Subgroups — C₃×#986 #64256

Order 288.

89 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	2	yes	maximal, complemented
3	#986	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₃×AGL(1,ℤ/8)	—	3	1	no	maximal
5	C₃×D₁₂	—	4	1	no	maximal
6	C₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
7	C₃×#41	D₃	1	1	yes	characteristic, complemented
8	C₂×C₃×D₄	—	3	1	no	—
9	#41⋊C₃	—	2	1	no	—
10	#41⋊C₃	C₂×C₃	1	1	yes	characteristic, complemented
11	GL(2,𝔽₃)	C₂×C₃	1	2	yes	complemented
12	C₃×SD₁₆	—	3	2	no	—
13	C₃×D₈	—	3	2	no	—
14	C₃×M₁₆	—	3	1	no	—
15	C₂×C₃×D₃	—	4	2	no	—
16	C₃²×C₄	—	4	1	no	—
17	AGL(1,ℤ/8)	—	3	1	no	Sylow
18	C₂³×C₃	—	6	1	no	—
19	C₃×Q₈	C₂×D₃	1	1	yes	characteristic
20	C₃×D₄	—	3	1	no	—
21	C₃×D₄	—	3	2	no	—
22	C₃×D₄	—	6	1	no	—
23	C₂×C₃×C₄	—	3	1	no	—
24	D₁₂	—	4	1	no	—
25	SL(2,𝔽₃)	—	2	1	no	—
26	SL(2,𝔽₃)	C₂²×C₃	1	1	yes	characteristic, complemented
27	C₃×C₈	—	3	2	no	—
28	C₂×C₃²	—	4	1	no	—
29	C₃×D₃	—	8	2	no	—
30	#41	C₃×D₃	1	1	yes	characteristic, complemented
31	C₂×D₄	—	3	1	no	—
32	SD₁₆	—	3	2	no	—
33	D₈	—	3	2	no	—
34	M₁₆	—	3	1	no	—
35	C₂²×C₃	—	3	1	no	—
36	C₂²×C₃	—	6	2	no	—
37	C₂²×C₃	—	12	2	no	—
38	C₂×D₃	—	4	2	no	—
39	C₃×C₄	—	3	1	no	—
40	C₃×C₄	—	4	1	no	—
41	C₃×C₄	—	8	1	no	—
42	C₃×C₄	S₄	1	1	yes	characteristic
43	C₃²	—	4	1	no	Sylow
44	C₂³	—	6	1	no	—
45	Q₈	C₂×C₃×D₃	1	1	yes	characteristic
46	D₄	—	3	1	no	—
47	D₄	—	3	2	no	—
48	D₄	—	6	1	no	—
49	C₂×C₄	—	3	1	no	—
50	C₈	—	3	2	no	—
51	C₂×C₃	—	4	1	no	—
52	C₂×C₃	—	6	1	no	—
53	C₂×C₃	—	8	1	no	—
54	C₂×C₃	—	12	2	no	—
55	C₂×C₃	C₂×S₄	1	1	yes	characteristic
56	D₃	—	8	2	no	—
57	C₂²	—	3	1	no	—
58	C₂²	—	6	2	no	—
59	C₂²	—	12	2	no	—
60	C₄	—	3	1	no	—
61	C₄	C₃×S₄	1	1	yes	characteristic
62	C₃	—	4	1	no	—
63	C₃	—	8	1	no	—
64	C₃	#986	1	1	yes	characteristic, direct product factor
65	C₂	—	6	1	no	—
66	C₂	—	12	2	no	—
67	C₂	C₂×C₃×S₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₂×S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂²×C₃
Frattini	C₂	C₂×C₃×S₄
Fitting	C₃×#41	D₃
hypercenter	C₃×C₄	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄

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. #986	C₃	direct (N × H)
6. C₃×SL(2,𝔽₃)	C₂²	semidirect
7. C₃×#41	D₃	semidirect
10. #41⋊C₃	C₂×C₃	semidirect
11. GL(2,𝔽₃)	C₂×C₃	semidirect
26. SL(2,𝔽₃)	C₂²×C₃	semidirect
30. #41	C₃×D₃	semidirect
64. C₃	#986	direct (N × H)

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
#986	3
C₃×AGL(1,ℤ/8)	3
C₃×D₁₂	4

Subgroups — C3×#986 #64256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#986 #64256