Finite Group Databasebeta

← A4×#817 #11921320

Subgroups — A4×#817 #11921320

Order 1152.

342 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 75 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
2A4×C4⋊Dic3C212yesmaximal
3C22×#817C311yesmaximal, characteristic, complemented
4A4×#106—31nomaximal
5C2×C3×C4×A4C2211yescharacteristic
6C2×Dic3×A4—22no—
7C3×C8×A4C412yescomplemented
8C3×#817—41nomaximal
9C23×C3×C8C2×C311yescharacteristic
10C22×C4⋊Dic3C2×C312yes—
11C2×C8×A4—21no—
12C2×C8×A4D311yescharacteristic
13A4×C4⋊C4—32no—
14C2×#817—31no—
15C2×#817—31no—
16C2×#817—32no—
17C22×C3×A4D411yescharacteristic
18C3×C4×A4C2×C411yescharacteristic
19C3×C4×A4Q811yescharacteristic
20Dic3×A4—42no—
21C2×C32×C8—41no—
22C3×C4⋊Dic3—42no—
23C22×#106—31noSylow
24C23×C3×C4C22×C311yescharacteristic
25C23×Dic3—22no—
26C2×C4×A4—21no—
27C2×C4×A4—62no—
28C2×C4×A4C2×D311yescharacteristic
29C22×C3×C8—31no—
30C22×C3×C8—32no—
31C22×C3×C8—31no—
32C22×C3×C8C3×C412yescomplemented
33C2×C4⋊Dic3—32no—
34C2×C4⋊Dic3—32no—
35C2×C4⋊Dic3—62no—
36C8×A4—22no—
37C8×A4Dic312yescomplemented
38C3×#106—121no—
39#817—32no—
40#817—31no—
41#817—32no—
42#817A411yescharacteristic, direct product factor
43C2×C3×A4C4⋊C411yescharacteristic
44C2×C3×A4SD1612yes—
45C2×C32×C4—41no—
46C2×C3×Dic3—82no—
47C32×C8—42no—
48C23×C8C3×D311yescharacteristic
49C22×C4⋊C4—32no—
50C2×#106—91no—
51C2×#106—92no—
52C2×#106—91no—
53C24×C3C3×D411yescharacteristic
54C22×A4—21no—
55C22×A4D1211yescharacteristic
56C22×C3×C4—31no—
57C22×C3×C4—31no—
58C22×C3×C4—31no—
59C22×C3×C4—31no—
60C22×C3×C4C2×C3×C411yescharacteristic
61C22×C3×C4C3×Q811yescharacteristic
62C22×Dic3—42no—
63C22×Dic3—62no—
64C22×Dic3—62no—
65C22×Dic3—122no—
66C4×A4—21no—
67C4×A4—21no—
68C4×A4—122no—
69C4×A4Dic611yescharacteristic
70C4×A4C2×Dic311yescharacteristic
71C2×C3×C8—32no—
72C2×C3×C8—31no—
73C2×C3×C8—32no—
74C2×C3×C8—32no—
75C2×C3×C8—32no—
76C2×C3×C8—41no—
77C2×C3×C8—81no—
78C2×C3×C8C2×A411yescharacteristic
79C3×C4⋊C4—122no—
80C4⋊Dic3—32no—
81C4⋊Dic3—62no—
82C4⋊Dic3—62no—
83C4⋊Dic3C2×A412yes—
84C22×C32—41no—
85C3×A4#10611yescharacteristic, complemented
86C32×C4—41no—
87C32×C4—41no—
88C3×Dic3—162no—
89C23×C4—62no—
90C23×C4C2×C3×D311yescharacteristic
91C22×C8—31no—
92C22×C8—31no—
93C22×C8—32no—
94C22×C8C3×Dic312yescomplemented
95C2×C4⋊C4—92no—
96C2×C4⋊C4—92no—
97C2×C4⋊C4—182no—
98#106—31no—
99#106—92no—
100#106—92no—
101#106—91no—
102C23×C3—32no—
103C23×C3—31no—
104C23×C3—31no—
105C23×C3C3×C4⋊C411yescharacteristic
106C23×C3C3×SD1612yes—
107C2×A4—22no—
108C2×A4—21no—
109C2×A4#20412yes—
110C2×A4C4⋊Dic311yescharacteristic
111C2×C3×C4—32no—
112C2×C3×C4—31no—
113C2×C3×C4—31no—
114C2×C3×C4—31no—
115C2×C3×C4—31no—
116C2×C3×C4—31no—
117C2×C3×C4—31no—
118C2×C3×C4—31no—
119C2×C3×C4—41no—
120C2×C3×C4—81no—
121C2×C3×C4—242no—
122C2×C3×C4C22×A411yescharacteristic
123C2×Dic3—22no—
