Finite Group Databasebeta

← A4×#808 #11921255

Subgroups — A4×#808 #11921255

Order 1152.

346 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C3×A4×C4⋊C4C211yesmaximal, characteristic
2A4×C4⋊Dic3C211yesmaximal, characteristic
3C2×A4×C3⋊C8C211yesmaximal, characteristic
4C22×#808C311yesmaximal, characteristic, complemented
5A4×#106—31nomaximal
6C2×C3×C4×A4—21no—
7C2×C3×C4×A4C2211yescharacteristic
8C2×Dic3×A4—21no—
9A4×C3⋊C8C412yescomplemented
10C3×#808—41nomaximal
11C22×C3×C4⋊C4C2×C311yescharacteristic
12C22×C4⋊Dic3C2×C311yescharacteristic
13C23×C3⋊C8C2×C311yescharacteristic
14C2×C8×A4—31no—
15A4×C4⋊C4—21no—
16A4×C4⋊C4—31no—
17A4×C4⋊C4D311yescharacteristic
18C2×#808—31no—
19C2×#808—31no—
20C2×#808—31no—
21C2×#808—31no—
22C22×C3×A4D411yescharacteristic
23C3×C4×A4—41no—
24C3×C4×A4C2×C411yescharacteristic
25C3×C4×A4Q811yescharacteristic
26Dic3×A4—41no—
27C32×C4⋊C4—41no—
28C3×C4⋊Dic3—41no—
29C2×C3×C3⋊C8—41no—
30C22×#106—31noSylow
31C23×C3×C4—21no—
32C23×C3×C4C22×C311yescharacteristic
33C23×Dic3—21no—
34C2×C4×A4—22no—
35C2×C4×A4—21no—
36C2×C4×A4—21no—
37C2×C4×A4—61no—
38C2×C4×A4C2×D311yescharacteristic
39C2×C3×C4⋊C4—31no—
40C2×C3×C4⋊C4—31no—
41C2×C3×C4⋊C4—61no—
42C2×C4⋊Dic3—31no—
43C2×C4⋊Dic3—31no—
44C2×C4⋊Dic3—61no—
45C22×C3⋊C8—31no—
46C22×C3⋊C8—31no—
47C22×C3⋊C8—32no—
48C22×C3⋊C8C3×C412yescomplemented
49C8×A4—32no—
50C3×#106—121no—
51#808—32no—
52#808—31no—
53#808—31no—
54#808—31no—
55#808A411yescharacteristic, direct product factor
56C2×C3×A4C4⋊C411yescharacteristic
57C2×C3×A4SD1611yescharacteristic
58C2×C3×A4SD1611yescharacteristic
59C2×C32×C4—41no—
60C2×C32×C4—81no—
61C2×C3×Dic3—81no—
62C3×C3⋊C8—42no—
63C23×C8—31no—
64C22×C4⋊C4—31no—
65C22×C4⋊C4C3×D311yescharacteristic
66C2×#106—91no—
67C2×#106—91no—
68C2×#106—91no—
69C2×#106—91no—
70C24×C3C3×D411yescharacteristic
71C22×A4—21no—
72C22×A4#6711yescharacteristic
73C22×C3×C4—31no—
74C22×C3×C4—31no—
75C22×C3×C4—31no—
76C22×C3×C4—31no—
77C22×C3×C4—41no—
78C22×C3×C4—61no—
79C22×C3×C4—61no—
80C22×C3×C4—121no—
81C22×C3×C4C2×C3×C411yescharacteristic
82C22×C3×C4C3×Q811yescharacteristic
83C22×Dic3—41no—
84C22×Dic3—61no—
85C22×Dic3—61no—
86C22×Dic3—121no—
87C4×A4—21no—
88C4×A4—21no—
89C4×A4—42no—
90C4×A4—41no—
91C4×A4—121no—
92C4×A4Dic611yescharacteristic
93C4×A4C4×D311yescharacteristic
94C2×C3×C8—121no—
95C3×C4⋊C4—31no—
96C3×C4⋊C4—41no—
97C3×C4⋊C4—61no—
98C3×C4⋊C4—61no—
99C3×C4⋊C4—81no—
