Subgroups — #10624908

Order 729.

309 conjugacy classes of subgroups.

The subgroup lattice has 173 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6900	C₃	1	1	yes	maximal, characteristic, complemented
2	#6899	C₃	1	2	yes	maximal, complemented
3	#6899	C₃	1	2	yes	maximal, complemented
4	#6899	C₃	1	2	yes	maximal, complemented
5	#6899	C₃	1	2	yes	maximal, complemented
6	#6898	C₃	1	1	yes	maximal, characteristic, complemented
7	#6898	C₃	1	2	yes	maximal, complemented
8	#6897	C₃	1	2	yes	maximal, complemented
9	#6897	C₃	1	2	yes	maximal, complemented
10	#6897	C₃	1	2	yes	maximal, complemented
11	#6896	C₃	1	1	yes	maximal, characteristic, complemented
12	#6896	C₃	1	2	yes	maximal, complemented
13	#6896	C₃	1	2	yes	maximal, complemented
14	#6896	C₃	1	1	yes	maximal, characteristic, complemented
15	#6895	C₃	1	1	yes	maximal, characteristic, complemented
16	#6894	C₃	1	1	yes	maximal, characteristic, complemented
17	#6894	C₃	1	2	yes	maximal, complemented
18	#6893	C₃	1	1	yes	maximal, characteristic
19	C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
20	C₉⋊H(3,𝔽₃)	C₃	1	2	yes	maximal
21	C₉×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
22	C₉×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
23	C₉×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
24	#6887	C₃	1	2	yes	maximal, complemented
25	C₃×3₋¹⁺²	C₃²	1	2	yes	—
26	C₃×3₋¹⁺²	C₃²	1	2	yes	—
27	C₃×3₋¹⁺²	C₃²	1	2	yes	—
28	C₃×3₋¹⁺²	C₃²	1	2	yes	—
29	C₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
30	C₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
31	C₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
32	C₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
33	C₃²×C₉	C₃²	1	2	yes	—
34	C₃²×C₉	C₃²	1	1	yes	characteristic
35	C₃²×C₉	C₃²	1	1	yes	characteristic
36	C₉⋊C₉	C₃²	1	2	yes	—
37	C₉⋊C₉	C₃²	1	2	yes	—
38	C₉⋊C₉	C₃²	1	2	yes	—
39	C₉⋊C₉	C₃²	1	2	yes	—
40	C₉⋊C₉	C₃²	1	2	yes	—
41	C₉⋊C₉	C₃²	1	2	yes	—
42	C₉⋊C₉	C₃²	1	2	yes	—
43	C₉⋊C₉	C₃²	1	2	yes	—
44	C₉⋊C₉	C₃²	1	2	yes	—
45	C₉⋊C₉	C₃²	1	2	yes	—
46	C₉⋊C₉	C₃²	1	2	yes	—
47	C₉⋊C₉	C₃²	1	2	yes	—
48	C₉⋊C₉	C₃²	1	2	yes	—
49	C₉⋊C₉	C₃²	1	2	yes	—
50	C₉⋊C₉	C₃²	1	2	yes	—
51	C₉⋊C₉	C₃²	1	2	yes	—
52	C₉⋊C₉	C₃²	1	2	yes	—
53	C₉⋊C₉	C₃²	1	2	yes	—
54	C₉⋊C₉	C₃²	1	2	yes	—
55	C₉⋊C₉	C₃²	1	2	yes	—
56	C₉⋊C₉	C₃²	1	2	yes	—
57	C₉⋊C₉	C₃²	1	1	yes	characteristic
58	C₉⋊C₉	C₃²	1	1	yes	characteristic
59	C₉⋊C₉	C₃²	1	2	yes	—
60	C₉⋊C₉	C₃²	1	1	yes	characteristic
61	C₉⋊C₉	C₃²	1	2	yes	—
62	C₉⋊C₉	C₃²	1	1	yes	characteristic
63	C₉⋊C₉	C₃²	1	1	yes	characteristic
64	C₉⋊C₉	C₃²	1	2	yes	—
65	C₉⋊C₉	C₃²	1	1	yes	characteristic
66	C₉⋊C₉	C₃²	1	2	yes	—
67	C₉⋊C₉	C₃²	1	2	yes	—
68	C₉⋊C₉	C₃²	1	2	yes	—
69	C₉⋊C₉	C₃²	1	2	yes	—
70	C₉⋊C₉	C₃²	1	2	yes	—
71	C₉⋊C₉	C₃²	1	2	yes	—
72	C₉⋊C₉	C₃²	1	2	yes	—
73	C₉⋊C₉	C₃²	1	1	yes	characteristic
74	C₃²⋊C₉	C₃²	1	2	yes	—
75	C₃²⋊C₉	C₃²	1	2	yes	—
76	C₃²⋊C₉	C₃²	1	2	yes	—
77	C₃²⋊C₉	C₃²	1	2	yes	—
78	C₃²⋊C₉	C₃²	1	2	yes	—
79	C₃²⋊C₉	C₃²	1	2	yes	—
