Subgroups — #64199

Order 288.

91 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊(#41⋊C₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊GL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×GL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	#986	—	3	1	no	maximal
5	#914	—	3	1	no	maximal
6	C₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
7	#626	—	4	1	no	maximal
8	#239	D₃	1	1	yes	characteristic, complemented
9	D₃×D₄	—	3	1	no	—
10	#41⋊C₃	—	3	1	no	—
11	GL(2,𝔽₃)	—	3	1	no	—
12	GL(2,𝔽₃)	—	6	1	no	—
13	GL(2,𝔽₃)	D₃	1	1	yes	characteristic, complemented
14	C₃×SD₁₆	—	3	1	no	—
15	#215	—	3	1	no	—
16	#213	—	3	1	no	—
17	D₂₄	—	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	AGL(1,ℤ/8)	—	9	1	no	Sylow
23	C₂²×D₃	—	6	1	no	—
24	C₃×Q₈	C₂×D₃	1	1	yes	characteristic
25	C₃×D₄	—	3	1	no	—
26	#67	—	6	1	no	—
27	D₁₂	—	3	1	no	—
28	D₁₂	—	3	1	no	—
29	D₁₂	—	12	1	no	—
30	C₄×D₃	—	3	1	no	—
31	SL(2,𝔽₃)	—	2	1	no	—
32	SL(2,𝔽₃)	C₂×D₃	1	1	yes	characteristic, complemented
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	—	3	1	no	—
39	C₂×D₄	—	9	1	no	—
40	SD₁₆	—	3	1	no	—
41	SD₁₆	—	9	1	no	—
42	D₈	—	9	1	no	—
43	D₈	—	9	1	no	—
44	M₁₆	—	9	1	no	—
45	C₂²×C₃	—	6	1	no	—
46	C₂×D₃	—	3	1	no	—
47	C₂×D₃	—	4	1	no	—
48	C₂×D₃	—	6	1	no	—
49	C₂×D₃	—	12	1	no	—
50	C₂×D₃	—	12	2	no	—
51	C₂×D₃	—	24	1	no	—
52	C₃×C₄	—	3	1	no	—
53	C₃×C₄	—	12	1	no	—
54	Dic₃	S₄	1	1	yes	characteristic
55	C₃²	—	4	1	no	Sylow
56	C₂³	—	18	1	no	—
57	Q₈	D₃²	1	1	yes	characteristic
58	D₄	—	3	1	no	—
59	D₄	—	9	1	no	—
60	D₄	—	9	1	no	—
61	D₄	—	18	1	no	—
62	C₂×C₄	—	9	1	no	—
63	C₈	—	3	1	no	—
64	C₈	—	9	1	no	—
65	C₂×C₃	—	4	1	no	—
66	C₂×C₃	—	8	1	no	—
67	C₂×C₃	—	12	1	no	—
68	C₂×C₃	C₂×S₄	1	1	yes	characteristic
69	D₃	—	6	1	no	—
70	D₃	—	8	1	no	—
71	D₃	—	12	1	no	—
72	D₃	—	24	1	no	—
73	D₃	—	24	2	no	—
74	C₂²	—	6	1	no	—
75	C₂²	—	9	1	no	—
76	C₂²	—	18	1	no	—
77	C₂²	—	36	2	no	—
78	C₄	—	3	1	no	—
79	C₄	—	3	1	no	—
80	C₃	—	4	1	no	—
81	C₃	—	8	1	no	—
82	C₃	#986	1	1	yes	characteristic, complemented
83	C₂	—	12	1	no	—
84	C₂	—	18	1	no	—
85	C₂	—	36	1	no	—
86	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. C₃⋊(#41⋊C₃)	C₂	semidirect
2. C₃⋊GL(2,𝔽₃)	C₂	semidirect
3. C₃×GL(2,𝔽₃)	C₂	semidirect
6. C₃×SL(2,𝔽₃)	C₂²	semidirect
8. #239	D₃	semidirect
13. GL(2,𝔽₃)	D₃	semidirect
32. SL(2,𝔽₃)	C₂×D₃	semidirect
82. C₃	#986	semidirect

Maximal subgroups

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

subgroup	index
C₃⋊(#41⋊C₃)	2
C₃⋊GL(2,𝔽₃)	2
C₃×GL(2,𝔽₃)	2
#986	3
#914	3
#626	4