Subgroups — C₂×C₃×GL(2,𝔽₃) #64252

Order 288.

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

	subgroup	G/H
center, socle, hypercenter	C₂²×C₃	S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂²×C₃
Frattini	C₂	C₂×C₃×S₄
Fitting	C₂×C₃×Q₈	D₃
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
1. C₂×C₃×SL(2,𝔽₃)	C₂	semidirect
2. C₃×GL(2,𝔽₃)	C₂	direct (N × H)
3. C₂×GL(2,𝔽₃)	C₃	direct (N × H)
6. C₃×SL(2,𝔽₃)	C₂²	semidirect
7. C₂×C₃×Q₈	D₃	semidirect
10. C₂×SL(2,𝔽₃)	C₂×C₃	semidirect
11. GL(2,𝔽₃)	C₂×C₃	direct (N × H)
22. C₃×Q₈	C₂×D₃	semidirect
27. SL(2,𝔽₃)	C₂²×C₃	semidirect
32. C₂×Q₈	C₃×D₃	semidirect
49. Q₈	C₂×C₃×D₃	semidirect
59. C₂×C₃	GL(2,𝔽₃)	direct (N × H)
69. C₃	C₂×GL(2,𝔽₃)	direct (N × H)
71. 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×C₃×SD₁₆	3
C₂²×C₃×D₃	4

Subgroups — C2×C3×GL(2,𝔽3) #64252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×GL(2,𝔽₃) #64252