Subgroups — C₃²×C₉⋊H(3,𝔽₃) #14555254

Order 2187.

8594 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×3₋¹⁺²	C₃	1	3	yes	maximal, complemented
2	C₃³×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃⁴×C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉⋊H(3,𝔽₃)	C₃	1	108	yes	maximal, direct product factor
5	C₃²×C₃²⋊C₉	C₃	1	6	yes	maximal, complemented
6	C₃²×C₃²⋊C₉	C₃	1	2	yes	maximal, complemented
7	C₃⁵	C₃²	1	3	yes	—
8	C₃⁵	C₃²	1	1	yes	characteristic
9	C₃²×3₋¹⁺²	—	3	81	no	—
10	C₃²×3₋¹⁺²	C₃²	1	108	yes	complemented
11	C₃²×H(3,𝔽₃)	—	3	27	no	—
12	C₃²×H(3,𝔽₃)	C₃²	1	36	yes	—
13	C₃³×C₉	—	3	27	no	—
14	C₃³×C₉	C₃²	1	2	yes	—
15	C₃³×C₉	C₃²	1	6	yes	—
16	C₃³×C₉	C₃²	1	1	yes	characteristic
17	C₃³×C₉	C₃²	1	36	yes	complemented
18	C₉⋊H(3,𝔽₃)	C₃²	1	729	yes	direct product factor
19	C₃×C₃²⋊C₉	C₃²	1	216	yes	complemented
20	C₃×C₃²⋊C₉	C₃²	1	72	yes	complemented
21	C₃⁴	—	3	9	no	—
22	C₃⁴	—	3	18	no	—
23	C₃⁴	—	3	27	no	—
24	C₃⁴	—	3	27	no	—
25	C₃⁴	—	3	54	no	—
26	C₃⁴	3₋¹⁺²	1	27	yes	complemented
27	C₃⁴	C₃³	1	36	yes	—
28	C₃⁴	C₃³	1	12	yes	—
29	C₃⁴	C₃³	1	1	yes	characteristic
30	C₃×3₋¹⁺²	—	3	972	no	—
31	C₃×3₋¹⁺²	C₃³	1	243	yes	complemented
32	C₃×H(3,𝔽₃)	—	3	324	no	—
33	C₃×H(3,𝔽₃)	C₃³	1	81	yes	—
34	C₃²×C₉	—	3	18	no	—
35	C₃²×C₉	—	3	54	no	—
36	C₃²×C₉	—	3	324	no	—
37	C₃²×C₉	H(3,𝔽₃)	1	27	yes	complemented
38	C₃²×C₉	C₃³	1	24	yes	—
39	C₃²×C₉	C₃³	1	72	yes	—
40	C₃²×C₉	C₃³	1	12	yes	—
41	C₃²×C₉	C₃³	1	81	yes	complemented
42	C₃²⋊C₉	C₃³	1	486	yes	complemented
43	C₃²⋊C₉	C₃³	1	162	yes	complemented
44	C₃³	—	3	36	no	—
45	C₃³	—	3	72	no	—
46	C₃³	—	3	243	no	—
47	C₃³	—	3	108	no	—
48	C₃³	—	3	108	no	—
49	C₃³	—	3	216	no	—
50	C₃³	—	9	243	no	—
51	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
52	C₃³	C₃×3₋¹⁺²	1	9	yes	—
53	C₃³	C₃×3₋¹⁺²	1	108	yes	complemented
54	C₃³	#737	1	18	yes	—
55	C₃³	C₃⁴	1	27	yes	—
56	C₃³	C₃⁴	1	9	yes	—
57	C₃³	C₃⁴	1	4	yes	—
58	3₋¹⁺²	—	3	729	no	—
59	H(3,𝔽₃)	—	3	243	no	—
60	C₃×C₉	—	3	72	no	—
61	C₃×C₉	—	3	216	no	—
62	C₃×C₉	—	3	243	no	—
63	C₃×C₉	C₃×H(3,𝔽₃)	1	108	yes	complemented
64	C₃×C₉	C₃⁴	1	18	yes	—
65	C₃×C₉	C₃⁴	1	54	yes	—
66	C₃×C₉	C₃⁴	1	9	yes	—
67	C₃²	—	3	324	no	—
68	C₃²	—	3	9	no	—
69	C₃²	—	3	18	no	—
70	C₃²	—	3	27	no	—
71	C₃²	—	3	27	no	—
72	C₃²	—	3	54	no	—
73	C₃²	—	9	324	no	—
74	C₃²	C₉⋊H(3,𝔽₃)	1	81	yes	direct product factor
75	C₃²	C₃²×H(3,𝔽₃)	1	12	yes	—
76	C₃²	C₃²×3₋¹⁺²	1	12	yes	—
77	C₃²	C₃²×3₋¹⁺²	1	27	yes	complemented
78	C₃²	C₃×#737	1	24	yes	—
79	C₃²	C₃⁵	1	1	yes	characteristic
80	C₉	—	3	18	no	—
81	C₉	—	3	54	no	—
82	C₉	C₃²×H(3,𝔽₃)	1	27	yes	complemented
83	C₃	—	3	27	no	—
84	C₃	—	9	27	no	—
85	C₃	C₃×C₉⋊H(3,𝔽₃)	1	36	yes	direct product factor
86	C₃	C₃³×H(3,𝔽₃)	1	1	yes	characteristic
87	C₃	C₃³×3₋¹⁺²	1	1	yes	characteristic
88	C₃	C₃²×#737	1	2	yes	—

	subgroup	G/H
center, socle	C₃⁴	C₃³
commutator, Frattini, cosocle	C₃²	C₃⁵
prime radical	C₃³×H(3,𝔽₃)	C₃
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃⁴ ◁ G
LECS	G ▷ C₃² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₃³×3₋¹⁺²	C₃	semidirect
3. C₃⁴×C₉	C₃	semidirect
4. C₃×C₉⋊H(3,𝔽₃)	C₃	direct (N × H)
5. C₃²×C₃²⋊C₉	C₃	semidirect
6. C₃²×C₃²⋊C₉	C₃	semidirect
10. C₃²×3₋¹⁺²	C₃²	semidirect
17. C₃³×C₉	C₃²	semidirect
18. C₉⋊H(3,𝔽₃)	C₃²	direct (N × H)
19. C₃×C₃²⋊C₉	C₃²	semidirect
20. C₃×C₃²⋊C₉	C₃²	semidirect
26. C₃⁴	3₋¹⁺²	semidirect
31. C₃×3₋¹⁺²	C₃³	semidirect
37. C₃²×C₉	H(3,𝔽₃)	semidirect
41. C₃²×C₉	C₃³	semidirect
42. C₃²⋊C₉	C₃³	semidirect
43. C₃²⋊C₉	C₃³	semidirect
53. C₃³	C₃×3₋¹⁺²	semidirect
63. C₃×C₉	C₃×H(3,𝔽₃)	semidirect
74. C₃²	C₉⋊H(3,𝔽₃)	direct (N × H)
77. C₃²	C₃²×3₋¹⁺²	semidirect
82. C₉	C₃²×H(3,𝔽₃)	semidirect
85. C₃	C₃×C₉⋊H(3,𝔽₃)	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
C₃³×H(3,𝔽₃)	3
C₃⁴×C₉	3
C₃×C₉⋊H(3,𝔽₃)	3
C₃²×C₃²⋊C₉	3

Subgroups — C32×C9⋊H(3,𝔽3) #14555254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₉⋊H(3,𝔽₃) #14555254