Subgroups — #14552344

Order 2187.

695 conjugacy classes of subgroups.

The subgroup lattice has 255 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	#10624577	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624576	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624576	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624574	C₃	1	3	yes	maximal, complemented
6	#10624573	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624573	C₃	1	1	yes	maximal, characteristic, complemented
8	#10624573	C₃	1	3	yes	maximal, complemented
9	#10624573	C₃	1	1	yes	maximal, characteristic, complemented
10	#10624570	C₃	1	1	yes	maximal, characteristic
11	#10624566	C₃	1	3	yes	maximal
12	#10624565	C₃	1	1	yes	maximal, characteristic
13	#10624565	C₃	1	1	yes	maximal, characteristic
14	#10624561	C₃	1	3	yes	maximal, complemented
15	#10624559	C₃	1	3	yes	maximal, complemented
16	#10624559	C₃	1	3	yes	maximal, complemented
17	#10624559	C₃	1	3	yes	maximal, complemented
18	#10624557	C₃	1	3	yes	maximal
19	#10624556	C₃	1	3	yes	maximal
20	#10624553	C₃	1	1	yes	maximal, characteristic, complemented
21	#10624551	C₃	1	1	yes	maximal, characteristic, complemented
22	#10624544	C₃	1	1	yes	maximal, characteristic, complemented
23	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
24	C₃³×C₉	C₃²	1	1	yes	characteristic
25	C₉×3₋¹⁺²	—	3	9	no	—
26	C₃×C₉⋊C₉	C₃²	1	3	yes	—
27	C₃×C₉⋊C₉	C₃²	1	3	yes	—
28	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
29	C₃×C₉⋊C₉	C₃²	1	3	yes	—
30	C₃×C₉⋊C₉	C₃²	1	3	yes	—
31	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
32	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
33	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
34	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
35	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
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	3	yes	—
40	C₃×C₉⋊C₉	C₃²	1	3	yes	—
41	C₃×C₉⋊C₉	C₃²	1	3	yes	—
42	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
43	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
44	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
45	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
46	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
47	C₃×C₉²	C₃²	1	3	yes	—
48	C₃×C₉²	C₃²	1	1	yes	characteristic
49	C₃×C₉²	C₃²	1	1	yes	characteristic
50	C₃×C₉²	C₃²	1	1	yes	characteristic
51	#6855	C₃²	1	3	yes	—
52	#6855	C₃²	1	1	yes	characteristic
53	#6855	C₃²	1	1	yes	characteristic
54	#6855	C₃²	1	3	yes	—
55	#6855	C₃²	1	1	yes	characteristic
56	#6855	C₃²	1	3	yes	—
57	#6855	C₃²	1	3	yes	—
58	#6855	C₃²	1	1	yes	characteristic
59	#6855	C₃²	1	1	yes	characteristic
