Finite Group Databasebeta

← C3×C4×#120⋊C3 #11919412

Subgroups — C3×C4×#120⋊C3 #11919412

Order 1152.

568 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C3×C4×C22⋊A4C211yesmaximal, characteristic, complemented
2C2×C3×#120⋊C3C211yesmaximal, characteristic
3C3×#379⋊C3C211yesmaximal, characteristic, complemented
4C3×C4×#120C311yesmaximal, characteristic, complemented
5C4×#120⋊C3C313yesmaximal, direct product factor
6C2×C3×C22⋊A4C2211yescharacteristic
7C2×C3×C4×A4—41nomaximal
8C3×#120⋊C3C412yesdirect product factor
9C24×C3×C4C2×C311yescharacteristic, complemented
10C4×C22⋊A4C2×C313yescomplemented
11C2×C3×#120C2×C311yescharacteristic
12C2×C3×C4×D4—31no—
13C2×#120⋊C3C2×C313yes—
14C3×#386—31no—
15C3×#385—31no—
16C3×#379C2×C311yescharacteristic, complemented
17C3×C4×C22⋊C4—31no—
18#379⋊C3C2×C313yescomplemented
19C3×C22⋊A4C2×C411yescharacteristic, complemented
20C22×C3×A4—41no—
21C3×C4×A4—41no—
22C3×C4×A4—41no—
23C3×C4×A4—82no—
24C4×#120C3211yescharacteristic, complemented, Sylow
25C25×C3C22×C311yescharacteristic
26C2×C22⋊A4C22×C313yes—
27C22×C3×D4—31no—
28C23×C3×C4—31no—
29C23×C3×C4—31no—
30C23×C3×C4—62no—
31C23×C3×C4—62no—
32C23×C3×C4A411yescharacteristic, complemented
33C2×C4×A4—43no—
34C3×#120—32no—
35C3×#120C3×C412yescomplemented
36C3×C4×D4—62no—
37C3×C4×D4—62no—
38C2×C3×C4⋊C4—31no—
39C2×C3×C22⋊C4—31no—
40C2×C3×C22⋊C4—31no—
41C2×C3×C22⋊C4—31no—
42C2×C3×C22⋊C4—31no—
43C2×C3×C42—31no—
44#120⋊C3C3×C416yesdirect product factor
45C3×#95—32no—
46C2×C3×A4—42no—
47C2×C3×A4—41no—
48C2×C3×A4—82no—
49C2×C32×C4—161no—
50C24×C4C2×C3211yescharacteristic, complemented
51C2×#120C2×C3211yescharacteristic
52C2×C4×D4—31no—
53#386—31no—
54#385—31no—
55#379C2×C3211yescharacteristic, complemented
56C4×C22⋊C4—31no—
57C24×C3—31no—
58C24×C3—31no—
59C24×C3—62no—
60C24×C3—62no—
61C24×C3C2×C3×C411yescharacteristic, complemented
62C24×C3C2×A411yescharacteristic
63C22⋊A4C2×C3×C413yescomplemented
64C22×A4—43no—
65C2×C3×D4—32no—
66C2×C3×D4—32no—
67C2×C3×D4—62no—
68C2×C3×D4—62no—
69C22×C3×C4—22no—
70C22×C3×C4—32no—
71C22×C3×C4—32no—
72C22×C3×C4—31no—
73C22×C3×C4—31no—
74C22×C3×C4—31no—
75C22×C3×C4—61no—
76C22×C3×C4—62no—
77C22×C3×C4—62no—
78C22×C3×C4—64no—
79C22×C3×C4—61no—
80C22×C3×C4—62no—
81C22×C3×C4—62no—
82C22×C3×C4—64no—
83C22×C3×C4—62no—
84C22×C3×C4—62no—
85C22×C3×C4C2×A411yescharacteristic
86C22×C3×C4C2×A411yescharacteristic
87C4×A4—43no—
88C4×A4—43no—
89C4×A4—86no—
90C3×C4⋊C4—62no—
91C3×C22⋊C4—32no—
