Finite Group Databasebeta

← A4×#815 #11921319

Subgroups — A4×#815 #11921319

Order 1152.

382 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×C3×C8×A4C211yesmaximal, characteristic
2C4×Dic3×A4C211yesmaximal, characteristic
3C2×A4×C3⋊C8C211yesmaximal, characteristic
4C22×#815C311yesmaximal, characteristic, complemented
5A4×#97—31nomaximal
6C2×C3×C4×A4C2211yescharacteristic
7C2×Dic3×A4C412yes—
8C3×C8×A4C412yescomplemented
9A4×C3⋊C8C412yescomplemented
10C3×#815—41nomaximal
11C23×C3×C8C2×C311yescharacteristic
12C22×C4×Dic3C2×C311yescharacteristic
13C23×C3⋊C8C2×C311yescharacteristic
14C2×C8×A4—21no—
15C2×C8×A4—31no—
16C2×C8×A4D311yescharacteristic
17C42×A4—31no—
18C2×#815—31no—
19C2×#815—31no—
20C2×#815—31no—
21C2×#815—31no—
22C22×C3×A4C2×C411yescharacteristic
23C3×C4×A4C2×C411yescharacteristic
24C3×C4×A4C2×C411yescharacteristic
25Dic3×A4—22no—
26C2×C32×C8—41no—
27C3×C4×Dic3—41no—
28C2×C3×C3⋊C8—41no—
29C22×#97—31noSylow
30C23×C3×C4C22×C311yescharacteristic
31C23×Dic3C3×C412yes—
32C2×C4×A4—21no—
33C2×C4×A4—32no—
34C2×C4×A4C2×D311yescharacteristic
35C22×C3×C8—31no—
36C22×C3×C8—32no—
37C22×C3×C8—31no—
38C22×C3×C8C3×C412yescomplemented
39C2×C4×Dic3—31no—
40C2×C4×Dic3—32no—
41C2×C4×Dic3—31no—
42C22×C3⋊C8—31no—
43C22×C3⋊C8—32no—
44C22×C3⋊C8—31no—
45C22×C3⋊C8C3×C412yescomplemented
46C8×A4—22no—
47C8×A4—32no—
48C8×A4Dic312yescomplemented
49C3×#97—121no—
50#815—32no—
51#815—31no—
52#815—31no—
53#815—31no—
54#815A411yescharacteristic, direct product factor
55C2×C3×A4C4211yescharacteristic
56C2×C3×A4M1612yes—
57C2×C32×C4—41no—
58C2×C3×Dic3—42no—
59C32×C8—42no—
60C3×C3⋊C8—42no—
61C23×C8—31no—
62C23×C8C3×D311yescharacteristic
63C22×C42—31no—
64C2×#97—91no—
65C2×#97—91no—
66C2×#97—91no—
67C2×#97—91no—
68C24×C3C2×C3×C411yescharacteristic
69C22×A4—21no—
70C22×A4C4×D311yescharacteristic
71C22×C3×C4—31no—
72C22×C3×C4—31no—
73C22×C3×C4—31no—
74C22×C3×C4—31no—
75C22×C3×C4C2×C3×C411yescharacteristic
76C22×C3×C4C2×C3×C411yescharacteristic
77C22×Dic3—22no—
78C22×Dic3—32no—
79C22×Dic3—32no—
80C22×Dic3—62no—
81C4×A4—21no—
82C4×A4—21no—
83C4×A4—62no—
84C4×A4C4×D311yescharacteristic
85C4×A4C2×Dic311yescharacteristic
86C2×C3×C8—32no—
87C2×C3×C8—31no—
88C2×C3×C8—32no—
89C2×C3×C8—32no—
90C2×C3×C8—32no—
91C2×C3×C8—41no—
92C2×C3×C8—81no—
93C2×C3×C8—121no—
94C2×C3×C8C2×A411yescharacteristic
95C3×C42—121no—
96C4×Dic3—32no—
97C4×Dic3—31no—
98C4×Dic3—32no—
99C4×Dic3C2×A411yescharacteristic
100C2×C3⋊C8—32no—
101C2×C3⋊C8—31no—
102C2×C3⋊C8—32no—
103C2×C3⋊C8—32no—
104C2×C3⋊C8—32no—
105C2×C3⋊C8C2×A411yescharacteristic
106C22×C32—41no—
107C3×A4#9711yescharacteristic, complemented
108C32×C4—41no—
109C32×C4—41no—
110C3×Dic3—82no—
111C23×C4—32no—
112C23×C4C2×C3×D311yescharacteristic
113C22×C8—31no—
114C22×C8—31no—
115C22×C8—32no—
116C22×C8—32no—
117C22×C8—91no—
118C22×C8—91no—
119C22×C8—92no—
120C22×C8C3×Dic312yescomplemented
121C2×C42—91no—
122C2×C42—91no—
123C2×C42—92no—
124#97—31no—
125#97—91no—
126#97—91no—
127#97—92no—
128#97—91no—
129C23×C3—32no—
130C23×C3—31no—
131C23×C3—31no—
132C23×C3C3×C4211yescharacteristic
133C23×C3C3×M1612yes—
134C2×A4—22no—
135C2×A4—21no—
136C2×A4ΓL(1,𝔽25)12yes—
137C2×A4C4×Dic311yescharacteristic
138C2×C3×C4—32no—
139C2×C3×C4—31no—
140C2×C3×C4—31no—
141C2×C3×C4—31no—
