Subgroups — C₃×#10624894 #14555257

Order 2187.

7304 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#737	C₃	1	3	yes	maximal, complemented
2	C₃³×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	#10624894	C₃	1	81	yes	maximal, direct product factor
4	C₃×#6893	C₃	1	24	yes	maximal, complemented
5	C₃×#6891	C₃	1	8	yes	maximal, complemented
6	C₃²×C₃²⋊C₉	C₃	1	4	yes	maximal, complemented
7	C₃⁵	C₃²	1	4	yes	—
8	C₃×#737	—	3	81	no	—
9	C₃×#737	C₃²	1	81	yes	complemented
10	C₃²×3₋¹⁺²	C₃²	1	24	yes	complemented
11	C₃²×H(3,𝔽₃)	—	3	27	no	—
12	C₃²×H(3,𝔽₃)	C₃²	1	8	yes	—
13	C₃²×H(3,𝔽₃)	C₃²	1	27	yes	—
14	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
15	C₃³×C₉	C₃²	1	12	yes	complemented
16	#6893	C₃²	1	648	yes	complemented
17	#6891	C₃²	1	216	yes	complemented
18	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
19	C₃×C₃²⋊C₉	C₃²	1	8	yes	—
20	C₃×C₃²⋊C₉	C₃²	1	108	yes	complemented
21	C₃×C₃²⋊C₉	C₃²	1	24	yes	complemented
22	C₃×C₃²⋊C₉	C₃²	1	48	yes	complemented
23	C₃⁴	—	3	36	no	—
24	C₃⁴	—	3	36	no	—
25	C₃⁴	—	9	24	no	—
26	C₃⁴	C₃³	1	8	yes	—
27	C₃⁴	C₃³	1	4	yes	—
28	C₃⁴	C₃³	1	1	yes	characteristic
29	C₃⁴	C₃³	1	36	yes	—
30	#737	—	3	729	no	—
31	C₃×3₋¹⁺²	—	3	216	no	—
32	C₃×3₋¹⁺²	C₃³	1	216	yes	complemented
33	C₃×H(3,𝔽₃)	—	3	72	no	—
34	C₃×H(3,𝔽₃)	—	3	243	no	—
35	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	27	yes	—
36	C₃×H(3,𝔽₃)	C₃³	1	72	yes	—
37	C₃×H(3,𝔽₃)	C₃³	1	9	yes	—
38	C₃²×C₉	—	3	108	no	—
39	C₃²×C₉	C₃³	1	3	yes	—
40	C₃²×C₉	C₃³	1	24	yes	—
41	C₃²×C₉	C₃³	1	108	yes	complemented
42	C₃²⋊C₉	C₃³	1	9	yes	—
43	C₃²⋊C₉	C₃³	1	72	yes	—
44	C₃²⋊C₉	C₃³	1	216	yes	complemented
45	C₃²⋊C₉	C₃³	1	432	yes	complemented
46	C₃³	—	3	3	no	—
47	C₃³	—	3	12	no	—
48	C₃³	—	3	6	no	—
49	C₃³	—	3	24	no	—
50	C₃³	—	3	108	no	—
51	C₃³	—	3	24	no	—
52	C₃³	—	3	24	no	—
53	C₃³	—	3	108	no	—
54	C₃³	—	3	48	no	—
55	C₃³	—	9	72	no	—
56	C₃³	—	9	324	no	—
57	C₃³	C₃×H(3,𝔽₃)	1	36	yes	—
58	C₃³	#737	1	9	yes	complemented
59	C₃³	C₃⁴	1	24	yes	—
60	C₃³	C₃⁴	1	12	yes	—
61	C₃³	C₃⁴	1	3	yes	—
62	C₃³	C₃⁴	1	1	yes	characteristic
63	3₋¹⁺²	—	3	648	no	—
64	H(3,𝔽₃)	—	3	216	no	—
65	H(3,𝔽₃)	C₃×H(3,𝔽₃)	1	81	yes	—
66	C₃×C₉	—	3	9	no	—
67	C₃×C₉	—	3	72	no	—
68	C₃×C₉	—	3	324	no	—
69	C₃×C₉	C₃⁴	1	9	yes	—
70	C₃×C₉	C₃⁴	1	72	yes	—
71	C₃²	—	3	27	no	—
72	C₃²	—	3	108	no	—
73	C₃²	—	3	3	no	—
74	C₃²	—	3	12	no	—
75	C₃²	—	3	6	no	—
76	C₃²	—	3	24	no	—
77	C₃²	—	3	24	no	—
78	C₃²	—	3	24	no	—
79	C₃²	—	3	48	no	—
80	C₃²	—	9	324	no	—
81	C₃²	—	9	72	no	—
82	C₃²	C₃²×H(3,𝔽₃)	1	36	yes	—
83	C₃²	C₃²×H(3,𝔽₃)	1	3	yes	—
84	C₃²	C₃×#737	1	3	yes	—
85	C₃²	C₃×#737	1	9	yes	complemented
86	C₃²	3₋¹⁺⁴	1	6	yes	—
87	C₃²	C₃⁵	1	1	yes	characteristic
88	C₉	—	3	9	no	—
89	C₉	—	3	72	no	—
90	C₃	—	3	9	no	—
91	C₃	—	3	36	no	—
92	C₃	—	9	24	no	—
93	C₃	#10624894	1	9	yes	direct product factor
94	C₃	C₃³×H(3,𝔽₃)	1	1	yes	characteristic
95	C₃	C₃²×#737	1	1	yes	characteristic
96	C₃	C₃×3₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₃³	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁵
prime radical	C₃³×H(3,𝔽₃)	C₃
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. C₃²×#737	C₃	semidirect
3. #10624894	C₃	direct (N × H)
4. C₃×#6893	C₃	semidirect
5. C₃×#6891	C₃	semidirect
6. C₃²×C₃²⋊C₉	C₃	semidirect
9. C₃×#737	C₃²	semidirect
10. C₃²×3₋¹⁺²	C₃²	semidirect
15. C₃³×C₉	C₃²	semidirect
16. #6893	C₃²	semidirect
17. #6891	C₃²	semidirect
20. C₃×C₃²⋊C₉	C₃²	semidirect
21. C₃×C₃²⋊C₉	C₃²	semidirect
22. C₃×C₃²⋊C₉	C₃²	semidirect
32. C₃×3₋¹⁺²	C₃³	semidirect
41. C₃²×C₉	C₃³	semidirect
44. C₃²⋊C₉	C₃³	semidirect
45. C₃²⋊C₉	C₃³	semidirect
58. C₃³	#737	semidirect
85. C₃²	C₃×#737	semidirect
93. C₃	#10624894	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×#737	3
C₃³×H(3,𝔽₃)	3
#10624894	3
C₃×#6893	3
C₃×#6891	3
C₃²×C₃²⋊C₉	3

Subgroups — C3×#10624894 #14555257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10624894 #14555257