Subgroups — #14548233

Order 2187.

662 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624732	C₃	1	3	yes	maximal, complemented
3	#10624732	C₃	1	9	yes	maximal, complemented
4	C₃³×C₉	C₃²	1	1	yes	characteristic
5	C₃³×C₉	C₃²	1	3	yes	—
6	C₃×#732	—	3	27	no	—
7	C₃×#732	—	3	9	no	—
8	C₃²⋊H(3,𝔽₃)	C₃²	1	9	yes	—
9	C₃×C₉⋊C₉	—	3	9	no	—
10	C₃×C₉⋊C₉	—	3	3	no	—
11	C₃⁴	C₃³	1	1	yes	characteristic
12	C₃×H(3,𝔽₃)	—	3	9	no	—
13	C₃×H(3,𝔽₃)	—	3	27	no	—
14	C₃²×C₉	—	3	3	no	—
15	C₃²×C₉	—	3	9	no	—
16	C₃²×C₉	—	3	3	no	—
17	C₃²×C₉	—	3	6	no	—
18	C₃²×C₉	—	9	9	no	—
19	C₃²×C₉	H(3,𝔽₃)	1	9	yes	—
20	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
21	#732	—	9	81	no	—
22	#732	—	9	27	no	—
23	C₉⋊C₉	—	3	27	no	—
24	C₃³	—	3	3	no	—
25	C₃³	—	3	9	no	—
26	C₃³	—	9	9	no	—
27	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
28	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
29	H(3,𝔽₃)	—	9	27	no	—
30	H(3,𝔽₃)	—	9	81	no	—
31	C₃×C₉	—	3	27	no	—
32	C₃×C₉	—	3	3	no	—
33	C₃×C₉	—	3	6	no	—
34	C₃×C₉	—	3	9	no	—
35	C₃×C₉	—	3	27	no	—
36	C₃×C₉	—	9	9	no	—
37	C₃×C₉	—	9	18	no	—
38	C₃×C₉	#732	1	9	yes	complemented
39	C₃²	—	3	27	no	—
40	C₃²	—	3	3	no	—
41	C₃²	—	3	3	no	—
42	C₃²	—	3	6	no	—
43	C₃²	—	27	9	no	—
44	C₃²	—	27	27	no	—
45	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
46	C₃²	C₃×#732	1	3	yes	—
47	C₃²	#6911	1	9	yes	—
48	C₉	—	3	9	no	—
49	C₉	—	9	9	no	—
50	C₃	—	3	3	no	—
51	C₃	—	3	9	no	—
52	C₃	—	81	9	no	—
53	C₃	#10624732	1	1	yes	characteristic
54	C₃	#10624737	1	3	yes	—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
LCS	G ▷ C₃⁴ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ 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.

subgroup	index
C₃²×C₉⋊C₉	3
#10624732	3