Finite Group Databasebeta

← C22×C3×#120⋊C3 #11919690

Subgroups — C22×C3×#120⋊C3 #11919690

Order 1152.

1824 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1824 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊A4C211yesmaximal, characteristic, complemented
2C2×C3×#120⋊C3C216yesmaximal, direct product factor
3C22×C3×#120C311yesmaximal, characteristic, complemented
4C22×#120⋊C3C313yesmaximal, direct product factor
5C2×C3×C22⋊A4C2213yescomplemented
6C23×C3×A4—41nomaximal
7C3×#120⋊C3C2214yesdirect product factor
8C26×C3C2×C311yescharacteristic, complemented
9C22×C22⋊A4C2×C313yescomplemented
10C23×C3×D4—31no—
11C2×C3×#120—36no—
12C2×C3×#120C2×C316yescomplemented
13C22×C3×C22⋊C4—31no—
14C2×#120⋊C3C2×C3118yesdirect product factor
15C3×C22⋊A4C2311yescharacteristic, complemented
16C22×C3×A4—41no—
17C22×C3×A4—46no—
18C22×C3×A4—82no—
19C22×#120C3211yescharacteristic, complemented, Sylow
20C25×C3—31no—
21C25×C3—33no—
22C25×C3—62no—
23C25×C3—66no—
24C25×C3A411yescharacteristic, complemented
25C25×C3C22×C313yescomplemented
26C2×C22⋊A4C22×C319yescomplemented
27C23×A4—43no—
28C22×C3×D4—36no—
29C22×C3×D4—36no—
30C22×C3×D4—62no—
31C22×C3×D4—66no—
32C23×C3×C4—31no—
33C3×#120—38no—
34C3×#120—312no—
35C3×#120C22×C314yescomplemented
36C2×C3×C22⋊C4—36no—
37C2×C3×C22⋊C4—36no—
38#120⋊C3C22×C3112yesdirect product factor
39C23×C32—161no—
40C2×C3×A4—43no—
41C2×C3×A4—44no—
42C2×C3×A4—86no—
43C26C2×C3211yescharacteristic, complemented
44C23×D4—31no—
45C2×#120—36no—
46C2×#120C2×C3216yescomplemented
47C22×C22⋊C4—31no—
48C24×C3—22no—
49C24×C3—36no—
50C24×C3—32no—
51C24×C3—33no—
52C24×C3—33no—
53C24×C3—32no—
54C24×C3—33no—
55C24×C3—66no—
56C24×C3—66no—
57C24×C3—61no—
58C24×C3—66no—
59C24×C3—66no—
60C24×C3—62no—
61C24×C3—66no—
62C24×C3—612no—
63C24×C3—66no—
64C24×C3—612no—
65C24×C3—63no—
66C24×C3—66no—
67C24×C3—62no—
68C24×C3—64no—
69C24×C3—612no—
70C24×C3—612no—
71C24×C3C2×A411yescharacteristic
72C24×C3C2×A416yescomplemented
73C24×C3C23×C311yescharacteristic, complemented
74C22⋊A4C23×C313yescomplemented
75C22×A4—43no—
76C22×A4—418no—
77C22×A4—86no—
78C2×C3×D4—34no—
79C2×C3×D4—312no—
80C2×C3×D4—612no—
81C2×C3×D4—612no—
82C2×C3×D4—612no—
83C2×C3×D4—612no—
84C22×C3×C4—36no—
85C22×C3×C4—63no—
86C22×C3×C4—61no—
87C3×C22⋊C4—312no—
88C3×C22⋊C4—34no—
89C22×C32—166no—
90C22×C32—161no—
91C3×A4—41no—
92C3×A4—82no—
93C25—31no—
94C25—33no—
95C25—62no—
96C25—66no—
97C25C3×A411yescharacteristic, complemented
98C25C22×C3213yescomplemented
99C22×D4—36no—
100C22×D4—36no—
101C22×D4—62no—
102C22×D4—66no—
103C23×C4—31no—
104#120—38no—
105#120—312no—
106#120C22×C3214yescomplemented
107C2×C22⋊C4—36no—
108C2×C22⋊C4—36no—
109C23×C3—26no—
110C23×C3—33no—
111C23×C3—31no—
112C23×C3—36no—
113C23×C3—36no—
114C23×C3—36no—
115C23×C3—36no—
116C23×C3—33no—
117C23×C3—31no—
118C23×C3—41no—
119C23×C3—62no—
120C23×C3—66no—
121C23×C3—66no—
122C23×C3—66no—
123C23×C3—66no—
124C23×C3—612no—
125C23×C3—66no—
126C23×C3—66no—
127C23×C3—612no—
128C23×C3—66no—
129C23×C3—66no—
130C23×C3—612no—
131C23×C3—612no—
132C23×C3—66no—
133C23×C3—66no—
134C23×C3—66no—
135C23×C3—66no—
136C23×C3—66no—
137C23×C3—612no—
138C23×C3—612no—
139C23×C3—624no—
140C23×C3—624no—
141C23×C3—612no—
142C23×C3—612no—
143C23×C3—62no—
144C23×C3—123no—
145C23×C3—122no—
146C23×C3—163no—
147C23×C3C22×A413yes—
148C23×C3C22×A414yescomplemented
149C2×A4—49no—
150C2×A4—412no—
151C2×A4—818no—
152C3×D4—68no—
153C3×D4—624no—
154C2×C3×C4—34no—
155C2×C3×C4—66no—
156C2×C3×C4—66no—
157C2×C32—164no—
158C2×C32—163no—
159C24—22no—
160C24—36no—
161C24—32no—
162C24—33no—
163C24—33no—
164C24—32no—
165C24—33no—
166C24—66no—
167C24—66no—
168C24—61no—
169C24—66no—
170C24—66no—
171C24—62no—
