Subgroups — #64197

Order 288.

83 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, complemented
2	C₃⋊#226	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×GL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	#985	—	3	1	no	maximal
5	#916	—	3	1	no	maximal
6	C₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	D₃×Dic₃	—	4	1	no	maximal
8	#239	D₃	1	1	yes	characteristic, complemented
9	#237	—	3	1	no	—
10	#41⋊C₃	—	3	1	no	—
11	GL(2,𝔽₃)	D₃	1	1	yes	characteristic, complemented
12	#226	—	3	1	no	—
13	#226	—	6	1	no	—
14	C₃×SD₁₆	—	3	1	no	—
15	#216	—	3	1	no	—
16	#213	—	3	1	no	—
17	#204	—	3	1	no	—
18	D₃×C₈	—	3	1	no	—
19	C₂×C₃×D₃	—	4	1	no	—
20	C₃⋊Dic₃	—	4	1	no	—
21	C₃×Dic₃	—	4	1	no	—
22	#135	—	9	1	no	Sylow
23	C₃×Q₈	C₂×D₃	1	1	yes	characteristic
24	C₃×D₄	—	3	1	no	—
25	#67	—	6	1	no	—
26	C₂×Dic₃	—	6	1	no	—
27	D₁₂	—	3	1	no	—
28	C₄×D₃	—	3	1	no	—
29	C₄×D₃	—	12	1	no	—
30	Dic₆	—	3	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	C₃×D₃	—	4	2	no	—
37	#41	—	3	1	no	—
38	#41	—	9	1	no	—
39	Q₁₆	—	9	1	no	—
40	SD₁₆	—	3	1	no	—
41	SD₁₆	—	9	1	no	—
42	D₈	—	9	1	no	—
43	C₂×C₈	—	9	1	no	—
44	C₂²×C₃	—	6	1	no	—
45	C₂×D₃	—	3	1	no	—
46	C₂×D₃	—	4	1	no	—
47	C₃×C₄	—	3	1	no	—
48	C₃×C₄	—	12	1	no	—
49	Dic₃	—	6	1	no	—
50	Dic₃	—	12	1	no	—
51	Dic₃	—	24	1	no	—
52	Dic₃	S₄	1	1	yes	characteristic
53	C₃²	—	4	1	no	Sylow
54	Q₈	—	9	1	no	—
55	Q₈	D₃²	1	1	yes	characteristic
56	D₄	—	3	1	no	—
57	D₄	—	9	1	no	—
58	D₄	—	18	1	no	—
59	C₂×C₄	—	9	1	no	—
60	C₂×C₄	—	18	1	no	—
61	C₈	—	3	1	no	—
62	C₈	—	9	1	no	—
63	C₂×C₃	—	4	1	no	—
64	C₂×C₃	—	8	1	no	—
65	C₂×C₃	—	12	1	no	—
66	C₂×C₃	C₂×S₄	1	1	yes	characteristic
67	D₃	—	4	2	no	—
68	D₃	—	6	1	no	—
69	C₂²	—	6	1	no	—
70	C₂²	—	9	1	no	—
71	C₄	—	3	1	no	—
72	C₄	—	3	1	no	—
73	C₄	—	18	1	no	—
74	C₃	—	4	1	no	—
75	C₃	—	8	1	no	—
76	C₃	#985	1	1	yes	characteristic, complemented
77	C₂	—	12	1	no	—
78	C₂	—	18	1	no	—
79	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
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
1. C₃⋊(#41⋊C₃)	C₂	semidirect
2. C₃⋊#226	C₂	semidirect
3. C₃×GL(2,𝔽₃)	C₂	semidirect
8. #239	D₃	semidirect
11. GL(2,𝔽₃)	D₃	semidirect
76. C₃	#985	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×GL(2,𝔽₃)	2
#985	3
#916	3
D₃×Dic₃	4