Subgroups — A₄⋊C₆₄ #11710970

Order 768.

47 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	A₄×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊C₆₄	—	3	1	no	maximal, Sylow
3	A₄×C₁₆	C₄	1	1	yes	characteristic
4	C₃⋊C₆₄	—	4	1	no	maximal
5	C₂²×C₃₂	D₃	1	1	yes	characteristic
6	C₂×C₆₄	—	3	1	no	—
7	C₂×C₆₄	—	3	1	no	—
8	C₈×A₄	C₈	1	1	yes	characteristic
9	C₃×C₃₂	—	4	1	no	—
10	C₂²×C₁₆	Dic₃	1	1	yes	characteristic
11	C₂×C₃₂	—	3	1	no	—
12	C₂×C₃₂	—	3	1	no	—
13	C₆₄	—	6	1	no	—
14	C₆₄	—	6	1	no	—
15	C₄×A₄	C₁₆	1	1	yes	characteristic
16	C₃×C₁₆	—	4	1	no	—
17	C₂²×C₈	C₃⋊C₈	1	1	yes	characteristic
18	C₂×C₁₆	—	3	1	no	—
19	C₂×C₁₆	—	3	1	no	—
20	C₃₂	—	3	1	no	—
21	C₃₂	S₄	1	1	yes	characteristic
22	C₂×A₄	C₃₂	1	1	yes	characteristic
23	C₃×C₈	—	4	1	no	—
24	C₂²×C₄	C₃⋊C₁₆	1	1	yes	characteristic
25	C₂×C₈	—	3	1	no	—
26	C₂×C₈	—	3	1	no	—
27	C₁₆	—	3	1	no	—
28	C₁₆	SL(2,ℤ/4)	1	1	yes	characteristic
29	A₄	C₆₄	1	1	yes	characteristic, complemented
30	C₃×C₄	—	4	1	no	—
31	C₂³	C₃⋊C₃₂	1	1	yes	characteristic
32	C₂×C₄	—	3	1	no	—
33	C₂×C₄	—	3	1	no	—
34	C₈	—	3	1	no	—
35	C₈	A₄⋊C₈	1	1	yes	characteristic
36	C₂×C₃	—	4	1	no	—
37	C₂²	—	3	1	no	—
38	C₂²	—	3	1	no	—
39	C₂²	C₃⋊C₆₄	1	1	yes	characteristic, complemented
40	C₄	—	3	1	no	—
41	C₄	A₄⋊C₁₆	1	1	yes	characteristic
42	C₃	—	4	1	no	Sylow
43	C₂	—	3	1	no	—
44	C₂	—	3	1	no	—
45	C₂	A₄⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₃₂	S₄
commutator, hypocenter	A₄	C₆₄
Fitting	C₂²×C₃₂	D₃
socle	C₂³	C₃⋊C₃₂
prime radical	C₂×A₄	C₃₂
cosocle	A₄×C₃₂	C₂
LCS	G ▷ A₄
UCS	C₁ ◁ C₃₂

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

N	H ≅ G/N	type
29. A₄	C₆₄	semidirect
39. C₂²	C₃⋊C₆₄	semidirect

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