Finite Group Databasebeta

← C2×C3×#63553 #13158086

Subgroups — C2×C3×#63553 #13158086

Order 1728.

962 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2C22×C32×C3⋊C8C212yesmaximal
3C3×#63553C214yesmaximal, direct product factor
4C2×C3×C8×Dic3—32nomaximal
5C2×#63553C311yesmaximal, characteristic, direct product factor
6C22×C33×C4C2211yescharacteristic
7C22×C3×C3⋊Dic3C412yes—
8C3×C4×C3⋊Dic3C412yes—
9C3×C4×C3⋊Dic3C2212yes—
10C2×C32×C3⋊C8C418yescomplemented
11C2×C32×C3⋊C8C2214yes—
12C22×C32×C8—32no—
13C2×C4×C3⋊Dic3C2×C311yescharacteristic
14C2×C3×C4×Dic3—32no—
15C2×C3×C4×Dic3—61no—
16C22×C3×C3⋊C8—22no—
17C22×C3×C3⋊C8D312yes—
18C22×C3×C3⋊C8C2×C312yes—
19C3×C8×Dic3—38no—
20#63553C2×C314yesdirect product factor
21C23×C33C2×C411yescharacteristic
22C2×C33×C4C2×C411yescharacteristic
23C2×C33×C4C2×C414yes—
24C2×C33×C4C2311yescharacteristic
25C2×C3×C3⋊Dic3C818yescomplemented
26C2×C3×C3⋊Dic3C2×C414yes—
27C32×C3⋊C8C2×C418yescomplemented
28C2×C3×C4×C8—91no—
29C2×C8×Dic3—32no—
30C22×C32×C4—21no—
31C22×C32×C4—22no—
32C22×C32×C4—41no—
33C22×C32×C4C2×D312yes—
34C22×C32×C4C22×C311yescharacteristic
35C22×C3⋊Dic3C3×C412yes—
36C22×C3×Dic3—34no—
37C22×C3×Dic3—62no—
38C2×C32×C8—38no—
39C2×C32×C8—34no—
40C4×C3⋊Dic3C3×C412yes—
41C4×C3⋊Dic3C22×C312yes—
42C3×C4×Dic3—34no—
43C3×C4×Dic3—34no—
44C3×C4×Dic3—62no—
45C3×C4×Dic3—62no—
46C2×C3×C3⋊C8—28no—
47C2×C3×C3⋊C8—24no—
48C2×C3×C3⋊C8Dic318yescomplemented
49C2×C3×C3⋊C8C3×C418yescomplemented
50C2×C3×C3⋊C8C2×D314yes—
51C2×C3×C3⋊C8C22×C314yes—
52C22×C33C4212yes—
53C22×C33C2×C814yes—
54C22×C33C22×C411yescharacteristic
55C33×C4C4212yes—
56C33×C4C22×C412yes—
57C3×C3⋊Dic3C2×C818yescomplemented
58C22×C3×C8—32no—
59C22×C3×C8—32no—
60C22×C3×C8—62no—
61C2×C3×C42—91no—
62C2×C4×Dic3—32no—
63C2×C4×Dic3—61no—
64C22×C3⋊C8C3×D312yes—
65C3×C4×C8—94no—
66C8×Dic3—38no—
67C23×C32—21no—
68C23×C32—22no—
69C23×C32—41no—
70C23×C32C4×D312yes—
71C23×C32C2×C3×C411yescharacteristic
72C2×C32×C4—24no—
73C2×C32×C4—21no—
74C2×C32×C4—24no—
75C2×C32×C4—22no—
76C2×C32×C4—24no—
77C2×C32×C4—21no—
78C2×C32×C4—22no—
79C2×C32×C4—44no—
80C2×C32×C4—41no—
81C2×C32×C4—41no—
82C2×C32×C4C4×D314yes—
83C2×C32×C4C4×D312yes—
84C2×C32×C4C2×Dic314yes—
85C2×C32×C4C2×C3×C411yescharacteristic
86C2×C32×C4C2×C3×C414yes—
87C2×C32×C4C22×D312yes—
88C2×C32×C4C23×C311yescharacteristic
89C2×C3⋊Dic3C3×C818yescomplemented
90C2×C3⋊Dic3C2×C3×C414yes—
91C2×C3×Dic3—316no—
92C2×C3×Dic3—38no—
93C2×C3×Dic3—64no—
94C2×C3×Dic3—68no—
95C32×C8—38no—
96C3×C3⋊C8—28no—
97C3×C3⋊C8C2×Dic318yescomplemented
98C3×C3⋊C8C2×C3×C418yescomplemented
