Subgroups — #14546451

Order 2187.

87 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624509	C₃	1	2	yes	maximal
2	#10624506	C₃	1	1	yes	maximal, characteristic
3	#10624452	C₃	1	1	yes	maximal, characteristic
4	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
5	C₃×C₃²⋊C₉	—	3	1	no	—
6	#6866	—	3	1	no	—
7	#6866	—	3	2	no	—
8	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
9	C₃×3₋¹⁺²	—	3	1	no	—
10	C₃×3₋¹⁺²	—	3	2	no	—
11	C₃×3₋¹⁺²	—	9	3	no	—
12	C₃²×C₉	—	3	1	no	—
13	C₃²×C₉	—	9	1	no	—
14	C₃²⋊C₉	—	9	3	no	—
15	C₃²⋊C₉	—	9	1	no	—
16	C₃²⋊C₉	—	9	2	no	—
17	C₃³	—	9	1	no	—
18	C₃³	—	9	1	no	—
19	C₃³	—	9	1	no	—
20	C₃³	—	9	1	no	—
21	C₃³	#731	1	1	yes	characteristic
22	C₃³	#731	1	1	yes	characteristic
23	C₃³	#731	1	2	yes	—
24	3₋¹⁺²	—	9	3	no	—
25	3₋¹⁺²	—	9	6	no	—
26	C₃×C₉	—	9	1	no	—
27	C₃×C₉	—	9	3	no	—
28	C₃×C₉	—	9	3	no	—
29	C₃×C₉	—	27	1	no	—
30	C₃×C₉	—	27	1	no	—
31	C₃×C₉	—	27	1	no	—
32	C₃×C₉	—	27	1	no	—
33	C₃×C₉	—	27	2	no	—
34	C₃²	—	3	1	no	—
35	C₃²	—	3	1	no	—
36	C₃²	—	3	1	no	—
37	C₃²	—	3	1	no	—
38	C₃²	—	3	1	no	—
39	C₃²	—	3	2	no	—
40	C₃²	—	9	1	no	—
41	C₃²	—	9	2	no	—
42	C₃²	—	27	1	no	—
43	C₃²	—	27	1	no	—
44	C₃²	—	27	1	no	—
45	C₃²	#6862	1	1	yes	characteristic
46	C₉	—	9	3	no	—
47	C₉	—	27	3	no	—
48	C₉	—	81	1	no	—
49	C₉	—	81	2	no	—
50	C₃	—	3	1	no	—
51	C₃	—	3	1	no	—
52	C₃	—	3	1	no	—
53	C₃	—	3	1	no	—
54	C₃	—	9	1	no	—
55	C₃	—	9	2	no	—
56	C₃	#10624496	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃	#10624496
commutator, Frattini, cosocle	C₃²×3₋¹⁺²	C₃²
prime radical	C₃⁴	H(3,𝔽₃)
LCS	G ▷ C₃²×3₋¹⁺² ▷ C₃⁴ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺² ◁ G
LECS	G ▷ C₃²×3₋¹⁺² ▷ C₃⁴ ▷ C₃² ▷ C₃ ▷ C₁

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