Finite Group Databasebeta

← A4×#816 #11921325

Subgroups — A4×#816 #11921325

Order 1152.

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

normal subgroups only

This group has 434 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
1C2×A4×Dic6C211yesmaximal, characteristic
2C2×C3×C8×A4C211yesmaximal, characteristic
3A4×C4⋊Dic3C211yesmaximal, characteristic
4C22×#816C311yesmaximal, characteristic, complemented
5A4×#103—31nomaximal
6C2×C3×C4×A4C2211yescharacteristic
7C2×Dic3×A4—21no—
8C2×Dic3×A4—21no—
9A4×Dic6—21no—
10A4×Dic6C412yescomplemented
11C3×C8×A4—21no—
12C3×#816—41nomaximal
13C23×Dic6C2×C311yescharacteristic
14C2×Q8×A4—31no—
15C23×C3×C8C2×C311yescharacteristic
16C22×C4⋊Dic3C2×C311yescharacteristic
17C2×C8×A4—21no—
18C2×C8×A4D311yescharacteristic
19A4×C4⋊C4—31no—
20C2×#816—31no—
21C2×#816—31no—
22C2×#816—31no—
23C2×#816—31no—
24C22×C3×A4D411yescharacteristic
25C2×C3×Dic6—41no—
26C3×C4×A4C2×C411yescharacteristic
27C3×C4×A4D411yescharacteristic
28Dic3×A4—22no—
29Dic3×A4—41no—
30C2×C32×C8—41no—
31C3×C4⋊Dic3—41no—
32C22×#103—31noSylow
33C23×C3×C4C22×C311yescharacteristic
34C23×Dic3—21no—
35C23×Dic3—21no—
36C22×Dic6—21no—
37C22×Dic6—31no—
38C22×Dic6—32no—
39C22×Dic6—31no—
40C22×Dic6—61no—
41C22×Dic6—61no—
42C22×Dic6C3×C412yescomplemented
43Q8×A4—32no—
44Q8×A4—61no—
45C2×C4×A4—21no—
46C2×C4×A4—61no—
47C2×C4×A4—61no—
48C2×C4×A4C2×D311yescharacteristic
49C22×C3×C8—21no—
50C22×C3×C8—31no—
51C22×C3×C8—31no—
52C22×C3×C8—61no—
53C2×C4⋊Dic3—31no—
54C2×C4⋊Dic3—31no—
55C2×C4⋊Dic3—61no—
56C8×A4—22no—
57C8×A4—21no—
58C3×#103—121no—
59#816—32no—
60#816—31no—
61#816—31no—
62#816—31no—
63#816A411yescharacteristic, direct product factor
64C2×C3×A4C22⋊C411yescharacteristic
65C2×C3×A4SD1611yescharacteristic
66C2×C3×A4Q1611yescharacteristic
67C2×C32×C4—41no—
68C2×C3×Dic3—81no—
69C2×C3×Dic3—81no—
70C3×Dic6—42no—
71C3×Dic6—81no—
72C32×C8—81no—
73C23×Q8—31no—
74C23×C8C3×D311yescharacteristic
75C22×C4⋊C4—31no—
76C2×#103—91no—
77C2×#103—91no—
78C2×#103—91no—
79C2×#103—91no—
80C24×C3C3×D411yescharacteristic
81C22×A4—21no—
82C22×A4D1211yescharacteristic
83C2×C3×Q8—121no—
84C22×C3×C4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—31no—
87C22×C3×C4—31no—
88C22×C3×C4C2×C3×C411yescharacteristic
89C22×C3×C4C3×D411yescharacteristic
90C22×Dic3—22no—
91C22×Dic3—41no—
92C22×Dic3—61no—
93C22×Dic3—61no—
94C22×Dic3—62no—
95C22×Dic3—61no—
96C22×Dic3—61no—
97C22×Dic3—121no—
98C2×Dic6—32no—
99C2×Dic6—31no—
100C2×Dic6—32no—
101C2×Dic6—32no—
102C2×Dic6—32no—
103C2×Dic6—61no—
104C2×Dic6—61no—
105C2×Dic6—61no—
106C2×Dic6—64no—
107C2×Dic6—61no—
