Finite Group Databasebeta

Subgroups — #14547720

Order 2187.

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

normal subgroups only

This group has 452 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	#10624880	C₃	1	1	yes	maximal, characteristic
2	#10624820	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624818	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624818	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624805	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624805	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624804	C₃	1	1	yes	maximal, characteristic, complemented
8	#10624661	C₃	1	1	yes	maximal, characteristic, complemented
9	#10624658	C₃	1	1	yes	maximal, characteristic, complemented
10	#10624657	C₃	1	1	yes	maximal, characteristic, complemented
11	#10624640	C₃	1	1	yes	maximal, characteristic, complemented
12	#10624639	C₃	1	1	yes	maximal, characteristic, complemented
13	#10624635	C₃	1	1	yes	maximal, characteristic, complemented
14	C₃²×H(3,𝔽₃)	—	3	1	no	—
15	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
16	#6891	C₃²	1	1	yes	characteristic, complemented
17	#6891	C₃²	1	1	yes	characteristic, complemented
18	#6891	C₃²	1	1	yes	characteristic, complemented
19	#6891	C₃²	1	1	yes	characteristic, complemented
20	#6891	C₃²	1	1	yes	characteristic, complemented
21	#6891	C₃²	1	1	yes	characteristic, complemented
22	C₃²⋊H(3,𝔽₃)	—	3	2	no	—
23	C₃²⋊H(3,𝔽₃)	—	3	3	no	—
24	C₃²⋊H(3,𝔽₃)	—	3	3	no	—
25	C₃²⋊H(3,𝔽₃)	—	3	3	no	—
26	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
27	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
28	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
29	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic, complemented
30	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic, complemented
31	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic, complemented
32	H(3,𝔽₃)⋊C₉	—	3	3	no	—
33	H(3,𝔽₃)⋊C₉	—	3	3	no	—
34	H(3,𝔽₃)⋊C₉	—	3	3	no	—
35	#6866	—	3	1	no	—
36	#6866	—	3	2	no	—
37	#6866	—	3	1	no	—
38	#6866	—	3	1	no	—
39	#6866	—	3	2	no	—
40	#6866	—	3	2	no	—
41	#6859	—	3	3	no	—
42	#6859	—	3	3	no	—
43	#6859	—	3	3	no	—
44	#6857	—	3	3	no	—
45	#6857	—	3	3	no	—
46	#6857	—	3	3	no	—
47	C₃⁴	—	3	3	no	—
48	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
49	C₃⁴	H(3,𝔽₃)	1	2	yes	—
50	C₃⁴	C₃³	1	1	yes	characteristic
51	C₃×3₋¹⁺²	—	3	1	no	—
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×H(3,𝔽₃)	—	3	6	no	—
58	C₃×H(3,𝔽₃)	—	3	1	no	—
59	C₃×H(3,𝔽₃)	—	3	1	no	—
60	C₃×H(3,𝔽₃)	—	3	1	no	—
61	C₃×H(3,𝔽₃)	—	3	1	no	—
62	C₃×H(3,𝔽₃)	—	3	1	no	—
63	C₃×H(3,𝔽₃)	—	3	1	no	—
64	C₃×H(3,𝔽₃)	—	3	6	no	—
65	C₃×H(3,𝔽₃)	—	3	6	no	—
66	C₃×H(3,𝔽₃)	—	3	6	no	—
67	C₃×H(3,𝔽₃)	—	3	1	no	—
68	C₃×H(3,𝔽₃)	—	3	1	no	—
69	C₃×H(3,𝔽₃)	—	3	1	no	—
70	C₃×H(3,𝔽₃)	—	3	1	no	—
71	C₃×H(3,𝔽₃)	—	3	1	no	—
72	C₃×H(3,𝔽₃)	—	3	1	no	—
73	C₃×H(3,𝔽₃)	—	9	9	no	—
74	C₃×H(3,𝔽₃)	—	9	3	no	—
75	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
76	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
77	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
78	C₃²×C₉	—	3	1	no	—
79	C₃²×C₉	—	3	1	no	—
80	C₃²×C₉	—	3	1	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	C₃³	—	3	1	no	—
109	C₃³	—	3	3	no	—
110	C₃³	—	3	2	no	—
111	C₃³	—	3	3	no	—
112	C₃³	—	3	3	no	—
113	C₃³	—	3	3	no	—
114	C₃³	—	9	1	no	—
115	C₃³	—	9	1	no	—
116	C₃³	—	9	1	no	—
117	C₃³	—	9	1	no	—
