Subgroups — #14550710

Order 2187.

528 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624880	C₃	1	1	yes	maximal, characteristic
2	#10624698	C₃	1	8	yes	maximal, complemented
3	C₃×H(3,𝔽₃)⋊C₉	C₃	1	4	yes	maximal, complemented
4	C₃²×H(3,𝔽₃)	C₃²	1	4	yes	—
5	C₃³×C₉	C₃²	1	1	yes	characteristic
6	C₃²⋊H(3,𝔽₃)	—	3	12	no	—
7	C₃×C₃²⋊C₉	C₃²	1	8	yes	—
8	H(3,𝔽₃)⋊C₉	—	3	12	no	—
9	H(3,𝔽₃)⋊C₉	—	3	24	no	—
10	C₃⁴	—	3	4	no	—
11	C₃⁴	C₃³	1	1	yes	characteristic
12	C₃×H(3,𝔽₃)	—	3	36	no	—
13	C₃×H(3,𝔽₃)	—	3	8	no	—
14	C₃×H(3,𝔽₃)	—	3	18	no	—
15	C₃×H(3,𝔽₃)	—	3	9	no	—
16	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	12	yes	—
17	C₃²×C₉	—	3	4	no	—
18	C₃²×C₉	—	3	8	no	—
19	C₃²×C₉	—	9	3	no	—
20	C₃²⋊C₉	—	3	8	no	—
21	C₃²⋊C₉	—	3	16	no	—
22	C₃³	—	3	12	no	—
23	C₃³	—	9	1	no	—
24	C₃³	—	9	2	no	—
25	C₃³	—	9	12	no	—
26	C₃³	—	27	4	no	—
27	C₃³	—	27	4	no	—
28	C₃³	—	27	4	no	—
29	C₃³	C₃×H(3,𝔽₃)	1	4	yes	—
30	C₃³	#737	1	9	yes	—
31	H(3,𝔽₃)	—	3	36	no	—
32	H(3,𝔽₃)	—	9	24	no	—
33	H(3,𝔽₃)	—	27	18	no	—
34	H(3,𝔽₃)	—	27	9	no	—
35	C₃×C₉	—	9	1	no	—
36	C₃×C₉	—	9	12	no	—
37	C₃×C₉	—	9	8	no	—
38	C₃×C₉	—	27	8	no	—
39	C₃²	—	3	4	no	—
40	C₃²	—	3	4	no	—
41	C₃²	—	3	4	no	—
42	C₃²	—	9	9	no	—
43	C₃²	—	9	12	no	—
44	C₃²	—	9	12	no	—
45	C₃²	—	9	12	no	—
46	C₃²	—	9	12	no	—
47	C₃²	—	27	36	no	—
48	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
49	C₃²	#6893	1	12	yes	—
50	C₉	—	9	3	no	—
51	C₉	—	27	8	no	—
52	C₃	—	3	12	no	—
53	C₃	—	27	12	no	—
54	C₃	#10624564	1	1	yes	characteristic
55	C₃	#10624760	1	2	yes	—
56	C₃	#10624761	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624880	C₃
LCS	G ▷ C₃⁴ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624698	C₃	semidirect
3. C₃×H(3,𝔽₃)⋊C₉	C₃	semidirect

Maximal subgroups

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

subgroup	index
#10624880	3
#10624698	3
C₃×H(3,𝔽₃)⋊C₉	3