Finite Group Databasebeta

← A4×#810 #11921257

Subgroups — A4×#810 #11921257

Order 1152.

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

normal subgroups only

This group has 438 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
2C3×A4×C4⋊C4C211yesmaximal, characteristic
3C2×A4×C3⋊C8C211yesmaximal, characteristic
4C22×#810C311yesmaximal, characteristic, complemented
5A4×#103—31nomaximal
6C2×C3×C4×A4—21no—
7C2×C3×C4×A4C2211yescharacteristic
8C2×Dic3×A4—21no—
9A4×Dic6—21no—
10A4×Dic6C412yescomplemented
11A4×C3⋊C8—21no—
12C3×#810—41nomaximal
13C23×Dic6C2×C311yescharacteristic
14C2×Q8×A4—31no—
15C22×C3×C4⋊C4C2×C311yescharacteristic
16C23×C3⋊C8C2×C311yescharacteristic
17C2×C8×A4—31no—
18A4×C4⋊C4—21no—
19A4×C4⋊C4D311yescharacteristic
20C2×#810—31no—
21C2×#810—31no—
22C2×#810—31no—
23C2×#810—31no—
24C22×C3×A4D411yescharacteristic
25C2×C3×Dic6—41no—
26C3×C4×A4—41no—
27C3×C4×A4C2×C411yescharacteristic
28C3×C4×A4D411yescharacteristic
29Dic3×A4—22no—
30C32×C4⋊C4—41no—
31C2×C3×C3⋊C8—41no—
32C22×#103—31noSylow
33C23×C3×C4—21no—
34C23×C3×C4C22×C311yescharacteristic
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—22no—
46C2×C4×A4—21no—
47C2×C4×A4—21no—
48C2×C4×A4—61no—
49C2×C4×A4C2×D311yescharacteristic
50C2×C3×C4⋊C4—31no—
51C2×C3×C4⋊C4—31no—
52C2×C3×C4⋊C4—61no—
53C22×C3⋊C8—21no—
54C22×C3⋊C8—31no—
55C22×C3⋊C8—31no—
56C22×C3⋊C8—61no—
57C8×A4—61no—
58C3×#103—121no—
59#810—32no—
60#810—31no—
61#810—31no—
62#810—31no—
63#810A411yescharacteristic, direct product factor
64C2×C3×A4C22⋊C411yescharacteristic
65C2×C3×A4SD1611yescharacteristic
66C2×C3×A4Q1611yescharacteristic
67C2×C32×C4—41no—
68C2×C32×C4—81no—
69C2×C3×Dic3—81no—
70C3×Dic6—42no—
71C3×Dic6—81no—
72C3×C3⋊C8—81no—
73C23×Q8—31no—
74C23×C8—31no—
75C22×C4⋊C4C3×D311yescharacteristic
76C2×#103—91no—
77C2×#103—91no—
78C2×#103—91no—
79C2×#103—91no—
80C24×C3C3×D411yescharacteristic
81C22×A4—21no—
82C22×A4#6711yescharacteristic
83C2×C3×Q8—121no—
84C22×C3×C4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—31no—
87C22×C3×C4—31no—
88C22×C3×C4—41no—
89C22×C3×C4—61no—
90C22×C3×C4—61no—
91C22×C3×C4—121no—
92C22×C3×C4C2×C3×C411yescharacteristic
93C22×C3×C4C3×D411yescharacteristic
94C22×Dic3—22no—
95C22×Dic3—61no—
96C22×Dic3—62no—
97C22×Dic3—61no—
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—42no—
116C4×A4—41no—
117C4×A4—62no—
118C4×A4C4×D311yescharacteristic
119C4×A4D1211yescharacteristic
120C2×C3×C8—121no—
121C3×C4⋊C4—31no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—61no—
124C3×C4⋊C4—61no—
125C3×C4⋊C4—81no—
126C3×C4⋊C4C2×A411yescharacteristic
127C2×C3⋊C8—31no—
128C2×C3⋊C8—61no—
129C2×C3⋊C8—61no—
130C2×C3⋊C8—61no—
131C2×C3⋊C8—61no—
132C2×C3⋊C8C2×A411yescharacteristic
133C22×C32—41no—
134C3×A4#10311yescharacteristic, complemented
135C32×C4—41no—
136C32×C4—41no—
137C32×C4—161no—
138C3×Dic3—82no—
139C22×Q8—32no—
140C22×Q8—61no—
141C22×Q8—91no—
142C22×Q8—91no—
143C22×Q8—92no—
144C22×Q8—181no—
145C22×Q8—181no—
146C23×C4—21no—
147C23×C4—61no—
148C23×C4C2×C3×D311yescharacteristic
149C22×C8—61no—
150C22×C8—91no—
151C22×C8—91no—
152C22×C8—181no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—61no—
156#103—31no—
157#103—91no—
158#103—91no—
159#103—92no—
160#103—91no—
161C23×C3—31no—
162C23×C3—31no—
163C23×C3—31no—
164C23×C3—31no—
165C23×C3C3×C22⋊C411yescharacteristic
166C23×C3C3×SD1611yescharacteristic
167C23×C3C3×Q1611yescharacteristic
168C2×A4—21no—
169C2×A4—21no—
170C2×A4—21no—
171C2×A4(C2×D3)⋊C411yescharacteristic
172C2×A4#21411yescharacteristic
173C2×A4#21611yescharacteristic
174C3×Q8—122no—
175C3×Q8—241no—
176C2×C3×C4—21no—
177C2×C3×C4—32no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—41no—
