Subgroups — C₃⋊D₃×SL(2,𝔽₃) #88979

Order 432.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×C₃⋊D₃	C₃	1	1	yes	maximal, characteristic, complemented
3	D₃×SL(2,𝔽₃)	—	3	4	no	maximal
4	C₂×C₃×C₃⋊D₃	—	4	1	no	maximal
5	C₃²×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
6	C₄×C₃⋊D₃	—	3	1	no	—
7	#634	—	3	1	no	—
8	C₃×SL(2,𝔽₃)	—	2	4	no	—
9	C₃×SL(2,𝔽₃)	D₃	1	4	yes	complemented
10	C₂×C₃³	—	4	1	no	—
11	C₃×C₃⋊D₃	—	4	2	no	—
12	D₃×Q₈	—	3	4	no	—
13	C₂×SL(2,𝔽₃)	—	9	1	no	—
14	C₂×C₃⋊D₃	A₄	1	1	yes	characteristic
15	C₂×C₃×D₃	—	12	4	no	—
16	C₃²×C₄	—	3	1	no	—
17	C₃⋊Dic₃	—	3	1	no	—
18	C₃³	—	4	1	no	Sylow
19	C₃×Q₈	C₃×D₃	1	4	yes	complemented
20	C₄×D₃	—	9	4	no	—
21	Dic₆	—	9	4	no	—
22	SL(2,𝔽₃)	—	2	4	no	—
23	SL(2,𝔽₃)	C₃⋊D₃	1	1	yes	characteristic, direct product factor
24	C₂×C₃²	—	4	4	no	—
25	C₂×C₃²	—	8	4	no	—
26	C₂×C₃²	C₂×A₄	1	1	yes	characteristic
27	C₃⋊D₃	SL(2,𝔽₃)	1	2	yes	direct product factor
28	C₃×D₃	—	12	8	no	—
29	C₂×Q₈	—	9	1	no	Sylow
30	C₂²×C₃	—	36	1	no	—
31	C₂×D₃	—	3	4	no	—
32	C₃×C₄	—	3	4	no	—
33	Dic₃	—	9	4	no	—
34	C₃²	—	4	4	no	—
35	C₃²	—	8	4	no	—
36	C₃²	C₂×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
37	Q₈	—	27	1	no	—
38	Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
39	C₂×C₄	—	27	1	no	—
40	C₂×C₃	—	4	1	no	—
41	C₂×C₃	—	8	4	no	—
42	C₂×C₃	—	36	2	no	—
43	C₂×C₃	D₃×A₄	1	4	yes	—
44	D₃	—	3	8	no	—
45	C₂²	—	9	1	no	—
46	C₄	—	3	1	no	—
47	C₄	—	27	1	no	—
48	C₃	—	4	1	no	—
49	C₃	—	8	4	no	—
50	C₃	D₃×SL(2,𝔽₃)	1	4	yes	complemented
51	C₂	—	9	2	no	—
52	C₂	A₄×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	A₄×C₃⋊D₃
commutator, Fitting, cosocle, hypocenter	C₃²×Q₈	C₂×C₃
socle	C₂×C₃²	C₂×A₄
LCS	G ▷ C₃²×Q₈
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₃²×SL(2,𝔽₃)	C₂	semidirect
2. Q₈×C₃⋊D₃	C₃	semidirect
5. C₃²×Q₈	C₂×C₃	semidirect
9. C₃×SL(2,𝔽₃)	D₃	semidirect
19. C₃×Q₈	C₃×D₃	semidirect
23. SL(2,𝔽₃)	C₃⋊D₃	direct (N × H)
27. C₃⋊D₃	SL(2,𝔽₃)	direct (N × H)
36. C₃²	C₂×SL(2,𝔽₃)	semidirect
38. Q₈	C₃×C₃⋊D₃	semidirect
50. C₃	D₃×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
C₃²×SL(2,𝔽₃)	2
Q₈×C₃⋊D₃	3
D₃×SL(2,𝔽₃)	3
C₂×C₃×C₃⋊D₃	4

Subgroups — C3⋊D3×SL(2,𝔽3) #88979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊D₃×SL(2,𝔽₃) #88979