108C2×Dic6—62no—
109C2×Dic6—61no—
110C2×Dic6—61no—
111C2×Dic6—62no—
112C2×Dic6C2×A411yescharacteristic
113C4×A4—21no—
114C4×A4—21no—
115C4×A4—62no—
116C4×A4—121no—
117C4×A4C4×D311yescharacteristic
118C4×A4#6711yescharacteristic
119C2×C3×C8—31no—
120C2×C3×C8—41no—
121C2×C3×C8—61no—
122C2×C3×C8—61no—
123C2×C3×C8—61no—
124C2×C3×C8—61no—
125C2×C3×C8—81no—
126C2×C3×C8C2×A411yescharacteristic
127C3×C4⋊C4—121no—
128C4⋊Dic3—31no—
129C4⋊Dic3—61no—
130C4⋊Dic3—61no—
131C4⋊Dic3C2×A411yescharacteristic
132C22×C32—41no—
133C3×A4#10311yescharacteristic, complemented
134C32×C4—41no—
135C32×C4—41no—
136C3×Dic3—82no—
137C3×Dic3—161no—
138C22×Q8—32no—
139C22×Q8—61no—
140C22×Q8—91no—
141C22×Q8—91no—
142C22×Q8—92no—
143C22×Q8—181no—
144C22×Q8—181no—
145C23×C4—61no—
146C23×C4—61no—
147C23×C4C2×C3×D311yescharacteristic
148C22×C8—21no—
149C22×C8—31no—
150C22×C8—31no—
151C22×C8—61no—
152C2×C4⋊C4—91no—
153C2×C4⋊C4—91no—
154C2×C4⋊C4—181no—
155#103—31no—
156#103—91no—
157#103—91no—
158#103—92no—
159#103—91no—
160C23×C3—31no—
161C23×C3—31no—
162C23×C3—31no—
163C23×C3—31no—
164C23×C3C3×C22⋊C411yescharacteristic
165C23×C3C3×SD1611yescharacteristic
166C23×C3C3×Q1611yescharacteristic
167C2×A4—21no—
168C2×A4—21no—
169C2×A4—21no—
170C2×A4#20411yescharacteristic
171C2×A4Dic1211yescharacteristic
172C2×A4(C2×D3)⋊C411yescharacteristic
173C3×Q8—122no—
174C3×Q8—241no—
175C2×C3×C4—32no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—41no—
184C2×C3×C4—81no—
185C2×C3×C4—241no—
186C2×C3×C4—241no—
187C2×C3×C4C22×A411yescharacteristic
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—62no—
191C2×Dic3—62no—
192C2×Dic3—62no—
193C2×Dic3—61no—
194C2×Dic3—62no—
195C2×Dic3—61no—
196C2×Dic3—121no—
197C2×Dic3—121no—
198C2×Dic3—121no—
199C2×Dic3—121no—
200Dic6—21no—
201Dic6—32no—
202Dic6—62no—
203Dic6—62no—
204Dic6—62no—
205Dic6—61no—
206Dic6—62no—
207Dic6C4×A412yescomplemented
208C3×C8—21no—
209C3×C8—61no—
210C3×C8—81no—
211C3×C8—82no—
212C2×C32—41no—
213C2×C32—41no—
214C2×C32—41no—
215C24C3×D1211yescharacteristic
216C2×Q8—31no—
217C2×Q8—92no—
218C2×Q8—92no—
219C2×Q8—92no—
220C2×Q8—92no—
221C2×Q8—91no—
222C2×Q8—181no—
223C2×Q8—181no—
224C2×Q8—181no—
225C2×Q8—182no—
226C2×Q8—181no—
227C2×Q8—182no—
228C2×Q8—181no—
229C2×Q8—184no—
230C2×Q8—181no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—62no—
236C22×C4—121no—
237C22×C4—181no—
238C22×C4—181no—
239C22×C4—182no—
240C22×C4—181no—
241C22×C4—181no—
242C22×C4—361no—
243C22×C4C3×C4×D311yescharacteristic
244C22×C4C3×#6711yescharacteristic
245C2×C8—31no—
246C2×C8—61no—
247C2×C8—61no—
248C2×C8—61no—
249C2×C8—61no—
