Finite Group Databasebeta

← C2×C3×#63449 #13152387

Subgroups — C2×C3×#63449 #13152387

Order 1728.

986 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Dic18C211yesmaximal, characteristic
2C2×C3×C9×C4⋊C4C211yesmaximal, characteristic
3C2×C3×Dic9⋊C4C212yesmaximal
4C2×C3×C4×Dic9C211yesmaximal, characteristic
5C2×C3×C4×Dic9C212yesmaximal
6C3×#63449C218yesmaximal, direct product factor
7C2×C3×#887—31nomaximal
8C2×#63449C311yesmaximal, characteristic, direct product factor
9C22×C3×C4×C9C2212yes—
10C22×C3×C4×C9C2211yescharacteristic
11C22×C3×Dic9C2211yescharacteristic
12C22×C3×Dic9C2212yes—
13C22×C3×Dic9C2211yescharacteristic
14C2×C3×Dic18C418yescomplemented
15C2×C3×Dic18C2214yes—
16C3×C9×C4⋊C4C2214yes—
17C3×Dic9⋊C4C2218yes—
18C3×C4×Dic9C2214yes—
19C3×C4×Dic9C2218yes—
20C22×C3×Dic6—31no—
21C2×C32×C4⋊C4D311yescharacteristic
22C2×C3×Dic3⋊C4—32no—
23C2×C3×C4×Dic3—32no—
24C2×C3×C4×Dic3—31no—
25C3×#887—38no—
26C22×Dic18C2×C311yescharacteristic
27C2×C9×C4⋊C4—21no—
28C2×C9×C4⋊C4C2×C311yescharacteristic
29C2×Dic9⋊C4C2×C312yes—
30C2×C4×Dic9C2×C311yescharacteristic
31C2×C4×Dic9C2×C312yes—
32#63449C2×C318yesdirect product factor
33C23×C3×C9C2311yescharacteristic
34C2×C3×C4×C9—24no—
35C2×C3×C4×C9C2×C414yes—
36C2×C3×C4×C9C2314yes—
37C2×C3×C4×C9C2312yes—
38C2×C3×Dic9—22no—
39C2×C3×Dic9C2×C418yes—
40C2×C3×Dic9Q814yescomplemented
41C2×C3×Dic9C2312yes—
42C2×C3×Dic9C2314yes—
43C2×C3×Dic9C2312yes—
44C3×Dic18C2×C4116yescomplemented
45C2×C3×C4×Q8—91no—
46C2×#887—31no—
47C22×C32×C4C2×D312yes—
48C22×C32×C4C2×D311yescharacteristic
49C22×C3×Dic3—31no—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—31no—
52C2×C3×Dic6—38no—
53C2×C3×Dic6—34no—
54C32×C4⋊C4C2×D314yes—
55C3×Dic3⋊C4—38no—
56C3×C4×Dic3—38no—
57C3×C4×Dic3—34no—
58C22×C4×C9—21no—
59C22×C4×C9—22no—
60C22×C4×C9C22×C312yes—
61C22×C4×C9C22×C311yescharacteristic
62C22×Dic9C22×C311yescharacteristic
63C22×Dic9C22×C312yes—
64C22×Dic9C22×C311yescharacteristic
65C2×Dic18C3×C418yescomplemented
66C2×Dic18C22×C314yes—
67C9×C4⋊C4—24no—
68C9×C4⋊C4C22×C314yes—
69Dic9⋊C4C22×C318yes—
70C4×Dic9C22×C314yes—
71C4×Dic9C22×C318yes—
72C22×C3×C9C22×C412yes—
73C22×C3×C9C2×Q812yes—
74C22×C3×C9#4112yes—
75C22×C3×C9C2411yescharacteristic
76C3×C4×C9—24no—
77C3×C4×C9C22×C414yes—
78C3×Dic9—22no—
79C3×Dic9C22×C418yes—
80C3×Dic9C2×Q814yescomplemented
81C22×C3×Q8—91no—
82C22×Dic6—31no—
83C3×C4×Q8—98no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—92no—
86C2×C3×C4⋊C4D911yescharacteristic
87C2×C3×C4⋊C4C3×D311yescharacteristic
88C2×C3×C42—92no—
89C2×C3×C42—91no—
90C2×Dic3⋊C4—32no—
91C2×C4×Dic3—32no—
92C2×C4×Dic3—31no—
93#887—38no—
94C23×C32C22×D311yescharacteristic
95C2×C32×C4—24no—
96C2×C32×C4C4×D314yes—
97C2×C32×C4C22×D314yes—
98C2×C32×C4C22×D312yes—
99C2×C3×Dic3—38no—