100C3×C4⋊C4—121no—
101C3×C4⋊C4C2×A411yescharacteristic
102C4⋊Dic3—31no—
103C4⋊Dic3—61no—
104C4⋊Dic3—61no—
105C4⋊Dic3C2×A411yescharacteristic
106C2×C3⋊C8—32no—
107C2×C3⋊C8—31no—
108C2×C3⋊C8—32no—
109C2×C3⋊C8—32no—
110C2×C3⋊C8—32no—
111C2×C3⋊C8C2×A411yescharacteristic
112C22×C32—41no—
113C3×A4#10611yescharacteristic, complemented
114C32×C4—41no—
115C32×C4—41no—
116C32×C4—161no—
117C3×Dic3—161no—
118C23×C4—21no—
119C23×C4—61no—
120C23×C4C2×C3×D311yescharacteristic
121C22×C8—32no—
122C22×C8—91no—
123C22×C8—92no—
124C22×C8—91no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4—61no—
128C2×C4⋊C4—91no—
129C2×C4⋊C4—91no—
130C2×C4⋊C4—181no—
131#106—31no—
132#106—91no—
133#106—91no—
134#106—92no—
135#106—91no—
136C23×C3—31no—
137C23×C3—31no—
138C23×C3—31no—
139C23×C3—31no—
140C23×C3C3×C4⋊C411yescharacteristic
141C23×C3C3×SD1611yescharacteristic
142C23×C3C3×SD1611yescharacteristic
143C2×A4—21no—
144C2×A4—21no—
145C2×A4—21no—
146C2×A4Dic3⋊C411yescharacteristic
147C2×A4#21411yescharacteristic
148C2×A4#21511yescharacteristic
149C2×C3×C4—21no—
150C2×C3×C4—32no—
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—31no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—41no—
159C2×C3×C4—61no—
160C2×C3×C4—81no—
161C2×C3×C4—82no—
162C2×C3×C4—81no—
163C2×C3×C4—121no—
164C2×C3×C4—121no—
165C2×C3×C4—121no—
166C2×C3×C4—121no—
167C2×C3×C4—241no—
168C2×C3×C4C22×A411yescharacteristic
169C2×Dic3—21no—
170C2×Dic3—61no—
171C2×Dic3—121no—
172C2×Dic3—121no—
173C2×Dic3—121no—
174C2×Dic3—121no—
175C3×C8—122no—
176C3⋊C8—32no—
177C3⋊C8C4×A412yescomplemented
178C2×C32—41no—
179C2×C32—41no—
180C2×C32—41no—
181C24C3×#6711yescharacteristic
182C22×C4—31no—
183C22×C4—31no—
184C22×C4—31no—
185C22×C4—31no—
186C22×C4—41no—
187C22×C4—61no—
188C22×C4—61no—
189C22×C4—121no—
190C22×C4—121no—
191C22×C4—181no—
192C22×C4—181no—
193C22×C4—361no—
194C22×C4C3×Dic611yescharacteristic
195C22×C4C3×C4×D311yescharacteristic
196C2×C8—31no—
197C2×C8—92no—
198C2×C8—92no—
199C2×C8—92no—
200C2×C8—91no—
201C2×C8—92no—
202C4⋊C4—31no—
203C4⋊C4—31no—
204C4⋊C4—61no—
205C4⋊C4—61no—
206C4⋊C4—91no—
207C4⋊C4—181no—
208C4⋊C4—181no—
209C4⋊C4D3×A411yescharacteristic
210C22×C3—31no—
211C22×C3—31no—
212C22×C3—31no—
213C22×C3—32no—
214C22×C3—31no—
215C22×C3—31no—
216C22×C3—31no—
217C22×C3—31no—
218C22×C3—31no—
219C22×C3—31no—
220C22×C3—41no—
221C22×C3—81no—
222C22×C3C3×#10611yescharacteristic, complemented