124C2×Dic3—62no—
125C2×Dic3—122no—
126C2×Dic3—122no—
127C2×Dic3—122no—
128C2×Dic3—122no—
129C3×C8—32no—
130C3×C8—42no—
131C3×C8—82no—
132C3×C8C4×A412yescomplemented
133C2×C32—41no—
134C2×C32—42no—
135C24C3×D1211yescharacteristic
136C22×C4—31no—
137C22×C4—31no—
138C22×C4—31no—
139C22×C4—31no—
140C22×C4—122no—
141C22×C4—182no—
142C22×C4—182no—
143C22×C4—362no—
144C22×C4C3×Dic611yescharacteristic
145C22×C4C2×C3×Dic311yescharacteristic
146C2×C8—32no—
147C2×C8—31no—
148C2×C8—32no—
149C2×C8—32no—
150C2×C8—32no—
151C2×C8D3×A411yescharacteristic
152C4⋊C4—32no—
153C4⋊C4—92no—
154C4⋊C4—182no—
155C4⋊C4—182no—
156C22×C3—31no—
157C22×C3—32no—
158C22×C3—32no—
159C22×C3—31no—
160C22×C3—31no—
161C22×C3—32no—
162C22×C3—32no—
163C22×C3—41no—
164C22×C3—81no—
165C22×C3C3×#10611yescharacteristic, complemented
166C22×C3D4×A411yescharacteristic
167A4—21no—
168A4#81711yescharacteristic, direct product factor
169C3×C4—31no—
170C3×C4—31no—
171C3×C4—41no—
172C3×C4—41no—
173C3×C4—81no—
174C3×C4—81no—
175C3×C4—482no—
176C3×C4C2×C4×A411yescharacteristic
177C3×C4Q8×A411yescharacteristic
178Dic3—42no—
179Dic3—122no—
180C32—41noSylow
181C23—31no—
182C23—32no—
183C23—31no—
184C23C3×#20412yes—
185C23C3×C4⋊Dic311yescharacteristic
186C2×C4—32no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—31no—
190C2×C4—31no—
191C2×C4—31no—
192C2×C4—31no—
193C2×C4—31no—
194C2×C4—62no—
195C2×C4—182no—
196C2×C4—362no—
197C2×C4—362no—
198C2×C4—362no—
199C2×C4—362no—
200C2×C4C2×D3×A411yescharacteristic
201C8—32no—
202C8Dic3×A412yescomplemented
203C2×C3—31no—
204C2×C3—32no—
205C2×C3—31no—
206C2×C3—42no—
207C2×C3—41no—
208C2×C3—82no—
209C2×C3—81no—
210C2×C3A4×C4⋊C411yescharacteristic
211C2×C3A4×SD1612yes—
212C22—32no—
213C22—31no—
214C22—32no—
215C22—32no—
216C22—32no—
217C22—31no—
218C22—31no—
219C22C3×#81711yescharacteristic, complemented
220C22A4×D1211yescharacteristic
221C4—31no—
222C4—31no—
223C4—122no—
224C4—362no—
225C4A4×Dic611yescharacteristic
226C4C2×Dic3×A411yescharacteristic
227C3—41no—
228C3—81no—
229C3A4×#10611yescharacteristic, complemented
230C2—31no—
231C2—32no—
232C2—31no—
233C2A4×#20412yes—
234C2A4×C4⋊Dic311yescharacteristic
subgroupG/H
centerC22A4×D12
commutatorC22×C3×C4C2×C3×C4
FrattiniC2×C4C2×D3×A4
FittingC23×C3×C8C2×C3
socleC24×C3C3×D4
prime radicalC22×C3×A4D4
cosocleC23×C3×C4C22×C3
hypercenterC2×C8D3×A4
hypocenterC22×C3C3×#106
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. C22×#817C3semidirect
7. C3×C8×A4C4semidirect
32. C22×C3×C8C3×C4semidirect
37. C8×A4Dic3semidirect
42. #817A4direct (N × H)
85. C3×A4#106semidirect
94. C22×C8C3×Dic3semidirect
132. C3×C8C4×A4semidirect
165. C22×C3C3×#106semidirect
168. A4#817direct (N × H)
202. C8Dic3×A4semidirect
219. C22C3×#817semidirect
229. C3A4×#106semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C8×A42
A4×C4⋊Dic32
C22×#8173
A4×#1063
C3×#8174