Finite Group Databasebeta

← C2×C3×#63873 #13174405

Subgroups — C2×C3×#63873 #13174405

Order 1728.

1634 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
1C22×C32×Dic6C212yesmaximal
2C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
3C2×C3×#3724C214yesmaximal
4C3×#63873C218yesmaximal, direct product factor
5C2×C3×#944—32nomaximal
6C2×#63873C311yesmaximal, characteristic, direct product factor
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3C2212yes—
9C22×C32×Dic3C2214yes—
10C2×C32×Dic6—28no—
11C2×C32×Dic6C2218yes—
12C3×C4×C3⋊Dic3C2214yes—
13C3×#3724C22116yes—
14C22×C32×Q8—32no—
15C22×C3×Dic6—22no—
16C22×C3×Dic6D312yes—
17C22×C3×Dic6C2×C312yes—
18C2×C4×C3⋊Dic3C2×C311yescharacteristic
19C3×#944—316no—
20C2×C3×Dic3⋊C4—38no—
21C2×C3×C4×Dic3—32no—
22C2×C3×C4×Dic3—61no—
23C2×#3724C2×C314yes—
24#63873C2×C318yesdirect product factor
25C23×C33C2311yescharacteristic
26C2×C33×C4D414yes—
27C2×C33×C4C2312yes—
28C2×C3×C3⋊Dic3Q818yescomplemented
29C2×C3×C3⋊Dic3C2314yes—
30C2×C32×Dic3—28no—
31C2×C32×Dic3C2318yes—
32C32×Dic6—216no—
33C2×C3×C4⋊Q8—91no—
34C2×#944—32no—
35C2×C32×Q8—38no—
36C2×C32×Q8—68no—
37C22×C32×C4—21no—
38C22×C32×C4—22no—
39C22×C32×C4—34no—
40C22×C32×C4—41no—
41C22×C32×C4C2×D312yes—
42C22×C32×C4C22×C311yescharacteristic
43C22×C3⋊Dic3C22×C312yes—
44C22×C3×Dic3—24no—
45C22×C3×Dic3—34no—
46C22×C3×Dic3—62no—
47C22×C3×Dic3C2×D314yes—
48C22×C3×Dic3C22×C314yes—
49C2×C3×Dic6—216no—
50C2×C3×Dic6—28no—
51C2×C3×Dic6—28no—
52C2×C3×Dic6—28no—
53C2×C3×Dic6C2×D318yes—
54C2×C3×Dic6C22×C318yes—
55C4×C3⋊Dic3C22×C314yes—
56C3×Dic3⋊C4—332no—
57C3×C4×Dic3—38no—
58C3×C4×Dic3—64no—
59#3724C22×C3116yes—
60C22×C33C2×D412yes—
61C22×C33C2×Q814yes—
62C22×C33C2411yescharacteristic
63C33×C4C2×D414yes—
64C3×C3⋊Dic3C2×Q818yescomplemented
65C32×Dic3—28no—
66C22×C3×Q8—32no—
67C22×C3×Q8—32no—
68C22×C3×Q8—62no—
69C22×Dic6C3×D312yes—
70C3×C4⋊Q8—98no—
71C2×C3×C4⋊C4—94no—
72C2×C3×C42—91no—
73#944—316no—
74C2×Dic3⋊C4—38no—
75C2×C4×Dic3—32no—
76C2×C4×Dic3—61no—
77C23×C32—21no—
78C23×C32—22no—
79C23×C32—41no—
80C23×C32C22×D312yes—
81C23×C32C23×C311yescharacteristic
82C32×Q8—616no—
83C2×C32×C4—24no—
84C2×C32×C4—28no—
85C2×C32×C4—22no—
86C2×C32×C4—24no—
87C2×C32×C4—38no—
88C2×C32×C4—44no—
89C2×C32×C4—42no—
90C2×C32×C4—68no—
91C2×C32×C4#6718yes—
92C2×C32×C4C3×D414yes—
93C2×C32×C4C22×D314yes—
94C2×C32×C4C23×C312yes—
95C2×C3⋊Dic3C3×Q818yescomplemented
96C2×C3⋊Dic3C23×C314yes—
