Finite Group Databasebeta

Subgroups — #14550286

Order 2187.

2399 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 2399 conjugacy classes of subgroups — too many to draw. Only its 188 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	C₃³×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×H(3,𝔽₃)⋊C₉	C₃	1	9	yes	maximal, complemented
4	#10624547	C₃	1	2	yes	maximal, complemented
5	C₃⁵	—	3	1	no	—
6	C₃⁵	C₃²	1	1	yes	characteristic
7	C₃²×H(3,𝔽₃)	—	3	2	no	—
8	C₃²×H(3,𝔽₃)	—	3	6	no	—
9	C₃²×H(3,𝔽₃)	—	3	9	no	—
10	C₃²×H(3,𝔽₃)	—	3	18	no	—
11	C₃²×H(3,𝔽₃)	C₉	1	9	yes	complemented
12	C₃²×H(3,𝔽₃)	C₃²	1	3	yes	—
13	C₃³×C₉	C₃²	1	3	yes	complemented
14	C₃×C₃²⋊C₉	—	3	2	no	—
15	C₃×C₃²⋊C₉	—	3	1	no	—
16	C₃×C₃²⋊C₉	—	3	9	no	—
17	C₃×C₃²⋊C₉	—	3	4	no	—
18	C₃×C₃²⋊C₉	—	3	2	no	—
19	C₃×C₃²⋊C₉	C₃²	1	6	yes	complemented
20	H(3,𝔽₃)⋊C₉	—	3	27	no	—
21	C₃⁴	—	3	9	no	—
22	C₃⁴	—	3	18	no	—
23	C₃⁴	—	3	9	no	—
24	C₃⁴	—	3	3	no	—
25	C₃⁴	—	3	3	no	—
26	C₃⁴	—	3	3	no	—
27	C₃⁴	—	3	3	no	—
28	C₃⁴	—	3	3	no	—
29	C₃⁴	—	3	6	no	—
30	C₃⁴	—	9	2	no	—
31	C₃⁴	—	9	2	no	—
32	C₃⁴	—	9	2	no	—
33	C₃⁴	—	9	6	no	—
34	C₃⁴	—	9	6	no	—
35	C₃⁴	—	9	6	no	—
36	C₃⁴	—	9	3	no	—
37	C₃⁴	—	9	3	no	—
38	C₃⁴	—	9	3	no	—
39	C₃⁴	C₃×C₉	1	3	yes	—
40	C₃⁴	H(3,𝔽₃)	1	2	yes	—
41	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
42	C₃⁴	3₋¹⁺²	1	6	yes	—
43	C₃⁴	C₃³	1	1	yes	characteristic
44	C₃×H(3,𝔽₃)	—	3	18	no	—
45	C₃×H(3,𝔽₃)	—	3	36	no	—
46	C₃×H(3,𝔽₃)	—	3	162	no	—
47	C₃×H(3,𝔽₃)	—	3	18	no	—
48	C₃×H(3,𝔽₃)	—	3	6	no	—
49	C₃×H(3,𝔽₃)	—	3	12	no	—
50	C₃×H(3,𝔽₃)	—	3	6	no	—
51	C₃×H(3,𝔽₃)	—	3	81	no	—
52	C₃×H(3,𝔽₃)	C₃×C₉	1	9	yes	—
53	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
54	C₃×H(3,𝔽₃)	3₋¹⁺²	1	18	yes	—
55	C₃²×C₉	—	3	3	no	—
56	C₃²×C₉	—	3	9	no	—
57	C₃²×C₉	—	3	9	no	—
58	C₃²×C₉	—	3	6	no	—
59	C₃²×C₉	—	9	6	no	—
60	C₃²⋊C₉	—	3	18	no	—
61	C₃²⋊C₉	—	3	9	no	—
62	C₃²⋊C₉	—	3	18	no	—
63	C₃²⋊C₉	—	9	12	no	—
64	C₃²⋊C₉	—	9	6	no	—
65	C₃³	—	3	1	no	—
66	C₃³	—	3	2	no	—
67	C₃³	—	3	1	no	—
68	C₃³	—	3	1	no	—
69	C₃³	—	3	1	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	81	no	—
77	C₃³	—	3	9	no	—
78	C₃³	—	3	18	no	—
79	C₃³	—	3	9	no	—
80	C₃³	—	3	9	no	—
81	C₃³	—	3	2	no	—
82	C₃³	—	3	9	no	—
83	C₃³	—	3	4	no	—
84	C₃³	—	3	18	no	—
85	C₃³	—	3	18	no	—
86	C₃³	—	3	2	no	—
87	C₃³	—	3	2	no	—
88	C₃³	—	3	2	no	—
89	C₃³	—	3	4	no	—
90	C₃³	—	3	4	no	—
91	C₃³	—	3	4	no	—
92	C₃³	—	9	6	no	—
93	C₃³	—	9	12	no	—
94	C₃³	—	9	18	no	—
95	C₃³	—	9	18	no	—
96	C₃³	—	9	18	no	—
97	C₃³	—	9	9	no	—
98	C₃³	—	9	9	no	—
99	C₃³	—	9	9	no	—
100	C₃³	—	9	54	no	—
101	C₃³	—	9	54	no	—
102	C₃³	—	9	54	no	—
103	C₃³	—	9	54	no	—
104	C₃³	—	9	27	no	—
105	C₃³	—	9	27	no	—
106	C₃³	—	9	27	no	—
107	C₃³	—	9	27	no	—
108	C₃³	—	9	3	no	—
109	C₃³	—	9	3	no	—
110	C₃³	—	9	3	no	—
111	C₃³	—	9	6	no	—
