Subgroups — #10589067

Order 576.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₆×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₃⋊C₃₂	C₂	1	2	yes	maximal, complemented
3	D₃×C₃₂	—	3	2	no	maximal
4	C₈×C₃⋊D₃	C₄	1	1	yes	characteristic
5	C₃²×C₁₆	C₂²	1	1	yes	characteristic
6	C₃⋊(C₃⋊C₁₆)	C₄	1	1	yes	characteristic
7	D₃×C₁₆	—	3	2	no	—
8	D₃×C₁₆	—	6	1	no	—
9	C₃×C₃₂	—	3	2	no	—
10	C₃⋊C₃₂	D₃	1	2	yes	complemented
11	C₄×C₃⋊D₃	C₈	1	1	yes	characteristic
12	C₃²×C₈	C₂×C₄	1	1	yes	characteristic
13	C₃⋊(C₃⋊C₈)	C₈	1	1	yes	characteristic
14	C₂×C₃₂	—	9	1	no	Sylow
15	D₃×C₈	—	3	2	no	—
16	D₃×C₈	—	6	1	no	—
17	C₃×C₁₆	—	2	1	no	—
18	C₃×C₁₆	C₂×D₃	1	2	yes	—
19	C₃⋊C₁₆	—	3	2	no	—
20	C₃⋊C₁₆	—	6	1	no	—
21	C₂×C₃⋊D₃	C₁₆	1	1	yes	characteristic
22	C₃²×C₄	C₂×C₈	1	1	yes	characteristic
23	C₃⋊Dic₃	C₁₆	1	1	yes	characteristic
24	C₂×C₁₆	—	9	1	no	—
25	C₃₂	—	3	2	no	—
26	C₄×D₃	—	3	2	no	—
27	C₄×D₃	—	6	1	no	—
28	C₃×C₈	—	2	1	no	—
29	C₃×C₈	C₄×D₃	1	2	yes	—
30	C₃⋊C₈	—	3	2	no	—
31	C₃⋊C₈	—	6	1	no	—
32	C₂×C₃²	C₂×C₁₆	1	1	yes	characteristic
33	C₃⋊D₃	C₃₂	1	2	yes	complemented
34	C₂×C₈	—	9	1	no	—
35	C₁₆	—	9	1	no	—
36	C₁₆	D₃²	1	1	yes	characteristic
37	C₂×D₃	—	3	2	no	—
38	C₂×D₃	—	6	1	no	—
39	C₃×C₄	—	2	1	no	—
40	C₃×C₄	D₃×C₈	1	2	yes	—
41	Dic₃	—	3	2	no	—
42	Dic₃	—	6	1	no	—
43	C₃²	C₂×C₃₂	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₄	—	9	1	no	—
45	C₈	—	9	1	no	—
46	C₈	#624	1	1	yes	characteristic
47	C₂×C₃	—	2	1	no	—
48	C₂×C₃	D₃×C₁₆	1	2	yes	—
49	D₃	—	3	4	no	—
50	D₃	—	6	2	no	—
51	C₂²	—	9	1	no	—
52	C₄	—	9	1	no	—
53	C₄	#3710	1	1	yes	characteristic
54	C₃	—	2	1	no	—
55	C₃	D₃×C₃₂	1	2	yes	complemented
56	C₂	—	9	2	no	—
57	C₂	#63542	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₁₆	D₃²
commutator, hypocenter	C₃²	C₂×C₃₂
Fitting, cosocle	C₃²×C₁₆	C₂²
socle	C₂×C₃²	C₂×C₁₆
prime radical	C₂×C₃⋊D₃	C₁₆
LCS	G ▷ C₃²
UCS	C₁ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₃⋊C₃₂	C₂	semidirect
10. C₃⋊C₃₂	D₃	semidirect
33. C₃⋊D₃	C₃₂	semidirect
43. C₃²	C₂×C₃₂	semidirect
55. C₃	D₃×C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₁₆×C₃⋊D₃	2
C₃×C₃⋊C₃₂	2
D₃×C₃₂	3