Subgroups — Dic₃×SL(2,𝔽₃) #63761

Order 288.

67 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	Q₈×Dic₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₄×SL(2,𝔽₃)	—	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	C₂×C₃	1	1	yes	characteristic
7	C₂×SL(2,𝔽₃)	—	2	1	no	—
8	C₂×SL(2,𝔽₃)	D₃	1	1	yes	characteristic
9	C₄⋊Dic₃	—	3	1	no	—
10	C₄×Dic₃	—	3	1	no	—
11	C₂²×C₃²	—	4	1	no	—
12	C₃×Dic₃	—	4	2	no	—
13	C₄×Q₈	—	3	1	no	Sylow
14	C₃×Q₈	—	3	1	no	—
15	C₃×Q₈	C₃×C₄	1	1	yes	characteristic, complemented
16	C₂×C₃×C₄	—	3	1	no	—
17	C₂×C₃×C₄	—	12	1	no	—
18	C₂×Dic₃	—	3	1	no	—
19	C₂×Dic₃	A₄	1	1	yes	characteristic
20	SL(2,𝔽₃)	—	2	1	no	—
21	SL(2,𝔽₃)	Dic₃	1	1	yes	characteristic, direct product factor
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	C₄⋊C₄	—	9	1	no	—
27	C₄²	—	9	1	no	—
28	C₂²×C₃	—	4	1	no	—
29	C₂²×C₃	—	8	1	no	—
30	C₂²×C₃	C₂×A₄	1	1	yes	characteristic
31	C₃×C₄	—	3	1	no	—
32	C₃×C₄	—	3	1	no	—
33	C₃×C₄	—	12	2	no	—
34	Dic₃	—	6	1	no	—
35	Dic₃	SL(2,𝔽₃)	1	2	yes	direct product factor
36	C₃²	—	4	1	no	Sylow
37	Q₈	—	3	1	no	—
38	Q₈	C₃×Dic₃	1	1	yes	characteristic, complemented
39	C₂×C₄	—	3	1	no	—
40	C₂×C₄	—	3	1	no	—
41	C₂×C₄	—	9	1	no	—
42	C₂×C₃	—	4	1	no	—
43	C₂×C₃	—	4	1	no	—
44	C₂×C₃	—	4	1	no	—
45	C₂×C₃	—	8	1	no	—
46	C₂×C₃	—	8	1	no	—
47	C₂×C₃	—	8	1	no	—
48	C₂×C₃	C₄×A₄	1	1	yes	characteristic
49	C₂×C₃	C₂×SL(2,𝔽₃)	1	1	yes	characteristic
50	C₂×C₃	#41⋊C₃	1	1	yes	characteristic
51	C₂²	D₃×A₄	1	1	yes	characteristic
52	C₄	—	3	2	no	—
53	C₄	—	3	1	no	—
54	C₄	—	3	1	no	—
55	C₄	—	18	1	no	—
56	C₃	—	4	1	no	—
57	C₃	—	8	1	no	—
58	C₃	C₄×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
59	C₂	C₃⋊(#41⋊C₃)	1	1	yes	characteristic
60	C₂	D₃×SL(2,𝔽₃)	1	1	yes	characteristic
61	C₂	Dic₃×A₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₃×A₄
commutator, hypocenter	C₃×Q₈	C₃×C₄
Fitting, cosocle	C₂×C₃×Q₈	C₂×C₃
socle	C₂²×C₃	C₂×A₄
prime radical	C₂×C₃×SL(2,𝔽₃)	C₂
LCS	G ▷ C₃×Q₈
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₃	C₃	semidirect
5. C₃×SL(2,𝔽₃)	C₄	semidirect
15. C₃×Q₈	C₃×C₄	semidirect
21. SL(2,𝔽₃)	Dic₃	direct (N × H)
35. Dic₃	SL(2,𝔽₃)	direct (N × H)
38. Q₈	C₃×Dic₃	semidirect
58. C₃	C₄×SL(2,𝔽₃)	semidirect

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
Q₈×Dic₃	3
C₄×SL(2,𝔽₃)	3
C₂×C₃×Dic₃	4

Subgroups — Dic3×SL(2,𝔽3) #63761

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₃×SL(2,𝔽₃) #63761