Subgroups — #63547

Order 288.

57 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Dic₁₂	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	#828	—	3	1	no	maximal
5	#827	—	3	1	no	maximal
6	C₃×D₁₂	—	2	1	no	—
7	C₃×Dic₆	—	2	1	no	—
8	C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₃×Q₁₆	—	3	1	no	—
10	C₃×D₈	—	3	1	no	—
11	Dic₁₂	D₃	1	1	yes	characteristic, complemented
12	D₂₄	D₃	1	1	yes	characteristic
13	C₃⋊C₁₆	—	3	1	no	—
14	C₃⋊C₁₆	—	3	1	no	—
15	C₃⋊C₁₆	—	6	1	no	—
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×Q₈	—	6	1	no	—
21	C₃×D₄	—	6	1	no	—
22	D₁₂	—	2	1	no	—
23	Dic₆	—	2	1	no	—
24	C₃×C₈	—	2	1	no	—
25	C₃×C₈	C₂×D₃	1	1	yes	characteristic
26	C₃×C₈	C₂×D₃	1	1	yes	characteristic
27	C₂×C₃²	D₈	1	1	yes	characteristic
28	C₃×D₃	—	8	1	no	—
29	Q₁₆	—	3	1	no	—
30	D₈	—	3	1	no	—
31	C₁₆	—	9	1	no	—
32	C₂²×C₃	—	12	1	no	—
33	C₂×D₃	—	4	1	no	—
34	C₃×C₄	—	2	1	no	—
35	C₃×C₄	—	12	1	no	—
36	C₃×C₄	#67	1	1	yes	characteristic
37	C₃×C₄	#67	1	1	yes	characteristic
38	Dic₃	—	4	1	no	—
39	C₃²	SD₃₂	1	1	yes	characteristic, complemented, Sylow
40	Q₈	—	6	1	no	—
41	D₄	—	6	1	no	—
42	C₈	D₃²	1	1	yes	characteristic
43	C₂×C₃	—	2	1	no	—
44	C₂×C₃	—	24	1	no	—
45	C₂×C₃	#213	1	1	yes	characteristic
46	C₂×C₃	#213	1	1	yes	characteristic
47	D₃	—	8	1	no	—
48	C₂²	—	12	1	no	—
49	C₄	—	12	1	no	—
50	C₄	#625	1	1	yes	characteristic
51	C₃	—	2	1	no	—
52	C₃	#827	1	1	yes	characteristic, complemented
53	C₃	#828	1	1	yes	characteristic, complemented
54	C₂	—	24	1	no	—
55	C₂	#3713	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#3713
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. C₃×Dic₁₂	C₂	semidirect
3. C₃⋊(C₃⋊C₁₆)	C₂	semidirect
11. Dic₁₂	D₃	semidirect
39. C₃²	SD₃₂	semidirect
52. C₃	#827	semidirect
53. C₃	#828	semidirect

Maximal subgroups

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

subgroup	index
C₃×Dic₁₂	2
C₃×D₂₄	2
C₃⋊(C₃⋊C₁₆)	2
#828	3
#827	3