186C2×C3×C4—61no—
187C2×C3×C4—81no—
188C2×C3×C4—82no—
189C2×C3×C4—81no—
190C2×C3×C4—121no—
191C2×C3×C4—121no—
192C2×C3×C4—121no—
193C2×C3×C4—121no—
194C2×C3×C4—241no—
195C2×C3×C4C22×A411yescharacteristic
196C2×Dic3—21no—
197C2×Dic3—62no—
198C2×Dic3—62no—
199C2×Dic3—62no—
200C2×Dic3—61no—
201C2×Dic3—62no—
202Dic6—21no—
203Dic6—32no—
204Dic6—62no—
205Dic6—62no—
206Dic6—62no—
207Dic6—61no—
208Dic6—62no—
209Dic6C4×A412yescomplemented
210C3×C8—241no—
211C3⋊C8—21no—
212C3⋊C8—61no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—41no—
216C24C3×#6711yescharacteristic
217C2×Q8—31no—
218C2×Q8—92no—
219C2×Q8—92no—
220C2×Q8—92no—
221C2×Q8—92no—
222C2×Q8—91no—
223C2×Q8—181no—
224C2×Q8—181no—
225C2×Q8—181no—
226C2×Q8—182no—
227C2×Q8—181no—
228C2×Q8—182no—
229C2×Q8—181no—
230C2×Q8—184no—
231C2×Q8—181no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—31no—
236C22×C4—41no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—62no—
240C22×C4—121no—
241C22×C4—181no—
242C22×C4—182no—
243C22×C4—181no—
244C22×C4C3×C4×D311yescharacteristic
245C22×C4C3×D1211yescharacteristic
246C2×C8—31no—
247C2×C8—91no—
248C2×C8—181no—
249C2×C8—181no—
250C2×C8—181no—
251C2×C8—181no—
252C4⋊C4—31no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4D3×A411yescharacteristic
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—31no—
259C22×C3—32no—
260C22×C3—31no—
261C22×C3—31no—
262C22×C3—31no—
263C22×C3—31no—
264C22×C3—31no—
265C22×C3—31no—
266C22×C3—41no—
267C22×C3—81no—
268C22×C3C3×#10311yescharacteristic, complemented
269C22×C3D4×A411yescharacteristic
270A4—21no—
271A4#81011yescharacteristic, direct product factor
272C3×C4—31no—
273C3×C4—31no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—81no—
278C3×C4—81no—
279C3×C4—121no—
280C3×C4—161no—
281C3×C4—162no—
282C3×C4—242no—
283C3×C4C2×C4×A411yescharacteristic
284C3×C4D4×A411yescharacteristic
285Dic3—22no—
286Dic3—62no—
287C32—41noSylow
288C23—31no—
289C23—31no—
290C23—31no—
291C23—31no—
292C23C3×(C2×D3)⋊C411yescharacteristic
293C23C3×#21411yescharacteristic
294C23C3×#21611yescharacteristic
295Q8—32no—
296Q8—61no—
297Q8—92no—
298Q8—182no—
299Q8—182no—
300Q8—182no—
301Q8—182no—
302Q8—181no—
303C2×C4—21no—
304C2×C4—32no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—182no—
319C2×C4—182no—
320C2×C4—182no—
321C2×C4—181no—
322C2×C4—182no—
323C2×C4C2×D3×A411yescharacteristic
324C8—61no—
325C8—181no—
326C2×C3—31no—
327C2×C3—31no—
328C2×C3—31no—
329C2×C3—31no—
330C2×C3—41no—
331C2×C3—41no—
332C2×C3—41no—
333C2×C3—81no—
334C2×C3—81no—
335C2×C3—81no—
336C2×C3A4×C22⋊C411yescharacteristic
337C2×C3A4×SD1611yescharacteristic
338C2×C3A4×Q1611yescharacteristic
339C22—31no—
340C22—31no—
341C22—31no—
342C22—32no—
343C22—31no—
344C22—31no—
345C22—31no—
346C22—31no—
347C22—31no—
348C22—31no—
349C22C3×#81011yescharacteristic, complemented
350C22A4×#6711yescharacteristic
351C4—31no—
352C4—31no—
353C4—41no—
354C4—62no—
355C4—121no—
356C4—182no—
357C4C4×D3×A411yescharacteristic
358C4A4×D1211yescharacteristic
359C3—41no—
360C3—81no—
361C3A4×#10311yescharacteristic, complemented
362C2—31no—
363C2—31no—
364C2—31no—
365C2—31no—
366C2A4×(C2×D3)⋊C411yescharacteristic
367C2A4×#21411yescharacteristic
368C2A4×#21611yescharacteristic
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×#103
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×#810C3semidirect
10. A4×Dic6C4semidirect
42. C22×Dic6C3×C4semidirect
63. #810A4direct (N × H)
134. C3×A4#103semidirect
209. Dic6C4×A4semidirect
268. C22×C3C3×#103semidirect
271. A4#810direct (N × H)
349. C22C3×#810semidirect
361. 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
C3×A4×C4⋊C42
C2×A4×C3⋊C82
C22×#8103
A4×#1033
C3×#8104