112	C₃³	—	9	6	no	—
113	C₃³	—	9	6	no	—
114	C₃³	C₃²⋊C₉	1	6	yes	—
115	C₃³	C₃²⋊C₉	1	3	yes	—
116	C₃³	AΣL(1,𝔽₂₇)	1	3	yes	—
117	C₃³	#731	1	3	yes	—
118	C₃³	#732	1	3	yes	—
119	C₃³	C₃²×C₉	1	1	yes	characteristic
120	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
121	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
122	C₃³	C₃×3₋¹⁺²	1	2	yes	—
123	C₃³	C₃×3₋¹⁺²	1	6	yes	—
124	H(3,𝔽₃)	—	3	162	no	—
125	H(3,𝔽₃)	—	3	18	no	—
126	H(3,𝔽₃)	—	3	18	no	—
127	H(3,𝔽₃)	—	3	36	no	—
128	H(3,𝔽₃)	C₃²⋊C₉	1	27	yes	complemented
129	C₃×C₉	—	3	9	no	—
130	C₃×C₉	—	9	3	no	—
131	C₃×C₉	—	9	6	no	—
132	C₃×C₉	—	9	27	no	—
133	C₃×C₉	—	9	6	no	—
134	C₃×C₉	—	9	6	no	—
135	C₃×C₉	—	9	12	no	—
136	C₃²	—	3	6	no	—
137	C₃²	—	3	6	no	—
138	C₃²	—	3	6	no	—
139	C₃²	—	3	6	no	—
140	C₃²	—	3	6	no	—
141	C₃²	—	3	6	no	—
142	C₃²	—	3	3	no	—
143	C₃²	—	3	1	no	—
144	C₃²	—	3	2	no	—
145	C₃²	—	3	6	no	—
146	C₃²	—	3	2	no	—
147	C₃²	—	3	4	no	—
148	C₃²	—	3	3	no	—
149	C₃²	—	3	3	no	—
150	C₃²	—	3	3	no	—
151	C₃²	—	3	3	no	—
152	C₃²	—	3	3	no	—
153	C₃²	—	3	3	no	—
154	C₃²	—	3	6	no	—
155	C₃²	—	3	27	no	—
156	C₃²	—	3	6	no	—
157	C₃²	—	3	12	no	—
158	C₃²	—	9	27	no	—
159	C₃²	—	9	54	no	—
160	C₃²	—	9	243	no	—
161	C₃²	—	9	4	no	—
162	C₃²	—	9	18	no	—
163	C₃²	—	9	4	no	—
164	C₃²	—	9	18	no	—
165	C₃²	—	9	4	no	—
166	C₃²	—	9	18	no	—
167	C₃²	—	9	9	no	—
168	C₃²	—	9	18	no	—
169	C₃²	—	9	2	no	—
170	C₃²	—	9	9	no	—
171	C₃²	—	9	2	no	—
172	C₃²	—	9	9	no	—
173	C₃²	—	9	2	no	—
174	C₃²	—	9	9	no	—
175	C₃²	H(3,𝔽₃)⋊C₉	1	9	yes	complemented
176	C₃²	C₃×C₃²⋊C₉	1	3	yes	—
177	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
178	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
179	C₃²	#6891	1	6	yes	—
180	C₃²	#6891	1	2	yes	—
181	C₃²	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
182	C₃²	C₃×#731	1	1	yes	characteristic
183	C₃²	C₃×#732	1	1	yes	characteristic
184	C₃²	#6909	1	2	yes	—
185	C₃²	#6910	1	2	yes	—
186	C₃²	#6911	1	2	yes	—
187	C₉	—	9	9	no	—
188	C₉	—	27	6	no	—
189	C₃	—	3	9	no	—
190	C₃	—	3	3	no	—
191	C₃	—	3	6	no	—
192	C₃	—	9	6	no	—
193	C₃	—	9	27	no	—
194	C₃	#10624547	1	1	yes	characteristic
195	C₃	C₃×H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
196	C₃	#10624698	1	2	yes	—
197	C₃	#10624740	1	1	yes	characteristic
198	C₃	#10624741	1	1	yes	characteristic
199	C₃	#10624742	1	1	yes	characteristic
200	C₃	#10624777	1	2	yes	—
201	C₃	#10624778	1	2	yes	—
202	C₃	#10624779	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃²×C₃²⋊C₉	C₃	semidirect
3. C₃×H(3,𝔽₃)⋊C₉	C₃	semidirect
4. #10624547	C₃	semidirect
11. C₃²×H(3,𝔽₃)	C₉	semidirect
13. C₃³×C₉	C₃²	semidirect
19. C₃×C₃²⋊C₉	C₃²	semidirect
128. H(3,𝔽₃)	C₃²⋊C₉	semidirect
175. C₃²	H(3,𝔽₃)⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃³×H(3,𝔽₃)	3
C₃²×C₃²⋊C₉	3
C₃×H(3,𝔽₃)⋊C₉	3
#10624547	3