Subgroups — C₃×#859 #63751

Order 288.

63 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#859	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×#103	—	3	1	no	maximal
4	C₂×C₃×Dic₃	—	4	1	no	maximal
5	C₃×SL(2,𝔽₃)	C₄	1	1	yes	characteristic, complemented
6	C₂×C₃×Q₈	D₃	1	1	yes	characteristic
7	C₂×SL(2,𝔽₃)	—	2	1	no	—
8	C₂×SL(2,𝔽₃)	C₂×C₃	1	1	yes	characteristic
9	C₂×C₃×C₈	—	3	1	no	—
10	C₃×C₄⋊C₄	—	3	1	no	—
11	C₂²×C₃²	—	4	1	no	—
12	C₃×Dic₃	—	4	2	no	—
13	#103	—	3	1	no	Sylow
14	C₃×Q₈	—	3	1	no	—
15	C₃×Q₈	Dic₃	1	1	yes	characteristic, complemented
16	C₂×C₃×C₄	—	3	1	no	—
17	C₂×C₃×C₄	—	6	1	no	—
18	C₂×Dic₃	—	4	1	no	—
19	SL(2,𝔽₃)	—	2	1	no	—
20	SL(2,𝔽₃)	C₃×C₄	1	1	yes	characteristic, complemented
21	C₃×C₈	—	6	1	no	—
22	C₂×C₃²	—	4	1	no	—
23	C₂×C₃²	—	4	1	no	—
24	C₂×C₃²	—	4	1	no	—
25	C₂×Q₈	C₃×D₃	1	1	yes	characteristic
26	C₂×C₈	—	3	1	no	—
27	C₄⋊C₄	—	3	1	no	—
28	C₂²×C₃	—	4	1	no	—
29	C₂²×C₃	—	8	1	no	—
30	C₂²×C₃	S₄	1	1	yes	characteristic
31	C₃×C₄	—	3	1	no	—
32	C₃×C₄	—	3	1	no	—
33	C₃×C₄	—	12	1	no	—
34	Dic₃	—	4	2	no	—
35	C₃²	—	4	1	no	Sylow
36	Q₈	—	3	1	no	—
37	Q₈	C₃×Dic₃	1	1	yes	characteristic, complemented
38	C₂×C₄	—	3	1	no	—
39	C₂×C₄	—	6	1	no	—
40	C₈	—	6	1	no	—
41	C₂×C₃	—	4	1	no	—
42	C₂×C₃	—	4	1	no	—
43	C₂×C₃	—	4	1	no	—
44	C₂×C₃	—	8	1	no	—
45	C₂×C₃	—	8	1	no	—
46	C₂×C₃	—	8	1	no	—
47	C₂×C₃	#226	1	1	yes	characteristic
48	C₂×C₃	GL(2,𝔽₃)	1	1	yes	characteristic
49	C₂×C₃	SL(2,ℤ/4)	1	1	yes	characteristic
50	C₂²	C₃×S₄	1	1	yes	characteristic
51	C₄	—	3	1	no	—
52	C₄	—	3	1	no	—
53	C₄	—	12	1	no	—
54	C₃	—	4	1	no	—
55	C₃	—	8	1	no	—
56	C₃	#859	1	1	yes	characteristic, direct product factor
57	C₂	C₃×#226	1	1	yes	characteristic
58	C₂	C₃×GL(2,𝔽₃)	1	1	yes	characteristic
59	C₂	C₃×SL(2,ℤ/4)	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂²×C₃	S₄
commutator, hypocenter	SL(2,𝔽₃)	C₃×C₄
Frattini	C₂²	C₃×S₄
Fitting	C₂×C₃×Q₈	D₃
prime radical	C₂×C₃×SL(2,𝔽₃)	C₂
cosocle	C₂×SL(2,𝔽₃)	C₂×C₃
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₃

Complements

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

N	H ≅ G/N	type
2. #859	C₃	direct (N × H)
5. C₃×SL(2,𝔽₃)	C₄	semidirect
15. C₃×Q₈	Dic₃	semidirect
20. SL(2,𝔽₃)	C₃×C₄	semidirect
37. Q₈	C₃×Dic₃	semidirect
56. C₃	#859	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×SL(2,𝔽₃)	2
#859	3
C₃×#103	3
C₂×C₃×Dic₃	4

Subgroups — C3×#859 #63751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#859 #63751