80	C₃²⋊C₉	C₃²	1	2	yes	—
81	C₃²⋊C₉	C₃²	1	2	yes	—
82	C₃²⋊C₉	C₃²	1	2	yes	—
83	C₃²⋊C₉	C₃²	1	2	yes	—
84	C₃²⋊C₉	C₃²	1	2	yes	—
85	C₃²⋊C₉	C₃²	1	2	yes	—
86	C₃²⋊C₉	C₃²	1	2	yes	—
87	C₃²⋊C₉	C₃²	1	1	yes	characteristic
88	C₃²⋊C₉	C₃²	1	2	yes	—
89	C₃²⋊C₉	C₃²	1	1	yes	characteristic
90	C₃²⋊C₉	C₃²	1	2	yes	—
91	C₃²⋊C₉	C₃²	1	1	yes	characteristic
92	C₉²	C₃²	1	1	yes	characteristic
93	C₉²	C₃²	1	2	yes	—
94	C₉²	C₃²	1	2	yes	—
95	C₉²	C₃²	1	2	yes	—
96	C₉²	C₃²	1	2	yes	—
97	C₉²	C₃²	1	1	yes	characteristic
98	C₉²	C₃²	1	2	yes	—
99	C₃³	C₃³	1	2	yes	—
100	C₃³	C₃³	1	1	yes	characteristic
101	C₃³	C₃³	1	1	yes	characteristic
102	3₋¹⁺²	—	3	6	no	—
103	3₋¹⁺²	—	3	6	no	—
104	3₋¹⁺²	—	3	6	no	—
105	3₋¹⁺²	—	9	2	no	—
106	3₋¹⁺²	—	9	1	no	—
107	3₋¹⁺²	—	9	1	no	—
108	3₋¹⁺²	—	9	1	no	—
109	H(3,𝔽₃)	—	9	1	no	—
110	C₃×C₉	—	3	6	no	—
111	C₃×C₉	—	9	1	no	—
112	C₃×C₉	—	9	1	no	—
113	C₃×C₉	C₃³	1	2	yes	—
114	C₃×C₉	C₃³	1	2	yes	—
115	C₃×C₉	C₃³	1	2	yes	—
116	C₃×C₉	C₃³	1	2	yes	—
117	C₃×C₉	C₃³	1	2	yes	—
118	C₃×C₉	C₃³	1	2	yes	—
119	C₃×C₉	C₃³	1	2	yes	—
120	C₃×C₉	C₃³	1	2	yes	—
121	C₃×C₉	C₃³	1	2	yes	—
122	C₃×C₉	C₃³	1	1	yes	characteristic
123	C₃×C₉	C₃³	1	1	yes	characteristic
124	C₃×C₉	C₃³	1	2	yes	—
125	C₃×C₉	C₃³	1	2	yes	—
126	C₃×C₉	C₃³	1	2	yes	—
127	C₃×C₉	C₃³	1	2	yes	—
128	C₃×C₉	C₃³	1	1	yes	characteristic
129	C₃×C₉	C₃³	1	1	yes	characteristic
130	C₃×C₉	C₃³	1	2	yes	—
131	C₃×C₉	C₃³	1	1	yes	characteristic
132	C₃×C₉	C₃³	1	1	yes	characteristic
133	C₃×C₉	C₃³	1	2	yes	—
134	C₃²	—	3	2	no	—
135	C₃²	—	3	2	no	—
136	C₃²	—	3	2	no	—
137	C₃²	—	3	2	no	—
138	C₃²	—	3	1	no	—
139	C₃²	—	3	1	no	—
140	C₃²	—	3	2	no	—
141	C₃²	—	3	1	no	—
142	C₃²	—	3	2	no	—
143	C₃²	—	3	1	no	—
144	C₃²	C₃⁴	1	1	yes	characteristic
145	C₉	—	3	2	no	—
146	C₉	—	3	2	no	—
147	C₉	—	3	2	no	—
148	C₉	—	3	2	no	—
149	C₉	—	3	2	no	—
150	C₉	—	3	2	no	—
151	C₉	—	3	2	no	—
152	C₉	—	3	2	no	—
153	C₉	—	3	2	no	—
154	C₉	—	3	2	no	—
155	C₉	—	3	1	no	—
156	C₉	—	3	1	no	—
157	C₉	—	3	1	no	—
158	C₉	—	3	2	no	—
159	C₉	—	3	2	no	—
160	C₉	—	3	1	no	—
161	C₉	—	3	1	no	—
162	C₉	—	3	2	no	—
163	C₉	—	3	2	no	—
164	C₉	—	3	1	no	—
165	C₉	#737	1	6	yes	—
166	C₃	—	9	2	no	—
167	C₃	—	9	1	no	—
168	C₃	—	9	1	no	—
169	C₃	C₃×#737	1	2	yes	—
170	C₃	3₋¹⁺⁴	1	1	yes	characteristic
171	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. #6900	C₃	semidirect
2. #6899	C₃	semidirect
3. #6899	C₃	semidirect
4. #6899	C₃	semidirect
5. #6899	C₃	semidirect
6. #6898	C₃	semidirect
7. #6898	C₃	semidirect
8. #6897	C₃	semidirect
9. #6897	C₃	semidirect
10. #6897	C₃	semidirect
11. #6896	C₃	semidirect
12. #6896	C₃	semidirect
13. #6896	C₃	semidirect
14. #6896	C₃	semidirect
15. #6895	C₃	semidirect
16. #6894	C₃	semidirect
17. #6894	C₃	semidirect
21. C₉×3₋¹⁺²	C₃	semidirect
22. C₉×3₋¹⁺²	C₃	semidirect
23. C₉×3₋¹⁺²	C₃	semidirect
24. #6887	C₃	semidirect

Maximal subgroups

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

subgroup	index
#6900	3
#6899	3
#6898	3
#6897	3
#6896	3
#6895	3
#6894	3
#6893	3
C₉⋊H(3,𝔽₃)	3
C₉×3₋¹⁺²	3
#6887	3