Finite Group Databasebeta

← A4×#123 #85586

Subgroups — A4×#123 #85586

Order 384.

458 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 66 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×D4×A4C211yesmaximal, characteristic
2C22×C4×A4C211yesmaximal, characteristic, complemented
3A4×C4⋊C4C212yesmaximal, complemented
4A4×C22⋊C4C212yesmaximal, complemented
5A4×C22⋊C4C211yesmaximal, characteristic, complemented
6C22×#123C311yesmaximal, characteristic, complemented, Sylow
7C23×A4C2211yescharacteristic
8C23×A4C2211yescharacteristic
9D4×A4—22no—
10C2×C4×A4—22no—
11C2×C4×A4C2212yes—
12C2×C4×A4C2211yescharacteristic
13C2×C4×A4C2212yes—
14C3×#123—41nomaximal
15C23×D4C2×C311yescharacteristic
16C24×C4C2×C311yescharacteristic, complemented
17C2×#123—31no—
18C2×#123—31no—
19C2×#123—32no—
20C2×#123—32no—
21C2×#123—31no—
22C2×#123—31no—
23C22×C4⋊C4C2×C312yescomplemented
24C22×C22⋊C4C2×C311yescharacteristic, complemented
25C22×C22⋊C4C2×C312yescomplemented
26C22×A4—22no—
27C22×A4—21no—
28C22×A4—22no—
29C22×A4D412yes—
30C22×A4C2311yescharacteristic
31C2×C3×D4—41no—
32C22×C3×C4—41no—
33C4×A4—22no—
34C4×A4—21no—
35C4×A4—22no—
36C3×C4⋊C4—42no—
37C3×C22⋊C4—41no—
38C3×C22⋊C4—42no—
39C25C22×C311yescharacteristic
40C25C22×C311yescharacteristic
41C22×D4—22no—
42C22×D4—31no—
43C22×D4—31no—
44C22×D4—31no—
45C22×D4—31no—
46C22×D4—62no—
47C23×C4—22no—
48C23×C4—31no—
49C23×C4—31no—
50C23×C4—32no—
51C23×C4—62no—
52C23×C4C22×C312yes—
53C23×C4C22×C312yes—
54C23×C4C22×C311yescharacteristic
55#123—32no—
56#123—32no—
57#123—34no—
58#123—32no—
59#123—34no—
60#123—31no—
61#123—31no—
62#123—32no—
63#123—32no—
64#123—31no—
65#123A411yescharacteristic, direct product factor
66C2×C4⋊C4—32no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4—32no—
69C2×C4⋊C4—32no—
70C2×C22⋊C4—31no—
71C2×C22⋊C4—32no—
72C2×C22⋊C4—31no—
73C2×C22⋊C4—32no—
74C2×C22⋊C4—32no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4—32no—
77C23×C3—41no—
78C23×C3—41no—
79C2×A4—22no—
80C2×A4—41no—
81C2×A4C2×D411yescharacteristic
82C2×A4#4112yes—
83C3×D4—82no—
84C2×C3×C4—42no—
85C2×C3×C4—42no—
86C2×C3×C4—41no—
87C2×C3×C4—82no—
88C24—22no—
89C24—21no—
90C24—22no—
91C24—31no—
92C24—32no—
93C24—31no—
94C24—31no—
95C24—31no—
96C24—62no—
97C24—61no—
98C24—62no—
99C24C3×D412yes—
100C24C23×C311yescharacteristic
101C2×D4—32no—
102C2×D4—31no—
103C2×D4—31no—
104C2×D4—31no—
105C2×D4—62no—
106C2×D4—62no—
107C2×D4—62no—
108C2×D4—62no—
109C2×D4—62no—
110C2×D4—62no—
111C2×D4—62no—
112C2×D4—62no—
113C2×D4C2×A411yescharacteristic
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—32no—
118C22×C4—31no—
119C22×C4—32no—
120C22×C4—32no—
121C22×C4—31no—
122C22×C4—32no—
123C22×C4—32no—
124C22×C4—32no—
125C22×C4—31no—
126C22×C4—62no—
127C22×C4—62no—
128C22×C4—64no—
129C22×C4—64no—
130C22×C4—62no—
131C22×C4—62no—
132C22×C4—62no—
133C22×C4—62no—
134C22×C4—61no—
135C22×C4C2×A411yescharacteristic, complemented
136C4⋊C4—34no—
