Finite Group Databasebeta

Subgroups — #14552077

Order 2187.

1382 conjugacy classes of subgroups; only the 494 normal subgroups are shown.

normal subgroups only

This group has 1382 conjugacy classes of subgroups — too many to draw. Only its 494 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6891	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₃²⋊C₉	C₃	1	2	yes	maximal
3	#10624560	C₃	1	9	yes	maximal, complemented
4	#10624557	C₃	1	9	yes	maximal, complemented
5	#10624556	C₃	1	12	yes	maximal, complemented
6	#10624547	C₃	1	1	yes	maximal, characteristic
7	C₉×C₃²⋊C₉	C₃	1	6	yes	maximal, complemented
8	C₃⁵	C₃²	1	1	yes	characteristic
9	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
10	C₃³×C₉	C₃²	1	6	yes	—
11	#6891	C₉	1	27	yes	complemented
12	C₃×C₉⋊C₉	C₃²	1	18	yes	complemented
13	C₃×C₃²⋊C₉	—	3	18	no	—
14	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
15	C₃×C₃²⋊C₉	C₃²	1	4	yes	—
16	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
17	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
18	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
19	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
20	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
21	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
22	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
23	C₃×C₉²	C₃²	1	9	yes	complemented
24	#6855	C₃²	1	36	yes	complemented
25	#6855	C₃²	1	18	yes	complemented
26	C₃⁴	—	3	6	no	—
27	C₃⁴	—	3	6	no	—
28	C₃⁴	—	3	6	no	—
29	C₃⁴	—	9	1	no	—
30	C₃⁴	—	9	2	no	—
31	C₃⁴	—	9	1	no	—
32	C₃⁴	—	9	1	no	—
33	C₃⁴	—	9	1	no	—
34	C₃⁴	C₃×C₉	1	9	yes	—
35	C₃⁴	C₃³	1	1	yes	characteristic
36	C₃⁴	C₃³	1	1	yes	characteristic
37	C₃⁴	C₃³	1	2	yes	—
38	C₃×3₋¹⁺²	—	9	3	no	—
39	C₃×3₋¹⁺²	—	9	3	no	—
40	C₃×3₋¹⁺²	—	9	3	no	—
41	C₃×3₋¹⁺²	C₃×C₉	1	27	yes	complemented
42	C₃²×C₉	—	3	18	no	—
43	C₃²×C₉	—	3	18	no	—
44	C₃²×C₉	—	9	6	no	—
45	C₃²×C₉	—	9	6	no	—
46	C₃²×C₉	C₃³	1	12	yes	—
47	C₃²×C₉	C₃³	1	9	yes	—
48	C₃²×C₉	C₃³	1	9	yes	—
49	C₃²×C₉	C₃³	1	6	yes	—
50	C₉⋊C₉	—	9	18	no	—
51	C₃²⋊C₉	—	3	12	no	—
52	C₃²⋊C₉	—	3	18	no	—
53	C₃²⋊C₉	—	3	18	no	—
54	C₃²⋊C₉	—	3	18	no	—
55	C₃²⋊C₉	—	9	3	no	—
56	C₃²⋊C₉	—	9	3	no	—
57	C₃²⋊C₉	C₃×C₉	1	27	yes	complemented
58	C₃²⋊C₉	C₃×C₉	1	54	yes	complemented
59	C₃²⋊C₉	3₋¹⁺²	1	18	yes	complemented
60	C₉²	—	9	9	no	—
61	C₃³	—	3	2	no	—
62	C₃³	—	3	2	no	—
63	C₃³	—	3	2	no	—
64	C₃³	—	3	2	no	—
65	C₃³	—	3	2	no	—
66	C₃³	—	3	2	no	—
67	C₃³	—	3	2	no	—
68	C₃³	—	3	2	no	—
69	C₃³	—	3	2	no	—
70	C₃³	—	3	2	no	—
71	C₃³	—	3	2	no	—
72	C₃³	—	3	2	no	—
73	C₃³	—	3	2	no	—
74	C₃³	—	3	2	no	—
75	C₃³	—	3	2	no	—
76	C₃³	—	3	1	no	—
77	C₃³	—	3	1	no	—
78	C₃³	—	3	1	no	—
79	C₃³	—	3	1	no	—
80	C₃³	—	3	1	no	—
81	C₃³	—	3	1	no	—
82	C₃³	—	3	1	no	—
83	C₃³	—	3	2	no	—
84	C₃³	—	3	2	no	—
85	C₃³	—	3	1	no	—
86	C₃³	—	3	54	no	—
