Finite Group Databasebeta

← C2×C3×#63872 #13174403

Subgroups — C2×C3×#63872 #13174403

Order 1728.

1250 conjugacy classes of subgroups.

The subgroup lattice has 296 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×Dic3⋊Dic3C214yesmaximal
4C3×#63872C218yesmaximal, direct product factor
5C2×C3×#890—32nomaximal
6C2×#63872C311yesmaximal, characteristic, direct product factor
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3C2212yes—
9C22×C32×Dic3C2214yes—
10C3×C4×C3⋊Dic3C2214yes—
11C32×C4⋊Dic3C2218yes—
12C3×Dic3⋊Dic3C22116yes—
13C2×C32×C4⋊C4—32no—
14C2×C4×C3⋊Dic3C2×C311yescharacteristic
15C2×C3×C4⋊Dic3—22no—
16C2×C3×C4⋊Dic3—34no—
17C2×C3×C4⋊Dic3D312yes—
18C2×C3×C4⋊Dic3C2×C312yes—
19C2×C3×Dic3⋊C4—34no—
20C2×C3×C4×Dic3—32no—
21C2×C3×C4×Dic3—61no—
22C3×#890—316no—
23C2×Dic3⋊Dic3C2×C314yes—
24#63872C2×C318yesdirect product factor
25C23×C33C2311yescharacteristic
26C2×C33×C4Q814yes—
27C2×C33×C4C2312yes—
28C2×C3×C3⋊Dic3—24no—
29C2×C3×C3⋊Dic3C2314yes—
30C2×C32×Dic3—28no—
31C2×C32×Dic3C2318yes—
32C2×C3×#125—91no—
33C2×#890—32no—
34C22×C32×C4—21no—
35C22×C32×C4—22no—
36C22×C32×C4—34no—
37C22×C32×C4—41no—
38C22×C32×C4C2×D312yes—
39C22×C32×C4C22×C311yescharacteristic
40C22×C3⋊Dic3C22×C312yes—
41C22×C3×Dic3—24no—
42C22×C3×Dic3—32no—
43C22×C3×Dic3—32no—
44C22×C3×Dic3—62no—
45C22×C3×Dic3C2×D314yes—
46C22×C3×Dic3C22×C314yes—
47C32×C4⋊C4—38no—
48C4×C3⋊Dic3C22×C314yes—
49C3×C4⋊Dic3—28no—
50C3×C4⋊Dic3—316no—
51C3×C4⋊Dic3C2×D318yes—
52C3×C4⋊Dic3C22×C318yes—
53C3×Dic3⋊C4—316no—
54C3×C4×Dic3—38no—
55C3×C4×Dic3—64no—
56Dic3⋊Dic3C22×C3116yes—
57C22×C33C2×Q812yes—
58C22×C33#4114yes—
59C22×C33C2411yescharacteristic
60C33×C4C2×Q814yes—
61C3×C3⋊Dic3—24no—
62C32×Dic3—28no—
63C3×#125—98no—
64C2×C3×C4⋊C4—32no—
65C2×C3×C4⋊C4—32no—
66C2×C3×C4⋊C4—62no—
67C2×C3×C4⋊C4—94no—
68C2×C3×C42—91no—
69C2×C4⋊Dic3—34no—
70C2×C4⋊Dic3C3×D312yes—
71C2×Dic3⋊C4—34no—
72C2×C4×Dic3—32no—
73C2×C4×Dic3—61no—
74#890—316no—
75C23×C32—21no—
76C23×C32—22no—
77C23×C32—41no—
78C23×C32C22×D312yes—
79C23×C32C23×C311yescharacteristic
80C2×C32×C4—24no—
81C2×C32×C4—28no—
82C2×C32×C4—22no—
83C2×C32×C4—24no—
84C2×C32×C4—38no—
85C2×C32×C4—44no—
86C2×C32×C4—42no—
87C2×C32×C4—68no—
88C2×C32×C4Dic618yes—
89C2×C32×C4C3×Q814yes—
90C2×C32×C4C22×D314yes—
91C2×C32×C4C23×C312yes—
92C2×C3⋊Dic3—24no—
93C2×C3⋊Dic3C23×C314yes—
94C2×C3×Dic3—28no—
95C2×C3×Dic3—28no—
