Subgroups — #14552224

Order 2187.

1151 conjugacy classes of subgroups.

The subgroup lattice has 263 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6894	C₃	1	3	yes	maximal, complemented
2	C₃×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×C₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₃²×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624566	C₃	1	6	yes	maximal, complemented
6	#10624566	C₃	1	6	yes	maximal, complemented
7	#10624565	C₃	1	6	yes	maximal, complemented
8	#10624556	C₃	1	1	yes	maximal, characteristic, complemented
9	#10624556	C₃	1	1	yes	maximal, characteristic, complemented
10	#10624556	C₃	1	1	yes	maximal, characteristic, complemented
11	#10624556	C₃	1	1	yes	maximal, characteristic, complemented
12	#10624556	C₃	1	1	yes	maximal, characteristic, complemented
13	#10624556	C₃	1	1	yes	maximal, characteristic, complemented
14	#10624556	C₃	1	1	yes	maximal, characteristic, complemented
15	#10624556	C₃	1	6	yes	maximal, complemented
16	#10624555	C₃	1	1	yes	maximal, characteristic
17	#10624555	C₃	1	1	yes	maximal, characteristic
18	#10624548	C₃	1	1	yes	maximal, characteristic, complemented
19	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
20	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
21	C₃³×C₉	C₃²	1	1	yes	characteristic
22	C₃³×C₉	C₃²	1	1	yes	characteristic
23	C₃³×C₉	C₃²	1	1	yes	characteristic
24	C₃³×C₉	C₃²	1	1	yes	characteristic
25	C₃³×C₉	C₃²	1	3	yes	—
26	#6894	—	3	27	no	—
27	C₉⋊H(3,𝔽₃)	—	3	9	no	—
28	C₉×H(3,𝔽₃)	—	3	9	no	—
29	C₃×C₉⋊C₉	—	3	9	no	—
30	C₃×C₉⋊C₉	C₃²	1	3	yes	—
31	C₃×C₉⋊C₉	C₃²	1	2	yes	—
32	C₃×C₉⋊C₉	C₃²	1	3	yes	—
33	C₃×C₉⋊C₉	C₃²	1	3	yes	—
34	C₃×C₉⋊C₉	C₃²	1	3	yes	—
35	C₃×C₉⋊C₉	C₃²	1	3	yes	—
36	C₃×C₉⋊C₉	C₃²	1	3	yes	—
37	C₃×C₉⋊C₉	C₃²	1	3	yes	—
38	C₃×C₉⋊C₉	C₃²	1	3	yes	—
39	C₃×C₉⋊C₉	C₃²	1	2	yes	—
40	C₃×C₉⋊C₉	C₃²	1	2	yes	—
41	C₃×C₉⋊C₉	C₃²	1	2	yes	—
42	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
43	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
44	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
45	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
46	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
47	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
48	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
49	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
50	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
51	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
52	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
53	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
54	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
55	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
56	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
57	#6855	C₃²	1	6	yes	—
58	#6855	C₃²	1	6	yes	—
59	#6855	C₃²	1	6	yes	—