87	C₃³	—	9	9	no	—
88	C₃³	—	9	9	no	—
89	C₃³	—	9	18	no	—
90	C₃³	—	9	9	no	—
91	C₃³	—	9	18	no	—
92	C₃³	—	9	9	no	—
93	C₃³	—	9	18	no	—
94	C₃³	—	9	9	no	—
95	C₃³	C₃²×C₉	1	3	yes	—
96	C₃³	C₃²×C₉	1	3	yes	—
97	C₃³	C₃²×C₉	1	6	yes	—
98	C₃³	C₃×3₋¹⁺²	1	6	yes	—
99	C₃³	#737	1	6	yes	—
100	C₃³	#737	1	6	yes	—
101	C₃³	C₃⁴	1	1	yes	characteristic
102	3₋¹⁺²	—	9	27	no	—
103	C₃×C₉	—	3	6	no	—
104	C₃×C₉	—	3	6	no	—
105	C₃×C₉	—	3	9	no	—
106	C₃×C₉	—	3	6	no	—
107	C₃×C₉	—	3	9	no	—
108	C₃×C₉	—	3	9	no	—
109	C₃×C₉	—	3	12	no	—
110	C₃×C₉	—	3	18	no	—
111	C₃×C₉	—	3	12	no	—
112	C₃×C₉	—	3	9	no	—
113	C₃×C₉	—	3	12	no	—
114	C₃×C₉	—	3	9	no	—
115	C₃×C₉	—	3	12	no	—
116	C₃×C₉	—	9	54	no	—
117	C₃×C₉	C₃²×C₉	1	27	yes	—
118	C₃×C₉	C₃×3₋¹⁺²	1	18	yes	—
119	C₃²	—	3	6	no	—
120	C₃²	—	3	6	no	—
121	C₃²	—	3	6	no	—
122	C₃²	—	3	6	no	—
123	C₃²	—	3	6	no	—
124	C₃²	—	3	6	no	—
125	C₃²	—	3	6	no	—
126	C₃²	—	3	6	no	—
127	C₃²	—	3	6	no	—
128	C₃²	—	3	6	no	—
129	C₃²	—	3	6	no	—
130	C₃²	—	3	6	no	—
131	C₃²	—	3	6	no	—
132	C₃²	—	3	6	no	—
133	C₃²	—	3	3	no	—
134	C₃²	—	3	3	no	—
135	C₃²	—	3	3	no	—
136	C₃²	—	3	3	no	—
137	C₃²	—	9	81	no	—
138	C₃²	—	9	2	no	—
139	C₃²	—	9	2	no	—
140	C₃²	—	9	1	no	—
141	C₃²	—	9	1	no	—
142	C₃²	—	9	2	no	—
143	C₃²	—	9	2	no	—
144	C₃²	—	9	1	no	—
145	C₃²	—	9	1	no	—
146	C₃²	—	9	2	no	—
147	C₃²	—	9	2	no	—
148	C₃²	—	9	1	no	—
149	C₃²	—	9	1	no	—
150	C₃²	C₉×3₋¹⁺²	1	18	yes	—
151	C₃²	C₃³×C₉	1	1	yes	characteristic
152	C₃²	C₃²×3₋¹⁺²	1	2	yes	—
153	C₃²	C₃×#737	1	2	yes	—
154	C₃²	C₃×#737	1	2	yes	—
155	C₃²	H(5,𝔽₃)	1	1	yes	characteristic
156	C₃²	3₋¹⁺⁴	1	1	yes	characteristic
157	C₃²	3₋¹⁺⁴	1	1	yes	characteristic
158	C₃²	3₋¹⁺⁴	1	2	yes	—
159	C₃²	3₋¹⁺⁴	1	1	yes	characteristic
160	C₉	—	3	27	no	—
161	C₉	—	3	18	no	—
162	C₉	—	9	12	no	—
163	C₉	—	9	9	no	—
164	C₃	—	3	18	no	—
165	C₃	—	9	3	no	—
166	C₃	—	9	3	no	—
167	C₃	C₃×C₉×3₋¹⁺²	1	2	yes	—
168	C₃	#10624862	1	1	yes	characteristic
169	C₃	#10624863	1	1	yes	characteristic
170	C₃	#10624897	1	2	yes	—
171	C₃	3₋¹⁺²²	1	1	yes	characteristic
172	C₃	#10624899	1	2	yes	—
173	C₃	#10624902	1	1	yes	characteristic
174	C₃	#10624903	1	2	yes	—
175	C₃	#10624905	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle, cosocle	C₃³	C₃⁴
commutator	C₃²	C₃³×C₉
prime radical	C₃⁵	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³ ◁ G
LECS	G ▷ C₃³ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #10624560	C₃	semidirect
4. #10624557	C₃	semidirect
5. #10624556	C₃	semidirect
7. C₉×C₃²⋊C₉	C₃	semidirect
11. #6891	C₉	semidirect
12. C₃×C₉⋊C₉	C₃²	semidirect
23. C₃×C₉²	C₃²	semidirect
24. #6855	C₃²	semidirect
25. #6855	C₃²	semidirect
41. C₃×3₋¹⁺²	C₃×C₉	semidirect
57. C₃²⋊C₉	C₃×C₉	semidirect
58. C₃²⋊C₉	C₃×C₉	semidirect
59. C₃²⋊C₉	3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6891	3
C₃²×C₃²⋊C₉	3
#10624560	3
#10624557	3
#10624556	3
#10624547	3
C₉×C₃²⋊C₉	3