Subgroups — A₄⋊C₃₂ #68005

Order 384.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, Frattini, hypercenter	C₁₆	S₄
commutator, hypocenter	A₄	C₃₂
Fitting	C₂²×C₁₆	D₃
socle	C₂³	C₃⋊C₁₆
prime radical	C₂×A₄	C₁₆
cosocle	A₄×C₁₆	C₂
LCS	G ▷ A₄
UCS	C₁ ◁ C₁₆

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

N	H ≅ G/N	type
22. A₄	C₃₂	semidirect
32. C₂²	C₃⋊C₃₂	semidirect

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