Finite Group Databasebeta

← C4⋊(C3⋊(#120⋊C3)) #11921357

Subgroups — C4⋊(C3⋊(#120⋊C3)) #11921357

Order 1152.

529 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 529 conjugacy classes of subgroups — too many to draw. Only its 33 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⋊C3)C212yesmaximal, complemented
3#84032C311yesmaximal, characteristic, complemented
4C4⋊(#120⋊C3)—31nomaximal
5C2×C3×C22⋊A4C2211yescharacteristic
6A4×D12—41nomaximal
7C3⋊(#120⋊C3)—22no—
8C24×C3×C4C2×C311yescharacteristic, complemented
9C4×C22⋊A4—21no—
10C4×C22⋊A4D311yescharacteristic, complemented
11C2×#953C2×C312yescomplemented
12C2×#930—31no—
13C2×#120⋊C3—32no—
14#5213—32no—
15#5212—31no—
16C3×C22⋊A4D411yescharacteristic, complemented
17C2×D3×A4—42no—
18C3×C4×A4—41no—
19C3×C4×A4—82no—
20#2847—31noSylow
21C25×C3C22×C311yescharacteristic
22C2×C22⋊A4—21no—
23C2×C22⋊A4C2×D311yescharacteristic
24C23×C3×C4—31no—
25C23×C3×C4—31no—
26C23×C3×C4—62no—
27C23×C3×C4—62no—
28C22×#67—32no—
29C22×D12A411yescharacteristic, complemented
30D4×A4—121no—
31#953—22no—
32#953—62no—
33C2×#217—32no—
34#930—62no—
35#930—62no—
36C2×(C2×D3)⋊C4—32no—
37C2×C4⋊Dic3—31no—
38#120⋊C3—62no—
39C3⋊#95—32no—
40C2×C3×A4—41no—
41C2×C3×A4—82no—
42D3×A4—82no—
43C3×D12—161no—
44C24×C4C3×D311yescharacteristic, complemented
45C2×#121—91no—
46C2×#120—32no—
47#386—92no—
48#380—91no—
49C24×C3—31no—
50C24×C3—31no—
51C24×C3—62no—
52C24×C3—62no—
53C24×C3C3×D411yescharacteristic, complemented
54C23×D3C2×A412yes—
55C22⋊A4—21no—
56C22⋊A4D1211yescharacteristic, complemented
57C22×A4—122no—
58C22×C3×C4—22no—
59C22×C3×C4—32no—
60C22×C3×C4—32no—
61C22×C3×C4—31no—
62C22×C3×C4—62no—
63C22×C3×C4—62no—
64C22×C3×C4—64no—
65C22×C3×C4—62no—
66C22×C3×C4—62no—
67C22×C3×C4—64no—
68C22×C3×C4—62no—
69C22×C3×C4—62no—
70C22×C3×C4C2×A411yescharacteristic
71C2×#67—62no—
72C2×#67—62no—
73C2×#67—64no—
74C2×#67—64no—
75C22×Dic3—32no—
76C2×D12—61no—
77C2×D12—61no—
78C4×A4—41no—
79C4×A4—82no—
80C4×A4—84no—
81C4×A4—81no—
82#217—62no—
83#217—62no—
84(C2×D3)⋊C4—64no—
85C4⋊Dic3—61no—
86C4⋊Dic3—61no—
87C2×C3×D3—162no—
88C3×A4—41no—
89C3×A4—82no—
90C32×C4—161no—
91C25C2×C3×D311yescharacteristic
92C22×D4—31no—
93C22×D4—92no—
94C23×C4—31no—
95C23×C4—31no—
96C23×C4—62no—
97C23×C4—62no—
98#121—182no—
99#121—182no—
100#120—62no—
101#120—182no—
102C2×C4⋊C4—91no—
103C2×C22⋊C4—92no—
104C2×C22⋊C4—92no—
105#95—92no—
106C23×C3—22no—
107C23×C3—32no—
108C23×C3—32no—
109C23×C3—31no—
110C23×C3—31no—
111C23×C3—62no—
112C23×C3—62no—
113C23×C3—62no—
114C23×C3—62no—
115C23×C3—62no—
116C23×C3—64no—
117C23×C3—62no—
118C23×C3—62no—
119C23×C3—64no—
120C23×C3C22×A411yescharacteristic
121C22×D3—22no—
122C22×D3—62no—
123C22×D3—62no—
124C2×A4—41no—
125C2×A4—82no—
126C2×A4—84no—
127C2×A4—81no—
128C2×A4—242no—
129C3×D4—481no—
130C2×C3×C4—31no—
131C2×C3×C4—31no—
132C2×C3×C4—32no—
133C2×C3×C4—62no—
134C2×C3×C4—64no—
135C2×C3×C4—62no—
136C2×C3×C4—64no—
137C2×C3×C4—62no—
138C2×C3×C4—62no—
139C2×C3×C4—62no—
140#67—124no—
141#67—124no—
142C2×Dic3—62no—
143C2×Dic3—62no—
144C2×Dic3—62no—
145D12—41no—
