Finite Group Databasebeta

Subgroups — #14548467

Order 2187.

311 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 50 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	#10624899	C₃	1	1	yes	maximal, characteristic
2	#10624781	C₃	1	1	yes	maximal, characteristic
3	#10624776	C₃	1	1	yes	maximal, characteristic
4	#10624675	C₃	1	1	yes	maximal, characteristic
5	#10624671	C₃	1	1	yes	maximal, characteristic
6	#10624659	C₃	1	1	yes	maximal, characteristic
7	#10624647	C₃	1	1	yes	maximal, characteristic
8	#10624633	C₃	1	1	yes	maximal, characteristic
9	#10624633	C₃	1	1	yes	maximal, characteristic
10	#10624612	C₃	1	1	yes	maximal, characteristic
11	#10624607	C₃	1	1	yes	maximal, characteristic
12	#10624607	C₃	1	1	yes	maximal, characteristic
13	#10624594	C₃	1	1	yes	maximal, characteristic
14	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
15	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
16	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
17	#6899	—	3	3	no	—
18	#6896	—	3	3	no	—
19	#6894	—	3	2	no	—
20	#6894	—	3	2	no	—
21	#6891	C₃²	1	1	yes	characteristic
22	#6891	C₃²	1	1	yes	characteristic
23	#6891	C₃²	1	1	yes	characteristic
24	#6891	C₃²	1	1	yes	characteristic
25	#6891	C₃²	1	1	yes	characteristic
26	#6891	C₃²	1	1	yes	characteristic
27	#6891	C₃²	1	1	yes	characteristic
28	#6891	C₃²	1	1	yes	characteristic
29	#6891	C₃²	1	1	yes	characteristic
30	C₉×3₋¹⁺²	—	3	1	no	—
31	C₉×3₋¹⁺²	—	3	1	no	—
32	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
33	3₋¹⁺²⋊C₉	—	3	3	no	—
34	#6868	—	3	3	no	—
35	#6862	—	3	3	no	—
36	#6860	—	3	1	no	—
37	#6860	—	3	1	no	—
38	#6860	—	3	2	no	—
39	#6860	—	3	2	no	—
40	#6860	—	3	1	no	—
41	#6860	—	3	2	no	—
42	#6860	—	3	1	no	—
43	#6860	—	3	2	no	—
44	#6860	—	3	1	no	—
45	#6860	—	3	2	no	—
46	#6860	—	3	3	no	—
47	#6858	—	3	1	no	—
48	#6858	—	3	2	no	—
49	#6858	—	3	3	no	—
50	#6858	—	3	3	no	—
51	C₃⁴	C₃³	1	1	yes	characteristic
52	C₃×3₋¹⁺²	—	3	1	no	—
53	C₃×3₋¹⁺²	—	3	1	no	—
54	C₃×3₋¹⁺²	—	3	1	no	—
55	C₃×3₋¹⁺²	—	3	1	no	—
56	C₃×3₋¹⁺²	—	3	1	no	—
57	C₃×3₋¹⁺²	—	3	1	no	—
58	C₃×3₋¹⁺²	—	3	1	no	—
59	C₃×3₋¹⁺²	—	3	1	no	—
60	C₃×3₋¹⁺²	—	3	1	no	—
61	C₃×3₋¹⁺²	—	3	1	no	—
62	C₃×3₋¹⁺²	—	3	1	no	—
63	C₃×3₋¹⁺²	—	3	1	no	—
64	C₃×3₋¹⁺²	—	3	1	no	—
65	C₃×3₋¹⁺²	—	3	1	no	—
66	C₃×3₋¹⁺²	—	3	1	no	—
67	C₃×3₋¹⁺²	—	3	1	no	—
68	C₃×3₋¹⁺²	—	3	1	no	—
69	C₃×3₋¹⁺²	—	9	3	no	—
70	C₃×3₋¹⁺²	—	9	3	no	—
71	C₃×3₋¹⁺²	—	9	3	no	—
72	C₃×3₋¹⁺²	H(3,𝔽₃)	1	1	yes	characteristic
73	C₃×3₋¹⁺²	H(3,𝔽₃)	1	2	yes	—
74	C₃²×C₉	—	3	1	no	—
75	C₃²×C₉	—	3	1	no	—
76	C₃²×C₉	—	3	1	no	—
77	C₃²×C₉	—	3	1	no	—
78	C₉⋊C₉	—	3	6	no	—
79	C₉⋊C₉	—	3	6	no	—
80	C₉⋊C₉	—	3	12	no	—
81	C₃²⋊C₉	—	3	1	no	—
82	C₃²⋊C₉	—	3	1	no	—
83	C₃²⋊C₉	—	3	1	no	—
84	C₃²⋊C₉	—	3	1	no	—
85	C₃²⋊C₉	—	3	1	no	—
86	C₃²⋊C₉	—	3	1	no	—
87	C₃²⋊C₉	—	3	1	no	—
88	C₃²⋊C₉	—	3	1	no	—
89	C₃²⋊C₉	—	3	1	no	—
90	C₃²⋊C₉	—	3	1	no	—
91	C₃²⋊C₉	—	3	1	no	—
92	C₃²⋊C₉	—	3	1	no	—
93	C₃²⋊C₉	—	3	1	no	—
94	C₃²⋊C₉	—	3	1	no	—
