Subgroups — #14548140

Order 2187.

408 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₉)	C₃	1	1	yes	maximal, characteristic
2	#10624811	C₃	1	2	yes	maximal, complemented
3	#10624811	C₃	1	2	yes	maximal, complemented
4	#10624661	C₃	1	4	yes	maximal, complemented
5	#10624657	C₃	1	4	yes	maximal, complemented
6	C₃³×C₉	C₃²	1	1	yes	characteristic, complemented
7	#6891	C₃²	1	4	yes	complemented
8	#6891	C₃²	1	4	yes	complemented
9	C₃²⋊H(3,𝔽₃)	—	3	6	no	—
10	C₃²⋊H(3,𝔽₃)	—	3	6	no	—
11	C₃²⋊H(3,𝔽₃)	C₃²	1	2	yes	—
12	C₃²⋊H(3,𝔽₃)	C₃²	1	2	yes	—
13	H(3,𝔽₃)⋊C₉	—	3	6	no	—
14	H(3,𝔽₃)⋊C₉	—	3	6	no	—
15	#6859	—	3	12	no	—
16	#6857	—	3	12	no	—
17	C₃⁴	—	3	3	no	—
18	C₃⁴	C₃³	1	1	yes	characteristic
19	C₃×3₋¹⁺²	—	3	4	no	—
20	C₃×3₋¹⁺²	—	3	4	no	—
21	C₃×H(3,𝔽₃)	—	3	12	no	—
22	C₃×H(3,𝔽₃)	—	3	6	no	—
23	C₃×H(3,𝔽₃)	—	3	6	no	—
24	C₃×H(3,𝔽₃)	—	3	2	no	—
25	C₃×H(3,𝔽₃)	—	3	2	no	—
26	C₃×H(3,𝔽₃)	—	3	2	no	—
27	C₃×H(3,𝔽₃)	—	3	2	no	—
28	C₃×H(3,𝔽₃)	—	3	2	no	—
29	C₃×H(3,𝔽₃)	—	3	2	no	—
30	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	6	yes	—
31	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	6	yes	—
32	C₃²×C₉	—	3	2	no	—
33	C₃²×C₉	—	3	2	no	—
34	C₃²×C₉	—	9	3	no	—
35	C₃²⋊C₉	—	3	4	no	—
36	C₃²⋊C₉	—	3	4	no	—
37	C₃²⋊C₉	—	3	8	no	—
38	C₃²⋊C₉	—	3	4	no	—
39	C₃²⋊C₉	—	3	4	no	—
40	C₃³	—	3	6	no	—
41	C₃³	—	3	6	no	—
42	C₃³	—	9	1	no	—
43	C₃³	—	9	2	no	—
44	C₃³	—	9	1	no	—
45	C₃³	—	27	6	no	—
46	C₃³	—	27	3	no	—
47	C₃³	—	27	3	no	—
48	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
49	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
50	3₋¹⁺²	—	27	4	no	—
51	3₋¹⁺²	—	27	4	no	—
52	H(3,𝔽₃)	—	9	6	no	—
53	H(3,𝔽₃)	—	9	6	no	—
54	H(3,𝔽₃)	—	27	2	no	—
55	H(3,𝔽₃)	—	27	2	no	—
56	H(3,𝔽₃)	—	27	2	no	—
57	H(3,𝔽₃)	—	27	2	no	—
58	H(3,𝔽₃)	—	27	2	no	—
59	H(3,𝔽₃)	—	27	2	no	—
60	H(3,𝔽₃)	—	27	12	no	—
61	H(3,𝔽₃)	—	27	6	no	—
62	H(3,𝔽₃)	—	27	6	no	—
63	C₃×C₉	—	9	1	no	—
64	C₃×C₉	—	9	6	no	—
65	C₃×C₉	—	9	6	no	—
66	C₃×C₉	—	9	4	no	—
67	C₃×C₉	—	9	4	no	—
68	C₃²	—	3	2	no	—
69	C₃²	—	3	2	no	—
70	C₃²	—	3	2	no	—
71	C₃²	—	3	2	no	—
72	C₃²	—	3	2	no	—
73	C₃²	—	3	2	no	—
74	C₃²	—	3	2	no	—
75	C₃²	—	3	2	no	—
76	C₃²	—	9	9	no	—
77	C₃²	—	9	6	no	—
78	C₃²	—	9	6	no	—
79	C₃²	—	9	6	no	—
80	C₃²	—	9	6	no	—
81	C₃²	—	9	6	no	—
82	C₃²	—	9	6	no	—
83	C₃²	—	9	6	no	—
84	C₃²	—	9	6	no	—
85	C₃²	—	27	27	no	—
86	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
87	C₉	—	9	3	no	—
88	C₉	—	27	4	no	—
89	C₉	—	27	4	no	—
90	C₃	—	9	2	no	—
91	C₃	—	9	2	no	—
92	C₃	—	27	6	no	—
93	C₃	—	27	6	no	—
94	C₃	#10624748	1	2	yes	—
95	C₃	#10624760	1	1	yes	characteristic
96	C₃	#10624761	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	H(3,𝔽₉)	C₃
LCS	G ▷ C₃⁴ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624811	C₃	semidirect
3. #10624811	C₃	semidirect
4. #10624661	C₃	semidirect
5. #10624657	C₃	semidirect
6. C₃³×C₉	C₃²	semidirect
7. #6891	C₃²	semidirect
8. #6891	C₃²	semidirect

Maximal subgroups

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

subgroup	index
H(3,𝔽₉)	3
#10624811	3
#10624661	3
#10624657	3