Subgroups — C₃×#6909 #10624924

Order 729.

598 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
2	C₃²×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	#6909	C₃	1	27	yes	maximal, direct product factor
5	C₃×AΣL(1,𝔽₂₇)	C₃	1	9	yes	maximal, complemented
6	C₃⁴	—	3	1	no	—
7	C₃⁴	C₃²	1	1	yes	characteristic
8	C₃⁴	C₃²	1	3	yes	complemented
9	C₃×3₋¹⁺²	—	3	12	no	—
10	C₃×3₋¹⁺²	—	3	2	no	—
11	C₃×3₋¹⁺²	—	3	2	no	—
12	C₃×3₋¹⁺²	C₃²	1	6	yes	complemented
13	C₃×3₋¹⁺²	C₃²	1	18	yes	complemented
14	C₃×H(3,𝔽₃)	—	3	6	no	—
15	C₃×H(3,𝔽₃)	—	3	6	no	—
16	C₃×H(3,𝔽₃)	—	3	2	no	—
17	C₃×H(3,𝔽₃)	—	3	1	no	—
18	C₃×H(3,𝔽₃)	—	3	2	no	—
19	C₃×H(3,𝔽₃)	C₃²	1	3	yes	—
20	C₃×H(3,𝔽₃)	C₃²	1	9	yes	complemented
21	C₃×H(3,𝔽₃)	C₃²	1	9	yes	complemented
22	C₃²×C₉	—	3	2	no	—
23	AΣL(1,𝔽₂₇)	C₃²	1	81	yes	complemented
24	C₃³	—	3	6	no	—
25	C₃³	—	3	9	no	—
26	C₃³	—	3	3	no	—
27	C₃³	—	3	3	no	—
28	C₃³	—	9	3	no	—
29	C₃³	—	9	9	no	—
30	C₃³	—	9	9	no	—
31	C₃³	H(3,𝔽₃)	1	2	yes	—
32	C₃³	H(3,𝔽₃)	1	1	yes	characteristic
33	C₃³	H(3,𝔽₃)	1	6	yes	—
34	C₃³	C₃³	1	1	yes	characteristic
35	C₃³	C₃³	1	3	yes	—
36	C₃³	C₃³	1	9	yes	complemented
37	3₋¹⁺²	—	3	18	no	—
38	3₋¹⁺²	—	3	12	no	—
39	3₋¹⁺²	—	3	18	no	—
40	3₋¹⁺²	C₃³	1	18	yes	complemented
41	H(3,𝔽₃)	—	3	6	no	—
42	H(3,𝔽₃)	—	3	18	no	—
43	H(3,𝔽₃)	—	3	9	no	—
44	H(3,𝔽₃)	—	3	18	no	—
45	H(3,𝔽₃)	C₃³	1	9	yes	—
46	C₃×C₉	—	3	18	no	—
47	C₃×C₉	—	3	6	no	—
48	C₃²	—	3	3	no	—
49	C₃²	—	3	3	no	—
50	C₃²	—	3	6	no	—
51	C₃²	—	3	9	no	—
52	C₃²	—	3	9	no	—
53	C₃²	—	9	6	no	—
54	C₃²	—	9	3	no	—
55	C₃²	—	9	9	no	—
56	C₃²	—	9	18	no	—
57	C₃²	—	9	27	no	—
58	C₃²	—	9	9	no	—
59	C₃²	—	9	9	no	—
60	C₃²	C₃×H(3,𝔽₃)	1	6	yes	—
61	C₃²	C₃×H(3,𝔽₃)	1	2	yes	—
62	C₃²	C₃×H(3,𝔽₃)	1	3	yes	—
63	C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
64	C₃²	C₃⁴	1	1	yes	characteristic
65	C₉	—	3	18	no	—
66	C₃	—	3	3	no	—
67	C₃	—	3	2	no	—
68	C₃	—	3	1	no	—
69	C₃	—	3	6	no	—
70	C₃	—	9	9	no	—
71	C₃	—	9	9	no	—
72	C₃	#6909	1	3	yes	direct product factor
73	C₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃⁴
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃² ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₃²×3₋¹⁺²	C₃	semidirect
2. C₃²×H(3,𝔽₃)	C₃	semidirect
3. C₃²×H(3,𝔽₃)	C₃	semidirect
4. #6909	C₃	direct (N × H)
5. C₃×AΣL(1,𝔽₂₇)	C₃	semidirect
8. C₃⁴	C₃²	semidirect
12. C₃×3₋¹⁺²	C₃²	semidirect
13. C₃×3₋¹⁺²	C₃²	semidirect
20. C₃×H(3,𝔽₃)	C₃²	semidirect
21. C₃×H(3,𝔽₃)	C₃²	semidirect
23. AΣL(1,𝔽₂₇)	C₃²	semidirect
36. C₃³	C₃³	semidirect
40. 3₋¹⁺²	C₃³	semidirect
72. C₃	#6909	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×3₋¹⁺²	3
C₃²×H(3,𝔽₃)	3
#6909	3
C₃×AΣL(1,𝔽₂₇)	3

Subgroups — C3×#6909 #10624924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#6909 #10624924