Subgroups — #63548

Order 288.

61 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3746	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₂₄	C₂	1	1	yes	maximal, characteristic
3	C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#827	—	3	1	no	maximal
5	#800	—	3	1	no	maximal
6	#634	—	2	1	no	—
7	C₃×D₁₂	—	2	1	no	—
8	C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₃×D₈	—	3	1	no	—
10	Dic₁₂	—	3	1	no	—
11	Dic₁₂	—	3	1	no	—
12	Dic₁₂	—	6	1	no	—
13	D₂₄	D₃	1	1	yes	characteristic
14	C₃×C₁₆	—	3	1	no	—
15	C₃⋊C₁₆	D₃	1	1	yes	characteristic, complemented
16	C₂×C₃×D₃	—	4	1	no	—
17	C₃²×C₄	D₄	1	1	yes	characteristic
18	C₃⋊Dic₃	—	4	1	no	—
19	SD₃₂	—	9	1	no	Sylow
20	C₃×D₄	—	6	1	no	—
21	D₁₂	—	2	1	no	—
22	Dic₆	—	6	2	no	—
23	Dic₆	—	6	1	no	—
24	Dic₆	—	6	1	no	—
25	C₃×C₈	—	2	1	no	—
26	C₃×C₈	C₂×D₃	1	1	yes	characteristic
27	C₃×C₈	C₂×D₃	1	1	yes	characteristic
28	C₂×C₃²	D₈	1	1	yes	characteristic
29	C₃×D₃	—	8	1	no	—
30	Q₁₆	—	9	1	no	—
31	D₈	—	3	1	no	—
32	C₁₆	—	3	1	no	—
33	C₂²×C₃	—	12	1	no	—
34	C₂×D₃	—	4	1	no	—
35	C₃×C₄	—	2	1	no	—
36	C₃×C₄	D₁₂	1	1	yes	characteristic
37	C₃×C₄	#67	1	1	yes	characteristic
38	Dic₃	—	12	2	no	—
39	Dic₃	—	12	1	no	—
40	Dic₃	—	12	1	no	—
41	C₃²	SD₃₂	1	1	yes	characteristic, complemented, Sylow
42	Q₈	—	18	1	no	—
43	D₄	—	6	1	no	—
44	C₈	D₃²	1	1	yes	characteristic
45	C₂×C₃	—	2	1	no	—
46	C₂×C₃	—	24	1	no	—
47	C₂×C₃	D₂₄	1	1	yes	characteristic
48	C₂×C₃	#213	1	1	yes	characteristic
49	D₃	—	8	1	no	—
50	C₂²	—	12	1	no	—
51	C₄	—	36	1	no	—
52	C₄	#626	1	1	yes	characteristic
53	C₃	—	2	1	no	—
54	C₃	#800	1	1	yes	characteristic, complemented
55	C₃	#827	1	1	yes	characteristic, complemented
56	C₂	—	24	1	no	—
57	C₂	#3714	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#3714
commutator, Fitting, cosocle	C₃²×C₈	C₂²
Frattini, hypercenter	C₈	D₃²
socle	C₂×C₃²	D₈
prime radical	C₃×D₂₄	C₂
hypocenter	C₃²	SD₃₂
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
1. #3746	C₂	semidirect
3. C₃×C₃⋊C₁₆	C₂	semidirect
15. C₃⋊C₁₆	D₃	semidirect
41. C₃²	SD₃₂	semidirect
54. C₃	#800	semidirect
55. C₃	#827	semidirect

Maximal subgroups

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

subgroup	index
#3746	2
C₃×D₂₄	2
C₃×C₃⋊C₁₆	2
#827	3
#800	3