Subgroups — C₃×#10624900 #14555304

Order 2187.

4256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#737	C₃	1	1	yes	maximal, characteristic
2	#10624900	C₃	1	81	yes	maximal, direct product factor
3	C₃×#6895	C₃	1	8	yes	maximal, complemented
4	C₃×#6894	C₃	1	8	yes	maximal, complemented
5	C₃×#6894	C₃	1	16	yes	maximal, complemented
6	C₃×C₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
7	C₃×C₉×H(3,𝔽₃)	C₃	1	2	yes	maximal
8	C₃²×C₉⋊C₉	C₃	1	4	yes	maximal, complemented
9	C₃×#737	—	3	27	no	—
10	C₃×#737	C₃²	1	27	yes	—
11	C₃²×3₋¹⁺²	C₃²	1	8	yes	—
12	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
13	C₃³×C₉	C₃²	1	4	yes	—
14	C₃³×C₉	C₃²	1	4	yes	—
15	C₃³×C₉	C₃²	1	8	yes	—
16	#6895	C₃²	1	216	yes	complemented
17	#6894	C₃²	1	216	yes	complemented
18	#6894	C₃²	1	432	yes	complemented
19	C₉×H(3,𝔽₃)	C₃²	1	27	yes	—
20	C₉×H(3,𝔽₃)	C₃²	1	54	yes	—
21	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
22	C₃×C₉⋊C₉	C₃²	1	16	yes	—
23	C₃×C₉⋊C₉	C₃²	1	8	yes	—
24	C₃×C₉⋊C₉	C₃²	1	8	yes	—
25	C₃×C₉⋊C₉	C₃²	1	48	yes	—
26	C₃×C₉⋊C₉	C₃²	1	108	yes	complemented
27	C₃×C₃²⋊C₉	C₃²	1	8	yes	—
28	C₃×C₃²⋊C₉	C₃²	1	16	yes	—
29	C₃⁴	C₃³	1	4	yes	—
30	#737	—	3	243	no	—
31	C₃×3₋¹⁺²	—	3	72	no	—
32	C₃×3₋¹⁺²	C₃³	1	72	yes	—
33	C₃×H(3,𝔽₃)	3₋¹⁺²	1	27	yes	—
34	C₃×H(3,𝔽₃)	C₃³	1	9	yes	—
35	C₃²×C₉	—	3	36	no	—
36	C₃²×C₉	—	3	36	no	—
37	C₃²×C₉	—	9	24	no	—
38	C₃²×C₉	C₃³	1	36	yes	—
39	C₃²×C₉	C₃³	1	36	yes	—
40	C₃²×C₉	C₃³	1	1	yes	characteristic
41	C₃²×C₉	C₃³	1	2	yes	—
42	C₃²×C₉	C₃³	1	8	yes	—
43	C₃²×C₉	C₃³	1	8	yes	—
44	C₃²×C₉	C₃³	1	16	yes	—
45	C₃²×C₉	C₃³	1	72	yes	—
46	C₃²×C₉	C₃³	1	1	yes	characteristic
47	C₉⋊C₉	C₃³	1	9	yes	—
48	C₉⋊C₉	C₃³	1	144	yes	—
49	C₉⋊C₉	C₃³	1	72	yes	—
50	C₉⋊C₉	C₃³	1	72	yes	—
51	C₉⋊C₉	C₃³	1	432	yes	—
52	C₃²⋊C₉	C₃³	1	72	yes	—
53	C₃²⋊C₉	C₃³	1	144	yes	—
54	C₃³	—	3	12	no	—
55	C₃³	—	3	24	no	—
56	C₃³	C₃×3₋¹⁺²	1	36	yes	—
57	C₃³	C₃⁴	1	12	yes	—
58	C₃³	C₃⁴	1	1	yes	characteristic
59	3₋¹⁺²	—	3	216	no	—
60	H(3,𝔽₃)	C₃×3₋¹⁺²	1	81	yes	—
61	C₃×C₉	—	3	3	no	—
62	C₃×C₉	—	3	24	no	—
63	C₃×C₉	—	3	108	no	—
64	C₃×C₉	—	3	24	no	—
65	C₃×C₉	—	3	6	no	—
66	C₃×C₉	—	3	48	no	—
67	C₃×C₉	—	3	108	no	—
68	C₃×C₉	—	9	72	no	—
69	C₃×C₉	#737	1	9	yes	complemented
70	C₃×C₉	C₃⁴	1	24	yes	—
71	C₃×C₉	C₃⁴	1	48	yes	—
72	C₃×C₉	C₃⁴	1	3	yes	—
73	C₃×C₉	C₃⁴	1	6	yes	—
74	C₃×C₉	C₃⁴	1	24	yes	—
75	C₃×C₉	C₃⁴	1	3	yes	—
76	C₃²	—	3	108	no	—
77	C₃²	—	3	12	no	—
78	C₃²	—	3	24	no	—
79	C₃²	C₃²×3₋¹⁺²	1	36	yes	—
80	C₃²	C₃²×3₋¹⁺²	1	3	yes	—
81	C₃²	C₃×#737	1	3	yes	—
82	C₃²	3₋¹⁺⁴	1	6	yes	—
83	C₃²	C₃⁵	1	1	yes	characteristic
84	C₉	—	3	3	no	—
85	C₉	—	3	24	no	—
86	C₉	—	3	24	no	—
87	C₉	—	3	6	no	—
88	C₉	—	3	48	no	—
89	C₉	C₃×#737	1	9	yes	complemented
90	C₃	—	3	36	no	—
91	C₃	#10624900	1	9	yes	direct product factor
92	C₃	C₃³×3₋¹⁺²	1	1	yes	characteristic
93	C₃	C₃²×#737	1	1	yes	characteristic
94	C₃	C₃×3₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₃³	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁵
prime radical	C₃²×H(3,𝔽₃)	C₃²
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
2. #10624900	C₃	direct (N × H)
3. C₃×#6895	C₃	semidirect
4. C₃×#6894	C₃	semidirect
5. C₃×#6894	C₃	semidirect
8. C₃²×C₉⋊C₉	C₃	semidirect
16. #6895	C₃²	semidirect
17. #6894	C₃²	semidirect
18. #6894	C₃²	semidirect
26. C₃×C₉⋊C₉	C₃²	semidirect
69. C₃×C₉	#737	semidirect
89. C₉	C₃×#737	semidirect
91. C₃	#10624900	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
#10624900	3
C₃×#6895	3
C₃×#6894	3
C₃×C₉×H(3,𝔽₃)	3
C₃²×C₉⋊C₉	3

Subgroups — C3×#10624900 #14555304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10624900 #14555304