92C3×C22⋊C4—32no—
93C3×C22⋊C4—62no—
94C3×C22⋊C4—62no—
95C3×C22⋊C4—62no—
96C3×C42—61no—
97C3×C42—61no—
98C22×C32—161no—
99C3×A4—41no—
100C3×A4—82no—
101C32×C4—161no—
102C32×C4—161no—
103C25C22×C3211yescharacteristic
104C22×D4—31no—
105C23×C4—31no—
106C23×C4—31no—
107C23×C4—62no—
108C23×C4—62no—
109C23×C4C3×A411yescharacteristic, complemented
110#120—32no—
111#120C32×C412yescomplemented
112C4×D4—62no—
113C4×D4—62no—
114C2×C4⋊C4—31no—
115C2×C22⋊C4—31no—
116C2×C22⋊C4—31no—
117C2×C22⋊C4—31no—
118C2×C22⋊C4—31no—
119C2×C42—31no—
120#95—32no—
121C23×C3—22no—
122C23×C3—32no—
123C23×C3—32no—
124C23×C3—31no—
125C23×C3—31no—
126C23×C3—62no—
127C23×C3—62no—
128C23×C3—61no—
129C23×C3—62no—
130C23×C3—62no—
131C23×C3—62no—
132C23×C3—64no—
133C23×C3—61no—
134C23×C3—62no—
135C23×C3—62no—
136C23×C3—64no—
137C23×C3C4×A412yescomplemented
138C23×C3C22×A411yescharacteristic
139C2×A4—43no—
140C2×A4—46no—
141C2×A4—86no—
142C3×D4—64no—
143C3×D4—64no—
144C2×C3×C4—31no—
145C2×C3×C4—31no—
146C2×C3×C4—32no—
147C2×C3×C4—32no—
148C2×C3×C4—41no—
149C2×C3×C4—62no—
150C2×C3×C4—64no—
151C2×C3×C4—61no—
152C2×C3×C4—61no—
153C2×C3×C4—62no—
154C2×C3×C4—61no—
155C2×C3×C4—64no—
156C2×C3×C4—62no—
157C2×C3×C4—62no—
158C2×C3×C4—62no—
159C2×C3×C4—61no—
160C2×C3×C4—62no—
161C2×C3×C4—121no—
162C2×C3×C4—163no—
163C2×C32—162no—
164C2×C32—161no—
165C24—31no—
166C24—31no—
167C24—62no—
168C24—62no—
169C24C2×C32×C411yescharacteristic, complemented
170C24C2×C3×A411yescharacteristic
171C2×D4—32no—
172C2×D4—32no—
173C2×D4—62no—
174C2×D4—62no—
175C22×C4—22no—
176C22×C4—32no—
177C22×C4—32no—
178C22×C4—31no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—61no—
182C22×C4—62no—
183C22×C4—62no—
184C22×C4—64no—
185C22×C4—61no—
186C22×C4—62no—
187C22×C4—62no—
188C22×C4—64no—
189C22×C4—62no—
190C22×C4—62no—
191C22×C4C2×C3×A411yescharacteristic
192C22×C4C2×C3×A411yescharacteristic
193C4⋊C4—62no—
194C22⋊C4—32no—
195C22⋊C4—32no—
196C22⋊C4—62no—
197C22⋊C4—62no—
198C22⋊C4—62no—
199C42—61no—
200C42—61no—
201C22×C3—22no—
202C22×C3—31no—
203C22×C3—31no—
204C22×C3—32no—
205C22×C3—32no—
206C22×C3—41no—
207C22×C3—62no—
208C22×C3—62no—
209C22×C3—62no—
210C22×C3—62no—
211C22×C3—64no—
212C22×C3—62no—
213C22×C3—62no—
214C22×C3—62no—
215C22×C3—64no—
216C22×C3—62no—
217C22×C3—121no—
218C22×C3—163no—
219C22×C3C2×C4×A411yescharacteristic
220A4—43no—
221A4—86no—
222C3×C4—31no—
223C3×C4—41no—
224C3×C4—62no—
225C3×C4—62no—
226C3×C4—121no—
227C3×C4—163no—
228C3×C4—163no—
