Subgroups — C₂×C₃×#226 #64251

Order 288.

85 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	C₃×#226	C₂	1	2	yes	maximal, direct product factor
3	C₂×#226	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₂×C₃×Q₁₆	—	3	1	no	maximal
5	C₂×C₃×Dic₃	—	4	1	no	maximal
6	C₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₂×C₃×Q₈	—	3	1	no	—
8	C₂×C₃×Q₈	D₃	1	1	yes	characteristic
9	C₂×SL(2,𝔽₃)	—	2	1	no	—
10	C₂×SL(2,𝔽₃)	C₂×C₃	1	1	yes	characteristic
11	#226	C₂×C₃	1	2	yes	direct product factor
12	C₃×Q₁₆	—	3	2	no	—
13	C₃×Q₁₆	—	3	2	no	—
14	C₂×C₃×C₈	—	3	1	no	—
15	C₂²×C₃²	—	4	1	no	—
16	C₃×Dic₃	—	4	2	no	—
17	C₂×Q₁₆	—	3	1	no	Sylow
18	C₃×Q₈	—	3	1	no	—
19	C₃×Q₈	—	3	2	no	—
20	C₃×Q₈	—	6	1	no	—
21	C₃×Q₈	C₂×D₃	1	1	yes	characteristic
22	C₂×C₃×C₄	—	3	1	no	—
23	C₂×C₃×C₄	—	6	1	no	—
24	C₂×Dic₃	—	4	1	no	—
25	SL(2,𝔽₃)	—	2	1	no	—
26	SL(2,𝔽₃)	C₂²×C₃	1	1	yes	characteristic
27	C₃×C₈	—	3	2	no	—
28	C₂×C₃²	—	4	2	no	—
29	C₂×C₃²	—	4	1	no	—
30	C₂×Q₈	—	3	1	no	—
31	C₂×Q₈	C₃×D₃	1	1	yes	characteristic
32	Q₁₆	—	3	2	no	—
33	Q₁₆	—	3	2	no	—
34	C₂×C₈	—	3	1	no	—
35	C₂²×C₃	—	4	1	no	—
36	C₂²×C₃	—	8	1	no	—
37	C₂²×C₃	S₄	1	1	yes	characteristic
38	C₃×C₄	—	3	1	no	—
39	C₃×C₄	—	3	1	no	—
40	C₃×C₄	—	6	2	no	—
41	Dic₃	—	4	2	no	—
42	C₃²	—	4	1	no	Sylow
43	Q₈	—	3	1	no	—
44	Q₈	—	3	2	no	—
45	Q₈	—	6	1	no	—
46	Q₈	C₂×C₃×D₃	1	1	yes	characteristic
47	C₂×C₄	—	3	1	no	—
48	C₂×C₄	—	6	1	no	—
49	C₈	—	3	2	no	—
50	C₂×C₃	—	4	1	no	—
51	C₂×C₃	—	4	2	no	—
52	C₂×C₃	—	8	1	no	—
53	C₂×C₃	—	8	2	no	—
54	C₂×C₃	#226	1	2	yes	direct product factor
55	C₂×C₃	C₂×S₄	1	1	yes	characteristic
56	C₂²	C₃×S₄	1	1	yes	characteristic
57	C₄	—	3	1	no	—
58	C₄	—	3	1	no	—
59	C₄	—	6	2	no	—
60	C₃	—	4	1	no	—
61	C₃	—	8	1	no	—
62	C₃	C₂×#226	1	1	yes	characteristic, direct product factor
63	C₂	C₃×#226	1	2	yes	direct product factor
64	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
prime radical	C₂×C₃×SL(2,𝔽₃)	C₂
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. C₃×#226	C₂	direct (N × H)
3. C₂×#226	C₃	direct (N × H)
11. #226	C₂×C₃	direct (N × H)
54. C₂×C₃	#226	direct (N × H)
62. C₃	C₂×#226	direct (N × H)
63. C₂	C₃×#226	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×#226	2
C₂×#226	3
C₂×C₃×Q₁₆	3
C₂×C₃×Dic₃	4

Subgroups — C2×C3×#226 #64251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#226 #64251