95	C₃²⋊C₉	—	3	1	no	—
96	C₃²⋊C₉	—	3	1	no	—
97	C₃²⋊C₉	—	3	1	no	—
98	C₃²⋊C₉	—	3	1	no	—
99	C₃²⋊C₉	—	3	1	no	—
100	C₃²⋊C₉	—	3	1	no	—
101	C₃²⋊C₉	—	3	1	no	—
102	C₃²⋊C₉	—	3	1	no	—
103	C₃²⋊C₉	—	3	1	no	—
104	C₃²⋊C₉	—	3	1	no	—
105	C₃²⋊C₉	—	3	1	no	—
106	C₃²⋊C₉	—	3	1	no	—
107	C₃²⋊C₉	—	3	1	no	—
108	C₃²⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
109	C₃²⋊C₉	H(3,𝔽₃)	1	2	yes	—
110	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
111	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
112	C₉²	—	3	3	no	—
113	C₃³	—	9	1	no	—
114	C₃³	—	9	1	no	—
115	C₃³	—	9	1	no	—
116	C₃³	—	9	1	no	—
117	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
118	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
119	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
120	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
121	3₋¹⁺²	—	3	3	no	—
122	3₋¹⁺²	—	3	3	no	—
123	3₋¹⁺²	—	9	3	no	—
124	3₋¹⁺²	—	9	3	no	—
125	3₋¹⁺²	—	9	3	no	—
126	3₋¹⁺²	—	9	2	no	—
127	3₋¹⁺²	—	9	3	no	—
128	3₋¹⁺²	—	9	3	no	—
129	3₋¹⁺²	—	9	3	no	—
130	3₋¹⁺²	—	9	3	no	—
131	3₋¹⁺²	—	9	3	no	—
132	3₋¹⁺²	—	9	2	no	—
133	3₋¹⁺²	—	27	1	no	—
134	3₋¹⁺²	—	27	1	no	—
135	3₋¹⁺²	—	27	1	no	—
136	3₋¹⁺²	—	27	1	no	—
137	3₋¹⁺²	—	27	1	no	—
138	3₋¹⁺²	—	27	1	no	—
139	3₋¹⁺²	—	27	1	no	—
140	3₋¹⁺²	—	27	1	no	—
141	C₃×C₉	—	3	1	no	—
142	C₃×C₉	—	3	3	no	—
143	C₃×C₉	—	3	1	no	—
144	C₃×C₉	—	3	3	no	—
145	C₃×C₉	—	3	3	no	—
146	C₃×C₉	—	3	2	no	—
147	C₃×C₉	—	3	2	no	—
148	C₃×C₉	—	9	3	no	—
149	C₃×C₉	—	9	2	no	—
150	C₃×C₉	—	9	1	no	—
151	C₃×C₉	—	9	1	no	—
152	C₃×C₉	—	9	3	no	—
153	C₃×C₉	—	9	1	no	—
154	C₃×C₉	—	9	1	no	—
155	C₃×C₉	—	9	1	no	—
156	C₃×C₉	—	9	1	no	—
157	C₃×C₉	—	9	1	no	—
158	C₃×C₉	—	9	1	no	—
159	C₃×C₉	—	9	1	no	—
160	C₃×C₉	—	27	1	no	—
161	C₃²	—	3	1	no	—
162	C₃²	—	3	1	no	—
163	C₃²	—	3	1	no	—
164	C₃²	—	3	1	no	—
165	C₃²	—	3	1	no	—
166	C₃²	—	3	1	no	—
167	C₃²	—	3	1	no	—
168	C₃²	—	3	1	no	—
169	C₃²	—	3	1	no	—
170	C₃²	—	3	1	no	—
171	C₃²	—	3	1	no	—
172	C₃²	—	3	1	no	—
173	C₃²	—	3	1	no	—
174	C₃²	—	3	1	no	—
175	C₃²	—	3	1	no	—
176	C₃²	—	3	1	no	—
177	C₃²	—	27	3	no	—
178	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
179	C₉	—	3	3	no	—
180	C₉	—	9	3	no	—
181	C₉	—	9	1	no	—
182	C₉	—	9	2	no	—
183	C₉	—	9	2	no	—
184	C₉	—	9	3	no	—
185	C₉	—	9	3	no	—
186	C₉	—	9	3	no	—
187	C₉	—	27	1	no	—
188	C₉	—	27	1	no	—
189	C₉	—	27	1	no	—
190	C₉	—	27	1	no	—
191	C₉	—	27	1	no	—
192	C₉	—	27	1	no	—
193	C₉	—	27	1	no	—
194	C₃	—	9	1	no	—
195	C₃	—	9	1	no	—
196	C₃	—	9	1	no	—
197	C₃	—	9	1	no	—
198	C₃	#10624752	1	1	yes	characteristic
199	C₃	#10624755	1	1	yes	characteristic
200	C₃	#10624758	1	1	yes	characteristic
201	C₃	#10624762	1	1	yes	characteristic

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

Maximal subgroups

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

subgroup	index
#10624899	3
#10624781	3
#10624776	3
#10624675	3
#10624671	3
#10624659	3
#10624647	3
#10624633	3
#10624612	3
#10624607	3
#10624594	3