223C22×C3D4×A411yescharacteristic
224A4—21no—
225A4#80811yescharacteristic, direct product factor
226C3×C4—31no—
227C3×C4—31no—
228C3×C4—41no—
229C3×C4—41no—
230C3×C4—41no—
231C3×C4—81no—
232C3×C4—81no—
233C3×C4—121no—
234C3×C4—161no—
235C3×C4—162no—
236C3×C4—481no—
237C3×C4C2×C4×A411yescharacteristic
238C3×C4Q8×A411yescharacteristic
239Dic3—41no—
240Dic3—121no—
241C32—41noSylow
242C23—31no—
243C23—31no—
244C23—31no—
245C23—31no—
246C23C3×Dic3⋊C411yescharacteristic
247C23C3×#21411yescharacteristic
248C23C3×#21511yescharacteristic
249C2×C4—21no—
250C2×C4—32no—
251C2×C4—31no—
252C2×C4—31no—
253C2×C4—31no—
254C2×C4—31no—
255C2×C4—31no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—121no—
261C2×C4—121no—
262C2×C4—121no—
263C2×C4—121no—
264C2×C4—181no—
265C2×C4—361no—
266C2×C4—361no—
267C2×C4—361no—
268C2×C4—361no—
269C2×C4C2×D3×A411yescharacteristic
270C8—32no—
271C8—92no—
272C2×C3—31no—
273C2×C3—31no—
274C2×C3—31no—
275C2×C3—31no—
276C2×C3—41no—
277C2×C3—41no—
278C2×C3—41no—
279C2×C3—81no—
280C2×C3—81no—
281C2×C3—81no—
282C2×C3A4×C4⋊C411yescharacteristic
283C2×C3A4×SD1611yescharacteristic
284C2×C3A4×SD1611yescharacteristic
285C22—31no—
286C22—31no—
287C22—31no—
288C22—32no—
289C22—31no—
290C22—31no—
291C22—31no—
292C22—31no—
293C22—31no—
294C22—31no—
295C22C3×#80811yescharacteristic, complemented
296C22A4×#6711yescharacteristic
297C4—31no—
298C4—31no—
299C4—41no—
300C4—121no—
301C4—121no—
302C4—361no—
303C4A4×Dic611yescharacteristic
304C4C4×D3×A411yescharacteristic
305C3—41no—
306C3—81no—
307C3A4×#10611yescharacteristic, complemented
308C2—31no—
309C2—31no—
310C2—31no—
311C2—31no—
312C2A4×Dic3⋊C411yescharacteristic
313C2A4×#21411yescharacteristic
314C2A4×#21511yescharacteristic
subgroupG/H
centerC22A4×#67
commutatorC22×C3×C4C2×C3×C4
FrattiniC2×C4C2×D3×A4
FittingC22×C3×C4⋊C4C2×C3
socleC24×C3C3×D4
prime radicalC22×C3×A4D4
cosocleC23×C3×C4C22×C3
hypercenterC4⋊C4D3×A4
hypocenterC22×C3C3×#106
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C22×#808C3semidirect
9. A4×C3⋊C8C4semidirect
48. C22×C3⋊C8C3×C4semidirect
55. #808A4direct (N × H)
113. C3×A4#106semidirect
177. C3⋊C8C4×A4semidirect
222. C22×C3C3×#106semidirect
225. A4#808direct (N × H)
295. C22C3×#808semidirect
307. C3A4×#106semidirect

Maximal subgroups

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

subgroupindex
C3×A4×C4⋊C42
A4×C4⋊Dic32
C2×A4×C3⋊C82
C22×#8083
A4×#1063
C3×#8084