Subgroups — #64196

Order 288.

77 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	1	yes	maximal, characteristic, complemented
3	C₃×#226	C₂	1	1	yes	maximal, characteristic, complemented
4	#984	—	3	1	no	maximal
5	#918	—	3	1	no	maximal
6	C₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	#627	—	4	1	no	maximal
8	#239	D₃	1	1	yes	characteristic
9	D₃×Q₈	—	3	1	no	—
10	#41⋊C₃	—	3	1	no	—
11	#226	—	3	1	no	—
12	#226	—	6	1	no	—
13	#226	D₃	1	1	yes	characteristic, complemented
14	C₃×Q₁₆	—	3	1	no	—
15	#216	—	3	1	no	—
16	#215	—	3	1	no	—
17	#204	—	3	1	no	—
18	ΓL(1,𝔽₂₅)	—	3	1	no	—
19	C₃⋊Dic₃	—	4	1	no	—
20	C₃×Dic₃	—	4	1	no	—
21	C₃×Dic₃	—	4	1	no	—
22	#137	—	9	1	no	Sylow
23	C₃×Q₈	—	3	1	no	—
24	C₃×Q₈	C₂×D₃	1	1	yes	characteristic
25	D₁₂	—	3	1	no	—
26	C₄×D₃	—	3	1	no	—
27	C₄×D₃	—	6	1	no	—
28	Dic₆	—	3	1	no	—
29	Dic₆	—	6	1	no	—
30	Dic₆	—	12	1	no	—
31	SL(2,𝔽₃)	—	2	1	no	—
32	SL(2,𝔽₃)	C₂×D₃	1	1	yes	characteristic
33	C₃×C₈	—	3	1	no	—
34	C₃⋊C₈	—	3	1	no	—
35	C₂×C₃²	—	4	1	no	—
36	#41	—	3	1	no	—
37	C₂×Q₈	—	9	1	no	—
38	Q₁₆	—	3	1	no	—
39	Q₁₆	—	9	1	no	—
40	SD₁₆	—	9	1	no	—
41	SD₁₆	—	9	1	no	—
42	M₁₆	—	9	1	no	—
43	C₂×D₃	—	3	1	no	—
44	C₃×C₄	—	3	1	no	—
45	C₃×C₄	—	6	1	no	—
46	C₃×C₄	—	12	1	no	—
47	Dic₃	—	4	1	no	—
48	Dic₃	—	6	1	no	—
49	Dic₃	—	12	1	no	—
50	Dic₃	—	24	1	no	—
51	Dic₃	S₄	1	1	yes	characteristic
52	C₃²	—	4	1	no	Sylow
53	Q₈	—	3	1	no	—
54	Q₈	—	9	1	no	—
55	Q₈	—	18	1	no	—
56	Q₈	D₃²	1	1	yes	characteristic
57	D₄	—	9	1	no	—
58	C₂×C₄	—	9	1	no	—
59	C₂×C₄	—	18	1	no	—
60	C₈	—	3	1	no	—
61	C₈	—	9	1	no	—
62	C₂×C₃	—	4	1	no	—
63	C₂×C₃	—	8	1	no	—
64	C₂×C₃	C₂×S₄	1	1	yes	characteristic
65	D₃	—	6	1	no	—
66	C₂²	—	9	1	no	—
67	C₄	—	3	1	no	—
68	C₄	—	3	1	no	—
69	C₄	—	6	1	no	—
70	C₄	—	18	1	no	—
71	C₃	—	4	1	no	—
72	C₃	—	8	1	no	—
73	C₃	#984	1	1	yes	characteristic, complemented
74	C₂	—	18	1	no	—
75	C₂	D₃×S₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×S₄
commutator, cosocle, hypocenter	C₃×SL(2,𝔽₃)	C₂²
Fitting	C₃×Q₈	C₂×D₃
socle	C₂×C₃	C₂×S₄
prime radical	C₃⋊(#41⋊C₃)	C₂
LCS	G ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₃⋊#226	C₂	semidirect
3. C₃×#226	C₂	semidirect
13. #226	D₃	semidirect
73. C₃	#984	semidirect

Maximal subgroups

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

subgroup	index
C₃⋊(#41⋊C₃)	2
C₃⋊#226	2
C₃×#226	2
#984	3
#918	3
#627	4