146D12—121no—
147C2×C32—161no—
148C3×D3—322no—
149C24—31no—
150C24—31no—
151C24—32no—
152C24—62no—
153C24—62no—
154C24C3×D1211yescharacteristic, complemented
155C2×D4—181no—
156C2×D4—181no—
157C2×D4—182no—
158C2×D4—182no—
159C2×D4—184no—
160C2×D4—184no—
161C22×C4—22no—
162C22×C4—32no—
163C22×C4—32no—
164C22×C4—31no—
165C22×C4—62no—
166C22×C4—62no—
167C22×C4—64no—
168C22×C4—62no—
169C22×C4—62no—
170C22×C4—64no—
171C22×C4—62no—
172C22×C4—62no—
173C22×C4—92no—
174C22×C4D3×A411yescharacteristic
175C4⋊C4—181no—
176C4⋊C4—181no—
177C22⋊C4—184no—
178C22⋊C4—182no—
179C22⋊C4—182no—
180C22×C3—22no—
181C22×C3—31no—
182C22×C3—31no—
183C22×C3—32no—
184C22×C3—32no—
185C22×C3—62no—
186C22×C3—62no—
187C22×C3—62no—
188C22×C3—64no—
189C22×C3—62no—
190C22×C3—62no—
191C22×C3—62no—
192C22×C3—64no—
193C22×C3—62no—
194C22×C3—482no—
195C22×C3D4×A411yescharacteristic
196C2×D3—42no—
197C2×D3—122no—
198C2×D3—122no—
199A4—41no—
200A4—82no—
201A4—84no—
202A4—81no—
203C3×C4—31no—
204C3×C4—62no—
205C3×C4—161no—
206C3×C4—321no—
207C3×C4#120⋊C311yescharacteristic, complemented
208Dic3—122no—
209C32—161noSylow
210C23—22no—
211C23—32no—
212C23—32no—
213C23—31no—
214C23—31no—
215C23—62no—
216C23—62no—
217C23—62no—
218C23—62no—
219C23—62no—
220C23—64no—
221C23—62no—
222C23—62no—
223C23—64no—
224C23—62no—
225C23—182no—
226C23—182no—
227C23C2×D3×A411yescharacteristic
228D4—121no—
229D4—361no—
230D4—364no—
231D4—364no—
232C2×C4—31no—
233C2×C4—31no—
234C2×C4—32no—
235C2×C4—62no—
236C2×C4—64no—
237C2×C4—62no—
238C2×C4—64no—
239C2×C4—62no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—182no—
243C2×C4—182no—
244C2×C4—182no—
245C2×C3—31no—
246C2×C3—31no—
247C2×C3—62no—
248C2×C3—62no—
249C2×C3—161no—
250C2×C3—321no—
251C2×C3—962no—
252C2×C3C2×#120⋊C311yescharacteristic
253D3—82no—
254C22—22no—
255C22—31no—
256C22—31no—
257C22—32no—
258C22—32no—
259C22—62no—
260C22—62no—
261C22—62no—
262C22—64no—
263C22—62no—
264C22—62no—
265C22—62no—
266C22—64no—
267C22—62no—
268C22—122no—
269C22—362no—
270C22—362no—
271C22A4×D1211yescharacteristic
272C4—31no—
273C4—62no—
274C4—362no—
275C4C3⋊(#120⋊C3)11yescharacteristic, complemented
276C3—161no—
277C3—321no—
278C3C4⋊(#120⋊C3)11yescharacteristic, complemented
279C2—31no—
280C2—31no—
281C2—62no—
282C2—62no—
283C2—242no—
284C2C2×C3⋊(#120⋊C3)11yescharacteristic
subgroupG/H
centerC2C2×C3⋊(#120⋊C3)
commutator, cosocleC25×C3C22×C3
FrattiniC23C2×D3×A4
FittingC24×C3×C4C2×C3
socleC23×C3C22×A4
hypercenterC4C3⋊(#120⋊C3)
hypocenterC24×C3C3×D4
LCSG ▷ C25×C3 ▷ C24×C3
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C3×C4×C22⋊A4C2semidirect
2. C2×C3⋊(#120⋊C3)C2semidirect
3. #84032C3semidirect
8. C24×C3×C4C2×C3semidirect
10. C4×C22⋊A4D3semidirect
11. C2×#953C2×C3semidirect
16. C3×C22⋊A4D4semidirect
29. C22×D12A4semidirect
44. C24×C4C3×D3semidirect
53. C24×C3C3×D4semidirect
56. C22⋊A4D12semidirect
154. C24C3×D12semidirect
207. C3×C4#120⋊C3semidirect
275. C4C3⋊(#120⋊C3)semidirect
278. C3C4⋊(#120⋊C3)semidirect

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⋊C3)2
#840323
C4⋊(#120⋊C3)3
A4×D124