Finite Group Databasebeta

← A4×#121 #85584

Subgroups — A4×#121 #85584

Order 384.

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

normal subgroups only

This group has 578 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×D4×A4C212yesmaximal, complemented
2C2×D4×A4C211yesmaximal, characteristic, complemented
3C22×C4×A4C211yesmaximal, characteristic, complemented
4A4×C4⋊C4C211yesmaximal, characteristic, complemented
5A4×C22⋊C4C212yesmaximal, complemented
6C22×#121C311yesmaximal, characteristic, complemented, Sylow
7C23×A4C2212yes—
8C23×A4C2211yescharacteristic
9D4×A4—24no—
10D4×A4—22no—
11C2×C4×A4—22no—
12C2×C4×A4C2211yescharacteristic
13C2×C4×A4C2212yes—
14C2×C4×A4C2211yescharacteristic
15C3×#121—41nomaximal
16C23×D4C2×C312yescomplemented
17C23×D4C2×C311yescharacteristic, complemented
18C24×C4C2×C311yescharacteristic, complemented
19C2×#121—31no—
20C2×#121—32no—
21C2×#121—31no—
22C2×#121—31no—
23C2×#121—32no—
24C2×#121—31no—
25C22×C4⋊C4C2×C311yescharacteristic, complemented
26C22×C22⋊C4C2×C312yescomplemented
27C22×A4—21no—
28C22×A4—21no—
29C22×A4—22no—
30C22×A4—22no—
31C22×A4—22no—
32C22×A4D412yescomplemented
33C22×A4C2311yescharacteristic
34C2×C3×D4—41no—
35C2×C3×D4—42no—
36C22×C3×C4—41no—
37C4×A4—21no—
38C4×A4—22no—
39C4×A4D412yescomplemented
40C3×C4⋊C4—41no—
41C3×C22⋊C4—42no—
42C25C22×C312yes—
43C25C22×C311yescharacteristic
44C22×D4—24no—
45C22×D4—22no—
46C22×D4—31no—
47C22×D4—32no—
48C22×D4—31no—
49C22×D4—32no—
50C22×D4—32no—
51C22×D4—32no—
52C22×D4—32no—
53C22×D4—62no—
54C22×D4—64no—
55C23×C4—22no—
56C23×C4—31no—
57C23×C4—31no—
58C23×C4—31no—
59C23×C4—31no—
60C23×C4—62no—
61C23×C4C22×C311yescharacteristic
62C23×C4C22×C312yes—
63C23×C4C22×C311yescharacteristic
64#121—32no—
65#121—32no—
66#121—34no—
67#121—34no—
68#121—32no—
69#121—31no—
70#121—32no—
71#121—31no—
72#121—31no—
73#121—32no—
74#121A411yescharacteristic, direct product factor
75C2×C4⋊C4—31no—
76C2×C4⋊C4—31no—
77C2×C4⋊C4—32no—
78C2×C22⋊C4—32no—
79C2×C22⋊C4—32no—
80C2×C22⋊C4—32no—
81C2×C22⋊C4—32no—
82C23×C3—41no—
83C23×C3—42no—
84C2×A4—22no—
85C2×A4—42no—
86C2×A4C2×D411yescharacteristic
87C2×A4C2×D411yescharacteristic
88C2×A4#4111yescharacteristic
89C3×D4—82no—
90C3×D4—84no—
91C2×C3×C4—41no—
92C2×C3×C4—41no—
93C2×C3×C4—42no—
94C2×C3×C4—82no—
95C24—21no—
96C24—21no—
97C24—22no—
98C24—22no—
99C24—22no—
100C24—31no—
101C24—32no—
102C24—31no—
103C24—32no—
104C24—32no—
105C24—61no—
106C24—61no—
107C24—62no—
108C24—62no—
109C24—62no—
110C24C3×D412yescomplemented
111C24C23×C311yescharacteristic
112C2×D4—32no—
113C2×D4—34no—
114C2×D4—32no—
115C2×D4—32no—
116C2×D4—32no—
117C2×D4—32no—
118C2×D4—31no—
119C2×D4—62no—
120C2×D4—62no—
121C2×D4—64no—
122C2×D4—62no—
123C2×D4—64no—
124C2×D4—62no—
125C2×D4—64no—
126C2×D4—64no—
127C2×D4—64no—
128C2×D4—64no—
129C2×D4—64no—
130C2×D4—64no—
131C2×D4—64no—
132C2×D4—64no—
133C2×D4C2×A412yescomplemented
134C2×D4C2×A411yescharacteristic, complemented
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—31no—
138C22×C4—31no—
139C22×C4—32no—
140C22×C4—32no—
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—32no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—31no—
147C22×C4—32no—
148C22×C4—62no—
149C22×C4—62no—
150C22×C4—62no—
151C22×C4—62no—
152C22×C4—62no—
153C22×C4—62no—
154C22×C4—62no—
155C22×C4—62no—
156C22×C4—61no—
157C22×C4—62no—
158C22×C4C3×D412yescomplemented
159C22×C4C2×A411yescharacteristic, complemented
160C4⋊C4—32no—
161C4⋊C4—32no—
