Finite Group Databasebeta

← A4×C22⋊Q8 #85585

Subgroups — A4×C22⋊Q8 #85585

Order 384.

430 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 72 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×Q8×A4C211yesmaximal, characteristic, complemented
2C22×C4×A4C211yesmaximal, characteristic
3A4×C4⋊C4C212yesmaximal, complemented
4A4×C4⋊C4C211yesmaximal, characteristic, complemented
5A4×C22⋊C4C212yesmaximal
6C22×C22⋊Q8C311yesmaximal, characteristic, complemented, Sylow
7C23×A4C2211yescharacteristic
8Q8×A4—22no—
9C2×C4×A4—22no—
10C2×C4×A4C2211yescharacteristic
11C2×C4×A4C2212yes—
12C2×C4×A4C2212yes—
13C2×C4×A4C2211yescharacteristic
14C3×C22⋊Q8—41nomaximal
15C23×Q8C2×C311yescharacteristic, complemented
16C24×C4C2×C311yescharacteristic
17C2×C22⋊Q8—31no—
18C2×C22⋊Q8—32no—
19C2×C22⋊Q8—31no—
20C2×C22⋊Q8—32no—
21C2×C22⋊Q8—31no—
22C2×C22⋊Q8—31no—
23C22×C4⋊C4C2×C311yescharacteristic, complemented
24C22×C4⋊C4C2×C312yescomplemented
25C22×C22⋊C4C2×C312yes—
26C22×A4—21no—
27C22×A4—21no—
28C22×A4Q812yescomplemented
29C22×A4C2311yescharacteristic
30C2×C3×Q8—41no—
31C22×C3×C4—41no—
32C4×A4—21no—
33C4×A4—22no—
34C4×A4—22no—
35C4×A4D412yes—
36C3×C4⋊C4—41no—
37C3×C4⋊C4—42no—
38C3×C22⋊C4—42no—
39C25C22×C311yescharacteristic
40C22×Q8—22no—
41C22×Q8—31no—
42C22×Q8—32no—
43C22×Q8—31no—
44C22×Q8—62no—
45C23×C4—22no—
46C23×C4—31no—
47C23×C4—31no—
48C23×C4—31no—
49C23×C4—31no—
50C23×C4—62no—
51C23×C4C22×C312yes—
52C23×C4C22×C312yes—
53C23×C4C22×C311yescharacteristic
54C23×C4C22×C311yescharacteristic
55C22⋊Q8—32no—
56C22⋊Q8—32no—
57C22⋊Q8—34no—
58C22⋊Q8—34no—
59C22⋊Q8—32no—
60C22⋊Q8—31no—
61C22⋊Q8—31no—
62C22⋊Q8—32no—
63C22⋊Q8—31no—
64C22⋊Q8—32no—
65C22⋊Q8A411yescharacteristic, direct product factor
66C2×C4⋊C4—31no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4—31no—
69C2×C4⋊C4—32no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—32no—
73C2×C22⋊C4—32no—
74C2×C22⋊C4—32no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4—32no—
77C23×C3—41no—
78C2×A4—22no—
79C2×A4C2×D411yescharacteristic
80C2×A4C2×Q811yescharacteristic
81C2×A4#4111yescharacteristic
82C3×Q8—82no—
83C2×C3×C4—41no—
84C2×C3×C4—42no—
85C2×C3×C4—42no—
86C2×C3×C4—41no—
87C2×C3×C4—82no—
88C24—21no—
89C24—21no—
90C24—31no—
91C24—32no—
92C24—31no—
93C24—61no—
94C24—61no—
95C24C3×Q812yescomplemented
96C24C23×C311yescharacteristic
97C2×Q8—32no—
98C2×Q8—32no—
99C2×Q8—31no—
100C2×Q8—62no—
101C2×Q8—62no—
102C2×Q8—62no—
103C2×Q8—64no—
104C2×Q8—62no—
105C2×Q8—64no—
106C2×Q8C2×A411yescharacteristic, complemented
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—31no—
111C22×C4—31no—
112C22×C4—32no—
113C22×C4—32no—
114C22×C4—31no—
115C22×C4—32no—
116C22×C4—32no—
117C22×C4—31no—
118C22×C4—31no—
119C22×C4—31no—
120C22×C4—32no—
121C22×C4—32no—
122C22×C4—31no—
123C22×C4—62no—
124C22×C4—62no—
125C22×C4—62no—
126C22×C4—62no—
127C22×C4—62no—
128C22×C4—62no—
129C22×C4—62no—
130C22×C4—62no—
131C22×C4—61no—
132C22×C4—62no—
133C22×C4—62no—
134C22×C4C3×D412yes—
135C22×C4C2×A411yescharacteristic
