Subgroups — A₄×Dic₆ #64270

Order 288.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×A₄	C₂	1	1	yes	maximal, characteristic
2	Dic₃×A₄	C₂	1	2	yes	maximal
3	C₂²×Dic₆	C₃	1	1	yes	maximal, characteristic, complemented
4	Q₈×A₄	—	3	1	no	maximal
5	C₂×C₃×A₄	C₂²	1	1	yes	characteristic
6	C₃×Dic₆	—	4	1	no	maximal
7	C₂²×C₃×C₄	C₂×C₃	1	1	yes	characteristic
8	C₂²×Dic₃	C₂×C₃	1	2	yes	—
9	C₂×Dic₆	—	3	1	no	—
10	C₂×Dic₆	—	3	1	no	—
11	C₂×Dic₆	—	3	2	no	—
12	C₄×A₄	—	2	1	no	—
13	C₄×A₄	—	3	2	no	—
14	C₄×A₄	D₃	1	1	yes	characteristic
15	C₃×A₄	Q₈	1	1	yes	characteristic, complemented
16	C₃²×C₄	—	4	1	no	—
17	C₃×Dic₃	—	4	2	no	—
18	C₂²×Q₈	—	3	1	no	Sylow
19	C₂³×C₃	C₂²×C₃	1	1	yes	characteristic
20	C₂×A₄	—	2	1	no	—
21	C₂×A₄	C₂×D₃	1	1	yes	characteristic
22	C₃×Q₈	—	12	1	no	—
23	C₂×C₃×C₄	—	3	1	no	—
24	C₂×C₃×C₄	—	3	1	no	—
25	C₂×Dic₃	—	3	2	no	—
26	C₂×Dic₃	—	3	2	no	—
27	Dic₆	—	3	2	no	—
28	Dic₆	—	3	1	no	—
29	Dic₆	—	3	2	no	—
30	Dic₆	A₄	1	1	yes	characteristic, direct product factor
31	C₂×C₃²	—	4	1	no	—
32	C₂×Q₈	—	9	1	no	—
33	C₂×Q₈	—	9	1	no	—
34	C₂×Q₈	—	9	2	no	—
35	C₂²×C₄	—	3	2	no	—
36	C₂²×C₄	C₃×D₃	1	1	yes	characteristic
37	C₂²×C₃	—	3	1	no	—
38	C₂²×C₃	—	3	1	no	—
39	C₂²×C₃	C₃×Q₈	1	1	yes	characteristic, complemented
40	A₄	—	2	1	no	—
41	A₄	Dic₆	1	1	yes	characteristic, direct product factor
42	C₃×C₄	—	3	1	no	—
43	C₃×C₄	—	4	1	no	—
44	C₃×C₄	—	8	1	no	—
45	C₃×C₄	—	12	2	no	—
46	C₃×C₄	C₂×A₄	1	1	yes	characteristic
47	Dic₃	—	3	2	no	—
48	Dic₃	C₂×A₄	1	2	yes	—
49	C₃²	—	4	1	no	Sylow
50	C₂³	C₂×C₃×D₃	1	1	yes	characteristic
51	Q₈	—	3	1	no	—
52	Q₈	—	9	2	no	—
53	Q₈	—	9	2	no	—
54	Q₈	—	9	1	no	—
55	C₂×C₄	—	3	1	no	—
56	C₂×C₄	—	3	1	no	—
57	C₂×C₄	—	9	2	no	—
58	C₂×C₄	—	9	2	no	—
59	C₂×C₃	—	3	1	no	—
60	C₂×C₃	—	3	1	no	—
61	C₂×C₃	—	4	1	no	—
62	C₂×C₃	—	8	1	no	—
63	C₂×C₃	C₂²×A₄	1	1	yes	characteristic
64	C₂²	—	3	1	no	—
65	C₂²	—	3	1	no	—
66	C₂²	C₃×Dic₆	1	1	yes	characteristic, complemented
67	C₄	—	3	1	no	—
68	C₄	—	3	2	no	—
69	C₄	—	9	2	no	—
70	C₄	D₃×A₄	1	1	yes	characteristic
71	C₃	—	4	1	no	—
72	C₃	—	8	1	no	—
73	C₃	Q₈×A₄	1	1	yes	characteristic, complemented
74	C₂	—	3	1	no	—
75	C₂	—	3	1	no	—
76	C₂	C₂×D₃×A₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×A₄
commutator, socle, cosocle	C₂³×C₃	C₂²×C₃
Fitting	C₂²×C₃×C₄	C₂×C₃
prime radical	C₂×C₃×A₄	C₂²
hypercenter	C₄	D₃×A₄
hypocenter	C₂²×C₃	C₃×Q₈
LCS	G ▷ C₂³×C₃ ▷ C₂²×C₃
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
3. C₂²×Dic₆	C₃	semidirect
15. C₃×A₄	Q₈	semidirect
30. Dic₆	A₄	direct (N × H)
39. C₂²×C₃	C₃×Q₈	semidirect
41. A₄	Dic₆	direct (N × H)
66. C₂²	C₃×Dic₆	semidirect
73. C₃	Q₈×A₄	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×A₄	2
Dic₃×A₄	2
C₂²×Dic₆	3
Q₈×A₄	3
C₃×Dic₆	4

Subgroups — A4×Dic6 #64270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — A₄×Dic₆ #64270