Subgroups — C₃×#10624880 #14555249

Order 2187.

10736 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×H(3,𝔽₃)	C₃	1	4	yes	maximal, complemented
2	#10624880	C₃	1	81	yes	maximal, direct product factor
3	C₃×C₃²⋊H(3,𝔽₃)	C₃	1	36	yes	maximal, complemented
4	C₃⁵	C₃²	1	1	yes	characteristic
5	C₃⁵	C₃²	1	12	yes	complemented
6	C₃²×H(3,𝔽₃)	—	3	108	no	—
7	C₃²×H(3,𝔽₃)	C₃²	1	36	yes	complemented
8	C₃²×H(3,𝔽₃)	C₃²	1	108	yes	complemented
9	C₃²×H(3,𝔽₃)	C₃²	1	81	yes	complemented
10	C₃²⋊H(3,𝔽₃)	C₃²	1	972	yes	complemented
11	C₃⁴	—	3	108	no	—
12	C₃⁴	—	9	9	no	—
13	C₃⁴	—	9	108	no	—
14	C₃⁴	H(3,𝔽₃)	1	27	yes	complemented
15	C₃⁴	C₃³	1	36	yes	—
16	C₃⁴	C₃³	1	4	yes	—
17	C₃⁴	C₃³	1	9	yes	—
18	C₃⁴	C₃³	1	108	yes	complemented
19	C₃×H(3,𝔽₃)	—	3	324	no	—
20	C₃×H(3,𝔽₃)	—	3	972	no	—
21	C₃×H(3,𝔽₃)	—	9	243	no	—
22	C₃×H(3,𝔽₃)	C₃³	1	324	yes	complemented
23	C₃×H(3,𝔽₃)	C₃³	1	729	yes	complemented
24	C₃³	—	3	36	no	—
25	C₃³	—	3	108	no	—
26	C₃³	—	3	324	no	—
27	C₃³	—	3	324	no	—
28	C₃³	—	9	81	no	—
29	C₃³	—	9	27	no	—
30	C₃³	—	9	324	no	—
31	C₃³	—	9	972	no	—
32	C₃³	C₃×H(3,𝔽₃)	1	36	yes	—
33	C₃³	C₃×H(3,𝔽₃)	1	81	yes	complemented
34	C₃³	C₃⁴	1	108	yes	—
35	C₃³	C₃⁴	1	12	yes	—
36	C₃³	C₃⁴	1	1	yes	characteristic
37	H(3,𝔽₃)	—	3	972	no	—
38	H(3,𝔽₃)	—	9	729	no	—
39	C₃²	—	3	108	no	—
40	C₃²	—	3	36	no	—
41	C₃²	—	3	108	no	—
42	C₃²	—	3	324	no	—
43	C₃²	—	9	81	no	—
44	C₃²	—	9	972	no	—
45	C₃²	—	9	324	no	—
46	C₃²	C₃²×H(3,𝔽₃)	1	36	yes	—
47	C₃²	C₃²×H(3,𝔽₃)	1	3	yes	—
48	C₃²	H(5,𝔽₃)	1	9	yes	—
49	C₃²	C₃⁵	1	1	yes	characteristic
50	C₃	—	3	36	no	—
51	C₃	—	9	108	no	—
52	C₃	#10624880	1	9	yes	direct product factor
53	C₃	C₃³×H(3,𝔽₃)	1	1	yes	characteristic
54	C₃	C₃×H(5,𝔽₃)	1	3	yes	—

	subgroup	G/H
center, socle	C₃³	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁵
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³ ◁ G
LECS	G ▷ C₃² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₃³×H(3,𝔽₃)	C₃	semidirect
2. #10624880	C₃	direct (N × H)
3. C₃×C₃²⋊H(3,𝔽₃)	C₃	semidirect
5. C₃⁵	C₃²	semidirect
7. C₃²×H(3,𝔽₃)	C₃²	semidirect
8. C₃²×H(3,𝔽₃)	C₃²	semidirect
9. C₃²×H(3,𝔽₃)	C₃²	semidirect
10. C₃²⋊H(3,𝔽₃)	C₃²	semidirect
14. C₃⁴	H(3,𝔽₃)	semidirect
18. C₃⁴	C₃³	semidirect
22. C₃×H(3,𝔽₃)	C₃³	semidirect
23. C₃×H(3,𝔽₃)	C₃³	semidirect
33. C₃³	C₃×H(3,𝔽₃)	semidirect
52. C₃	#10624880	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×H(3,𝔽₃)	3
#10624880	3
C₃×C₃²⋊H(3,𝔽₃)	3

Subgroups — C3×#10624880 #14555249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10624880 #14555249