Finite Group Databasebeta

← C4⋊(C3⋊(A4⋊C8)) #11921001

Subgroups — C4⋊(C3⋊(A4⋊C8)) #11921001

Order 1152.

406 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 107 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×C42×A4C211yesmaximal, characteristic
2C2×C3⋊(A4⋊C8)C212yesmaximal
3C4⋊(A4⋊C8)—33nomaximal
4#71530—31nomaximal
5C2×C3×C4×A4C411yescharacteristic
6C2×C3×C4×A4C411yescharacteristic
7C2×C3×C4×A4C2211yescharacteristic
8C3⋊(A4⋊C8)—22no—
9C4⋊(C3⋊(C3⋊C8))—41nomaximal
10C22×C3×C42D311yescharacteristic
11C42×A4D313yes—
12C2×A4⋊C8—36no—
13C2×#830—32no—
14C2×C4⋊(C3⋊C8)—31no—
15C2×C4⋊(C3⋊C8)—31no—
16#4524—32no—
17C22×C3×A4C2×C411yescharacteristic
18C3×C4×A4—21no—
19C3×C4×A4C812yescomplemented
20C3×C4×A4D411yescharacteristic
21C3×C4×A4Q811yescharacteristic
22C32×C42—41no—
23C2×C3⋊(C3⋊C8)—42no—
24#1797—91noSylow
25C23×C3×C4Dic311yescharacteristic
26C23×C3×C4Dic311yescharacteristic
27C23×C3×C4C2×D311yescharacteristic
28C2×C4×A4Dic313yes—
29C2×C4×A4Dic313yes—
30C2×C4×A4C2×D313yes—
31C2×C3×C42—31no—
32C2×C3×C42—31no—
33C2×C3×C42—31no—
34C2×C3×C42—31no—
35C22×C3⋊C8—32no—
36C22×C3⋊C8—32no—
37A4⋊C8—66no—
38#830—62no—
39#830—62no—
40C4⋊(C3⋊C8)—61no—
41C4⋊(C3⋊C8)—61no—
42C4⋊(C3⋊C8)—61no—
43C4⋊(C3⋊C8)—61no—
44C4⋊(C3⋊C8)—123no—
45C2×C3×A4C4⋊C411yescharacteristic
46C2×C3×A4C2×C811yescharacteristic
47C2×C3×A4M1611yescharacteristic
48C2×C32×C4—41no—
49C2×C32×C4—41no—
50C2×C32×C4—41no—
51C3⋊(C3⋊C8)—82no—
52C22×C42C3⋊D311yescharacteristic
53C2×C4⋊C8—91no—
54C2×C4⋊C8—91no—
55C2×C22⋊C8—92no—
56#336—92no—
57C24×C3C2×Dic311yescharacteristic
58C22×A4C2×Dic313yes—
59C22×C3×C4—21no—
60C22×C3×C4—31no—
61C22×C3×C4—31no—
62C22×C3×C4—31no—
63C22×C3×C4—31no—
64C22×C3×C4—31no—
65C22×C3×C4—31no—
66C22×C3×C4—32no—
67C22×C3×C4—31no—
68C22×C3×C4—31no—
69C22×C3×C4—61no—
70C22×C3×C4C3⋊C812yescomplemented
71C22×C3×C4Dic611yescharacteristic
72C22×C3×C4D1211yescharacteristic
73C4×A4—23no—
74C4×A4C3⋊C816yescomplemented
75C4×A4Dic613yes—
76C4×A4D1213yes—
77C3×C42—31no—
78C3×C42—31no—
79C3×C42—31no—
80C3×C42—43no—
81C3×C42—61no—
82C3×C42S411yescharacteristic
83C2×C3⋊C8—62no—
84C2×C3⋊C8—62no—
85C2×C3⋊C8—62no—
86C2×C3⋊C8—62no—
87C2×C3⋊C8—62no—
88C2×C3⋊C8—62no—
89C2×C3⋊C8—126no—
90C22×C32—41no—
91C3×A4C4⋊C811yescharacteristic, complemented
92C32×C4—42no—
93C32×C4—41no—
94C32×C4—41no—
95C32×C4—81no—
96C23×C4C3⋊Dic311yescharacteristic
97C23×C4C3⋊Dic311yescharacteristic
98C23×C4C2×C3⋊D311yescharacteristic
99C22×C8—92no—
100C22×C8—92no—
101C2×C42—31no—
102C2×C42—31no—
103C2×C42—31no—
104C2×C42—31no—
105C4⋊C8—181no—
106C4⋊C8—181no—
107C4⋊C8—181no—
108C4⋊C8—181no—
109C22⋊C8—182no—
110C22⋊C8—182no—
111C23×C3—31no—
112C23×C3—31no—
113C23×C3—31no—
114C23×C3—31no—
115C23×C3C2×C3⋊C811yescharacteristic
116C23×C3#20811yescharacteristic
117C23×C3C4⋊Dic311yescharacteristic
118C2×A4C2×C3⋊C813yes—
119C2×A4#20813yes—
120C2×A4C4⋊Dic313yes—
121C2×C3×C4—31no—
122C2×C3×C4—31no—
123C2×C3×C4—32no—
124C2×C3×C4—32no—
125C2×C3×C4—31no—
126C2×C3×C4—31no—
127C2×C3×C4—31no—
128C2×C3×C4—31no—
129C2×C3×C4—32no—
130C2×C3×C4—31no—
131C2×C3×C4—31no—
132C2×C3×C4—32no—
133C2×C3×C4—31no—
134C2×C3×C4—43no—
135C2×C3×C4—43no—
136C2×C3×C4—43no—
137C2×C3×C4—61no—
138C2×C3×C4—61no—
139C2×C3×C4—61no—
140C2×C3×C4—61no—
141C2×C3×C4—61no—
142C2×C3×C4SL(2,ℤ/4)11yescharacteristic
143C2×C3×C4SL(2,ℤ/4)11yescharacteristic