100C2×C3×Dic3—34no—
101C2×C3×Dic3—32no—
102C2×C3×Dic3—34no—
103C2×C3×Dic3—32no—
104C2×C3×Dic3—62no—
105C3×Dic6—316no—
106C23×C9—21no—
107C23×C9C23×C311yescharacteristic
108C2×C4×C9—24no—
109C2×C4×C9—24no—
110C2×C4×C9—22no—
111C2×C4×C9—24no—
112C2×C4×C9—44no—
113C2×C4×C9C2×C3×C414yes—
114C2×C4×C9C23×C314yes—
115C2×C4×C9C23×C312yes—
116C2×Dic9—22no—
117C2×Dic9C2×C3×C418yes—
118C2×Dic9C3×Q814yescomplemented
119C2×Dic9C23×C312yes—
120C2×Dic9C23×C314yes—
121C2×Dic9C23×C312yes—
122Dic18C2×C3×C4116yescomplemented
123C2×C4×Q8—91noSylow
124C2×C3×C9C4×Q814yescomplemented
125C2×C3×C9C23×C411yescharacteristic
126C2×C3×C9C22×Q811yescharacteristic
127C2×C3×C9C2×#4111yescharacteristic
128C2×C3×Q8—98no—
129C2×C3×Q8—94no—
130C22×C3×C4—21no—
131C22×C3×C4—22no—
132C22×C3×C4—91no—
133C22×C3×C4—92no—
134C22×C3×C4—91no—
135C22×C3×C4C2×D912yes—
136C22×C3×C4C2×D911yescharacteristic
137C22×C3×C4C2×C3×D312yes—
138C22×C3×C4C2×C3×D311yescharacteristic
139C22×Dic3—31no—
140C22×Dic3—32no—
141C22×Dic3—31no—
142C2×Dic6—38no—
143C2×Dic6—34no—
144C3×C4⋊C4—24no—
145C3×C4⋊C4—98no—
146C3×C4⋊C4C2×D914yes—
147C3×C4⋊C4C2×C3×D314yes—
148C3×C42—98no—
149C3×C42—94no—
150Dic3⋊C4—38no—
151C4×Dic3—38no—
152C4×Dic3—34no—
153C22×C32C2×C4×D312yes—
154C22×C32#23712yes—
155C22×C32D3×Q812yes—
156C22×C32C23×D311yescharacteristic
157C32×C4—24no—
158C32×C4C2×C4×D314yes—
159C3×Dic3—38no—
160C3×Dic3—34no—
161C3×Dic3—62no—
162C22×C9—21no—
163C22×C9—22no—
164C22×C9—22no—
165C22×C9—22no—
166C22×C9C22×C3×C412yes—
167C22×C9C2×C3×Q812yes—
168C22×C9C3×#4112yes—
169C22×C9C24×C311yescharacteristic
170C4×C9—24no—
171C4×C9—24no—
172C4×C9—44no—
173C4×C9C22×C3×C414yes—
174Dic9—22no—
175Dic9C22×C3×C418yes—
176Dic9C2×C3×Q814yescomplemented
177C22×Q8—91no—
178C4×Q8—98no—
179C2×C4⋊C4—92no—
180C2×C4⋊C4C3×D911yescharacteristic
181C2×C42—92no—
182C2×C42—91no—
183C3×C9C2×C4×Q811yescharacteristic, complemented, Sylow
184C23×C3—21no—
185C23×C3C22×D911yescharacteristic
186C23×C3C22×C3×D311yescharacteristic
187C3×Q8—916no—
188C2×C3×C4—24no—
189C2×C3×C4—24no—
190C2×C3×C4—24no—
191C2×C3×C4—22no—
192C2×C3×C4—24no—
193C2×C3×C4—44no—
194C2×C3×C4—98no—
195C2×C3×C4—94no—
196C2×C3×C4—92no—
197C2×C3×C4—94no—
198C2×C3×C4—92no—
199C2×C3×C4—182no—
200C2×C3×C4C4×D914yes—
201C2×C3×C4C22×D914yes—
202C2×C3×C4C22×D912yes—
203C2×C3×C4C3×C4×D314yes—
204C2×C3×C4C22×C3×D314yes—
205C2×C3×C4C22×C3×D312yes—
206C2×Dic3—32no—
207C2×Dic3—34no—
208C2×Dic3—38no—
209C2×Dic3—34no—
210C2×Dic3—32no—
211C2×Dic3—62no—
212Dic6—316no—
213C2×C32#88714yes—
214C2×C32C22×C4×D311yescharacteristic
215C2×C32C2×#23711yescharacteristic
216C2×C32C2×D3×Q811yescharacteristic
217C2×C9—24no—
218C2×C9—21no—
219C2×C9—21no—
220C2×C9—21no—
221C2×C9C3×C4×Q814yescomplemented
222C2×C9C23×C3×C411yescharacteristic
223C2×C9C22×C3×Q811yescharacteristic
