Finite Group Databasebeta

Subgroups — #14546646

Order 2187.

998 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 116 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×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624547	C₃	1	2	yes	maximal, complemented
4	#10624449	C₃	1	9	yes	maximal, complemented
5	C₃⁵	C₃²	1	1	yes	characteristic
6	C₃²×H(3,𝔽₃)	—	3	2	no	—
7	C₃²×H(3,𝔽₃)	—	3	1	no	—
8	C₃²×H(3,𝔽₃)	C₃²	1	3	yes	—
9	C₃³×C₉	—	3	1	no	—
10	C₃²⋊H(3,𝔽₃)	—	3	6	no	—
11	C₃²⋊H(3,𝔽₃)	C₉	1	9	yes	complemented
12	C₃×C₃²⋊C₉	—	3	2	no	—
13	C₃×C₃²⋊C₉	—	3	9	no	—
14	C₃×C₃²⋊C₉	—	3	2	no	—
15	C₃×C₃²⋊C₉	—	3	2	no	—
16	C₃×C₃²⋊C₉	—	3	2	no	—
17	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
18	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
19	C₃⁴	—	3	3	no	—
20	C₃⁴	—	3	3	no	—
21	C₃⁴	—	3	3	no	—
22	C₃⁴	—	3	3	no	—
23	C₃⁴	—	9	1	no	—
24	C₃⁴	—	9	2	no	—
25	C₃⁴	—	9	2	no	—
26	C₃⁴	—	9	2	no	—
27	C₃⁴	—	9	2	no	—
28	C₃⁴	C₃×C₉	1	3	yes	—
29	C₃⁴	H(3,𝔽₃)	1	2	yes	—
30	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
31	C₃⁴	3₋¹⁺²	1	6	yes	—
32	C₃⁴	C₃³	1	1	yes	characteristic
33	C₃×H(3,𝔽₃)	—	3	18	no	—
34	C₃×H(3,𝔽₃)	—	3	9	no	—
35	C₃×H(3,𝔽₃)	—	3	6	no	—
36	C₃×H(3,𝔽₃)	—	9	3	no	—
37	C₃×H(3,𝔽₃)	—	9	6	no	—
38	C₃×H(3,𝔽₃)	—	9	6	no	—
39	C₃×H(3,𝔽₃)	—	9	6	no	—
40	C₃×H(3,𝔽₃)	—	9	6	no	—
41	C₃×H(3,𝔽₃)	—	9	3	no	—
42	C₃×H(3,𝔽₃)	—	9	3	no	—
43	C₃×H(3,𝔽₃)	—	9	3	no	—
44	C₃×H(3,𝔽₃)	C₃×C₉	1	9	yes	complemented
45	C₃²×C₉	—	3	9	no	—
46	C₃²×C₉	—	3	3	no	—
47	C₃²×C₉	—	3	6	no	—
48	C₃²×C₉	—	9	3	no	—
49	C₃²×C₉	—	9	6	no	—
50	C₃²⋊C₉	—	3	18	no	—
51	C₃²⋊C₉	—	3	9	no	—
52	C₃²⋊C₉	—	9	6	no	—
53	C₃²⋊C₉	—	9	6	no	—
54	C₃²⋊C₉	—	9	6	no	—
55	C₃²⋊C₉	—	9	6	no	—
56	C₃³	—	3	1	no	—
57	C₃³	—	3	2	no	—
58	C₃³	—	3	2	no	—
59	C₃³	—	3	2	no	—
60	C₃³	—	3	2	no	—
61	C₃³	—	3	9	no	—
62	C₃³	—	3	18	no	—
63	C₃³	—	3	2	no	—
64	C₃³	—	3	1	no	—
65	C₃³	—	3	2	no	—
66	C₃³	—	3	1	no	—
67	C₃³	—	3	2	no	—
68	C₃³	—	3	1	no	—
69	C₃³	—	3	2	no	—
70	C₃³	—	3	2	no	—
71	C₃³	—	3	2	no	—
72	C₃³	—	3	1	no	—
73	C₃³	—	3	2	no	—
74	C₃³	—	3	2	no	—
75	C₃³	—	3	2	no	—
76	C₃³	—	3	2	no	—
77	C₃³	—	3	2	no	—
78	C₃³	—	3	2	no	—
79	C₃³	—	3	2	no	—
80	C₃³	—	3	2	no	—
81	C₃³	—	3	2	no	—
82	C₃³	—	3	2	no	—
83	C₃³	—	3	9	no	—
84	C₃³	—	9	3	no	—
85	C₃³	—	9	6	no	—
86	C₃³	—	9	6	no	—
87	C₃³	—	9	12	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	3	no	—
95	C₃³	—	9	3	no	—
96	C₃³	—	9	3	no	—
97	C₃³	—	9	6	no	—
98	C₃³	—	9	3	no	—
99	C₃³	—	9	6	no	—
100	C₃³	—	9	6	no	—
101	C₃³	—	9	6	no	—
102	C₃³	C₃²⋊C₉	1	6	yes	—
103	C₃³	C₃²⋊C₉	1	3	yes	—
104	C₃³	AΣL(1,𝔽₂₇)	1	3	yes	—
105	C₃³	#731	1	3	yes	—
106	C₃³	#732	1	3	yes	—