60	#6855	C₃²	1	6	yes	—
61	#6855	C₃²	1	6	yes	—
62	#6855	C₃²	1	6	yes	—
63	#6855	C₃²	1	6	yes	—
64	#6855	C₃²	1	6	yes	—
65	C₃⁴	C₃³	1	3	yes	—
66	C₃⁴	C₃³	1	1	yes	characteristic
67	C₃×3₋¹⁺²	—	3	9	no	—
68	C₃×3₋¹⁺²	—	3	9	no	—
69	C₃×3₋¹⁺²	—	9	6	no	—
70	C₃×H(3,𝔽₃)	—	3	3	no	—
71	C₃×H(3,𝔽₃)	—	3	3	no	—
72	C₃×H(3,𝔽₃)	—	3	3	no	—
73	C₃×H(3,𝔽₃)	3₋¹⁺²	1	9	yes	—
74	C₃²×C₉	—	3	3	no	—
75	C₃²×C₉	—	3	3	no	—
76	C₃²×C₉	—	3	6	no	—
77	C₃²×C₉	—	3	3	no	—
78	C₃²×C₉	—	3	3	no	—
79	C₃²×C₉	—	3	3	no	—
80	C₃²×C₉	—	3	3	no	—
81	C₃²×C₉	—	3	3	no	—
82	C₃²×C₉	—	3	3	no	—
83	C₃²×C₉	—	3	9	no	—
84	C₃²×C₉	—	3	9	no	—
85	C₃²×C₉	—	9	6	no	—
86	C₃²×C₉	—	9	2	no	—
87	C₃²×C₉	—	9	2	no	—
88	C₃²×C₉	—	9	2	no	—
89	C₃²×C₉	C₃³	1	2	yes	—
90	C₃²×C₉	C₃³	1	2	yes	—
91	C₃²×C₉	C₃³	1	3	yes	—
92	C₃²×C₉	C₃³	1	2	yes	—
93	C₃²×C₉	C₃³	1	3	yes	—
94	C₃²×C₉	C₃³	1	3	yes	—
95	C₃²×C₉	C₃³	1	3	yes	—
96	C₃²×C₉	C₃³	1	3	yes	—
97	C₃²×C₉	C₃³	1	3	yes	—
98	C₃²×C₉	C₃³	1	3	yes	—
99	C₃²×C₉	C₃³	1	1	yes	characteristic
100	C₃²×C₉	C₃³	1	1	yes	characteristic
101	C₃²×C₉	C₃³	1	1	yes	characteristic
102	C₃²×C₉	C₃³	1	1	yes	characteristic
103	C₃²×C₉	C₃³	1	3	yes	—
104	C₃²×C₉	C₃³	1	2	yes	—
105	C₉⋊C₉	—	3	6	no	—
106	C₉⋊C₉	—	3	6	no	—
107	C₉⋊C₉	—	3	6	no	—
108	C₉⋊C₉	—	3	6	no	—
109	C₉⋊C₉	—	3	9	no	—
110	C₉⋊C₉	—	3	9	no	—
111	C₉⋊C₉	—	3	9	no	—
112	C₉⋊C₉	—	3	9	no	—
113	C₉⋊C₉	—	3	9	no	—
114	C₉⋊C₉	—	3	9	no	—
115	C₉⋊C₉	—	3	9	no	—
116	C₉⋊C₉	—	3	9	no	—
117	C₉⋊C₉	H(3,𝔽₃)	1	9	yes	complemented
118	C₃²⋊C₉	—	3	3	no	—
119	C₃²⋊C₉	—	3	18	no	—
120	C₃²⋊C₉	—	3	3	no	—
121	C₃²⋊C₉	—	3	3	no	—
122	C₃²⋊C₉	—	3	3	no	—
123	C₃²⋊C₉	—	3	3	no	—
124	C₃²⋊C₉	—	3	3	no	—
125	C₃²⋊C₉	—	3	3	no	—
126	C₃²⋊C₉	—	3	9	no	—
127	C₃²⋊C₉	—	3	9	no	—
128	C₃²⋊C₉	—	3	18	no	—
129	C₃²⋊C₉	—	3	18	no	—
130	C₃²⋊C₉	—	9	6	no	—
131	C₃²⋊C₉	3₋¹⁺²	1	9	yes	complemented
132	C₃³	—	3	1	no	—
133	C₃³	—	3	2	no	—
134	C₃³	—	3	2	no	—
135	C₃³	—	3	2	no	—
136	C₃³	—	3	2	no	—
137	C₃³	—	3	3	no	—
138	C₃³	—	3	3	no	—
139	C₃³	—	3	6	no	—
140	C₃³	—	3	3	no	—
141	C₃³	—	3	3	no	—
142	C₃³	—	3	6	no	—
143	C₃³	—	3	6	no	—
144	C₃³	—	3	6	no	—
145	C₃³	C₃×3₋¹⁺²	1	9	yes	—
146	C₃³	C₃×3₋¹⁺²	1	3	yes	—
147	C₃³	C₃×3₋¹⁺²	1	3	yes	—
148	C₃³	#737	1	3	yes	—
149	C₃³	#737	1	3	yes	—
150	C₃³	C₃⁴	1	1	yes	characteristic
151	3₋¹⁺²	—	9	27	no	—
152	H(3,𝔽₃)	—	3	27	no	—
153	C₃×C₉	—	3	27	no	—
154	C₃×C₉	—	3	2	no	—
155	C₃×C₉	—	3	3	no	—
156	C₃×C₉	—	3	1	no	—
157	C₃×C₉	—	3	1	no	—
158	C₃×C₉	—	3	1	no	—
159	C₃×C₉	—	3	2	no	—
160	C₃×C₉	—	3	2	no	—
161	C₃×C₉	—	3	2	no	—
162	C₃×C₉	—	3	1	no	—
163	C₃×C₉	—	3	2	no	—
164	C₃×C₉	—	3	6	no	—
165	C₃×C₉	—	3	2	no	—
166	C₃×C₉	—	3	2	no	—
167	C₃×C₉	—	3	3	no	—
168	C₃×C₉	—	3	3	no	—
169	C₃×C₉	—	3	3	no	—
170	C₃×C₉	—	3	3	no	—
171	C₃×C₉	—	3	2	no	—
172	C₃×C₉	—	3	2	no	—
173	C₃×C₉	—	3	3	no	—
174	C₃×C₉	—	3	2	no	—
175	C₃×C₉	—	3	3	no	—
176	C₃×C₉	—	3	3	no	—
177	C₃×C₉	—	3	2	no	—
178	C₃×C₉	—	3	3	no	—
179	C₃×C₉	—	3	2	no	—
