Subgroups — #64265

Order 288.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#41⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#226	C₂	1	2	yes	maximal, complemented
3	#984	—	3	3	no	maximal
4	#951	—	3	1	no	maximal
5	#634	—	4	1	no	maximal
6	C₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₃×#41	D₃	1	1	yes	characteristic
8	C₂×Dic₆	—	3	1	no	—
9	#41⋊C₃	D₃	1	3	yes	—
10	#226	—	3	6	no	—
11	#216	—	3	2	no	—
12	#214	—	3	2	no	—
13	#208	—	3	1	no	—
14	C₃²×C₄	—	4	1	no	—
15	C₃⋊Dic₃	—	4	2	no	—
16	#137	—	9	1	no	Sylow
17	C₃×Q₈	C₂×D₃	1	1	yes	characteristic
18	C₃×D₄	—	3	1	no	—
19	C₂×C₃×C₄	—	3	1	no	—
20	C₂×Dic₃	—	6	1	no	—
21	Dic₆	—	3	2	no	—
22	Dic₆	—	6	1	no	—
23	Dic₆	—	12	3	no	—
24	SL(2,𝔽₃)	C₂×D₃	1	3	yes	—
25	C₃⋊C₈	—	3	2	no	—
26	C₂×C₃²	—	4	1	no	—
27	#41	C₃⋊D₃	1	1	yes	characteristic
28	C₂×Q₈	—	9	1	no	—
29	Q₁₆	—	9	2	no	—
30	SD₁₆	—	9	2	no	—
31	M₁₆	—	9	1	no	—
32	C₂²×C₃	—	3	1	no	—
33	C₃×C₄	—	3	1	no	—
34	C₃×C₄	—	4	3	no	—
35	C₃×C₄	S₄	1	1	yes	characteristic
36	Dic₃	—	6	2	no	—
37	Dic₃	—	12	6	no	—
38	C₃²	—	4	1	no	Sylow
39	Q₈	—	9	2	no	—
40	Q₈	—	18	1	no	—
41	Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
42	D₄	—	3	1	no	—
43	C₂×C₄	—	3	1	no	—
44	C₂×C₄	—	18	1	no	—
45	C₈	—	9	2	no	—
46	C₂×C₃	—	4	3	no	—
47	C₂×C₃	—	6	1	no	—
48	C₂×C₃	C₂×S₄	1	1	yes	characteristic
49	C₂²	—	3	1	no	—
50	C₄	—	3	1	no	—
51	C₄	—	18	2	no	—
52	C₄	C₃⋊S₄	1	1	yes	characteristic
53	C₃	—	4	3	no	—
54	C₃	#984	1	1	yes	characteristic, complemented
55	C₂	—	6	1	no	—
56	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	C₃×#41	D₃
socle	C₂×C₃	C₂×S₄
prime radical	C₃×#41⋊C₃	C₂
hypercenter	C₄	C₃⋊S₄
LCS	G ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄

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

N	H ≅ G/N	type
2. C₃⋊#226	C₂	semidirect
54. C₃	#984	semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.