60	#6855	C₃²	1	3	yes	—
61	#6855	C₃²	1	1	yes	characteristic
62	#6855	C₃²	1	3	yes	—
63	#6855	C₃²	1	3	yes	—
64	#6855	C₃²	1	1	yes	characteristic
65	#6855	C₃²	1	1	yes	characteristic
66	#6855	C₃²	1	1	yes	characteristic
67	#6855	C₃²	1	3	yes	—
68	#6855	C₃²	1	3	yes	—
69	#6855	C₃²	1	3	yes	—
70	#6855	C₃²	1	3	yes	—
71	#6855	C₃²	1	3	yes	—
72	#6855	C₃²	1	3	yes	—
73	#6855	C₃²	1	3	yes	—
74	#6855	C₃²	1	3	yes	—
75	#6855	C₃²	1	3	yes	—
76	#6855	C₃²	1	3	yes	—
77	#6855	C₃²	1	3	yes	—
78	#6855	C₃²	1	3	yes	—
79	#6855	C₃²	1	3	yes	—
80	#6855	C₃²	1	3	yes	—
81	#6855	C₃²	1	3	yes	—
82	C₃⁴	C₃³	1	1	yes	characteristic
83	C₃×3₋¹⁺²	—	3	3	no	—
84	C₃×3₋¹⁺²	—	3	3	no	—
85	C₃×3₋¹⁺²	—	3	3	no	—
86	C₃×3₋¹⁺²	—	3	3	no	—
87	C₃²×C₉	—	3	3	no	—
88	C₃²×C₉	—	9	3	no	—
89	C₃²×C₉	C₃³	1	1	yes	characteristic
90	C₃²×C₉	C₃³	1	1	yes	characteristic
91	C₃²×C₉	C₃³	1	3	yes	—
92	C₃²×C₉	C₃³	1	3	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	1	yes	characteristic
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	3	yes	—
103	C₃²×C₉	C₃³	1	1	yes	characteristic
104	C₃²×C₉	C₃³	1	1	yes	characteristic
105	C₃²×C₉	C₃³	1	1	yes	characteristic
106	C₃²×C₉	C₃³	1	1	yes	characteristic
107	C₃²×C₉	C₃³	1	1	yes	characteristic
108	C₃²×C₉	C₃³	1	1	yes	characteristic
109	C₃²×C₉	C₃³	1	3	yes	—
110	C₉⋊C₉	—	3	3	no	—
111	C₉⋊C₉	—	3	3	no	—
112	C₉⋊C₉	—	3	3	no	—
113	C₉⋊C₉	—	3	3	no	—
114	C₉⋊C₉	—	3	3	no	—
115	C₉⋊C₉	—	3	3	no	—
116	C₉⋊C₉	—	3	9	no	—
117	C₉⋊C₉	—	3	9	no	—
118	C₉⋊C₉	—	3	9	no	—
119	C₉⋊C₉	—	3	9	no	—
120	C₉⋊C₉	—	9	3	no	—
121	C₉⋊C₉	—	9	3	no	—
122	C₉⋊C₉	—	9	3	no	—
123	C₉⋊C₉	—	9	3	no	—
124	C₉⋊C₉	—	9	3	no	—
125	C₉⋊C₉	—	9	3	no	—
126	C₃²⋊C₉	—	3	9	no	—
127	C₃²⋊C₉	—	3	3	no	—
128	C₃²⋊C₉	—	3	3	no	—
129	C₃²⋊C₉	—	3	9	no	—
130	C₃²⋊C₉	—	9	3	no	—
131	C₉²	—	3	3	no	—
132	C₉²	—	3	3	no	—
133	C₉²	—	3	9	no	—
134	C₉²	3₋¹⁺²	1	9	yes	—
135	C₃³	—	3	1	no	—
136	C₃³	—	3	3	no	—
137	C₃³	—	3	1	no	—
138	C₃³	—	3	1	no	—
139	C₃³	—	3	3	no	—
140	C₃³	—	3	3	no	—
141	C₃³	#737	1	3	yes	—
142	C₃³	C₃⁴	1	1	yes	characteristic
143	3₋¹⁺²	—	3	27	no	—
144	C₃×C₉	—	3	1	no	—
145	C₃×C₉	—	3	1	no	—
146	C₃×C₉	—	3	3	no	—
147	C₃×C₉	—	3	3	no	—
148	C₃×C₉	—	3	1	no	—
149	C₃×C₉	—	3	1	no	—
150	C₃×C₉	—	3	1	no	—
151	C₃×C₉	—	3	1	no	—
152	C₃×C₉	—	3	1	no	—
153	C₃×C₉	—	3	1	no	—
154	C₃×C₉	—	3	1	no	—
155	C₃×C₉	—	3	3	no	—
156	C₃×C₉	—	3	3	no	—
157	C₃×C₉	—	3	3	no	—
158	C₃×C₉	—	3	3	no	—
159	C₃×C₉	—	3	1	no	—
160	C₃×C₉	—	3	1	no	—
161	C₃×C₉	—	3	1	no	—
162	C₃×C₉	—	3	1	no	—
163	C₃×C₉	—	3	1	no	—
164	C₃×C₉	—	3	1	no	—
165	C₃×C₉	—	3	3	no	—
166	C₃×C₉	—	3	3	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	3	no	—
172	C₃×C₉	—	3	3	no	—
173	C₃×C₉	—	3	3	no	—
174	C₃×C₉	—	3	3	no	—
175	C₃×C₉	—	3	3	no	—
176	C₃×C₉	—	3	3	no	—