229C3×C4#120⋊C311yescharacteristic, direct product factor
230C32—161noSylow
231C23—22no—
232C23—32no—
233C23—32no—
234C23—31no—
235C23—31no—
236C23—62no—
237C23—62no—
238C23—61no—
239C23—62no—
240C23—62no—
241C23—62no—
242C23—64no—
243C23—61no—
244C23—62no—
245C23—62no—
246C23—64no—
247C23C3×C4×A412yescomplemented
248C23C22×C3×A411yescharacteristic
249D4—64no—
250D4—64no—
251C2×C4—31no—
252C2×C4—31no—
253C2×C4—32no—
254C2×C4—32no—
255C2×C4—41no—
256C2×C4—62no—
257C2×C4—64no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—62no—
261C2×C4—61no—
262C2×C4—64no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—61no—
267C2×C4—62no—
268C2×C4—121no—
269C2×C3—31no—
270C2×C3—31no—
271C2×C3—42no—
272C2×C3—62no—
273C2×C3—62no—
274C2×C3—166no—
275C2×C3—163no—
276C2×C3C2×#120⋊C311yescharacteristic
277C22—22no—
278C22—31no—
279C22—31no—
280C22—32no—
281C22—32no—
282C22—41no—
283C22—62no—
284C22—62no—
285C22—62no—
286C22—62no—
287C22—64no—
288C22—62no—
289C22—62no—
290C22—62no—
291C22—64no—
292C22—62no—
293C22—121no—
294C22C2×C3×C4×A411yescharacteristic
295C4—31no—
296C4—41no—
297C4—62no—
298C4—62no—
299C4—121no—
300C4C3×#120⋊C311yescharacteristic, direct product factor
301C3—163no—
302C3C4×#120⋊C311yescharacteristic, direct product factor
303C2—31no—
304C2—31no—
305C2—42no—
306C2—62no—
307C2—62no—
308C2C2×C3×#120⋊C311yescharacteristic
subgroupG/H
center, hypercenterC3×C4#120⋊C3
commutator, hypocenterC24C2×C32×C4
FrattiniC23C22×C3×A4
FittingC3×C4×#120C3
socleC23×C3C22×A4
prime radicalC2×C3×#120⋊C3C2
cosocleC25C22×C32
LCSG ▷ C24
UCSC1 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×C4×C22⋊A4C2semidirect
3. C3×#379⋊C3C2semidirect
4. C3×C4×#120C3semidirect
5. C4×#120⋊C3C3direct (N × H)
8. C3×#120⋊C3C4direct (N × H)
9. C24×C3×C4C2×C3semidirect
10. C4×C22⋊A4C2×C3semidirect
16. C3×#379C2×C3semidirect
18. #379⋊C3C2×C3semidirect
19. C3×C22⋊A4C2×C4semidirect
24. C4×#120C32semidirect
32. C23×C3×C4A4semidirect
35. C3×#120C3×C4semidirect
44. #120⋊C3C3×C4direct (N × H)
50. C24×C4C2×C32semidirect
55. #379C2×C32semidirect
61. C24×C3C2×C3×C4semidirect
63. C22⋊A4C2×C3×C4semidirect
109. C23×C4C3×A4semidirect
111. #120C32×C4semidirect
137. C23×C3C4×A4semidirect
169. C24C2×C32×C4semidirect
229. C3×C4#120⋊C3direct (N × H)
247. C23C3×C4×A4semidirect
300. C4C3×#120⋊C3direct (N × H)
302. C3C4×#120⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×C22⋊A42
C2×C3×#120⋊C32
C3×#379⋊C32
C3×C4×#1203
C4×#120⋊C33
C2×C3×C4×A44