97C2×C3×Dic3—216no—
98C2×C3×Dic3—28no—
99C2×C3×Dic3—28no—
100C2×C3×Dic3—28no—
101C2×C3×Dic3—316no—
102C2×C3×Dic3—38no—
103C2×C3×Dic3—64no—
104C2×C3×Dic3—68no—
105C2×C3×Dic3C22×D318yes—
106C2×C3×Dic3C23×C318yes—
107C3×Dic6—232no—
108C3×Dic6—216no—
109C3×Dic6—216no—
110C2×C4⋊Q8—91noSylow
111C2×C33C4⋊Q814yescomplemented
112C2×C33C22×D411yescharacteristic
113C2×C33C22×Q812yes—
114C2×C3×Q8—38no—
115C2×C3×Q8—38no—
116C2×C3×Q8—68no—
117C2×C3×Q8—616no—
118C2×C3×Q8—68no—
119C2×C3×Q8—68no—
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4—34no—
123C22×C3×C4—34no—
124C22×C3×C4—41no—
125C22×C3×C4—64no—
126C22×C3×C4—92no—
127C22×C3×C4D3211yescharacteristic
128C22×C3×C4C2×C3×D312yes—
129C22×Dic3—34no—
130C22×Dic3—62no—
131C22×Dic3C2×C3×D314yes—
132C2×Dic6—28no—
133C2×Dic6C2×C3×D318yes—
134C3×C4⋊C4—916no—
135C3×C42—94no—
136Dic3⋊C4—332no—
137C4×Dic3—38no—
138C4×Dic3—64no—
139C22×C32—21no—
140C22×C32—22no—
141C22×C32—24no—
142C22×C32—22no—
143C22×C32—28no—
144C22×C32—24no—
145C22×C32—42no—
146C22×C32—44no—
147C22×C32—41no—
148C22×C32D3×Q818yes—
149C22×C32C2×#6714yes—
150C22×C32C2×C3×D412yes—
151C22×C32C2×C3×Q814yes—
152C22×C32C23×D312yes—
153C22×C32C24×C311yescharacteristic
154C32×C4—24no—
155C32×C4—28no—
156C32×C4—44no—
157C32×C4—68no—
158C32×C4C2×#6718yes—
159C32×C4C2×C3×D414yes—
160C3⋊Dic3C2×C3×Q818yescomplemented
161C3×Dic3—216no—
162C3×Dic3—28no—
163C3×Dic3—28no—
164C3×Dic3—316no—
165C3×Dic3—68no—
166C22×Q8—32no—
167C4⋊Q8—98no—
168C2×C4⋊C4—94no—
169C2×C42—91no—
170C33C2×C4⋊Q811yescharacteristic, complemented, Sylow
171C23×C3—22no—
172C23×C3—21no—
173C23×C3—41no—
174C23×C3C2×D3211yescharacteristic
175C23×C3C22×C3×D312yes—
176C3×Q8—632no—
177C3×Q8—616no—
178C3×Q8—616no—
179C2×C3×C4—28no—
180C2×C3×C4—24no—
181C2×C3×C4—24no—
182C2×C3×C4—22no—
183C2×C3×C4—38no—
184C2×C3×C4—38no—
185C2×C3×C4—44no—
186C2×C3×C4—42no—
187C2×C3×C4—68no—
188C2×C3×C4—616no—
189C2×C3×C4—68no—
190C2×C3×C4—68no—
191C2×C3×C4—94no—
192C2×C3×C4—98no—
193C2×C3×C4#62514yes—
194C2×C3×C4C3×#6718yes—
195C2×C3×C4C2×D3212yes—
196C2×C3×C4C22×C3×D314yes—
197C2×Dic3—28no—
198C2×Dic3—38no—
199C2×Dic3—316no—
200C2×Dic3—68no—
201C2×Dic3—64no—
202C2×Dic3C22×C3×D318yes—
203Dic6—216no—
204C2×C32—22no—
205C2×C32—28no—
206C2×C32—21no—
207C2×C32—24no—
208C2×C32—22no—
209C2×C32—24no—
210C2×C32—44no—
211C2×C32—41no—
212C2×C32—42no—
213C2×C32#94418yescomplemented
214C2×C32C3×C4⋊Q814yescomplemented
215C2×C32C2×D3×Q814yes—