177	C₃×C₉	—	3	3	no	—
178	C₃×C₉	—	3	3	no	—
179	C₃×C₉	—	3	3	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	3	no	—
185	C₃×C₉	—	3	3	no	—
186	C₃×C₉	—	3	3	no	—
187	C₃×C₉	—	3	3	no	—
188	C₃×C₉	—	3	3	no	—
189	C₃×C₉	—	3	3	no	—
190	C₃×C₉	—	3	3	no	—
191	C₃×C₉	—	3	3	no	—
192	C₃×C₉	—	3	3	no	—
193	C₃×C₉	—	3	3	no	—
194	C₃×C₉	—	3	3	no	—
195	C₃×C₉	—	3	3	no	—
196	C₃×C₉	—	3	3	no	—
197	C₃×C₉	—	3	3	no	—
198	C₃×C₉	—	3	3	no	—
199	C₃×C₉	—	3	3	no	—
200	C₃×C₉	—	3	3	no	—
201	C₃×C₉	—	3	3	no	—
202	C₃×C₉	—	3	3	no	—
203	C₃×C₉	—	9	9	no	—
204	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
205	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
206	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
207	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
208	C₃×C₉	#737	1	9	yes	—
209	C₃×C₉	#737	1	3	yes	—
210	C₃×C₉	#737	1	3	yes	—
211	C₃²	—	3	3	no	—
212	C₃²	—	3	9	no	—
213	C₃²	—	9	1	no	—
214	C₃²	—	9	1	no	—
215	C₃²	—	9	1	no	—
216	C₃²	—	9	3	no	—
217	C₃²	—	9	3	no	—
218	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
219	C₃²	C₃×#737	1	1	yes	characteristic
220	C₃²	C₃×#737	1	1	yes	characteristic
221	C₃²	C₃×#737	1	1	yes	characteristic
222	C₃²	C₃×#737	1	3	yes	—
223	C₃²	3₋¹⁺⁴	1	3	yes	—
224	C₃²	3₋¹⁺⁴	1	3	yes	—
225	C₉	—	3	3	no	—
226	C₉	—	3	3	no	—
227	C₉	—	3	3	no	—
228	C₉	—	3	3	no	—
229	C₉	—	3	3	no	—
230	C₉	—	3	3	no	—
231	C₉	—	3	9	no	—
232	C₉	—	9	1	no	—
233	C₉	—	9	1	no	—
234	C₉	—	9	1	no	—
235	C₉	—	9	1	no	—
236	C₉	—	9	1	no	—
237	C₉	—	9	1	no	—
238	C₉	—	9	3	no	—
239	C₉	—	9	3	no	—
240	C₉	—	9	3	no	—
241	C₉	—	9	3	no	—
242	C₉	—	9	3	no	—
243	C₉	—	9	3	no	—
244	C₉	—	9	3	no	—
245	C₉	—	9	3	no	—
246	C₃	—	9	3	no	—
247	C₃	#10624872	1	1	yes	characteristic
248	C₃	#10624879	1	1	yes	characteristic
249	C₃	#10624884	1	1	yes	characteristic
250	C₃	#10624887	1	1	yes	characteristic
251	C₃	#10624904	1	3	yes	—
252	C₃	#10624907	1	3	yes	—
253	C₃	#10624908	1	3	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃³	C₃⁴
prime radical	C₃⁴	C₃³
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
2. #10624577	C₃	semidirect
3. #10624576	C₃	semidirect
4. #10624576	C₃	semidirect
5. #10624574	C₃	semidirect
6. #10624573	C₃	semidirect
7. #10624573	C₃	semidirect
8. #10624573	C₃	semidirect
9. #10624573	C₃	semidirect
14. #10624561	C₃	semidirect
15. #10624559	C₃	semidirect
16. #10624559	C₃	semidirect
17. #10624559	C₃	semidirect
20. #10624553	C₃	semidirect
21. #10624551	C₃	semidirect
22. #10624544	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×3₋¹⁺²	3
#10624577	3
#10624576	3
#10624574	3
#10624573	3
#10624570	3
#10624566	3
#10624565	3
#10624561	3
#10624559	3
#10624557	3
#10624556	3
#10624553	3
#10624551	3
#10624544	3