Subgroups — C₄⋊(C₃⋊C₃₂) #67509

Order 384.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊C₃₂	C₂	1	2	yes	maximal
3	C₄⋊C₃₂	—	3	1	no	maximal, Sylow
4	C₂×C₃×C₁₆	C₄	1	1	yes	characteristic
5	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
6	C₃×C₄×C₈	C₄	1	1	yes	characteristic
7	C₃⋊C₃₂	—	2	2	no	—
8	C₂×C₃₂	—	3	2	no	—
9	C₄×C₁₆	D₃	1	1	yes	characteristic
10	C₂×C₃×C₈	C₈	1	1	yes	characteristic
11	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
12	C₃×C₄²	C₈	1	1	yes	characteristic
13	C₃×C₁₆	—	2	1	no	—
14	C₃×C₁₆	D₄	1	1	yes	characteristic
15	C₃×C₁₆	Q₈	1	1	yes	characteristic
16	C₂×C₁₆	Dic₃	1	1	yes	characteristic
17	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
18	C₄×C₈	Dic₃	1	1	yes	characteristic
19	C₃₂	—	6	2	no	—
20	C₂×C₃×C₄	C₁₆	1	1	yes	characteristic
21	C₂×C₃×C₄	C₁₆	1	1	yes	characteristic
22	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
23	C₃×C₈	—	2	1	no	—
24	C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
25	C₃×C₈	M₁₆	1	1	yes	characteristic
26	C₂×C₈	C₃⋊C₈	1	1	yes	characteristic
27	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
28	C₄²	C₃⋊C₈	1	1	yes	characteristic
29	C₁₆	—	2	1	no	—
30	C₁₆	Dic₆	1	1	yes	characteristic
31	C₁₆	D₁₂	1	1	yes	characteristic
32	C₂²×C₃	C₂×C₁₆	1	1	yes	characteristic
33	C₃×C₄	—	2	1	no	—
34	C₃×C₄	C₃₂	1	2	yes	complemented
35	C₃×C₄	C₄⋊C₈	1	1	yes	characteristic
36	C₃×C₄	M₃₂	1	1	yes	characteristic
37	C₂×C₄	C₃⋊C₁₆	1	1	yes	characteristic
38	C₂×C₄	C₃⋊C₁₆	1	1	yes	characteristic
39	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
40	C₈	—	2	1	no	—
41	C₈	#208	1	1	yes	characteristic
42	C₈	C₄⋊Dic₃	1	1	yes	characteristic
43	C₂×C₃	C₄⋊C₁₆	1	1	yes	characteristic
44	C₂×C₃	C₂×C₃₂	1	1	yes	characteristic
45	C₂×C₃	M₆₄	1	1	yes	characteristic
46	C₂²	C₂×C₃⋊C₁₆	1	1	yes	characteristic
47	C₄	—	2	1	no	—
48	C₄	C₃⋊C₃₂	1	2	yes	complemented
49	C₄	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
50	C₄	#812	1	1	yes	characteristic
51	C₃	C₄⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
52	C₂	C₄⋊(C₃⋊C₁₆)	1	1	yes	characteristic
53	C₂	C₂×C₃⋊C₃₂	1	1	yes	characteristic
54	C₂	#4499	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×D₃
commutator	C₂×C₃	C₂×C₃₂
Fitting	C₃×C₄×C₁₆	C₂
socle, prime radical	C₂²×C₃	C₂×C₁₆
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	C₄×C₁₆	D₃
hypocenter	C₃	C₄⋊C₃₂
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
34. C₃×C₄	C₃₂	semidirect
48. C₄	C₃⋊C₃₂	semidirect
51. C₃	C₄⋊C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₁₆	2
C₂×C₃⋊C₃₂	2
C₄⋊C₃₂	3

Subgroups — C4⋊(C3⋊C32) #67509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊(C₃⋊C₃₂) #67509