412yes—
133C22×C32C3×#4114yes—
134C32×C4—24no—
135C32×C4—28no—
136C32×C4—44no—
137C32×C4—68no—
138C32×C4C4×Dic318yes—
139C32×C4C3×C4214yes—
140C3⋊Dic3—24no—
141C3×Dic3—216no—
142C3×Dic3—28no—
143C3×Dic3—28no—
144C3×Dic3—68no—
145C3×Dic3—68no—
146C2×C42—32no—
147C2×C42—91no—
148#95—94no—
149C33#37611yescharacteristic, complemented, Sylow
150C23×C3—22no—
151C23×C3—21no—
152C23×C3—41no—
153C23×C3C2×D3211yescharacteristic
154C23×C3C22×C3×D312yes—
155C2×C3×C4—24no—
156C2×C3×C4—24no—
157C2×C3×C4—24no—
158C2×C3×C4—24no—
159C2×C3×C4—22no—
160C2×C3×C4—38no—
161C2×C3×C4—38no—
162C2×C3×C4—44no—
163C2×C3×C4—42no—
164C2×C3×C4—68no—
165C2×C3×C4—616no—
166C2×C3×C4—68no—
167C2×C3×C4—68no—
168C2×C3×C4—94no—
169C2×C3×C4—184no—
170C2×C3×C4D3×Dic314yes—
171C2×C3×C4#62412yes—
172C2×C3×C4C3×C4×D314yes—
173C2×C3×C4C3×C4×D314yes—
174C2×C3×C4C2×C3×Dic314yes—
175C2×Dic3—28no—
176C2×Dic3—38no—
177C2×Dic3—64no—
178C2×Dic3—68no—
179C2×Dic3—68no—
180C2×Dic3C2×C3×Dic318yes—
181C2×C32—22no—
182C2×C32—24no—
183C2×C32—24no—
184C2×C32—22no—
185C2×C32—21no—
186C2×C32—24no—
187C2×C32—24no—
188C2×C32—41no—
189C2×C32—44no—
190C2×C32—42no—
191C2×C32#87214yes—
192C2×C32#87214yes—
193C2×C32C2×C4×Dic312yes—
194C2×C32#92614yes—
195C2×C32C2×C3×C4211yescharacteristic
196C2×C32C3×#11714yes—
197C2×C32C3×#11712yes—
198C22×C4—34no—
199C22×C4—92no—
200C22×C4C3×D3211yescharacteristic
201C42—38no—
202C42—94no—
203C22×C3—24no—
204C22×C3—28no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3—21no—
209C22×C3—22no—
210C22×C3—44no—
211C22×C3—42no—
212C22×C3—41no—
213C22×C3#379814yes—
214C22×C3C2×D3×Dic312yes—
215C22×C3C2×#62411yescharacteristic
216C22×C3C2×C3×C4×D312yes—
217C22×C3C2×C3×C4×D312yes—
218C22×C3C3×#23518yes—
219C22×C3C22×C3×Dic312yes—
220C3×C4—28no—
221C3×C4—24no—
222C3×C4—44no—
223C3×C4—616no—
224C3×C4—68no—
225C3×C4—68no—
226C3×C4—184no—
227C3×C4Dic3214yes—
228C3×C4C3×C4×Dic318yes—
229Dic3—28no—
230Dic3—68no—
231Dic3—68no—
232C32—22no—
233C32—21no—
234C32—41no—
235C32#493212yescomplemented
236C32C3×#37611yescharacteristic, complemented
237C23C2×C3×D3211yescharacteristic
238C2×C4—38no—
239C2×C4—68no—
240C2×C4—94no—
241C2×C4—184no—
242C2×C4C3×D3×Dic314yes—
243C2×C4C3×#62412yes—
244C2×C3—22no—
245C2×C3—21no—
246C2×C3—24no—
247C2×C3—24no—
248C2×C3—24no—
249C2×C3—22no—
250C2×C3—24no—
251C2×C3—41no—
252C2×C3—44no—
253C2×C3—42no—
254C2×C3#6385514yes—
255C2×C3#6387412yes—
256C2×C3C2×Dic3211yescharacteristic
257C2×C3C3×#87214yes—
258C2×C3C3×#87214yes—
259C2×C3C2×C3×C4×Dic312yes—
260C2×C3C3×#92614yes—
261C22C3×#379814yes—
262C22C2×C3×D3×Dic312yes—
263C22C2×C3×#62411yescharacteristic
264C4—68no—
265C4—184no—
266C4C3×Dic3214yes—
267C3—22no—
268C3—21no—
269C3—41no—
270C3#1059122211yescharacteristic, direct product factor
271C3C3×#493212yescomplemented
272C2C3×#6385514yes—
273C2C3×#6387412yes—
274C2C2×C3×Dic3211yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4D32
commutatorC2×C32C2×C3×C42
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypocenterC32C3×#376
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
5. #10591222C3direct (N × H)
9. C3×C4×C3⋊Dic3C4semidirect
10. C32×C4×Dic3C4semidirect
42. C4×C3⋊Dic3C3×C4semidirect
46. C3×C4×Dic3Dic3semidirect
47. C3×C4×Dic3C3×C4semidirect
116. C4×Dic3C3×Dic3semidirect
149. C33#376semidirect
235. C32#4932semidirect
236. C32C3×#376semidirect
270. C3#10591222direct (N × H)
271. C3C3×#4932semidirect

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
C3×#635792
C3×#49323
#105912223