Subgroups — #10624907

Order 729.

311 conjugacy classes of subgroups.

The subgroup lattice has 149 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6899	C₃	1	3	yes	maximal, complemented
2	#6899	C₃	1	3	yes	maximal, complemented
3	#6898	C₃	1	3	yes	maximal, complemented
4	#6897	C₃	1	3	yes	maximal, complemented
5	#6897	C₃	1	3	yes	maximal, complemented
6	#6896	C₃	1	3	yes	maximal, complemented
7	#6895	C₃	1	1	yes	maximal, characteristic, complemented
8	#6895	C₃	1	3	yes	maximal, complemented
9	#6894	C₃	1	1	yes	maximal, characteristic, complemented
10	#6894	C₃	1	1	yes	maximal, characteristic, complemented
11	#6894	C₃	1	1	yes	maximal, characteristic, complemented
12	#6894	C₃	1	1	yes	maximal, characteristic, complemented
13	#6894	C₃	1	3	yes	maximal, complemented
14	#6893	C₃	1	1	yes	maximal, characteristic
15	C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
16	C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
17	C₉×3₋¹⁺²	C₃	1	3	yes	maximal, complemented
18	C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
19	C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
20	C₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
21	#6887	C₃	1	1	yes	maximal, characteristic, complemented
22	C₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
23	C₃×3₋¹⁺²	C₃²	1	3	yes	—
24	C₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
25	C₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
26	C₃×3₋¹⁺²	C₃²	1	3	yes	—
27	C₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
28	C₃²×C₉	C₃²	1	1	yes	characteristic
29	C₃²×C₉	C₃²	1	1	yes	characteristic
30	C₃²×C₉	C₃²	1	1	yes	characteristic
31	C₃²×C₉	C₃²	1	1	yes	characteristic
32	C₃²×C₉	C₃²	1	3	yes	—
33	C₉⋊C₉	C₃²	1	1	yes	characteristic
34	C₉⋊C₉	C₃²	1	1	yes	characteristic
35	C₉⋊C₉	C₃²	1	1	yes	characteristic
36	C₉⋊C₉	C₃²	1	1	yes	characteristic
37	C₉⋊C₉	C₃²	1	1	yes	characteristic
38	C₉⋊C₉	C₃²	1	1	yes	characteristic
39	C₉⋊C₉	C₃²	1	1	yes	characteristic
40	C₉⋊C₉	C₃²	1	1	yes	characteristic
41	C₉⋊C₉	C₃²	1	1	yes	characteristic
42	C₉⋊C₉	C₃²	1	3	yes	—
43	C₉⋊C₉	C₃²	1	3	yes	—
44	C₉⋊C₉	C₃²	1	3	yes	—
45	C₉⋊C₉	C₃²	1	3	yes	—
46	C₉⋊C₉	C₃²	1	3	yes	—
47	C₉⋊C₉	C₃²	1	3	yes	—
48	C₉⋊C₉	C₃²	1	3	yes	—
49	C₉⋊C₉	C₃²	1	3	yes	—
50	C₉⋊C₉	C₃²	1	3	yes	—
51	C₉⋊C₉	C₃²	1	3	yes	—
52	C₉⋊C₉	C₃²	1	3	yes	—
53	C₉⋊C₉	C₃²	1	3	yes	—
54	C₉⋊C₉	C₃²	1	3	yes	—
55	C₉⋊C₉	C₃²	1	3	yes	—
56	C₉⋊C₉	C₃²	1	3	yes	—
57	C₉⋊C₉	C₃²	1	3	yes	—
58	C₉⋊C₉	C₃²	1	3	yes	—
59	C₉⋊C₉	C₃²	1	3	yes	—
60	C₉⋊C₉	C₃²	1	3	yes	—
61	C₉⋊C₉	C₃²	1	3	yes	—
62	C₉⋊C₉	C₃²	1	3	yes	—
63	C₃²⋊C₉	C₃²	1	1	yes	characteristic
64	C₃²⋊C₉	C₃²	1	1	yes	characteristic
65	C₃²⋊C₉	C₃²	1	1	yes	characteristic
66	C₃²⋊C₉	C₃²	1	1	yes	characteristic