99C2×C4×C8—91noSylow
100C2×C33C4×C814yescomplemented
101C2×C33C2×C4211yescharacteristic
102C2×C33C22×C812yes—
103C22×C3×C4—21no—
104C22×C3×C4—22no—
105C22×C3×C4—41no—
106C22×C3×C4—92no—
107C22×C3×C4D3211yescharacteristic
108C22×C3×C4C2×C3×D312yes—
109C22×Dic3—34no—
110C22×Dic3—62no—
111C2×C3×C8—38no—
112C2×C3×C8—38no—
113C2×C3×C8—34no—
114C2×C3×C8—34no—
115C2×C3×C8—68no—
116C2×C3×C8—64no—
117C3×C42—92no—
118C3×C42—92no—
119C4×Dic3—34no—
120C4×Dic3—34no—
121C4×Dic3—62no—
122C4×Dic3—62no—
123C2×C3⋊C8C3×Dic318yescomplemented
124C2×C3⋊C8C2×C3×D314yes—
125C22×C32—24no—
126C22×C32—22no—
127C22×C32—28no—
128C22×C32—24no—
129C22×C32—21no—
130C22×C32—22no—
131C22×C32—42no—
132C22×C32—44no—
133C22×C32—41no—
134C22×C32D3×C818yes—
135C22×C32C4×Dic314yes—
136C22×C32C3×C4212yes—
137C22×C32C2×C3×C814yes—
138C22×C32C2×C4×D312yes—
139C22×C32C22×C3×C411yescharacteristic
140C32×C4—22no—
141C32×C4—24no—
142C32×C4—22no—
143C32×C4—22no—
144C32×C4—22no—
145C32×C4—42no—
146C32×C4—42no—
147C32×C4C4×Dic314yes—
148C32×C4C3×C4212yes—
149C32×C4C2×C4×D312yes—
150C32×C4C22×Dic312yes—
151C32×C4C22×C3×C412yes—
152C3⋊Dic3C2×C3×C818yescomplemented
153C3×Dic3—316no—
154C3×Dic3—68no—
155C22×C8—32no—
156C2×C42—91no—
157C4×C8—94no—
158C33C2×C4×C811yescharacteristic, complemented, Sylow
159C23×C3—22no—
160C23×C3—21no—
161C23×C3—41no—
162C23×C3#62411yescharacteristic
163C23×C3C3×C4×D312yes—
164C2×C3×C4—24no—
165C2×C3×C4—24no—
166C2×C3×C4—24no—
167C2×C3×C4—22no—
168C2×C3×C4—21no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—44no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—98no—
175C2×C3×C4—94no—
176C2×C3×C4D3×Dic314yes—
177C2×C3×C4#62411yescharacteristic
178C2×C3×C4C3×C4×D314yes—
179C2×C3×C4C3×C4×D312yes—
180C2×C3×C4C2×C3×Dic314yes—
181C2×C3×C4C2×D3211yescharacteristic
182C2×C3×C4C22×C3×D312yes—
183C2×Dic3—316no—
184C2×Dic3—38no—
185C2×Dic3—68no—
186C2×Dic3—64no—
187C3×C8—38no—
188C3×C8—38no—
189C3×C8—68no—
190C3⋊C8C2×C3×Dic318yescomplemented
191C2×C32—24no—
192C2×C32—22no—
193C2×C32—24no—
194C2×C32—22no—
195C2×C32—28no—
196C2×C32—21no—
197C2×C32—44no—
198C2×C32—41no—
199C2×C32—42no—
200C2×C32C8×Dic318yescomplemented
201C2×C32C3×C4×C814yescomplemented
202C2×C32C2×D3×C814yes—
203C2×C32C2×C4×Dic312yes—
204C2×C32C2×C3×C4211yescharacteristic
205C2×C32C22×C3×C812yes—
206C22×C4—92no—
207C22×C4C3×D3211yescharacteristic
208C2×C8—38no—
209C2×C8—34no—
210C42—92no—
211C42—92no—
212C22×C3—21no—
213C22×C3—22no—
214C22×C3—24no—
215C22×C3—28no—
216C22×C3—24no—
217C22×C3—22no—
218C22×C3—41no—
219C22×C3—44no—