144C2×C3×C4C2×S411yescharacteristic
145C3⋊C8—122no—
146C3⋊C8—122no—
147C3⋊C8—246no—
148C2×C32—41no—
149C2×C32—41no—
150C2×C32—41no—
151C24C2×C3⋊Dic311yescharacteristic
152C22×C4—21no—
153C22×C4—31no—
154C22×C4—31no—
155C22×C4—31no—
156C22×C4—31no—
157C22×C4—31no—
158C22×C4—31no—
159C22×C4—32no—
160C22×C4—31no—
161C22×C4—31no—
162C22×C4—61no—
163C22×C4C3⋊(C3⋊C8)12yescomplemented
164C22×C4#63411yescharacteristic
165C22×C4C4⋊(C3⋊D3)11yescharacteristic
166C2×C8—182no—
167C2×C8—182no—
168C2×C8—182no—
169C2×C8—182no—
170C2×C8—182no—
171C2×C8—182no—
172C42—31no—
173C42—31no—
174C42—31no—
175C42—61no—
176C42C3⋊S411yescharacteristic
177C22×C3—31no—
178C22×C3—31no—
179C22×C3—31no—
180C22×C3—31no—
181C22×C3—31no—
182C22×C3—31no—
183C22×C3—31no—
184C22×C3—31no—
185C22×C3—31no—
186C22×C3—43no—
187C22×C3—61no—
188C22×C3C4⋊(C3⋊C8)11yescharacteristic, complemented
189C22×C3C2×SL(2,ℤ/4)11yescharacteristic
190A4C4⋊(C3⋊C8)13yescomplemented
191C3×C4—21no—
192C3×C4—32no—
193C3×C4—31no—
194C3×C4—31no—
195C3×C4—43no—
196C3×C4—43no—
197C3×C4—46no—
198C3×C4—61no—
199C3×C4—83no—
200C3×C4A4⋊C812yescomplemented
201C3×C4A4⋊Q811yescharacteristic
202C3×C4C4⋊S411yescharacteristic
203C32—41noSylow
204C23—31no—
205C23—31no—
206C23—31no—
207C23—31no—
208C23C2×C3⋊(C3⋊C8)11yescharacteristic
209C23#374811yescharacteristic
210C23C4⋊(C3⋊Dic3)11yescharacteristic
211C2×C4—31no—
212C2×C4—31no—
213C2×C4—31no—
214C2×C4—31no—
215C2×C4—31no—
216C2×C4—31no—
217C2×C4—32no—
218C2×C4—32no—
219C2×C4—32no—
220C2×C4—31no—
221C2×C4—31no—
222C2×C4—32no—
223C2×C4—31no—
224C2×C4—61no—
225C2×C4—61no—
226C2×C4—61no—
227C2×C4—61no—
228C2×C4—61no—
229C2×C4C3⋊SL(2,ℤ/4)11yescharacteristic
230C2×C4C3⋊SL(2,ℤ/4)11yescharacteristic
231C2×C4C2×C3⋊S411yescharacteristic
232C8—362no—
233C8—362no—
234C2×C3—31no—
235C2×C3—31no—
236C2×C3—31no—
237C2×C3—31no—
238C2×C3—43no—
239C2×C3—43no—
240C2×C3—43no—
241C2×C3C2×A4⋊C811yescharacteristic
242C2×C3#540911yescharacteristic
243C2×C3C4⋊SL(2,ℤ/4)11yescharacteristic
244C22—31no—
245C22—31no—
246C22—31no—
247C22—31no—
248C22—31no—
249C22—31no—
250C22—31no—
251C22—31no—
252C22—31no—
253C22—61no—
254C22C4⋊(C3⋊(C3⋊C8))11yescharacteristic, complemented
255C22C2×C3⋊SL(2,ℤ/4)11yescharacteristic
256C4—21no—
257C4—32no—
258C4—31no—
259C4—31no—
260C4—61no—
261C4C3⋊(A4⋊C8)12yescomplemented
262C4#6425911yescharacteristic
263C4C4⋊(C3⋊S4)11yescharacteristic
264C3—43no—
265C3C4⋊(A4⋊C8)11yescharacteristic, complemented
266C2—31no—
267C2—31no—
268C2—31no—
269C2—31no—
270C2C2×C3⋊(A4⋊C8)11yescharacteristic
271C2#1059364311yescharacteristic
272C2C4⋊(C3⋊SL(2,ℤ/4))11yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×C3⋊S4
commutatorC2×C3×A4C2×C8
FittingC22×C3×C42D3
socleC24×C3C2×Dic3
prime radicalC22×C3×A4C2×C4
cosocleC2×C3×C4×A4C22
hypercenterC42C3⋊S4
hypocenterC3×A4C4⋊C8
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
19. C3×C4×A4C8semidirect
70. C22×C3×C4C3⋊C8semidirect
74. C4×A4C3⋊C8semidirect
91. C3×A4C4⋊C8semidirect
163. C22×C4C3⋊(C3⋊C8)semidirect
188. C22×C3C4⋊(C3⋊C8)semidirect
190. A4C4⋊(C3⋊C8)semidirect
200. C3×C4A4⋊C8semidirect
254. C22C4⋊(C3⋊(C3⋊C8))semidirect
261. C4C3⋊(A4⋊C8)semidirect
265. C3C4⋊(A4⋊C8)semidirect

Maximal subgroups

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

subgroupindex
C3×C42×A42
C2×C3⋊(A4⋊C8)2
C4⋊(A4⋊C8)3
#715303
C4⋊(C3⋊(C3⋊C8))4