162C4⋊C4—31no—
163C4⋊C4C2×A411yescharacteristic, complemented
164C22⋊C4—34no—
165C22⋊C4—32no—
166C22⋊C4—32no—
167C22⋊C4—32no—
168C22⋊C4C2×A412yescomplemented
169C22×C3—42no—
170C22×C3—41no—
171C22×C3—81no—
172C22×C3—81no—
173C22×C3—82no—
174C22×C3—82no—
175C22×C3—82no—
176A4#12111yescharacteristic, direct product factor
177C3×C4—42no—
178C3×C4—81no—
179C3×C4—82no—
180C23—22no—
181C23—31no—
182C23—31no—
183C23—31no—
184C23—31no—
185C23—32no—
186C23—32no—
187C23—32no—
188C23—32no—
189C23—31no—
190C23—32no—
191C23—42no—
192C23—61no—
193C23—61no—
194C23—61no—
195C23—61no—
196C23—62no—
197C23—62no—
198C23—62no—
199C23—62no—
200C23—62no—
201C23—62no—
202C23—62no—
203C23—61no—
204C23—61no—
205C23—62no—
206C23—62no—
207C23—61no—
208C23—61no—
209C23—64no—
210C23—62no—
211C23—62no—
212C23—62no—
213C23—62no—
214C23—62no—
215C23—62no—
216C23—122no—
217C23—122no—
218C23—122no—
219C23—124no—
220C23C2×C3×D411yescharacteristic
221C23C2×C3×D411yescharacteristic
222C23C3×#4111yescharacteristic
223C23C22×A412yes—
224C23C22×A411yescharacteristic
225D4—24no—
226D4—22no—
227D4—64no—
228D4—64no—
229D4—64no—
230D4—64no—
231D4—68no—
232D4—64no—
233D4—62no—
234C2×C4—22no—
235C2×C4—32no—
236C2×C4—31no—
237C2×C4—32no—
238C2×C4—32no—
239C2×C4—32no—
240C2×C4—31no—
241C2×C4—32no—
242C2×C4—62no—
243C2×C4—61no—
244C2×C4—62no—
245C2×C4—61no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4—64no—
249C2×C4—62no—
250C2×C4—61no—
251C2×C4—62no—
252C2×C4—61no—
253C2×C4—62no—
254C2×C4C22×A411yescharacteristic
255C2×C4C22×A412yes—
256C2×C4C22×A411yescharacteristic
257C2×C3—41no—
258C2×C3—41no—
259C2×C3—41no—
260C2×C3—82no—
261C2×C3—162no—
262C22—21no—
263C22—21no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—31no—
268C22—31no—
269C22—31no—
270C22—32no—
271C22—31no—
272C22—31no—
273C22—31no—
274C22—31no—
275C22—31no—
276C22—31no—
277C22—32no—
278C22—62no—
279C22—62no—
280C22—62no—
281C22—62no—
282C22—62no—
283C22—62no—
284C22—62no—
285C22—62no—
286C22—61no—
287C22—61no—
288C22—62no—
289C22—62no—
290C22—62no—
291C22—122no—
292C22—122no—
293C22—122no—
294C22—122no—
295C22—122no—
296C22—122no—
297C22—122no—
298C22—122no—
299C22C3×#12111yescharacteristic, complemented
300C22D4×A412yescomplemented
301C22C23×A411yescharacteristic
302C4—21no—
303C4—22no—
304C4—32no—
305C4—61no—
306C4—62no—
307C4D4×A412yescomplemented
308C3—41noSylow
309C2—22no—
310C2—31no—
311C2—31no—
312C2—31no—
313C2—31no—
314C2—42no—
315C2—62no—
316C2—122no—
317C2C2×D4×A411yescharacteristic
318C2C2×D4×A411yescharacteristic
319C2A4×#4111yescharacteristic
subgroupG/H
center, FrattiniC22C23×A4
commutator, socle, cosocleC24C23×C3
FittingC22×#121C3
hypercenter#121A4
hypocenterC22C3×#121
LCSG ▷ C24 ▷ C22
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×A4C2semidirect
2. C2×D4×A4C2semidirect
3. C22×C4×A4C2semidirect
4. A4×C4⋊C4C2semidirect
5. A4×C22⋊C4C2semidirect
6. C22×#121C3semidirect
16. C23×D4C2×C3semidirect
17. C23×D4C2×C3semidirect
18. C24×C4C2×C3semidirect
25. C22×C4⋊C4C2×C3semidirect
26. C22×C22⋊C4C2×C3semidirect
32. C22×A4D4semidirect
39. C4×A4D4semidirect
74. #121A4direct (N × H)
110. C24C3×D4semidirect
133. C2×D4C2×A4semidirect
134. C2×D4C2×A4semidirect
158. C22×C4C3×D4semidirect
159. C22×C4C2×A4semidirect
163. C4⋊C4C2×A4semidirect
168. C22⋊C4C2×A4semidirect
176. A4#121direct (N × H)
299. C22C3×#121semidirect
300. C22D4×A4semidirect
307. C4D4×A4semidirect

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×#1213
C3×#1214