Finite Group Databasebeta

Subgroups — #14554478

Order 2187.

1142 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 1142 conjugacy classes of subgroups — too many to draw. Only its 269 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	2	yes	maximal
2	C₃×#6891	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
4	#10624809	C₃	1	18	yes	maximal, complemented
5	#10624808	C₃	1	9	yes	maximal, complemented
6	#10624690	C₃	1	9	yes	maximal, complemented
7	C₃⁵	C₃²	1	1	yes	characteristic
8	C₃²×3₋¹⁺²	—	3	2	no	—
9	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
10	C₃³×C₉	—	3	1	no	—
11	#6891	—	3	2	no	—
12	#6891	—	3	2	no	—
13	#6891	—	3	2	no	—
14	#6891	—	3	2	no	—
15	#6891	—	3	2	no	—
16	#6891	—	3	2	no	—
17	#6891	—	3	1	no	—
18	#6891	—	3	2	no	—
19	#6891	—	3	2	no	—
20	#6891	—	3	2	no	—
21	#6891	—	3	1	no	—
22	#6891	—	3	1	no	—
23	#6891	C₃²	1	6	yes	—
24	#6891	C₃²	1	6	yes	—
25	#6891	C₃²	1	6	yes	—
26	C₃×C₃²⋊C₉	—	3	2	no	—
27	C₃×C₃²⋊C₉	—	3	2	no	—
28	C₃×C₃²⋊C₉	—	3	2	no	—
29	C₃×C₃²⋊C₉	—	3	1	no	—
30	C₃×C₃²⋊C₉	—	3	1	no	—
31	C₃×C₃²⋊C₉	—	3	2	no	—
32	C₃×C₃²⋊C₉	—	3	2	no	—
33	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
34	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
35	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
36	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
37	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
38	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
39	#6867	C₃²	1	81	yes	complemented
40	C₃⁴	—	3	9	no	—
41	C₃⁴	—	9	3	no	—
42	C₃⁴	—	9	6	no	—
43	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
44	C₃⁴	H(3,𝔽₃)	1	2	yes	—
45	C₃⁴	H(3,𝔽₃)	1	2	yes	—
46	C₃⁴	H(3,𝔽₃)	1	2	yes	—
47	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
48	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
49	C₃⁴	C₃³	1	2	yes	—
50	C₃⁴	C₃³	1	1	yes	characteristic
51	C₃⁴	C₃³	1	1	yes	characteristic
52	C₃×3₋¹⁺²	—	3	6	no	—
53	C₃×3₋¹⁺²	—	3	6	no	—
54	C₃×3₋¹⁺²	—	3	6	no	—
55	C₃×3₋¹⁺²	—	3	6	no	—
56	C₃×3₋¹⁺²	—	3	3	no	—
57	C₃×3₋¹⁺²	—	3	27	no	—
58	C₃×3₋¹⁺²	—	9	18	no	—
59	C₃²×C₉	—	3	3	no	—
60	C₃²×C₉	—	3	6	no	—
61	C₃²×C₉	—	3	3	no	—
62	C₃²×C₉	—	3	6	no	—
63	C₃²×C₉	—	9	9	no	—
64	C₃²×C₉	C₃³	1	9	yes	—
65	C₃²⋊C₉	—	3	3	no	—
66	C₃²⋊C₉	—	3	6	no	—
67	C₃²⋊C₉	—	3	6	no	—
68	C₃²⋊C₉	—	3	3	no	—
69	C₃²⋊C₉	—	3	6	no	—
70	C₃²⋊C₉	—	3	6	no	—
71	C₃²⋊C₉	—	3	6	no	—
72	C₃²⋊C₉	—	3	6	no	—
73	C₃²⋊C₉	—	3	6	no	—
74	C₃²⋊C₉	—	3	6	no	—
75	C₃²⋊C₉	—	3	6	no	—
76	C₃²⋊C₉	—	3	3	no	—
77	C₃²⋊C₉	—	3	6	no	—
78	C₃²⋊C₉	—	3	6	no	—
79	C₃²⋊C₉	—	3	3	no	—
80	C₃²⋊C₉	—	3	3	no	—
81	C₃²⋊C₉	—	3	6	no	—
82	C₃²⋊C₉	—	3	6	no	—
83	C₃²⋊C₉	—	3	6	no	—
84	C₃²⋊C₉	C₃³	1	9	yes	—
85	C₃²⋊C₉	C₃³	1	18	yes	—
86	C₃³	—	3	3	no	—
87	C₃³	—	3	3	no	—
88	C₃³	—	3	6	no	—
89	C₃³	—	3	3	no	—
90	C₃³	—	3	6	no	—
91	C₃³	—	3	6	no	—
92	C₃³	—	3	6	no	—
93	C₃³	—	3	6	no	—
94	C₃³	—	3	3	no	—