224C2×C9C2×C3×#4111yescharacteristic
225C2×Q8—98no—
226C2×Q8—94no—
227C22×C4—91no—
228C22×C4—92no—
229C22×C4—91no—
230C22×C4C2×C3×D912yes—
231C22×C4C2×C3×D911yescharacteristic
232C4⋊C4—98no—
233C4⋊C4C2×C3×D914yes—
234C42—98no—
235C42—94no—
236C22×C3—21no—
237C22×C3—22no—
238C22×C3—22no—
239C22×C3—22no—
240C22×C3C2×C4×D912yes—
241C22×C3#369912yes—
242C22×C3Q8×D912yes—
243C22×C3C23×D911yescharacteristic
244C22×C3C2×C3×C4×D312yes—
245C22×C3C3×#23712yes—
246C22×C3C3×D3×Q812yes—
247C22×C3C23×C3×D311yescharacteristic
248C3×C4—24no—
249C3×C4—24no—
250C3×C4—24no—
251C3×C4—44no—
252C3×C4—98no—
253C3×C4—94no—
254C3×C4—182no—
255C3×C4C2×C4×D914yes—
256C3×C4C2×C3×C4×D314yes—
257Dic3—38no—
258Dic3—34no—
259Dic3—62no—
260C32C2×#88711yescharacteristic
261C9—21no—
262C9C2×C3×C4×Q811yescharacteristic, complemented
263C23C22×C3×D911yescharacteristic
264Q8—916no—
265C2×C4—24no—
266C2×C4—92no—
267C2×C4—94no—
268C2×C4—98no—
269C2×C4—94no—
270C2×C4—92no—
271C2×C4—182no—
272C2×C4C3×C4×D914yes—
273C2×C4C22×C3×D914yes—
274C2×C4C22×C3×D912yes—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—24no—
279C2×C3#6344914yesdirect product factor
280C2×C3C22×C4×D911yescharacteristic
281C2×C3C2×#369911yescharacteristic
282C2×C3C2×Q8×D911yescharacteristic
283C2×C3C3×#88714yes—
284C2×C3C22×C3×C4×D311yescharacteristic
285C2×C3C2×C3×#23711yescharacteristic
286C2×C3C2×C3×D3×Q811yescharacteristic
287C22C2×C3×C4×D912yes—
288C22C3×#369912yes—
289C22C3×Q8×D912yes—
290C22C23×C3×D911yescharacteristic
291C4—24no—
292C4—98no—
293C4—94no—
294C4—182no—
295C4C2×C3×C4×D914yes—
296C3—21no—
297C3C2×#6344911yescharacteristic, direct product factor
298C3C2×C3×#88711yescharacteristic
299C2C3×#6344914yesdirect product factor
300C2C22×C3×C4×D911yescharacteristic
301C2C2×C3×#369911yescharacteristic
302C2C2×C3×Q8×D911yescharacteristic
subgroupG/H
centerC23×C3C22×D9
commutatorC2×C9C23×C3×C4
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socle, prime radicalC23×C32C22×D3
cosocleC22×C9C24×C3
hypercenterC2×C3×C4⋊C4D9
hypocenterC9C2×C3×C4×Q8
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. C3×#63449C2direct (N × H)
8. C2×#63449C3direct (N × H)
14. C2×C3×Dic18C4semidirect
32. #63449C2×C3direct (N × H)
40. C2×C3×Dic9Q8semidirect
44. C3×Dic18C2×C4semidirect
65. C2×Dic18C3×C4semidirect
80. C3×Dic9C2×Q8semidirect
118. C2×Dic9C3×Q8semidirect
122. Dic18C2×C3×C4semidirect
124. C2×C3×C9C4×Q8semidirect
176. Dic9C2×C3×Q8semidirect
183. C3×C9C2×C4×Q8semidirect
221. C2×C9C3×C4×Q8semidirect
262. C9C2×C3×C4×Q8semidirect
279. C2×C3#63449direct (N × H)
297. C3C2×#63449direct (N × H)
299. C2C3×#63449direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Dic182
C2×C3×C9×C4⋊C42
C2×C3×Dic9⋊C42
C2×C3×C4×Dic92
C3×#634492
C2×C3×#8873
C2×#634493