Subgroups — A₄×SL(2,𝔽₃) #64211

Order 288.

59 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊SL(2,𝔽₃)	C₃	1	2	yes	maximal, complemented
2	Q₈×A₄	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²×SL(2,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×A₄	—	4	1	no	maximal
5	C₃×SL(2,𝔽₃)	—	4	1	no	maximal
6	C₂×SL(2,𝔽₃)	—	3	1	no	—
7	C₄×A₄	—	3	1	no	—
8	C₃×A₄	—	4	1	no	—
9	C₂²×Q₈	C₃²	1	1	yes	characteristic, complemented, Sylow
10	C₂³×C₃	—	4	1	no	—
11	C₂×A₄	—	4	2	no	—
12	C₂×A₄	A₄	1	1	yes	characteristic
13	C₃×Q₈	—	4	1	no	—
14	SL(2,𝔽₃)	—	4	2	no	—
15	SL(2,𝔽₃)	—	12	2	no	—
16	SL(2,𝔽₃)	A₄	1	1	yes	characteristic, direct product factor
17	C₂×C₃²	—	16	1	no	—
18	C₂×Q₈	—	3	1	no	—
19	C₂×Q₈	—	9	1	no	—
20	C₂²×C₄	—	3	1	no	—
21	C₂²×C₃	—	4	1	no	—
22	C₂²×C₃	—	12	1	no	—
23	C₂²×C₃	—	12	1	no	—
24	A₄	—	4	2	no	—
25	A₄	SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor
26	C₃×C₄	—	12	1	no	—
27	C₃²	—	16	1	no	Sylow
28	C₂³	C₃×A₄	1	1	yes	characteristic
29	Q₈	—	3	2	no	—
30	Q₈	—	9	1	no	—
31	Q₈	C₃×A₄	1	1	yes	characteristic, complemented
32	C₂×C₄	—	9	1	no	—
33	C₂×C₄	—	9	1	no	—
34	C₂×C₃	—	4	1	no	—
35	C₂×C₃	—	4	1	no	—
36	C₂×C₃	—	12	1	no	—
37	C₂×C₃	—	12	1	no	—
38	C₂×C₃	—	16	2	no	—
39	C₂²	—	3	1	no	—
40	C₂²	—	3	1	no	—
41	C₂²	C₃×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
42	C₄	—	3	1	no	—
43	C₄	—	9	1	no	—
44	C₃	—	4	1	no	—
45	C₃	—	4	1	no	—
46	C₃	—	16	2	no	—
47	C₂	—	3	1	no	—
48	C₂	—	3	1	no	—
49	C₂	A₄²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	A₄²
commutator, Fitting, cosocle, hypocenter	C₂²×Q₈	C₃²
socle	C₂³	C₃×A₄
LCS	G ▷ C₂²×Q₈
UCS	C₁ ◁ C₂

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

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