Subgroups — C₂×C₃⋊GL(2,𝔽₃) #64263

Order 288.

144 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, complemented
2	C₃⋊GL(2,𝔽₃)	C₂	1	2	yes	maximal, direct product factor
3	C₂×GL(2,𝔽₃)	—	3	3	no	maximal
4	C₂×#215	—	3	1	no	maximal
5	C₂²×C₃⋊D₃	—	4	1	no	maximal
6	C₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
7	C₂×C₃×Q₈	D₃	1	1	yes	characteristic, complemented
8	C₂×D₁₂	—	3	1	no	—
9	C₂×SL(2,𝔽₃)	D₃	1	3	yes	complemented
10	GL(2,𝔽₃)	—	3	6	no	—
11	#215	—	3	2	no	—
12	#215	—	3	2	no	—
13	C₂×C₃⋊C₈	—	3	1	no	—
14	C₂²×C₃²	—	4	1	no	—
15	C₂×C₃⋊D₃	—	4	2	no	—
16	C₂×C₃⋊D₃	—	4	4	no	—
17	C₂×SD₁₆	—	9	1	no	Sylow
18	C₂²×D₃	—	6	1	no	—
19	C₂²×D₃	—	12	3	no	—
20	C₃×Q₈	—	3	1	no	—
21	C₃×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
22	C₂×C₃×C₄	—	3	1	no	—
23	D₁₂	—	3	2	no	—
24	D₁₂	—	6	1	no	—
25	SL(2,𝔽₃)	C₂×D₃	1	3	yes	complemented
26	C₃⋊C₈	—	3	2	no	—
27	C₂×C₃²	—	4	2	no	—
28	C₂×C₃²	—	4	1	no	—
29	C₃⋊D₃	—	4	4	no	—
30	C₂×Q₈	C₃⋊D₃	1	1	yes	characteristic, complemented
31	C₂×D₄	—	9	1	no	—
32	SD₁₆	—	9	2	no	—
33	SD₁₆	—	9	2	no	—
34	C₂×C₈	—	9	1	no	—
35	C₂²×C₃	—	4	3	no	—
36	C₂²×C₃	S₄	1	1	yes	characteristic
37	C₂×D₃	—	6	2	no	—
38	C₂×D₃	—	12	6	no	—
39	C₂×D₃	—	12	12	no	—
40	C₂×D₃	—	12	2	no	—
41	C₃×C₄	—	3	1	no	—
42	C₃×C₄	—	3	1	no	—
43	C₃²	—	4	1	no	Sylow
44	C₂³	—	18	1	no	—
45	Q₈	—	3	1	no	—
46	Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
47	D₄	—	9	2	no	—
48	D₄	—	18	1	no	—
49	C₂×C₄	—	3	1	no	—
50	C₈	—	9	2	no	—
51	C₂×C₃	—	4	3	no	—
52	C₂×C₃	—	4	6	no	—
53	C₂×C₃	GL(2,𝔽₃)	1	2	yes	complemented
54	C₂×C₃	C₂×S₄	1	1	yes	characteristic
55	D₃	—	12	12	no	—
56	D₃	—	12	2	no	—
57	C₂²	—	18	2	no	—
58	C₂²	—	36	2	no	—
59	C₂²	C₃⋊S₄	1	1	yes	characteristic
60	C₄	—	3	1	no	—
61	C₄	—	3	1	no	—
62	C₃	—	4	3	no	—
63	C₃	C₂×GL(2,𝔽₃)	1	1	yes	characteristic, complemented
64	C₂	—	36	2	no	—
65	C₂	C₃⋊GL(2,𝔽₃)	1	2	yes	direct product factor
66	C₂	C₂×C₃⋊S₄	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×SL(2,𝔽₃)	C₂	semidirect
2. C₃⋊GL(2,𝔽₃)	C₂	direct (N × H)
6. C₃×SL(2,𝔽₃)	C₂²	semidirect
7. C₂×C₃×Q₈	D₃	semidirect
9. C₂×SL(2,𝔽₃)	D₃	semidirect
21. C₃×Q₈	C₂×D₃	semidirect
25. SL(2,𝔽₃)	C₂×D₃	semidirect
30. C₂×Q₈	C₃⋊D₃	semidirect
46. Q₈	C₂×C₃⋊D₃	semidirect
53. C₂×C₃	GL(2,𝔽₃)	semidirect
63. C₃	C₂×GL(2,𝔽₃)	semidirect
65. C₂	C₃⋊GL(2,𝔽₃)	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
C₃⋊GL(2,𝔽₃)	2
C₂×GL(2,𝔽₃)	3
C₂×#215	3
C₂²×C₃⋊D₃	4

Subgroups — C2×C3⋊GL(2,𝔽3) #64263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃⋊GL(2,𝔽₃) #64263