Subgroups — C₃×#10624732 #14554254

Order 2187.

3374 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₉	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²×#732	C₃	1	9	yes	maximal, complemented
3	C₃×C₃²⋊H(3,𝔽₃)	C₃	1	3	yes	maximal, complemented
4	#10624732	C₃	1	27	yes	maximal, direct product factor
5	C₃⁵	C₃²	1	1	yes	characteristic
6	C₃²×H(3,𝔽₃)	—	3	9	no	—
7	C₃²×H(3,𝔽₃)	C₃²	1	9	yes	—
8	C₃³×C₉	—	3	6	no	—
9	C₃³×C₉	—	3	3	no	—
10	C₃³×C₉	—	3	27	no	—
11	C₃³×C₉	C₃²	1	3	yes	—
12	C₃³×C₉	C₃²	1	9	yes	complemented
13	C₃×#732	—	3	81	no	—
14	C₃×#732	C₃²	1	81	yes	complemented
15	C₃²⋊H(3,𝔽₃)	—	3	18	no	—
16	C₃²⋊H(3,𝔽₃)	C₃²	1	27	yes	complemented
17	C₃⁴	—	3	27	no	—
18	C₃⁴	—	3	9	no	—
19	C₃⁴	—	3	9	no	—
20	C₃⁴	H(3,𝔽₃)	1	6	yes	—
21	C₃⁴	H(3,𝔽₃)	1	3	yes	—
22	C₃⁴	C₃³	1	3	yes	—
23	C₃⁴	C₃³	1	1	yes	characteristic
24	C₃×H(3,𝔽₃)	—	3	81	no	—
25	C₃×H(3,𝔽₃)	—	3	81	no	—
26	C₃×H(3,𝔽₃)	—	3	18	no	—
27	C₃×H(3,𝔽₃)	—	9	81	no	—
28	C₃×H(3,𝔽₃)	C₃³	1	27	yes	—
29	C₃²×C₉	—	3	162	no	—
30	C₃²×C₉	—	3	81	no	—
31	C₃²×C₉	—	3	6	no	—
32	C₃²×C₉	—	3	18	no	—
33	C₃²×C₉	—	3	27	no	—
34	C₃²×C₉	—	3	3	no	—
35	C₃²×C₉	—	3	81	no	—
36	C₃²×C₉	H(3,𝔽₃)	1	27	yes	complemented
37	C₃²×C₉	C₃³	1	9	yes	—
38	#732	—	3	243	no	—
39	C₃³	—	3	243	no	—
40	C₃³	—	3	9	no	—
41	C₃³	—	3	18	no	—
42	C₃³	—	3	9	no	—
43	C₃³	—	3	18	no	—
44	C₃³	—	3	9	no	—
45	C₃³	—	3	54	no	—
46	C₃³	—	3	27	no	—
47	C₃³	—	3	27	no	—
48	C₃³	—	9	27	no	—
49	C₃³	—	9	81	no	—
50	C₃³	#732	1	27	yes	complemented
51	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
52	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
53	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
54	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
55	C₃³	C₃⁴	1	1	yes	characteristic
56	H(3,𝔽₃)	—	3	81	no	—
57	H(3,𝔽₃)	—	9	54	no	—
58	H(3,𝔽₃)	—	9	243	no	—
59	C₃×C₉	—	3	243	no	—
60	C₃×C₉	—	3	18	no	—
61	C₃×C₉	—	3	9	no	—
62	C₃×C₉	—	3	54	no	—
63	C₃×C₉	—	3	27	no	—
64	C₃×C₉	C₃×H(3,𝔽₃)	1	27	yes	complemented
65	C₃²	—	3	81	no	—
66	C₃²	—	3	162	no	—
67	C₃²	—	3	9	no	—
68	C₃²	—	3	6	no	—
69	C₃²	—	3	18	no	—
70	C₃²	—	3	3	no	—
71	C₃²	—	3	81	no	—
72	C₃²	—	3	27	no	—
73	C₃²	—	9	27	no	—
74	C₃²	—	9	81	no	—
75	C₃²	—	27	81	no	—
76	C₃²	C₃²⋊H(3,𝔽₃)	1	6	yes	—
77	C₃²	C₃²⋊H(3,𝔽₃)	1	3	yes	—
78	C₃²	C₃×#732	1	9	yes	—
79	C₃²	C₃×#732	1	27	yes	complemented
80	C₃²	C₃²×H(3,𝔽₃)	1	3	yes	—
81	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
82	C₉	—	3	27	no	—
83	C₃	—	3	3	no	—
84	C₃	—	3	6	no	—
85	C₃	—	3	27	no	—
86	C₃	—	27	27	no	—
87	C₃	#10624732	1	9	yes	direct product factor
88	C₃	C₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
89	C₃	C₃²×#732	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃⁴×C₉	C₃	semidirect
2. C₃²×#732	C₃	semidirect
3. C₃×C₃²⋊H(3,𝔽₃)	C₃	semidirect
4. #10624732	C₃	direct (N × H)
12. C₃³×C₉	C₃²	semidirect
14. C₃×#732	C₃²	semidirect
16. C₃²⋊H(3,𝔽₃)	C₃²	semidirect
36. C₃²×C₉	H(3,𝔽₃)	semidirect
50. C₃³	#732	semidirect
64. C₃×C₉	C₃×H(3,𝔽₃)	semidirect
79. C₃²	C₃×#732	semidirect
87. C₃	#10624732	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃⁴×C₉	3
C₃²×#732	3
C₃×C₃²⋊H(3,𝔽₃)	3
#10624732	3

Subgroups — C3×#10624732 #14554254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10624732 #14554254