118	C₃³	—	9	9	no	—
119	C₃³	—	9	2	no	—
120	C₃³	—	9	1	no	—
121	C₃³	—	9	1	no	—
122	C₃³	—	9	1	no	—
123	C₃³	—	9	2	no	—
124	C₃³	—	9	2	no	—
125	C₃³	—	9	2	no	—
126	C₃³	—	27	3	no	—
127	C₃³	—	27	3	no	—
128	C₃³	—	27	3	no	—
129	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
130	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
131	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
132	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
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	H(3,𝔽₃)	—	9	3	no	—
140	H(3,𝔽₃)	—	9	3	no	—
141	H(3,𝔽₃)	—	9	3	no	—
142	H(3,𝔽₃)	—	9	3	no	—
143	H(3,𝔽₃)	—	9	3	no	—
144	H(3,𝔽₃)	—	9	3	no	—
145	H(3,𝔽₃)	—	9	18	no	—
146	H(3,𝔽₃)	—	9	3	no	—
147	H(3,𝔽₃)	—	9	3	no	—
148	H(3,𝔽₃)	—	9	3	no	—
149	H(3,𝔽₃)	—	27	1	no	—
150	H(3,𝔽₃)	—	27	1	no	—
151	H(3,𝔽₃)	—	27	1	no	—
152	H(3,𝔽₃)	—	27	1	no	—
153	H(3,𝔽₃)	—	27	1	no	—
154	H(3,𝔽₃)	—	27	1	no	—
155	H(3,𝔽₃)	—	27	6	no	—
156	H(3,𝔽₃)	—	27	6	no	—
157	H(3,𝔽₃)	—	27	6	no	—
158	C₃×C₉	—	9	1	no	—
159	C₃×C₉	—	9	1	no	—
160	C₃×C₉	—	9	1	no	—
161	C₃×C₉	—	9	1	no	—
162	C₃×C₉	—	9	1	no	—
163	C₃×C₉	—	9	1	no	—
164	C₃×C₉	—	9	1	no	—
165	C₃×C₉	—	9	1	no	—
166	C₃×C₉	—	9	1	no	—
167	C₃×C₉	—	27	1	no	—
168	C₃×C₉	—	27	1	no	—
169	C₃×C₉	—	27	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	—	3	1	no	—
178	C₃²	—	3	1	no	—
179	C₃²	—	3	1	no	—
180	C₃²	—	3	1	no	—
181	C₃²	—	3	1	no	—
182	C₃²	—	3	1	no	—
183	C₃²	—	3	3	no	—
184	C₃²	—	3	1	no	—
185	C₃²	—	3	1	no	—
186	C₃²	—	3	1	no	—
187	C₃²	—	9	1	no	—
188	C₃²	—	9	1	no	—
189	C₃²	—	9	1	no	—
190	C₃²	—	9	3	no	—
191	C₃²	—	9	3	no	—
192	C₃²	—	9	3	no	—
193	C₃²	—	9	2	no	—
194	C₃²	—	9	2	no	—
195	C₃²	—	9	2	no	—
196	C₃²	—	9	3	no	—
197	C₃²	—	9	3	no	—
198	C₃²	—	9	3	no	—
199	C₃²	—	9	3	no	—
200	C₃²	—	9	3	no	—
201	C₃²	—	9	3	no	—
202	C₃²	—	9	2	no	—
203	C₃²	—	9	3	no	—
204	C₃²	—	9	3	no	—
205	C₃²	—	9	3	no	—
206	C₃²	—	27	3	no	—
207	C₃²	—	27	3	no	—
208	C₃²	—	27	6	no	—
209	C₃²	—	27	27	no	—
210	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
211	C₉	—	27	1	no	—
212	C₉	—	27	1	no	—
213	C₉	—	27	1	no	—
214	C₉	—	27	1	no	—
215	C₉	—	27	1	no	—
216	C₉	—	27	1	no	—
217	C₉	—	27	1	no	—
218	C₉	—	27	1	no	—
219	C₉	—	27	1	no	—
220	C₃	—	9	1	no	—
221	C₃	—	9	1	no	—
222	C₃	—	9	1	no	—
223	C₃	—	9	1	no	—
224	C₃	—	9	3	no	—
225	C₃	—	27	2	no	—
226	C₃	—	27	3	no	—
227	C₃	—	27	3	no	—
228	C₃	—	27	3	no	—
229	C₃	#10624748	1	1	yes	characteristic
230	C₃	#10624748	1	1	yes	characteristic
231	C₃	#10624748	1	1	yes	characteristic
232	C₃	#10624754	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624880	C₃
LCS	G ▷ C₃⁴ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624820	C₃	semidirect
3. #10624818	C₃	semidirect
4. #10624818	C₃	semidirect
5. #10624805	C₃	semidirect
6. #10624805	C₃	semidirect
7. #10624804	C₃	semidirect
8. #10624661	C₃	semidirect
9. #10624658	C₃	semidirect
10. #10624657	C₃	semidirect
11. #10624640	C₃	semidirect
12. #10624639	C₃	semidirect
13. #10624635	C₃	semidirect
16. #6891	C₃²	semidirect
17. #6891	C₃²	semidirect
18. #6891	C₃²	semidirect
19. #6891	C₃²	semidirect
20. #6891	C₃²	semidirect
21. #6891	C₃²	semidirect
29. C₃×C₃²⋊C₉	C₃²	semidirect
30. C₃×C₃²⋊C₉	C₃²	semidirect
31. C₃×C₃²⋊C₉	C₃²	semidirect

Maximal subgroups

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

subgroup	index
#10624880	3
#10624820	3
#10624818	3
#10624805	3
#10624804	3
#10624661	3
#10624658	3
#10624657	3
#10624640	3
#10624639	3
#10624635	3