96C2×C3×Dic3—28no—
97C2×C3×Dic3—34no—
98C2×C3×Dic3—34no—
99C2×C3×Dic3—48no—
100C2×C3×Dic3—64no—
101C2×C3×Dic3—68no—
102C2×C3×Dic3—64no—
103C2×C3×Dic3—64no—
104C2×C3×Dic3C22×D318yes—
105C2×C3×Dic3C23×C318yes—
106C2×#125—91noSylow
107C2×C33#12514yescomplemented
108C2×C33C22×Q811yescharacteristic
109C2×C33C2×#4112yes—
110C22×C3×C4—21no—
111C22×C3×C4—22no—
112C22×C3×C4—34no—
113C22×C3×C4—34no—
114C22×C3×C4—41no—
115C22×C3×C4—64no—
116C22×C3×C4—92no—
117C22×C3×C4D3211yescharacteristic
118C22×C3×C4C2×C3×D312yes—
119C22×Dic3—32no—
120C22×Dic3—32no—
121C22×Dic3—62no—
122C22×Dic3C2×C3×D314yes—
123C3×C4⋊C4—38no—
124C3×C4⋊C4—38no—
125C3×C4⋊C4—68no—
126C3×C4⋊C4—916no—
127C3×C42—94no—
128C4⋊Dic3—316no—
129C4⋊Dic3C2×C3×D318yes—
130Dic3⋊C4—316no—
131C4×Dic3—38no—
132C4×Dic3—64no—
133C22×C32—21no—
134C22×C32—22no—
135C22×C32—24no—
136C22×C32—22no—
137C22×C32—24no—
138C22×C32—24no—
139C22×C32—24no—
140C22×C32—42no—
141C22×C32—44no—
142C22×C32—41no—
143C22×C32C2×Dic614yes—
144C22×C32#23714yes—
145C22×C32#23914yes—
146C22×C32C2×C3×Q812yes—
147C22×C32C3×#4114yes—
148C22×C32C23×D312yes—
149C22×C32C24×C311yescharacteristic
150C32×C4—24no—
151C32×C4—28no—
152C32×C4—44no—
153C32×C4—68no—
154C32×C4C2×Dic618yes—
155C32×C4C2×C3×Q814yes—
156C3⋊Dic3—24no—
157C3×Dic3—28no—
158C3×Dic3—28no—
159C3×Dic3—48no—
160C3×Dic3—68no—
161C3×Dic3—64no—
162C3×Dic3—64no—
163#125—98no—
164C2×C4⋊C4—32no—
165C2×C4⋊C4—94no—
166C2×C42—91no—
167C33C2×#12511yescharacteristic, complemented, Sylow
168C23×C3—22no—
169C23×C3—21no—
170C23×C3—41no—
171C23×C3C2×D3211yescharacteristic
172C23×C3C22×C3×D312yes—
173C2×C3×C4—28no—
174C2×C3×C4—24no—
175C2×C3×C4—24no—
176C2×C3×C4—22no—
177C2×C3×C4—38no—
178C2×C3×C4—38no—
179C2×C3×C4—44no—
180C2×C3×C4—42no—
181C2×C3×C4—68no—
182C2×C3×C4—68no—
183C2×C3×C4—68no—
184C2×C3×C4—94no—
185C2×C3×C4—128no—
186C2×C3×C4—184no—
187C2×C3×C4#62714yes—
188C2×C3×C4C3×Dic618yes—
189C2×C3×C4C2×D3212yes—
190C2×C3×C4C22×C3×D314yes—
191C2×Dic3—28no—
192C2×Dic3—34no—
193C2×Dic3—34no—
194C2×Dic3—64no—
195C2×Dic3—68no—
196C2×Dic3—64no—
197C2×Dic3—64no—
198C2×Dic3C22×C3×D318yes—
199C2×C32—28no—
200C2×C32—22no—
201C2×C32—21no—
202C2×C32—22no—
203C2×C32—22no—
204C2×C32—22no—
205C2×C32—24no—
206C2×C32—44no—
207C2×C32—41no—
208C2×C32—42no—
209C2×C32#89018yescomplemented
210C2×C32C3×#12514yescomplemented
211C2×C32C22×Dic612yes—