137C4⋊C4—32no—
138C4⋊C4—32no—
139C4⋊C4—32no—
140C4⋊C4C2×A412yescomplemented
141C22⋊C4—32no—
142C22⋊C4—34no—
143C22⋊C4—31no—
144C22⋊C4—32no—
145C22⋊C4—32no—
146C22⋊C4—32no—
147C22⋊C4—32no—
148C22⋊C4C2×A411yescharacteristic, complemented
149C22⋊C4C2×A412yescomplemented
150C22×C3—42no—
151C22×C3—41no—
152C22×C3—82no—
153C22×C3—81no—
154C22×C3—82no—
155A4#12311yescharacteristic, direct product factor
156C3×C4—82no—
157C3×C4—81no—
158C3×C4—82no—
159C23—22no—
160C23—31no—
161C23—32no—
162C23—31no—
163C23—32no—
164C23—32no—
165C23—32no—
166C23—32no—
167C23—31no—
168C23—31no—
169C23—41no—
170C23—62no—
171C23—62no—
172C23—61no—
173C23—61no—
174C23—62no—
175C23—62no—
176C23—62no—
177C23—64no—
178C23—62no—
179C23—62no—
180C23—64no—
181C23—61no—
182C23—61no—
183C23—62no—
184C23—62no—
185C23—121no—
186C23—122no—
187C23—122no—
188C23C2×C3×D411yescharacteristic
189C23C3×#4112yes—
190C23C22×A411yescharacteristic
191C23C22×A411yescharacteristic
192D4—22no—
193D4—62no—
194D4—62no—
195D4—64no—
196D4—62no—
197C2×C4—22no—
198C2×C4—32no—
199C2×C4—32no—
200C2×C4—31no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—61no—
204C2×C4—62no—
205C2×C4—62no—
206C2×C4—64no—
207C2×C4—61no—
208C2×C4—64no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—61no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—61no—
216C2×C4C22×A412yes—
217C2×C4C22×A412yes—
218C2×C4C22×A411yescharacteristic
219C2×C3—41no—
220C2×C3—42no—
221C2×C3—82no—
222C2×C3—161no—
223C22—22no—
224C22—21no—
225C22—22no—
226C22—31no—
227C22—32no—
228C22—32no—
229C22—31no—
230C22—31no—
231C22—32no—
232C22—32no—
233C22—32no—
234C22—62no—
235C22—64no—
236C22—62no—
237C22—62no—
238C22—62no—
239C22—64no—
240C22—62no—
241C22—61no—
242C22—62no—
243C22—121no—
244C22—121no—
245C22—122no—
246C22—121no—
247C22—121no—
248C22—122no—
249C22C3×#12311yescharacteristic, complemented
250C22D4×A412yes—
251C22C23×A411yescharacteristic
252C4—22no—
253C4—21no—
254C4—22no—
255C4—62no—
256C4—61no—
257C4—62no—
258C3—41noSylow
259C2—22no—
260C2—31no—
261C2—32no—
262C2—31no—
263C2—41no—
264C2—62no—
265C2—121no—
266C2C2×D4×A411yescharacteristic
267C2A4×#4112yes—
subgroupG/H
center, FrattiniC22C23×A4
commutator, socle, cosocleC24C23×C3
FittingC22×#123C3
prime radicalC2×D4×A4C2
hypercenter#123A4
hypocenterC22C3×#123
LCSG ▷ C24 ▷ C22
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×A4C2semidirect
3. A4×C4⋊C4C2semidirect
4. A4×C22⋊C4C2semidirect
5. A4×C22⋊C4C2semidirect
6. C22×#123C3semidirect
16. C24×C4C2×C3semidirect
23. C22×C4⋊C4C2×C3semidirect
24. C22×C22⋊C4C2×C3semidirect
25. C22×C22⋊C4C2×C3semidirect
65. #123A4direct (N × H)
135. C22×C4C2×A4semidirect
140. C4⋊C4C2×A4semidirect
148. C22⋊C4C2×A4semidirect
149. C22⋊C4C2×A4semidirect
155. A4#123direct (N × H)
249. C22C3×#123semidirect

Maximal subgroups

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

subgroupindex
C2×D4×A42
C22×C4×A42
A4×C4⋊C42
A4×C22⋊C42
C22×#1233
C3×#1234