Subgroups — #10624906

Order 729.

322 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6898	C₃	1	9	yes	maximal, complemented
2	#6897	C₃	1	9	yes	maximal, complemented
3	#6894	C₃	1	12	yes	maximal, complemented
4	#6891	C₃	1	2	yes	maximal
5	C₉×3₋¹⁺²	C₃	1	6	yes	maximal, complemented
6	C₃×C₃²⋊C₉	C₃	1	2	yes	maximal
7	C₃⁴	C₃²	1	1	yes	characteristic
8	C₃×3₋¹⁺²	C₃²	1	6	yes	—
9	C₃²×C₉	C₃²	1	6	yes	—
10	C₉⋊C₉	C₃²	1	36	yes	complemented
11	C₉⋊C₉	C₃²	1	36	yes	complemented
12	C₃²⋊C₉	C₃²	1	2	yes	—
13	C₃²⋊C₉	C₃²	1	4	yes	—
14	C₃²⋊C₉	C₃²	1	6	yes	—
15	C₃²⋊C₉	C₃²	1	6	yes	—
16	C₃²⋊C₉	C₃²	1	6	yes	—
17	C₃²⋊C₉	C₃²	1	6	yes	—
18	C₃²⋊C₉	C₃²	1	6	yes	—
19	C₉²	C₃²	1	9	yes	complemented
20	C₃³	—	3	6	no	—
21	C₃³	—	9	2	no	—
22	C₃³	C₃³	1	1	yes	characteristic
23	C₃³	C₃³	1	1	yes	characteristic
24	C₃³	C₃³	1	2	yes	—
25	3₋¹⁺²	—	3	18	no	—
26	C₃×C₉	—	9	6	no	—
27	C₃×C₉	C₃³	1	6	yes	—
28	C₃×C₉	C₃³	1	18	yes	—
29	C₃×C₉	C₃³	1	12	yes	—
30	C₃²	—	3	2	no	—
31	C₃²	—	3	2	no	—
32	C₃²	—	3	2	no	—
33	C₃²	—	3	2	no	—
34	C₃²	—	3	2	no	—
35	C₃²	—	3	2	no	—
36	C₃²	—	3	2	no	—
37	C₃²	—	9	9	no	—
38	C₃²	#737	1	6	yes	—
39	C₃²	C₃⁴	1	1	yes	characteristic
40	C₉	—	3	18	no	—
41	C₉	—	3	6	no	—
42	C₉	—	3	12	no	—
43	C₃	—	3	6	no	—
44	C₃	—	9	1	no	—
45	C₃	—	9	1	no	—
46	C₃	C₃×#737	1	2	yes	—
47	C₃	3₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃²	C₃⁴
prime radical	C₃⁴	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ G
LECS	G ▷ C₃² ▷ 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.