142C2×C3×C4—31no—
143C2×C3×C4—31no—
144C2×C3×C4—31no—
145C2×C3×C4—31no—
146C2×C3×C4—41no—
147C2×C3×C4—81no—
148C2×C3×C4—122no—
149C2×C3×C4C22×A411yescharacteristic
150C2×Dic3—32no—
151C2×Dic3—62no—
152C2×Dic3—62no—
153C2×Dic3—62no—
154C2×Dic3—62no—
155C2×Dic3C4×A412yes—
156C3×C8—32no—
157C3×C8—42no—
158C3×C8—82no—
159C3×C8—122no—
160C3×C8C4×A412yescomplemented
161C3⋊C8—32no—
162C3⋊C8C4×A412yescomplemented
163C2×C32—41no—
164C2×C32—42no—
165C24C3×C4×D311yescharacteristic
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—62no—
171C22×C4—92no—
172C22×C4—92no—
173C22×C4—182no—
174C22×C4C3×C4×D311yescharacteristic
175C22×C4C2×C3×Dic311yescharacteristic
176C2×C8—32no—
177C2×C8—31no—
178C2×C8—31no—
179C2×C8—32no—
180C2×C8—32no—
181C2×C8—32no—
182C2×C8—92no—
183C2×C8—92no—
184C2×C8—92no—
185C2×C8—91no—
186C2×C8—92no—
187C2×C8D3×A411yescharacteristic
188C42—31no—
189C42—92no—
190C42—92no—
191C42—91no—
192C22×C3—31no—
193C22×C3—32no—
194C22×C3—32no—
195C22×C3—31no—
196C22×C3—31no—
197C22×C3—32no—
198C22×C3—32no—
199C22×C3—41no—
200C22×C3—81no—
201C22×C3C3×#9711yescharacteristic, complemented
202C22×C3C2×C4×A411yescharacteristic
203A4—21no—
204A4#81511yescharacteristic, direct product factor
205C3×C4—31no—
206C3×C4—31no—
207C3×C4—41no—
208C3×C4—41no—
209C3×C4—81no—
210C3×C4—81no—
211C3×C4—242no—
212C3×C4C2×C4×A411yescharacteristic
213C3×C4C2×C4×A411yescharacteristic
214Dic3—22no—
215Dic3—62no—
216C32—41noSylow
217C23—31no—
218C23—32no—
219C23—31no—
220C23C3×ΓL(1,𝔽25)12yes—
221C23C3×C4×Dic311yescharacteristic
222C2×C4—32no—
223C2×C4—31no—
224C2×C4—31no—
225C2×C4—31no—
226C2×C4—31no—
227C2×C4—31no—
228C2×C4—31no—
229C2×C4—31no—
230C2×C4—32no—
231C2×C4—92no—
232C2×C4—182no—
233C2×C4—182no—
234C2×C4—182no—
235C2×C4—182no—
236C2×C4C2×D3×A411yescharacteristic
237C8—32no—
238C8—32no—
239C8—92no—
240C8Dic3×A412yescomplemented
241C2×C3—31no—
242C2×C3—32no—
243C2×C3—31no—
244C2×C3—42no—
245C2×C3—41no—
246C2×C3—82no—
247C2×C3—81no—
248C2×C3C42×A411yescharacteristic
249C2×C3A4×M1612yes—
250C22—32no—
251C22—31no—
252C22—32no—
253C22—32no—
254C22—32no—
255C22—31no—
256C22—31no—
257C22C3×#81511yescharacteristic, complemented
258C22C4×D3×A411yescharacteristic
259C4—31no—
260C4—31no—
261C4—62no—
262C4—182no—
263C4C4×D3×A411yescharacteristic
264C4C2×Dic3×A411yescharacteristic
265C3—41no—
266C3—81no—
267C3A4×#9711yescharacteristic, complemented
268C2—31no—
269C2—32no—
270C2—31no—
271C2A4×ΓL(1,𝔽25)12yes—
272C2C4×Dic3×A411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×A4
commutatorC23×C3C3×C42
FittingC23×C3×C8C2×C3
socleC24×C3C2×C3×C4
prime radicalC22×C3×A4C2×C4
cosocleC23×C3×C4C22×C3
hypercenterC2×C8D3×A4
hypocenterC22×C3C3×#97
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. C22×#815C3semidirect
8. C3×C8×A4C4semidirect
9. A4×C3⋊C8C4semidirect
38. C22×C3×C8C3×C4semidirect
45. C22×C3⋊C8C3×C4semidirect
48. C8×A4Dic3semidirect
54. #815A4direct (N × H)
107. C3×A4#97semidirect
120. C22×C8C3×Dic3semidirect
160. C3×C8C4×A4semidirect
162. C3⋊C8C4×A4semidirect
201. C22×C3C3×#97semidirect
204. A4#815direct (N × H)
240. C8Dic3×A4semidirect
257. C22C3×#815semidirect
267. C3A4×#97semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C8×A42
C4×Dic3×A42
C2×A4×C3⋊C82
C22×#8153
A4×#973
C3×#8154