136C4⋊C4—32no—
137C4⋊C4—34no—
138C4⋊C4—32no—
139C4⋊C4—31no—
140C4⋊C4—32no—
141C4⋊C4—32no—
142C4⋊C4—32no—
143C4⋊C4C2×A411yescharacteristic, complemented
144C4⋊C4C2×A412yescomplemented
145C22⋊C4—34no—
146C22⋊C4—32no—
147C22⋊C4—32no—
148C22⋊C4—32no—
149C22⋊C4C2×A412yes—
150C22×C3—42no—
151C22×C3—41no—
152C22×C3—81no—
153C22×C3—81no—
154A4C22⋊Q811yescharacteristic, direct product factor
155C3×C4—42no—
156C3×C4—81no—
157C3×C4—82no—
158C3×C4—82no—
159C23—22no—
160C23—31no—
161C23—31no—
162C23—31no—
163C23—31no—
164C23—32no—
165C23—32no—
166C23—32no—
167C23—32no—
168C23—31no—
169C23—61no—
170C23—61no—
171C23—61no—
172C23—61no—
173C23—62no—
174C23—61no—
175C23—61no—
176C23—62no—
177C23—62no—
178C23—61no—
179C23—61no—
180C23—64no—
181C23C2×C3×D411yescharacteristic
182C23C2×C3×Q811yescharacteristic
183C23C3×#4111yescharacteristic
184C23C22×A411yescharacteristic
185Q8—22no—
186Q8—64no—
187Q8—64no—
188Q8—62no—
189C2×C4—22no—
190C2×C4—32no—
191C2×C4—31no—
192C2×C4—32no—
193C2×C4—32no—
194C2×C4—32no—
195C2×C4—32no—
196C2×C4—32no—
197C2×C4—31no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—61no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—61no—
204C2×C4—62no—
205C2×C4—62no—
206C2×C4—64no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—61no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—61no—
213C2×C4—62no—
214C2×C4C22×A412yes—
215C2×C4C22×A412yes—
216C2×C4C22×A411yescharacteristic
217C2×C4C22×A411yescharacteristic
218C2×C3—41no—
219C2×C3—41no—
220C2×C3—41no—
221C2×C3—82no—
222C22—21no—
223C22—21no—
224C22—31no—
225C22—31no—
226C22—31no—
227C22—32no—
228C22—31no—
229C22—31no—
230C22—31no—
231C22—31no—
232C22—31no—
233C22—31no—
234C22—32no—
235C22—62no—
236C22—62no—
237C22—62no—
238C22—62no—
239C22—62no—
240C22—62no—
241C22—62no—
242C22—62no—
243C22—61no—
244C22—61no—
245C22C3×C22⋊Q811yescharacteristic, complemented
246C22Q8×A412yescomplemented
247C22C23×A411yescharacteristic
248C4—21no—
249C4—22no—
250C4—22no—
251C4—32no—
252C4—61no—
253C4—62no—
254C4—62no—
255C4D4×A412yes—
256C3—41noSylow
257C2—22no—
258C2—31no—
259C2—31no—
260C2—31no—
261C2—31no—
262C2—62no—
263C2C2×D4×A411yescharacteristic
264C2C2×Q8×A411yescharacteristic
265C2A4×#4111yescharacteristic
subgroupG/H
center, FrattiniC22C23×A4
commutator, socle, cosocleC24C23×C3
FittingC22×C22⋊Q8C3
prime radicalC23×A4C22
hypercenterC22⋊Q8A4
hypocenterC22C3×C22⋊Q8
LCSG ▷ C24 ▷ C22
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×A4C2semidirect
3. A4×C4⋊C4C2semidirect
4. A4×C4⋊C4C2semidirect
6. C22×C22⋊Q8C3semidirect
15. C23×Q8C2×C3semidirect
23. C22×C4⋊C4C2×C3semidirect
24. C22×C4⋊C4C2×C3semidirect
28. C22×A4Q8semidirect
65. C22⋊Q8A4direct (N × H)
95. C24C3×Q8semidirect
106. C2×Q8C2×A4semidirect
143. C4⋊C4C2×A4semidirect
144. C4⋊C4C2×A4semidirect
154. A4C22⋊Q8direct (N × H)
245. C22C3×C22⋊Q8semidirect
246. C22Q8×A4semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×A42
C22×C4×A42
A4×C4⋊C42
A4×C22⋊C42
C22×C22⋊Q83
C3×C22⋊Q84