180	C₃×C₉	—	3	3	no	—
181	C₃×C₉	—	3	3	no	—
182	C₃×C₉	—	3	3	no	—
183	C₃×C₉	—	3	3	no	—
184	C₃×C₉	—	3	6	no	—
185	C₃×C₉	—	3	6	no	—
186	C₃×C₉	—	3	3	no	—
187	C₃×C₉	—	3	3	no	—
188	C₃×C₉	—	3	2	no	—
189	C₃×C₉	—	3	2	no	—
190	C₃×C₉	—	3	6	no	—
191	C₃×C₉	—	3	6	no	—
192	C₃×C₉	—	3	6	no	—
193	C₃×C₉	—	3	4	no	—
194	C₃×C₉	—	3	4	no	—
195	C₃×C₉	—	3	2	no	—
196	C₃×C₉	—	3	6	no	—
197	C₃×C₉	—	3	6	no	—
198	C₃×C₉	—	9	27	no	—
199	C₃×C₉	—	9	9	no	—
200	C₃×C₉	—	9	9	no	—
201	C₃×C₉	—	9	9	no	—
202	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
203	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
204	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
205	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
206	C₃×C₉	C₃×3₋¹⁺²	1	9	yes	—
207	C₃×C₉	#737	1	6	yes	—
208	C₃²	—	3	3	no	—
209	C₃²	—	3	6	no	—
210	C₃²	—	3	3	no	—
211	C₃²	—	3	3	no	—
212	C₃²	—	3	3	no	—
213	C₃²	—	3	6	no	—
214	C₃²	—	3	6	no	—
215	C₃²	—	3	6	no	—
216	C₃²	—	3	9	no	—
217	C₃²	—	3	9	no	—
218	C₃²	—	3	18	no	—
219	C₃²	—	9	3	no	—
220	C₃²	—	9	6	no	—
221	C₃²	—	9	3	no	—
222	C₃²	—	9	3	no	—
223	C₃²	—	9	6	no	—
224	C₃²	—	9	6	no	—
225	C₃²	#6894	1	9	yes	complemented
226	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
227	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
228	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
229	C₃²	C₃×#737	1	1	yes	characteristic
230	C₃²	C₃×#737	1	2	yes	—
231	C₃²	C₃×#737	1	1	yes	characteristic
232	C₃²	3₋¹⁺⁴	1	2	yes	—
233	C₃²	3₋¹⁺⁴	1	2	yes	—
234	C₃²	3₋¹⁺⁴	1	2	yes	—
235	C₉	—	3	3	no	—
236	C₉	—	3	6	no	—
237	C₉	—	3	3	no	—
238	C₉	—	3	3	no	—
239	C₉	—	3	9	no	—
240	C₉	—	3	3	no	—
241	C₉	—	9	3	no	—
242	C₉	—	9	3	no	—
243	C₉	—	9	3	no	—
244	C₉	—	9	2	no	—
245	C₉	—	9	2	no	—
246	C₉	—	9	3	no	—
247	C₉	—	9	3	no	—
248	C₉	—	9	3	no	—
249	C₉	—	9	3	no	—
250	C₉	—	9	2	no	—
251	C₃	—	3	9	no	—
252	C₃	—	9	9	no	—
253	C₃	#10624869	1	1	yes	characteristic
254	C₃	C₃×#6894	1	1	yes	characteristic
255	C₃	#10624891	1	2	yes	—
256	C₃	H(3,𝔽₃)×3₋¹⁺²	1	1	yes	characteristic
257	C₃	H(3,𝔽₃)×3₋¹⁺²	1	1	yes	characteristic
258	C₃	#10624896	1	1	yes	characteristic
259	C₃	#10624900	1	2	yes	—
260	C₃	#10624901	1	2	yes	—
261	C₃	#10624907	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃×#6894	C₃	semidirect
4. C₃²×C₉⋊C₉	C₃	semidirect
5. #10624566	C₃	semidirect
6. #10624566	C₃	semidirect
7. #10624565	C₃	semidirect
8. #10624556	C₃	semidirect
9. #10624556	C₃	semidirect
10. #10624556	C₃	semidirect
11. #10624556	C₃	semidirect
12. #10624556	C₃	semidirect
13. #10624556	C₃	semidirect
14. #10624556	C₃	semidirect
15. #10624556	C₃	semidirect
18. #10624548	C₃	semidirect
117. C₉⋊C₉	H(3,𝔽₃)	semidirect
131. C₃²⋊C₉	3₋¹⁺²	semidirect
225. C₃²	#6894	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6894	3
C₃×C₉⋊H(3,𝔽₃)	3
C₃×C₉×H(3,𝔽₃)	3
C₃²×C₉⋊C₉	3
#10624566	3
#10624565	3
#10624556	3
#10624555	3
#10624548	3