67	C₃²⋊C₉	C₃²	1	3	yes	—
68	C₃²⋊C₉	C₃²	1	3	yes	—
69	C₃²⋊C₉	C₃²	1	1	yes	characteristic
70	C₃²⋊C₉	C₃²	1	1	yes	characteristic
71	C₃²⋊C₉	C₃²	1	3	yes	—
72	C₃²⋊C₉	C₃²	1	3	yes	—
73	C₃²⋊C₉	C₃²	1	3	yes	—
74	C₃²⋊C₉	C₃²	1	3	yes	—
75	C₃²⋊C₉	C₃²	1	3	yes	—
76	C₃²⋊C₉	C₃²	1	3	yes	—
77	C₃²⋊C₉	C₃²	1	3	yes	—
78	C₉²	C₃²	1	3	yes	—
79	C₉²	C₃²	1	3	yes	—
80	C₉²	C₃²	1	3	yes	—
81	C₃³	C₃³	1	3	yes	—
82	C₃³	C₃³	1	1	yes	characteristic
83	3₋¹⁺²	—	3	3	no	—
84	3₋¹⁺²	—	3	3	no	—
85	3₋¹⁺²	—	3	9	no	—
86	3₋¹⁺²	—	9	3	no	—
87	H(3,𝔽₃)	—	3	3	no	—
88	C₃×C₉	—	3	3	no	—
89	C₃×C₉	—	3	3	no	—
90	C₃×C₉	—	9	1	no	—
91	C₃×C₉	—	9	3	no	—
92	C₃×C₉	—	9	1	no	—
93	C₃×C₉	C₃³	1	1	yes	characteristic
94	C₃×C₉	C₃³	1	3	yes	—
95	C₃×C₉	C₃³	1	1	yes	characteristic
96	C₃×C₉	C₃³	1	3	yes	—
97	C₃×C₉	C₃³	1	3	yes	—
98	C₃×C₉	C₃³	1	3	yes	—
99	C₃×C₉	C₃³	1	3	yes	—
100	C₃×C₉	C₃³	1	3	yes	—
101	C₃×C₉	C₃³	1	3	yes	—
102	C₃×C₉	C₃³	1	1	yes	characteristic
103	C₃×C₉	C₃³	1	1	yes	characteristic
104	C₃×C₉	C₃³	1	1	yes	characteristic
105	C₃×C₉	C₃³	1	1	yes	characteristic
106	C₃×C₉	C₃³	1	3	yes	—
107	C₃×C₉	C₃³	1	1	yes	characteristic
108	C₃×C₉	C₃³	1	1	yes	characteristic
109	C₃×C₉	C₃³	1	1	yes	characteristic
110	C₃×C₉	C₃³	1	1	yes	characteristic
111	C₃×C₉	C₃³	1	1	yes	characteristic
112	C₃×C₉	C₃³	1	1	yes	characteristic
113	C₃²	—	3	1	no	—
114	C₃²	—	3	1	no	—
115	C₃²	—	3	1	no	—
116	C₃²	—	3	3	no	—
117	C₃²	—	3	3	no	—
118	C₃²	—	3	3	no	—
119	C₃²	—	3	3	no	—
120	C₃²	#737	1	3	yes	—
121	C₃²	C₃⁴	1	1	yes	characteristic
122	C₉	—	3	1	no	—
123	C₉	—	3	1	no	—
124	C₉	—	3	1	no	—
125	C₉	—	3	1	no	—
126	C₉	—	3	1	no	—
127	C₉	—	3	3	no	—
128	C₉	—	3	3	no	—
129	C₉	—	3	3	no	—
130	C₉	—	3	3	no	—
131	C₉	—	3	1	no	—
132	C₉	—	3	3	no	—
133	C₉	—	3	1	no	—
134	C₉	—	3	1	no	—
135	C₉	—	3	1	no	—
136	C₉	—	3	3	no	—
137	C₉	—	3	1	no	—
138	C₉	—	3	1	no	—
139	C₉	—	3	3	no	—
140	C₉	—	3	3	no	—
141	C₉	#737	1	3	yes	—
142	C₃	—	3	3	no	—
143	C₃	—	9	3	no	—
144	C₃	C₃×#737	1	1	yes	characteristic
145	C₃	C₃×#737	1	1	yes	characteristic
146	C₃	3₋¹⁺⁴	1	1	yes	characteristic
147	C₃	3₋¹⁺⁴	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #6899	C₃	semidirect
2. #6899	C₃	semidirect
3. #6898	C₃	semidirect
4. #6897	C₃	semidirect
5. #6897	C₃	semidirect
6. #6896	C₃	semidirect
7. #6895	C₃	semidirect
8. #6895	C₃	semidirect
9. #6894	C₃	semidirect
10. #6894	C₃	semidirect
11. #6894	C₃	semidirect
12. #6894	C₃	semidirect
13. #6894	C₃	semidirect
17. C₉×3₋¹⁺²	C₃	semidirect
18. C₉×3₋¹⁺²	C₃	semidirect
19. C₉×3₋¹⁺²	C₃	semidirect
21. #6887	C₃	semidirect
22. C₃×C₉⋊C₉	C₃	semidirect

Maximal subgroups

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

subgroup	index
#6899	3
#6898	3
#6897	3
#6896	3
#6895	3
#6894	3
#6893	3
C₉⋊H(3,𝔽₃)	3
C₉×3₋¹⁺²	3
C₉×H(3,𝔽₃)	3
#6887	3
C₃×C₉⋊C₉	3