Subgroups — #64202

Order 288.

85 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊GL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#226	C₂	1	1	yes	maximal, characteristic, complemented
4	#983	—	3	1	no	maximal
5	#918	—	3	1	no	maximal
6	C₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	#626	—	4	1	no	maximal
8	#239	—	3	1	no	—
9	D₃×Q₈	D₃	1	1	yes	characteristic, complemented
10	C₂×SL(2,𝔽₃)	—	3	1	no	—
11	GL(2,𝔽₃)	—	3	1	no	—
12	GL(2,𝔽₃)	—	6	1	no	—
13	#226	D₃	1	1	yes	characteristic, complemented
14	C₃×Q₁₆	—	3	1	no	—
15	#216	—	3	1	no	—
16	#215	—	3	1	no	—
17	#204	—	3	1	no	—
18	ΓL(1,𝔽₂₅)	—	3	1	no	—
19	C₂×C₃⋊D₃	—	4	1	no	—
20	C₂×C₃×D₃	—	4	1	no	—
21	C₃×Dic₃	—	4	1	no	—
22	#137	—	9	1	no	Sylow
23	C₃×Q₈	—	3	1	no	—
24	C₃×Q₈	C₂×D₃	1	1	yes	characteristic
25	#67	—	12	1	no	—
26	D₁₂	—	3	1	no	—
27	D₁₂	—	6	1	no	—
28	C₄×D₃	—	3	1	no	—
29	C₄×D₃	—	6	1	no	—
30	Dic₆	—	3	1	no	—
31	SL(2,𝔽₃)	—	2	1	no	—
32	SL(2,𝔽₃)	C₂×D₃	1	1	yes	characteristic
33	C₃×C₈	—	3	1	no	—
34	C₃⋊C₈	—	3	1	no	—
35	C₂×C₃²	—	4	1	no	—
36	C₃⋊D₃	—	8	1	no	—
37	C₃×D₃	—	8	1	no	—
38	#41	—	9	1	no	—
39	C₂×Q₈	—	3	1	no	—
40	Q₁₆	—	3	1	no	—
41	Q₁₆	—	9	1	no	—
42	SD₁₆	—	9	1	no	—
43	SD₁₆	—	9	1	no	—
44	M₁₆	—	9	1	no	—
45	C₂²×C₃	—	12	1	no	—
46	C₂×D₃	—	6	1	no	—
47	C₂×D₃	—	12	1	no	—
48	C₂×D₃	—	24	1	no	—
49	C₂×D₃	S₄	1	1	yes	characteristic
50	C₃×C₄	—	3	1	no	—
51	C₃×C₄	—	6	1	no	—
52	Dic₃	—	3	1	no	—
53	Dic₃	—	4	1	no	—
54	C₃²	—	4	1	no	Sylow
55	Q₈	—	3	1	no	—
56	Q₈	—	9	1	no	—
57	Q₈	D₃²	1	1	yes	characteristic
58	D₄	—	9	1	no	—
59	D₄	—	18	1	no	—
60	C₂×C₄	—	9	1	no	—
61	C₂×C₄	—	18	1	no	—
62	C₈	—	3	1	no	—
63	C₈	—	9	1	no	—
64	C₂×C₃	—	4	1	no	—
65	C₂×C₃	—	8	1	no	—
66	C₂×C₃	—	24	1	no	—
67	C₂×C₃	C₂×S₄	1	1	yes	characteristic
68	D₃	—	2	1	no	—
69	D₃	—	12	1	no	—
70	D₃	—	24	1	no	—
71	D₃	—	24	2	no	—
72	C₂²	—	3	1	no	—
73	C₂²	—	18	1	no	—
74	C₄	—	3	1	no	—
75	C₄	—	6	1	no	—
76	C₄	—	9	1	no	—
77	C₃	—	4	1	no	—
78	C₃	—	8	1	no	—
79	C₃	#983	1	1	yes	characteristic, complemented
80	C₂	—	6	1	no	—
81	C₂	—	36	1	no	—
82	C₂	D₃×S₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×S₄
commutator, cosocle, hypocenter	C₃×SL(2,𝔽₃)	C₂²
Fitting	C₃×Q₈	C₂×D₃
socle	C₂×C₃	C₂×S₄
LCS	G ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. D₃×SL(2,𝔽₃)	C₂	semidirect
2. C₃⋊GL(2,𝔽₃)	C₂	semidirect
3. C₃×#226	C₂	semidirect
9. D₃×Q₈	D₃	semidirect
13. #226	D₃	semidirect
79. C₃	#983	semidirect

Maximal subgroups

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

subgroup	index
D₃×SL(2,𝔽₃)	2
C₃⋊GL(2,𝔽₃)	2
C₃×#226	2
#983	3
#918	3
#626	4