Subgroups — #64267

Order 288.

110 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⋊GL(2,𝔽₃)	C₂	1	2	yes	maximal, complemented
3	#986	—	3	3	no	maximal
4	#949	—	3	1	no	maximal
5	C₄⋊(C₃⋊D₃)	—	4	1	no	maximal
6	C₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
7	C₃×#41	D₃	1	1	yes	characteristic, complemented
8	C₂×D₁₂	—	3	1	no	—
9	#41⋊C₃	D₃	1	3	yes	complemented
10	GL(2,𝔽₃)	—	3	6	no	—
11	#215	—	3	2	no	—
12	#213	—	3	2	no	—
13	#208	—	3	1	no	—
14	C₂×C₃⋊D₃	—	4	2	no	—
15	C₃²×C₄	—	4	1	no	—
16	AGL(1,ℤ/8)	—	9	1	no	Sylow
17	C₂²×D₃	—	6	1	no	—
18	C₃×Q₈	C₂×D₃	1	1	yes	characteristic
19	C₃×D₄	—	3	1	no	—
20	C₂×C₃×C₄	—	3	1	no	—
21	D₁₂	—	3	2	no	—
22	D₁₂	—	6	1	no	—
23	D₁₂	—	12	3	no	—
24	SL(2,𝔽₃)	C₂×D₃	1	3	yes	complemented
25	C₃⋊C₈	—	3	2	no	—
26	C₂×C₃²	—	4	1	no	—
27	C₃⋊D₃	—	8	2	no	—
28	#41	C₃⋊D₃	1	1	yes	characteristic, complemented
29	C₂×D₄	—	9	1	no	—
30	SD₁₆	—	9	2	no	—
31	D₈	—	9	2	no	—
32	M₁₆	—	9	1	no	—
33	C₂²×C₃	—	3	1	no	—
34	C₂×D₃	—	6	2	no	—
35	C₂×D₃	—	12	6	no	—
36	C₂×D₃	—	12	2	no	—
37	C₃×C₄	—	3	1	no	—
38	C₃×C₄	—	4	3	no	—
39	C₃×C₄	S₄	1	1	yes	characteristic
40	C₃²	—	4	1	no	Sylow
41	C₂³	—	18	1	no	—
42	Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
43	D₄	—	3	1	no	—
44	D₄	—	9	2	no	—
45	D₄	—	18	1	no	—
46	C₂×C₄	—	3	1	no	—
47	C₈	—	9	2	no	—
48	C₂×C₃	—	4	3	no	—
49	C₂×C₃	—	6	1	no	—
50	C₂×C₃	C₂×S₄	1	1	yes	characteristic
51	D₃	—	12	2	no	—
52	D₃	—	24	6	no	—
53	C₂²	—	3	1	no	—
54	C₂²	—	18	2	no	—
55	C₂²	—	36	2	no	—
56	C₄	—	3	1	no	—
57	C₄	C₃⋊S₄	1	1	yes	characteristic
58	C₃	—	4	3	no	—
59	C₃	#986	1	1	yes	characteristic, complemented
60	C₂	—	6	1	no	—
61	C₂	—	36	2	no	—
62	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
hypercenter	C₄	C₃⋊S₄
LCS	G ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₃×#41⋊C₃	C₂	semidirect
2. C₃⋊GL(2,𝔽₃)	C₂	semidirect
6. C₃×SL(2,𝔽₃)	C₂²	semidirect
7. C₃×#41	D₃	semidirect
9. #41⋊C₃	D₃	semidirect
24. SL(2,𝔽₃)	C₂×D₃	semidirect
28. #41	C₃⋊D₃	semidirect
59. C₃	#986	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⋊GL(2,𝔽₃)	2
#986	3
#949	3
C₄⋊(C₃⋊D₃)	4