Subgroups — #92258

Order 486.

171 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²⋊H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃²×C₃⋊D₃	—	3	1	no	maximal
3	C₃⋊((C₃⋊D₃)⋊C₃)	C₃	1	4	yes	maximal, complemented
4	C₃⁴	D₃	1	1	yes	characteristic, complemented
5	C₃×H(3,𝔽₃)	—	2	4	no	—
6	C₃×H(3,𝔽₃)	C₂×C₃	1	4	yes	complemented
7	C₃⋊(C₃⋊D₃)	C₃²	1	1	yes	characteristic, complemented
8	C₃×C₃⋊D₃	—	3	4	no	—
9	C₃²×D₃	—	9	4	no	—
10	(C₃⋊D₃)⋊C₃	—	9	4	no	—
11	C₃³	—	2	4	no	—
12	C₃³	—	3	4	no	—
13	C₃³	—	6	4	no	—
14	C₃³	C₃×D₃	1	4	yes	—
15	C₃³	C₂×C₃²	1	1	yes	characteristic, complemented
16	H(3,𝔽₃)	—	3	4	no	—
17	H(3,𝔽₃)	—	6	4	no	—
18	H(3,𝔽₃)	—	6	8	no	—
19	H(3,𝔽₃)	—	6	4	no	—
20	C₂×C₃²	—	27	1	no	—
21	C₃⋊D₃	—	3	1	no	—
22	C₃⋊D₃	—	9	4	no	—
23	C₃×D₃	—	9	12	no	—
24	C₃×D₃	—	9	4	no	—
25	C₃²	—	2	4	no	—
26	C₃²	—	3	1	no	—
27	C₃²	—	3	12	no	—
28	C₃²	—	3	4	no	—
29	C₃²	—	6	6	no	—
30	C₃²	—	6	4	no	—
31	C₃²	—	6	3	no	—
32	C₃²	—	6	4	no	—
33	C₃²	—	6	12	no	—
34	C₃²	(C₃⋊D₃)⋊C₃	1	4	yes	complemented
35	C₃²	C₃²×D₃	1	1	yes	characteristic
36	C₂×C₃	—	27	4	no	—
37	D₃	—	9	4	no	—
38	D₃	—	81	1	no	—
39	C₃	—	3	4	no	—
40	C₃	—	6	4	no	—
41	C₃	—	9	1	no	—
42	C₃	—	18	4	no	—
43	C₃	C₃×(C₃⋊D₃)⋊C₃	1	4	yes	—
44	C₂	—	27	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃³	C₂×C₃²
Frattini, socle	C₃²	C₃²×D₃
Fitting	C₃²⋊H(3,𝔽₃)	C₂
LCS	G ▷ C₃³
UCS	C₁

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

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