Subgroups — #14519159

Order 1944.

226 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊(C₃⋊#1088)	C₂	1	2	yes	maximal
2	C₃²×#1088	C₂	1	1	yes	maximal, characteristic
3	C₃⋊SU(3,𝔽₂)	—	3	4	no	maximal
4	C₃²×#272	C₂²	1	1	yes	characteristic
5	C₃⋊#1088	—	3	8	no	—
6	C₃×#1088	D₃	1	4	yes	—
7	C₃²×H(3,𝔽₃)	Q₈	1	1	yes	characteristic, complemented, Sylow
8	C₃×#634	—	9	1	no	maximal
9	SU(3,𝔽₂)	—	9	1	no	—
10	C₃³×D₃	—	12	1	no	—
11	C₃×#272	C₂×D₃	1	4	yes	—
12	C₃³×C₄	—	9	1	no	—
13	C₃×C₃⋊Dic₃	—	9	2	no	—
14	#1088	—	9	2	no	—
15	#1088	C₃⋊D₃	1	1	yes	characteristic
16	C₃⁴	—	4	1	no	—
17	C₃×H(3,𝔽₃)	—	8	4	no	—
18	C₃×H(3,𝔽₃)	Dic₆	1	4	yes	complemented
19	#634	—	9	1	no	—
20	C₃×Dic₆	—	27	4	no	—
21	C₂×C₃³	—	9	1	no	—
22	C₃²×D₃	—	12	4	no	—
23	C₃²×D₃	—	12	1	no	—
24	C₃²×D₃	—	12	8	no	—
25	#272	C₂×C₃⋊D₃	1	1	yes	characteristic
26	C₃²×C₄	—	9	1	no	—
27	C₃²×C₄	—	9	4	no	—
28	C₃²×C₄	—	18	4	no	—
29	C₃⋊Dic₃	—	9	2	no	—
30	C₃×Dic₃	—	27	8	no	—
31	C₃³	—	4	4	no	—
32	C₃³	—	8	4	no	—
33	C₃³	—	12	8	no	—
34	C₃³	—	12	1	no	—
35	C₃³	PSU(3,𝔽₂)	1	1	yes	characteristic
36	H(3,𝔽₃)	—	8	4	no	—
37	H(3,𝔽₃)	—	8	6	no	—
38	H(3,𝔽₃)	#634	1	1	yes	characteristic, complemented
39	C₃×Q₈	—	81	1	no	—
40	Dic₆	—	27	4	no	—
41	C₂×C₃²	—	9	1	no	—
42	C₂×C₃²	—	9	4	no	—
43	C₂×C₃²	—	18	4	no	—
44	C₃×D₃	—	12	1	no	—
45	C₃×D₃	—	12	4	no	—
46	C₃×D₃	—	12	8	no	—
47	C₃×C₄	—	9	4	no	—
48	C₃×C₄	—	9	1	no	—
49	C₃×C₄	—	18	4	no	—
50	C₃×C₄	—	81	2	no	—
51	Dic₃	—	27	8	no	—
52	C₃²	—	2	4	no	—
53	C₃²	—	4	1	no	—
54	C₃²	—	8	4	no	—
55	C₃²	—	12	8	no	—
56	C₃²	—	12	4	no	—
57	C₃²	—	24	8	no	—
58	C₃²	—	24	4	no	—
59	C₃²	SU(3,𝔽₂)	1	1	yes	characteristic, complemented
60	C₃²	C₃⋊PSU(3,𝔽₂)	1	4	yes	—
61	Q₈	—	81	1	no	Sylow
62	C₂×C₃	—	9	1	no	—
63	C₂×C₃	—	9	4	no	—
64	C₂×C₃	—	18	4	no	—
65	D₃	—	12	1	no	—
66	C₄	—	9	1	no	—
67	C₄	—	81	2	no	—
68	C₃	—	2	4	no	—
69	C₃	—	12	1	no	—
70	C₃	—	24	4	no	—
71	C₃	C₃⋊SU(3,𝔽₂)	1	4	yes	complemented
72	C₃	#10620209	1	1	yes	characteristic
73	C₂	—	9	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₃	#10620209
commutator, prime radical, cosocle	C₃²×#272	C₂²
Fitting, hypocenter	C₃²×H(3,𝔽₃)	Q₈
socle	C₃³	PSU(3,𝔽₂)
LCS	G ▷ C₃²×#272 ▷ C₃²×H(3,𝔽₃)
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
7. C₃²×H(3,𝔽₃)	Q₈	semidirect
18. C₃×H(3,𝔽₃)	Dic₆	semidirect
38. H(3,𝔽₃)	#634	semidirect
59. C₃²	SU(3,𝔽₂)	semidirect
71. C₃	C₃⋊SU(3,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₃⋊(C₃⋊#1088)	2
C₃²×#1088	2
C₃⋊SU(3,𝔽₂)	3
C₃×#634	9