220C22×C3—42no—
221C22×C3#371014yes—
222C22×C3Dic3212yes—
223C22×C3C3×D3×C818yes—
224C22×C3C3×C4×Dic314yes—
225C22×C3C2×#62411yescharacteristic
226C22×C3C2×C3×C4×D312yes—
227C3×C4—22no—
228C3×C4—22no—
229C3×C4—22no—
230C3×C4—24no—
231C3×C4—22no—
232C3×C4—42no—
233C3×C4—42no—
234C3×C4—98no—
235C3×C4Dic3212yes—
236C3×C4C3×C4×Dic314yes—
237C3×C4C2×D3×Dic312yes—
238C3×C4C2×C3×C4×D312yes—
239C3×C4C22×C3×Dic312yes—
240Dic3—316no—
241Dic3—68no—
242C32—21no—
243C32—22no—
244C32—41no—
245C32C2×C8×Dic312yescomplemented
246C32C2×C3×C4×C811yescharacteristic, complemented
247C23C3×#62411yescharacteristic
248C2×C4—94no—
249C2×C4—98no—
250C2×C4C3×D3×Dic314yes—
251C2×C4C3×#62411yescharacteristic
252C2×C4C2×C3×D3211yescharacteristic
253C8—38no—
254C2×C3—22no—
255C2×C3—21no—
256C2×C3—22no—
257C2×C3—24no—
258C2×C3—28no—
259C2×C3—24no—
260C2×C3—41no—
261C2×C3—42no—
262C2×C3—44no—
263C2×C3#6355314yesdirect product factor
264C2×C3C3×C8×Dic318yescomplemented
265C2×C3C2×#371012yes—
266C2×C3C2×Dic3211yescharacteristic
267C2×C3C2×C3×D3×C814yes—
268C2×C3C2×C3×C4×Dic312yes—
269C22C3×#371014yes—
270C22C3×Dic3212yes—
271C22C2×C3×#62411yescharacteristic
272C4—98no—
273C4C3×Dic3212yes—
274C4C2×C3×D3×Dic312yes—
275C3—21no—
276C3—22no—
277C3—41no—
278C3C2×#6355311yescharacteristic, direct product factor
279C3C2×C3×C8×Dic312yescomplemented
280C2C3×#6355314yesdirect product factor
281C2C2×C3×#371012yes—
282C2C2×C3×Dic3211yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4D32
commutator, hypocenterC32C2×C3×C4×C8
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C2×C4
cosocleC2×C32×C4C23×C3
LCSG ▷ C32
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×#63553C2direct (N × H)
5. C2×#63553C3direct (N × H)
10. C2×C32×C3⋊C8C4semidirect
20. #63553C2×C3direct (N × H)
25. C2×C3×C3⋊Dic3C8semidirect
27. C32×C3⋊C8C2×C4semidirect
48. C2×C3×C3⋊C8Dic3semidirect
49. C2×C3×C3⋊C8C3×C4semidirect
57. C3×C3⋊Dic3C2×C8semidirect
89. C2×C3⋊Dic3C3×C8semidirect
97. C3×C3⋊C8C2×Dic3semidirect
98. C3×C3⋊C8C2×C3×C4semidirect
100. C2×C33C4×C8semidirect
123. C2×C3⋊C8C3×Dic3semidirect
152. C3⋊Dic3C2×C3×C8semidirect
158. C33C2×C4×C8semidirect
190. C3⋊C8C2×C3×Dic3semidirect
200. C2×C32C8×Dic3semidirect
201. C2×C32C3×C4×C8semidirect
245. C32C2×C8×Dic3semidirect
246. C32C2×C3×C4×C8semidirect
263. C2×C3#63553direct (N × H)
264. C2×C3C3×C8×Dic3semidirect
278. C3C2×#63553direct (N × H)
279. C3C2×C3×C8×Dic3semidirect
280. C2C3×#63553direct (N × H)

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
C22×C32×C3⋊C82
C3×#635532
C2×C3×C8×Dic33
C2×#635533