Subgroups — #63572

Order 288.

91 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₃⋊C₈	C₂	1	2	yes	maximal
3	Dic₃⋊C₈	—	3	2	no	maximal
4	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
5	C₂×C₃⋊Dic₃	C₄	1	1	yes	characteristic
6	C₂×C₃⋊Dic₃	C₄	1	1	yes	characteristic
7	C₃×C₃⋊C₈	—	2	2	no	—
8	C₂×C₃×C₈	—	3	2	no	—
9	C₄×Dic₃	—	3	2	no	—
10	C₄×Dic₃	—	6	1	no	—
11	C₂×C₃⋊C₈	D₃	1	2	yes	—
12	C₂²×C₃²	C₂×C₄	1	1	yes	characteristic
13	C₃²×C₄	D₄	1	1	yes	characteristic
14	C₃²×C₄	Q₈	1	1	yes	characteristic
15	C₃⋊Dic₃	—	2	1	no	—
16	C₃⋊Dic₃	C₈	1	2	yes	complemented
17	C₄⋊C₈	—	9	1	no	Sylow
18	C₂×C₃×C₄	—	2	1	no	—
19	C₂×C₃×C₄	C₂×D₃	1	2	yes	—
20	C₂×Dic₃	—	3	2	no	—
21	C₂×Dic₃	—	3	2	no	—
22	C₂×Dic₃	—	6	1	no	—
23	C₂×Dic₃	—	6	1	no	—
24	C₃×C₈	—	6	2	no	—
25	C₃⋊C₈	—	2	2	no	—
26	C₂×C₃²	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₃²	C₂×C₈	1	1	yes	characteristic
28	C₂×C₃²	M₁₆	1	1	yes	characteristic
29	C₂×C₈	—	3	2	no	—
30	C₄²	—	9	1	no	—
31	C₂²×C₃	—	2	1	no	—
32	C₂²×C₃	C₄×D₃	1	2	yes	—
33	C₃×C₄	—	2	1	no	—
34	C₃×C₄	—	2	1	no	—
35	C₃×C₄	Dic₆	1	2	yes	—
36	C₃×C₄	#67	1	2	yes	—
37	Dic₃	—	3	4	no	—
38	Dic₃	—	6	2	no	—
39	Dic₃	—	6	2	no	—
40	Dic₃	—	6	2	no	—
41	C₃²	C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₄	—	9	1	no	—
43	C₂×C₄	—	9	1	no	—
44	C₂×C₄	D₃²	1	1	yes	characteristic
45	C₈	—	6	2	no	—
46	C₂×C₃	—	2	1	no	—
47	C₂×C₃	—	2	1	no	—
48	C₂×C₃	—	2	1	no	—
49	C₂×C₃	D₃×C₈	1	2	yes	—
50	C₂×C₃	ΓL(1,𝔽₂₅)	1	2	yes	—
51	C₂×C₃	Dic₃⋊C₄	1	2	yes	—
52	C₂²	#624	1	1	yes	characteristic
53	C₄	—	9	2	no	—
54	C₄	—	18	1	no	—
55	C₄	#625	1	1	yes	characteristic
56	C₄	#627	1	1	yes	characteristic
57	C₃	—	2	1	no	—
58	C₃	Dic₃⋊C₈	1	2	yes	complemented
59	C₂	#3710	1	1	yes	characteristic
60	C₂	#3712	1	1	yes	characteristic
61	C₂	#3724	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₄	D₃²
commutator	C₂×C₃²	C₂×C₈
Fitting, cosocle	C₂×C₃²×C₄	C₂²
socle, prime radical	C₂²×C₃²	C₂×C₄
hypocenter	C₃²	C₄⋊C₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
16. C₃⋊Dic₃	C₈	semidirect
41. C₃²	C₄⋊C₈	semidirect
58. C₃	Dic₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃⋊Dic₃	2
C₂×C₃×C₃⋊C₈	2
Dic₃⋊C₈	3