107	C₃³	C₃²×C₉	1	1	yes	characteristic
108	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
109	C₃³	C₃×3₋¹⁺²	1	2	yes	—
110	H(3,𝔽₃)	—	9	9	no	—
111	H(3,𝔽₃)	—	9	18	no	—
112	H(3,𝔽₃)	—	9	54	no	—
113	H(3,𝔽₃)	—	9	27	no	—
114	C₃×C₉	—	3	9	no	—
115	C₃×C₉	—	9	27	no	—
116	C₃×C₉	—	9	3	no	—
117	C₃×C₉	—	9	6	no	—
118	C₃×C₉	—	9	6	no	—
119	C₃×C₉	—	9	6	no	—
120	C₃×C₉	—	9	6	no	—
121	C₃×C₉	—	9	6	no	—
122	C₃²	—	3	3	no	—
123	C₃²	—	3	6	no	—
124	C₃²	—	3	3	no	—
125	C₃²	—	3	6	no	—
126	C₃²	—	3	3	no	—
127	C₃²	—	3	6	no	—
128	C₃²	—	3	3	no	—
129	C₃²	—	3	6	no	—
130	C₃²	—	3	1	no	—
131	C₃²	—	3	2	no	—
132	C₃²	—	3	6	no	—
133	C₃²	—	3	6	no	—
134	C₃²	—	3	1	no	—
135	C₃²	—	3	2	no	—
136	C₃²	—	3	6	no	—
137	C₃²	—	3	6	no	—
138	C₃²	—	3	2	no	—
139	C₃²	—	3	4	no	—
140	C₃²	—	9	27	no	—
141	C₃²	—	9	54	no	—
142	C₃²	—	9	2	no	—
143	C₃²	—	9	1	no	—
144	C₃²	—	9	2	no	—
145	C₃²	—	9	2	no	—
146	C₃²	—	9	2	no	—
147	C₃²	—	9	2	no	—
148	C₃²	—	9	1	no	—
149	C₃²	—	9	2	no	—
150	C₃²	—	9	2	no	—
151	C₃²	—	9	2	no	—
152	C₃²	—	9	2	no	—
153	C₃²	—	9	1	no	—
154	C₃²	—	9	2	no	—
155	C₃²	—	9	2	no	—
156	C₃²	—	9	2	no	—
157	C₃²	—	9	9	no	—
158	C₃²	—	9	9	no	—
159	C₃²	—	9	18	no	—
160	C₃²	—	27	3	no	—
161	C₃²	—	27	6	no	—
162	C₃²	—	27	3	no	—
163	C₃²	—	27	6	no	—
164	C₃²	—	27	3	no	—
165	C₃²	—	27	6	no	—
166	C₃²	H(3,𝔽₃)⋊C₉	1	9	yes	—
167	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
168	C₃²	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
169	C₃²	C₃×#731	1	1	yes	characteristic
170	C₃²	C₃×#732	1	1	yes	characteristic
171	C₃²	#6909	1	1	yes	characteristic
172	C₃²	#6909	1	2	yes	—
173	C₃²	#6909	1	2	yes	—
174	C₃²	#6910	1	2	yes	—
175	C₃²	#6911	1	2	yes	—
176	C₉	—	9	9	no	—
177	C₉	—	27	6	no	—
178	C₃	—	3	9	no	—
179	C₃	—	9	1	no	—
180	C₃	—	9	2	no	—
181	C₃	—	9	6	no	—
182	C₃	—	27	9	no	—
183	C₃	#10624628	1	1	yes	characteristic
184	C₃	#10624629	1	2	yes	—
185	C₃	#10624635	1	1	yes	characteristic
186	C₃	#10624636	1	2	yes	—
187	C₃	#10624639	1	1	yes	characteristic
188	C₃	#10624640	1	2	yes	—
189	C₃	#10624689	1	1	yes	characteristic
190	C₃	C₃×H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
191	C₃	#10624698	1	2	yes	—

	subgroup	G/H
center, socle	C₃³	C₃×H(3,𝔽₃)
commutator	C₃³	C₃²×C₉
Frattini, cosocle	C₃⁴	C₃³
prime radical	C₃×C₃²⋊H(3,𝔽₃)	C₃
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³ ◁ C₃⁵ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₃²×C₃²⋊C₉	C₃	semidirect
3. #10624547	C₃	semidirect
4. #10624449	C₃	semidirect
11. C₃²⋊H(3,𝔽₃)	C₉	semidirect
44. C₃×H(3,𝔽₃)	C₃×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃²⋊H(3,𝔽₃)	3
C₃²×C₃²⋊C₉	3
#10624547	3
#10624449	3