250C2×C8D3×A411yescharacteristic
251C4⋊C4—31no—
252C4⋊C4—91no—
253C4⋊C4—181no—
254C4⋊C4—181no—
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—32no—
259C22×C3—31no—
260C22×C3—31no—
261C22×C3—31no—
262C22×C3—31no—
263C22×C3—31no—
264C22×C3—31no—
265C22×C3—41no—
266C22×C3—81no—
267C22×C3C3×#10311yescharacteristic, complemented
268C22×C3D4×A411yescharacteristic
269A4—21no—
270A4#81611yescharacteristic, direct product factor
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—81no—
276C3×C4—81no—
277C3×C4—242no—
278C3×C4—481no—
279C3×C4C2×C4×A411yescharacteristic
280C3×C4D4×A411yescharacteristic
281Dic3—22no—
282Dic3—41no—
283Dic3—62no—
284Dic3—121no—
285C32—41noSylow
286C23—31no—
287C23—31no—
288C23—31no—
289C23—31no—
290C23C3×#20411yescharacteristic
291C23C3×Dic1211yescharacteristic
292C23C3×(C2×D3)⋊C411yescharacteristic
293Q8—32no—
294Q8—61no—
295Q8—92no—
296Q8—182no—
297Q8—182no—
298Q8—182no—
299Q8—182no—
300Q8—181no—
301C2×C4—32no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—182no—
312C2×C4—182no—
313C2×C4—182no—
314C2×C4—181no—
315C2×C4—182no—
316C2×C4—181no—
317C2×C4—361no—
318C2×C4—361no—
319C2×C4—361no—
320C2×C4—361no—
321C2×C4C2×D3×A411yescharacteristic
322C8—21no—
323C8—61no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—31no—
328C2×C3—41no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—81no—
332C2×C3—81no—
333C2×C3—81no—
334C2×C3A4×C22⋊C411yescharacteristic
335C2×C3A4×SD1611yescharacteristic
336C2×C3A4×Q1611yescharacteristic
337C22—31no—
338C22—31no—
339C22—31no—
340C22—32no—
341C22—31no—
342C22—31no—
343C22—31no—
344C22—31no—
345C22—31no—
346C22—31no—
347C22C3×#81611yescharacteristic, complemented
348C22A4×D1211yescharacteristic
349C4—31no—
350C4—31no—
351C4—62no—
352C4—121no—
353C4—182no—
354C4—361no—
355C4C4×D3×A411yescharacteristic
356C4A4×#6711yescharacteristic
357C3—41no—
358C3—81no—
359C3A4×#10311yescharacteristic, complemented
360C2—31no—
361C2—31no—
362C2—31no—
363C2—31no—
364C2A4×#20411yescharacteristic
365C2A4×Dic1211yescharacteristic
366C2A4×(C2×D3)⋊C411yescharacteristic
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×#103
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
4. C22×#816C3semidirect
10. A4×Dic6C4semidirect
42. C22×Dic6C3×C4semidirect
63. #816A4direct (N × H)
133. C3×A4#103semidirect
207. Dic6C4×A4semidirect
267. C22×C3C3×#103semidirect
270. A4#816direct (N × H)
347. C22C3×#816semidirect
359. C3A4×#103semidirect

Maximal subgroups

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

subgroupindex
C2×A4×Dic62
C2×C3×C8×A42
A4×C4⋊Dic32
C22×#8163
A4×#1033
C3×#8164