Subgroups — #14547961

Order 2187.

452 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624880	C₃	1	1	yes	maximal, characteristic
2	#10624814	C₃	1	3	yes	maximal, complemented
3	#10624805	C₃	1	3	yes	maximal, complemented
4	#10624657	C₃	1	3	yes	maximal, complemented
5	#10624598	C₃	1	3	yes	maximal, complemented
6	C₃²×H(3,𝔽₃)	—	3	1	no	—
7	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
8	#6891	C₃²	1	3	yes	complemented
9	#6891	C₃²	1	3	yes	complemented
10	C₃²⋊H(3,𝔽₃)	—	3	2	no	—
11	C₃²⋊H(3,𝔽₃)	—	3	9	no	—
12	C₃²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
13	C₃×C₃²⋊C₉	C₃²	1	3	yes	complemented
14	H(3,𝔽₃)⋊C₉	—	3	9	no	—
15	#6866	—	3	3	no	—
16	#6866	—	3	6	no	—
17	#6859	—	3	9	no	—
18	#6857	—	3	9	no	—
19	C₃⁴	—	3	3	no	—
20	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃⁴	H(3,𝔽₃)	1	2	yes	—
22	C₃⁴	C₃³	1	1	yes	characteristic
23	C₃×3₋¹⁺²	—	3	3	no	—
24	C₃×3₋¹⁺²	—	3	3	no	—
25	C₃×H(3,𝔽₃)	—	3	6	no	—
26	C₃×H(3,𝔽₃)	—	3	3	no	—
27	C₃×H(3,𝔽₃)	—	3	3	no	—
28	C₃×H(3,𝔽₃)	—	3	18	no	—
29	C₃×H(3,𝔽₃)	—	3	3	no	—
30	C₃×H(3,𝔽₃)	—	3	3	no	—
31	C₃×H(3,𝔽₃)	—	9	9	no	—
32	C₃×H(3,𝔽₃)	—	9	3	no	—
33	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
34	C₃²×C₉	—	3	3	no	—
35	C₃²⋊C₉	—	3	3	no	—
36	C₃²⋊C₉	—	3	6	no	—
37	C₃²⋊C₉	—	3	3	no	—
38	C₃²⋊C₉	—	3	3	no	—
39	C₃²⋊C₉	—	3	3	no	—
40	C₃²⋊C₉	—	3	3	no	—
41	C₃²⋊C₉	—	3	6	no	—
42	C₃³	—	3	1	no	—
43	C₃³	—	3	3	no	—
44	C₃³	—	3	2	no	—
45	C₃³	—	3	9	no	—
46	C₃³	—	9	1	no	—
47	C₃³	—	9	3	no	—
48	C₃³	—	9	9	no	—
49	C₃³	—	9	2	no	—
50	C₃³	—	9	3	no	—
51	C₃³	—	9	6	no	—
52	C₃³	—	27	9	no	—
53	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
55	3₋¹⁺²	—	27	3	no	—
56	3₋¹⁺²	—	27	3	no	—
57	H(3,𝔽₃)	—	9	9	no	—
58	H(3,𝔽₃)	—	9	9	no	—
59	H(3,𝔽₃)	—	9	18	no	—
60	H(3,𝔽₃)	—	9	9	no	—
61	H(3,𝔽₃)	—	27	3	no	—
62	H(3,𝔽₃)	—	27	3	no	—
63	H(3,𝔽₃)	—	27	18	no	—
64	C₃×C₉	—	9	3	no	—
65	C₃×C₉	—	9	3	no	—
66	C₃×C₉	—	9	3	no	—
67	C₃×C₉	—	27	3	no	—
68	C₃²	—	3	3	no	—
69	C₃²	—	3	3	no	—
70	C₃²	—	3	3	no	—
71	C₃²	—	3	3	no	—
72	C₃²	—	3	3	no	—
73	C₃²	—	3	1	no	—
74	C₃²	—	3	3	no	—
75	C₃²	—	9	3	no	—
76	C₃²	—	9	9	no	—
77	C₃²	—	9	6	no	—
78	C₃²	—	9	9	no	—
79	C₃²	—	9	9	no	—
80	C₃²	—	9	2	no	—
81	C₃²	—	9	9	no	—
82	C₃²	—	27	3	no	—
83	C₃²	—	27	3	no	—
84	C₃²	—	27	6	no	—
85	C₃²	—	27	27	no	—
86	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
87	C₉	—	27	3	no	—
88	C₉	—	27	3	no	—
89	C₉	—	27	3	no	—
90	C₃	—	9	3	no	—
91	C₃	—	9	1	no	—
92	C₃	—	9	3	no	—
93	C₃	—	27	2	no	—
94	C₃	—	27	9	no	—
95	C₃	#10624748	1	3	yes	—
96	C₃	#10624754	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624880	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. #10624814	C₃	semidirect
3. #10624805	C₃	semidirect
4. #10624657	C₃	semidirect
5. #10624598	C₃	semidirect
8. #6891	C₃²	semidirect
9. #6891	C₃²	semidirect
13. C₃×C₃²⋊C₉	C₃²	semidirect

Maximal subgroups

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

subgroup	index
#10624880	3
#10624814	3
#10624805	3
#10624657	3
#10624598	3