212C2×C32C2×#23712yes—
213C2×C32C2×#23912yes—
214C2×C32C22×C3×Q811yescharacteristic
215C2×C32C2×C3×#4112yes—
216C22×C4—34no—
217C22×C4—92no—
218C22×C4C3×D3211yescharacteristic
219C4⋊C4—38no—
220C4⋊C4—916no—
221C42—94no—
222C22×C3—21no—
223C22×C3—22no—
224C22×C3—24no—
225C22×C3—24no—
226C22×C3—24no—
227C22×C3—24no—
228C22×C3—22no—
229C22×C3—41no—
230C22×C3—44no—
231C22×C3—42no—
232C22×C3#379614yes—
233C22×C3C2×#62712yes—
234C22×C3C2×C3×Dic614yes—
235C22×C3C3×#23714yes—
236C22×C3C3×#23914yes—
237C22×C3C22×D3211yescharacteristic
238C22×C3C23×C3×D312yes—
239C3×C4—28no—
240C3×C4—24no—
241C3×C4—44no—
242C3×C4—68no—
243C3×C4—68no—
244C3×C4—128no—
245C3×C4—184no—
246C3×C4C2×#62714yes—
247C3×C4C2×C3×Dic618yes—
248Dic3—28no—
249Dic3—68no—
250Dic3—64no—
251Dic3—64no—
252C32—22no—
253C32—21no—
254C32—41no—
255C32C2×#89012yescomplemented
256C32C2×C3×#12511yescharacteristic, complemented
257C23C2×C3×D3211yescharacteristic
258C2×C4—38no—
259C2×C4—68no—
260C2×C4—94no—
261C2×C4—184no—
262C2×C4C3×#62714yes—
263C2×C4C2×C3×D3212yes—
264C2×C3—22no—
265C2×C3—21no—
266C2×C3—28no—
267C2×C3—24no—
268C2×C3—22no—
269C2×C3—22no—
270C2×C3—22no—
271C2×C3—41no—
272C2×C3—44no—
273C2×C3—42no—
274C2×C3#6387214yesdirect product factor
275C2×C3C3×#89018yescomplemented
276C2×C3C2×#379612yes—
277C2×C3C22×#62711yescharacteristic
278C2×C3C22×C3×Dic612yes—
279C2×C3C2×C3×#23712yes—
280C2×C3C2×C3×#23912yes—
281C22C3×#379614yes—
282C22C2×C3×#62712yes—
283C22C22×C3×D3211yescharacteristic
284C4—68no—
285C4—184no—
286C4C2×C3×#62714yes—
287C3—22no—
288C3—21no—
289C3—41no—
290C3C2×#6387211yescharacteristic, direct product factor
291C3C2×C3×#89012yescomplemented
292C2C3×#6387214yesdirect product factor
293C2C2×C3×#379612yes—
294C2C22×C3×#62711yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×#125
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63872C2direct (N × H)
6. C2×#63872C3direct (N × H)
24. #63872C2×C3direct (N × H)
107. C2×C33#125semidirect
167. C33C2×#125semidirect
209. C2×C32#890semidirect
210. C2×C32C3×#125semidirect
255. C32C2×#890semidirect
256. C32C2×C3×#125semidirect
274. C2×C3#63872direct (N × H)
275. C2×C3C3×#890semidirect
290. C3C2×#63872direct (N × H)
291. C3C2×C3×#890semidirect
292. C2C3×#63872direct (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
C2×C32×C4⋊Dic32
C2×C3×Dic3⋊Dic32
C3×#638722
C2×C3×#8903
C2×#638723