95	C₃³	—	3	3	no	—
96	C₃³	—	3	6	no	—
97	C₃³	—	3	6	no	—
98	C₃³	—	3	6	no	—
99	C₃³	—	3	3	no	—
100	C₃³	—	3	6	no	—
101	C₃³	—	9	2	no	—
102	C₃³	—	9	2	no	—
103	C₃³	—	9	2	no	—
104	C₃³	—	9	2	no	—
105	C₃³	—	9	1	no	—
106	C₃³	—	9	1	no	—
107	C₃³	—	9	2	no	—
108	C₃³	—	9	2	no	—
109	C₃³	—	9	2	no	—
110	C₃³	—	9	2	no	—
111	C₃³	—	9	1	no	—
112	C₃³	—	9	2	no	—
113	C₃³	—	9	2	no	—
114	C₃³	—	9	2	no	—
115	C₃³	—	9	2	no	—
116	C₃³	—	9	81	no	—
117	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
118	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
119	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
120	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
121	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
122	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
123	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
124	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
125	C₃³	#737	1	9	yes	—
126	C₃³	C₃⁴	1	1	yes	characteristic
127	3₋¹⁺²	—	9	18	no	—
128	3₋¹⁺²	—	9	18	no	—
129	3₋¹⁺²	—	9	9	no	—
130	3₋¹⁺²	—	9	18	no	—
131	3₋¹⁺²	—	9	18	no	—
132	C₃×C₉	—	3	9	no	—
133	C₃×C₉	—	3	9	no	—
134	C₃×C₉	—	3	18	no	—
135	C₃×C₉	—	3	27	no	—
136	C₃×C₉	—	9	9	no	—
137	C₃×C₉	—	9	18	no	—
138	C₃×C₉	—	9	18	no	—
139	C₃×C₉	—	9	9	no	—
140	C₃²	—	3	27	no	—
141	C₃²	—	3	1	no	—
142	C₃²	—	3	2	no	—
143	C₃²	—	3	1	no	—
144	C₃²	—	3	2	no	—
145	C₃²	—	3	2	no	—
146	C₃²	—	3	2	no	—
147	C₃²	—	3	2	no	—
148	C₃²	—	3	2	no	—
149	C₃²	—	3	2	no	—
150	C₃²	—	3	2	no	—
151	C₃²	—	3	1	no	—
152	C₃²	—	3	2	no	—
153	C₃²	—	3	1	no	—
154	C₃²	—	3	2	no	—
155	C₃²	—	3	2	no	—
156	C₃²	—	3	1	no	—
157	C₃²	—	3	2	no	—
158	C₃²	—	3	2	no	—
159	C₃²	—	3	2	no	—
160	C₃²	—	3	2	no	—
161	C₃²	—	3	2	no	—
162	C₃²	—	3	2	no	—
163	C₃²	—	3	1	no	—
164	C₃²	—	3	2	no	—
165	C₃²	—	3	2	no	—
166	C₃²	—	3	1	no	—
167	C₃²	—	9	9	no	—
168	C₃²	—	9	18	no	—
169	C₃²	—	9	9	no	—
170	C₃²	—	9	18	no	—
171	C₃²	—	9	18	no	—
172	C₃²	—	9	18	no	—
173	C₃²	—	9	9	no	—
174	C₃²	—	9	9	no	—
175	C₃²	#6893	1	3	yes	—
176	C₃²	#6893	1	6	yes	—
177	C₃²	#6910	1	3	yes	—
178	C₃²	#6910	1	6	yes	—
179	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
180	C₃²	C₃×#737	1	3	yes	—
181	C₉	—	9	9	no	—
182	C₉	—	9	9	no	—
183	C₉	—	9	18	no	—
184	C₃	—	3	3	no	—
185	C₃	—	3	6	no	—
186	C₃	—	3	3	no	—
187	C₃	—	3	3	no	—
188	C₃	—	3	6	no	—
189	C₃	—	9	1	no	—
190	C₃	—	9	2	no	—
191	C₃	—	9	2	no	—
192	C₃	—	9	1	no	—
193	C₃	C₃×#6893	1	1	yes	characteristic
194	C₃	C₃×#6910	1	1	yes	characteristic
195	C₃	C₃×#6910	1	2	yes	—

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃⁵	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁵ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #10624809	C₃	semidirect
5. #10624808	C₃	semidirect
6. #10624690	C₃	semidirect
39. #6867	C₃²	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
#10624809	3
#10624808	3
#10624690	3