216C2×C32C22×#6712yes—
217C2×C32C22×C3×D411yescharacteristic
218C2×C32C22×C3×Q812yes—
219C2×Q8—38no—
220C2×Q8—68no—
221C22×C4—34no—
222C22×C4—92no—
223C22×C4C3×D3211yescharacteristic
224C4⋊C4—916no—
225C42—94no—
226C22×C3—21no—
227C22×C3—22no—
228C22×C3—24no—
229C22×C3—28no—
230C22×C3—24no—
231C22×C3—22no—
232C22×C3—41no—
233C22×C3—44no—
234C22×C3—42no—
235C22×C3#379714yes—
236C22×C3C2×#62512yes—
237C22×C3C3×D3×Q818yes—
238C22×C3C2×C3×#6714yes—
239C22×C3C22×D3211yescharacteristic
240C22×C3C23×C3×D312yes—
241C3×C4—28no—
242C3×C4—24no—
243C3×C4—44no—
244C3×C4—616no—
245C3×C4—68no—
246C3×C4—68no—
247C3×C4—98no—
248C3×C4C2×#62514yes—
249C3×C4C2×C3×#6718yes—
250Dic3—28no—
251Dic3—316no—
252Dic3—68no—
253C32—21no—
254C32—22no—
255C32—41no—
256C32C2×#94412yescomplemented
257C32C2×C3×C4⋊Q811yescharacteristic, complemented
258C23C2×C3×D3211yescharacteristic
259Q8—616no—
260C2×C4—38no—
261C2×C4—68no—
262C2×C4—94no—
263C2×C4—98no—
264C2×C4C3×#62514yes—
265C2×C4C2×C3×D3212yes—
266C2×C3—22no—
267C2×C3—21no—
268C2×C3—28no—
269C2×C3—24no—
270C2×C3—22no—
271C2×C3—24no—
272C2×C3—41no—
273C2×C3—44no—
274C2×C3—42no—
275C2×C3#6387314yesdirect product factor
276C2×C3C3×#94418yescomplemented
277C2×C3C2×#379712yes—
278C2×C3C22×#62511yescharacteristic
279C2×C3C2×C3×D3×Q814yes—
280C2×C3C22×C3×#6712yes—
281C22C3×#379714yes—
282C22C2×C3×#62512yes—
283C22C22×C3×D3211yescharacteristic
284C4—68no—
285C4—98no—
286C4C2×C3×#62514yes—
287C3—22no—
288C3—21no—
289C3—41no—
290C3C2×#6387311yescharacteristic, direct product factor
291C3C2×C3×#94412yescomplemented
292C2C3×#6387314yesdirect product factor
293C2C2×C3×#379712yes—
294C2C22×C3×#62511yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×C4⋊Q8
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×#63873C2direct (N × H)
6. C2×#63873C3direct (N × H)
24. #63873C2×C3direct (N × H)
28. C2×C3×C3⋊Dic3Q8semidirect
64. C3×C3⋊Dic3C2×Q8semidirect
95. C2×C3⋊Dic3C3×Q8semidirect
111. C2×C33C4⋊Q8semidirect
160. C3⋊Dic3C2×C3×Q8semidirect
170. C33C2×C4⋊Q8semidirect
213. C2×C32#944semidirect
214. C2×C32C3×C4⋊Q8semidirect
256. C32C2×#944semidirect
257. C32C2×C3×C4⋊Q8semidirect
275. C2×C3#63873direct (N × H)
276. C2×C3C3×#944semidirect
290. C3C2×#63873direct (N × H)
291. C3C2×C3×#944semidirect
292. C2C3×#63873direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×Dic62
C2×C3×C4×C3⋊Dic32
C2×C3×#37242
C3×#638732
C2×C3×#9443
C2×#638733