Subgroups — #64264

Order 288.

104 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	1	yes	maximal, characteristic, complemented
3	C₃⋊#226	C₂	1	1	yes	maximal, characteristic, complemented
4	#983	—	3	3	no	maximal
5	#942	—	3	1	no	maximal
6	#638	—	4	1	no	maximal
7	C₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
8	C₂×C₃×Q₈	D₃	1	1	yes	characteristic, complemented
9	#235	—	3	1	no	—
10	C₂×SL(2,𝔽₃)	D₃	1	3	yes	complemented
11	GL(2,𝔽₃)	—	3	3	no	—
12	#226	—	3	3	no	—
13	#216	—	3	1	no	—
14	#216	—	3	1	no	—
15	#215	—	3	1	no	—
16	#215	—	3	1	no	—
17	#208	—	3	1	no	—
18	C₂²×C₃²	—	4	1	no	—
19	C₂×C₃⋊D₃	—	4	1	no	—
20	C₃⋊Dic₃	—	4	1	no	—
21	#137	—	9	1	no	Sylow
22	C₃×Q₈	—	3	1	no	—
23	C₃×Q₈	C₂×D₃	1	1	yes	characteristic
24	C₂×C₃×C₄	—	3	1	no	—
25	#67	—	6	1	no	—
26	#67	—	12	3	no	—
27	D₁₂	—	3	1	no	—
28	C₄×D₃	—	6	1	no	—
29	Dic₆	—	3	1	no	—
30	SL(2,𝔽₃)	C₂×D₃	1	3	yes	—
31	C₃⋊C₈	—	3	1	no	—
32	C₃⋊C₈	—	3	1	no	—
33	C₂×C₃²	—	4	1	no	—
34	C₂×C₃²	—	8	1	no	—
35	C₃⋊D₃	—	8	1	no	—
36	#41	—	9	1	no	—
37	C₂×Q₈	C₃⋊D₃	1	1	yes	characteristic, complemented
38	Q₁₆	—	9	1	no	—
39	Q₁₆	—	9	1	no	—
40	SD₁₆	—	9	1	no	—
41	SD₁₆	—	9	1	no	—
42	M₁₆	—	9	1	no	—
43	C₂²×C₃	—	4	3	no	—
44	C₂²×C₃	S₄	1	1	yes	characteristic
45	C₂×D₃	—	6	1	no	—
46	C₂×D₃	—	12	3	no	—
47	C₃×C₄	—	3	1	no	—
48	C₃×C₄	—	3	1	no	—
49	Dic₃	—	6	1	no	—
50	Dic₃	—	12	3	no	—
51	C₃²	—	4	1	no	Sylow
52	Q₈	—	3	1	no	—
53	Q₈	—	9	1	no	—
54	Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
55	D₄	—	9	1	no	—
56	D₄	—	18	1	no	—
57	C₂×C₄	—	3	1	no	—
58	C₂×C₄	—	18	1	no	—
59	C₈	—	9	1	no	—
60	C₈	—	9	1	no	—
61	C₂×C₃	—	2	1	no	—
62	C₂×C₃	—	4	3	no	—
63	C₂×C₃	—	8	3	no	—
64	C₂×C₃	C₂×S₄	1	1	yes	characteristic
65	D₃	—	12	1	no	—
66	D₃	—	24	3	no	—
67	C₂²	—	18	1	no	—
68	C₂²	C₃⋊S₄	1	1	yes	characteristic
69	C₄	—	3	1	no	—
70	C₄	—	3	1	no	—
71	C₄	—	18	1	no	—
72	C₃	—	4	3	no	—
73	C₃	#983	1	1	yes	characteristic, complemented
74	C₂	—	2	1	no	—
75	C₂	—	36	1	no	—
76	C₂	C₂×C₃⋊S₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×C₃⋊S₄
commutator, cosocle, hypocenter	C₃×SL(2,𝔽₃)	C₂²
Fitting	C₂×C₃×Q₈	D₃
socle	C₂×C₃	C₂×S₄
hypercenter	C₂²	C₃⋊S₄
LCS	G ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ 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	semidirect
3. C₃⋊#226	C₂	semidirect
8. C₂×C₃×Q₈	D₃	semidirect
10. C₂×SL(2,𝔽₃)	D₃	semidirect
37. C₂×Q₈	C₃⋊D₃	semidirect
73. C₃	#983	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
C₃⋊GL(2,𝔽₃)	2
C₃⋊#226	2
#983	3
#942	3
#638	4