172C24—66no—
173C24—612no—
174C24—66no—
175C24—612no—
176C24—63no—
177C24—66no—
178C24—62no—
179C24—64no—
180C24—612no—
181C24—612no—
182C24C2×C3×A411yescharacteristic
183C24C2×C3×A416yescomplemented
184C24C23×C3211yescharacteristic, complemented
185C2×D4—34no—
186C2×D4—312no—
187C2×D4—612no—
188C2×D4—612no—
189C2×D4—612no—
190C2×D4—612no—
191C22×C4—36no—
192C22×C4—63no—
193C22×C4—61no—
194C22⋊C4—312no—
195C22⋊C4—34no—
196C22×C3—22no—
197C22×C3—33no—
198C22×C3—32no—
199C22×C3—33no—
200C22×C3—33no—
201C22×C3—32no—
202C22×C3—36no—
203C22×C3—46no—
204C22×C3—64no—
205C22×C3—62no—
206C22×C3—66no—
207C22×C3—612no—
208C22×C3—66no—
209C22×C3—612no—
210C22×C3—66no—
211C22×C3—66no—
212C22×C3—62no—
213C22×C3—66no—
214C22×C3—612no—
215C22×C3—64no—
216C22×C3—612no—
217C22×C3—66no—
218C22×C3—66no—
219C22×C3—126no—
220C22×C3—1618no—
221C22×C3—163no—
222C22×C3#120⋊C311yescharacteristic, direct product factor
223C22×C3C23×A411yescharacteristic
224A4—43no—
225A4—86no—
226C3×C4—64no—
227C32—161noSylow
228C23—26no—
229C23—33no—
230C23—31no—
231C23—36no—
232C23—36no—
233C23—36no—
234C23—36no—
235C23—31no—
236C23—33no—
237C23—41no—
238C23—62no—
239C23—66no—
240C23—66no—
241C23—66no—
242C23—66no—
243C23—612no—
244C23—66no—
245C23—66no—
246C23—612no—
247C23—66no—
248C23—66no—
249C23—612no—
250C23—612no—
251C23—66no—
252C23—66no—
253C23—66no—
254C23—66no—
255C23—66no—
256C23—612no—
257C23—612no—
258C23—624no—
259C23—624no—
260C23—612no—
261C23—612no—
262C23—62no—
263C23—123no—
264C23—122no—
265C23C22×C3×A413yes—
266C23C22×C3×A414yescomplemented
267D4—68no—
268D4—624no—
269C2×C4—34no—
270C2×C4—66no—
271C2×C4—66no—
272C2×C3—31no—
273C2×C3—33no—
274C2×C3—44no—
275C2×C3—62no—
276C2×C3—66no—
277C2×C3—1612no—
278C2×C3—169no—
279C2×C3C2×#120⋊C313yesdirect product factor
280C22—22no—
281C22—33no—
282C22—32no—
283C22—33no—
284C22—33no—
285C22—32no—
286C22—36no—
287C22—46no—
288C22—64no—
289C22—62no—
290C22—66no—
291C22—612no—
292C22—66no—
293C22—612no—
294C22—66no—
295C22—66no—
296C22—62no—
297C22—66no—
298C22—612no—
299C22—64no—
300C22—612no—
301C22—66no—
302C22—66no—
303C22—126no—
304C22C3×#120⋊C311yescharacteristic, direct product factor
305C22C23×C3×A411yescharacteristic
306C4—64no—
307C3—163no—
308C3C22×#120⋊C311yescharacteristic, direct product factor
309C2—31no—
310C2—33no—
311C2—44no—
312C2—62no—
313C2—66no—
314C2C2×C3×#120⋊C313yesdirect product factor
subgroupG/H
center, hypercenterC22×C3#120⋊C3
commutator, cosocle, hypocenterC24C23×C32
FrattiniC22C23×C3×A4
FittingC22×C3×#120C3
socleC24×C3C2×A4
LCSG ▷ C24
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×C22⋊A4C2semidirect
2. C2×C3×#120⋊C3C2direct (N × H)
3. C22×C3×#120C3semidirect
4. C22×#120⋊C3C3direct (N × H)
5. C2×C3×C22⋊A4C22semidirect
7. C3×#120⋊C3C22direct (N × H)
8. C26×C3C2×C3semidirect
9. C22×C22⋊A4C2×C3semidirect
12. C2×C3×#120C2×C3semidirect
14. C2×#120⋊C3C2×C3direct (N × H)
15. C3×C22⋊A4C23semidirect
19. C22×#120C32semidirect
24. C25×C3A4semidirect
25. C25×C3C22×C3semidirect
26. C2×C22⋊A4C22×C3semidirect
35. C3×#120C22×C3semidirect
38. #120⋊C3C22×C3direct (N × H)
43. C26C2×C32semidirect
46. C2×#120C2×C32semidirect
72. C24×C3C2×A4semidirect
73. C24×C3C23×C3semidirect
74. C22⋊A4C23×C3semidirect
97. C25C3×A4semidirect
98. C25C22×C32semidirect
106. #120C22×C32semidirect
148. C23×C3C22×A4semidirect
183. C24C2×C3×A4semidirect
184. C24C23×C32semidirect
222. C22×C3#120⋊C3direct (N × H)
266. C23C22×C3×A4semidirect
279. C2×C3C2×#120⋊C3direct (N × H)
304. C22C3×#120⋊C3direct (N × H)
308. C3C22×#120⋊C3direct (N × H)
314. C2C2×C3×#120⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊A42
C2×C3×#120⋊C32
C22×C3×#1203
C22×#120⋊C33
C23×C3×A44