Subgroups — #10624790

Order 729.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#732	C₃	1	1	yes	maximal, characteristic
2	C₉⋊H(3,𝔽₃)	C₃	1	3	yes	maximal, complemented
3	C₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	H(3,𝔽₃)⋊C₉	C₃	1	6	yes	maximal, complemented
5	#6868	C₃	1	2	yes	maximal, complemented
6	C₃×3₋¹⁺²	—	3	3	no	—
7	C₃×H(3,𝔽₃)	C₃²	1	3	yes	—
8	C₃²×C₉	C₃²	1	1	yes	characteristic
9	C₃²×C₉	C₃²	1	2	yes	—
10	C₃²×C₉	C₃²	1	1	yes	characteristic
11	#732	—	3	3	no	—
12	C₉⋊C₉	—	3	3	no	—
13	C₃²⋊C₉	—	3	6	no	—
14	C₃²⋊C₉	C₃²	1	6	yes	—
15	C₃³	—	3	3	no	—
16	C₃³	C₃³	1	1	yes	characteristic
17	3₋¹⁺²	—	9	9	no	—
18	H(3,𝔽₃)	—	3	3	no	—
19	H(3,𝔽₃)	—	3	6	no	—
20	C₃×C₉	—	3	1	no	—
21	C₃×C₉	—	3	2	no	—
22	C₃×C₉	—	3	6	no	—
23	C₃×C₉	—	3	2	no	—
24	C₃×C₉	—	9	1	no	—
25	C₃×C₉	—	9	2	no	—
26	C₃×C₉	H(3,𝔽₃)	1	3	yes	—
27	C₃×C₉	3₋¹⁺²	1	3	yes	complemented
28	C₃²	—	3	2	no	—
29	C₃²	—	3	1	no	—
30	C₃²	—	9	6	no	—
31	C₃²	—	9	3	no	—
32	C₃²	—	9	3	no	—
33	C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
34	C₃²	C₃×3₋¹⁺²	1	1	yes	characteristic
35	C₃²	#737	1	2	yes	—
36	C₉	—	3	3	no	—
37	C₉	—	3	3	no	—
38	C₉	—	9	2	no	—
39	C₉	—	9	6	no	—
40	C₃	—	3	1	no	—
41	C₃	—	3	2	no	—
42	C₃	—	27	3	no	—
43	C₃	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
44	C₃	#6911	1	1	yes	characteristic
45	C₃	#6913	1	2	yes	—

	subgroup	G/H
center, socle	C₃²	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³
prime radical	C₃×#732	C₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ 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.