Subgroups — C₃×#10624567 #14552155

Order 2187.

2834 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×3₋¹⁺²	C₃	1	2	yes	maximal
2	C₃²×C₃²⋊C₉	C₃	1	2	yes	maximal
3	#10624567	C₃	1	27	yes	maximal, direct product factor
4	C₃×#6855	C₃	1	9	yes	maximal, complemented
5	C₃⁵	C₃²	1	1	yes	characteristic
6	C₃²×3₋¹⁺²	—	3	18	no	—
7	C₃²×3₋¹⁺²	—	3	54	no	—
8	C₃²×3₋¹⁺²	C₃²	1	18	yes	—
9	C₃³×C₉	C₃²	1	6	yes	—
10	C₃³×C₉	C₃²	1	6	yes	—
11	C₃×C₃²⋊C₉	—	3	18	no	—
12	C₃×C₃²⋊C₉	C₃²	1	18	yes	—
13	#6855	C₃²	1	81	yes	complemented
14	C₃⁴	—	3	18	no	—
15	C₃⁴	—	3	18	no	—
16	C₃⁴	H(3,𝔽₃)	1	9	yes	—
17	C₃⁴	C₃³	1	3	yes	—
18	C₃⁴	C₃³	1	1	yes	characteristic
19	C₃×3₋¹⁺²	—	3	486	no	—
20	C₃×3₋¹⁺²	—	3	54	no	—
21	C₃×3₋¹⁺²	—	3	162	no	—
22	C₃²×C₉	—	3	18	no	—
23	C₃²×C₉	—	3	36	no	—
24	C₃²×C₉	—	3	18	no	—
25	C₃²×C₉	—	3	54	no	—
26	C₃²×C₉	3₋¹⁺²	1	54	yes	complemented
27	C₃²×C₉	C₃³	1	18	yes	—
28	C₃²×C₉	C₃³	1	18	yes	—
29	C₃²⋊C₉	—	3	54	no	—
30	C₃³	—	3	18	no	—
31	C₃³	—	3	18	no	—
32	C₃³	—	3	54	no	—
33	C₃³	—	3	54	no	—
34	C₃³	—	9	81	no	—
35	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
36	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
37	C₃³	C₃×3₋¹⁺²	1	18	yes	—
38	C₃³	#737	1	18	yes	—
39	C₃³	C₃⁴	1	1	yes	characteristic
40	3₋¹⁺²	—	3	486	no	—
41	C₃×C₉	—	3	18	no	—
42	C₃×C₉	—	3	36	no	—
43	C₃×C₉	—	3	162	no	—
44	C₃×C₉	—	3	18	no	—
45	C₃×C₉	—	3	54	no	—
46	C₃×C₉	—	9	54	no	—
47	C₃×C₉	C₃×3₋¹⁺²	1	54	yes	complemented
48	C₃²	—	3	18	no	—
49	C₃²	—	3	18	no	—
50	C₃²	—	9	81	no	—
51	C₃²	—	9	27	no	—
52	C₃²	#6891	1	18	yes	—
53	C₃²	#6893	1	18	yes	—
54	C₃²	#6896	1	81	yes	—
55	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
56	C₃²	C₃²×3₋¹⁺²	1	6	yes	—
57	C₃²	C₃×#737	1	6	yes	—
58	C₉	—	3	54	no	—
59	C₉	—	9	18	no	—
60	C₃	—	9	9	no	—
61	C₃	#10624567	1	27	yes	direct product factor
62	C₃	C₃×#6891	1	2	yes	—
63	C₃	C₃×#6893	1	2	yes	—
64	C₃	C₃×#6896	1	9	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #10624567	C₃	direct (N × H)
4. C₃×#6855	C₃	semidirect
13. #6855	C₃²	semidirect
26. C₃²×C₉	3₋¹⁺²	semidirect
47. C₃×C₉	C₃×3₋¹⁺²	semidirect
61. C₃	#10624567	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×3₋¹⁺²	3
C₃²×C₃²⋊C₉	3
#10624567	3
C₃×#6855	3

Subgroups — C3×#10624567 #14552155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10624567 #14552155