Subgroups — C₉×#5709 #13153228

Order 1728.

774 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#126	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#898	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#895	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#894	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#893	C₂	1	1	yes	maximal, characteristic, complemented
6	D₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#889	C₂	1	1	yes	maximal, characteristic, complemented
8	C₉×#886	C₂	1	1	yes	maximal, characteristic, complemented
9	C₉×#884	C₂	1	1	yes	maximal, characteristic, complemented
10	C₉×#884	C₂	1	1	yes	maximal, characteristic, complemented
11	C₉×#884	C₂	1	1	yes	maximal, characteristic, complemented
12	C₉×#67⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
13	C₉×#67⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₉×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
15	C₉×#872	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃×#5709	C₃	1	1	yes	maximal, characteristic
17	C₉×#553	—	3	1	no	maximal
18	C₂×C₉×#67	C₂²	1	1	yes	characteristic
19	C₂×C₉×#67	C₂²	1	1	yes	characteristic
20	C₂×C₉×#67	C₂²	1	1	yes	characteristic
21	C₂²×C₉×Dic₃	C₂²	1	1	yes	characteristic
22	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
23	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
24	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
30	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
31	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
33	C₃×C₉×C₂²⋊C₄	C₂²	1	1	yes	characteristic
34	C₃×C₉×C₂²⋊C₄	C₂²	1	1	yes	characteristic
35	C₃×C₉×C₂²⋊C₄	C₂²	1	1	yes	characteristic
36	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
37	C₉×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
38	C₉×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
39	C₉×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
40	C₉×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
41	C₉×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
42	C₉×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
43	C₉×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
44	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
45	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
46	C₉×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
47	C₉×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
48	C₉×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
49	C₉×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
50	C₉×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
51	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
52	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
53	C₃²×#126	C₂×C₃	1	1	yes	characteristic
54	C₃×#898	C₂×C₃	1	1	yes	characteristic
55	C₃×#895	C₂×C₃	1	1	yes	characteristic
56	C₃×#894	C₂×C₃	1	1	yes	characteristic
57	C₃×#893	C₂×C₃	1	1	yes	characteristic
58	C₃×D₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
59	C₃×#889	C₂×C₃	1	1	yes	characteristic
60	C₃×#886	C₂×C₃	1	1	yes	characteristic
61	C₃×#884	C₂×C₃	1	1	yes	characteristic
62	C₃×#884	C₂×C₃	1	1	yes	characteristic
63	C₃×#884	C₂×C₃	1	1	yes	characteristic
64	C₃×#67⋊C₄	C₂×C₃	1	1	yes	characteristic
65	C₃×#67⋊C₄	C₂×C₃	1	1	yes	characteristic
66	C₃×C₄×D₁₂	C₂×C₃	1	1	yes	characteristic
67	C₃×#872	C₂×C₃	1	1	yes	characteristic
68	C₉×#126	—	2	1	no	—
69	C₉×#126	—	3	1	no	—
70	C₉×#126	D₃	1	1	yes	characteristic, complemented
71	C₉×#125	—	3	1	no	—
72	C₉×#123	—	3	1	no	—
73	C₉×#123	—	3	1	no	—
74	C₉×#121	—	3	1	no	—
75	C₉×#121	—	3	1	no	—
76	C₉×#121	—	3	1	no	—
77	C₉×#121	—	3	1	no	—
78	C₄×D₄×C₉	—	3	1	no	—
79	C₄×D₄×C₉	—	3	1	no	—
80	C₄×D₄×C₉	—	3	1	no	—
81	C₄×D₄×C₉	—	3	1	no	—
82	C₉×#117	—	3	1	no	—
83	C₂×C₉×C₄⋊C₄	—	3	1	no	—
84	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
85	C₂²×D₃×C₉	C₂³	1	1	yes	characteristic
86	C₂²×D₃×C₉	C₂³	1	1	yes	characteristic
87	C₂²×D₃×C₉	C₂³	1	1	yes	characteristic
88	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
89	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
90	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
91	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
92	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
93	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
94	C₉×#67	—	2	2	no	—
95	C₉×#67	—	2	2	no	—
96	C₉×#67	—	4	1	no	—
97	C₂×C₉×Dic₃	—	4	1	no	—
98	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
99	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
100	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
101	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
102	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
103	C₉×D₁₂	—	2	2	no	—
104	C₉×D₁₂	—	2	2	no	—
105	C₉×D₁₂	—	4	1	no	—
106	C₄×D₃×C₉	—	2	2	no	—
107	C₄×D₃×C₉	—	2	2	no	—
108	C₄×D₃×C₉	—	4	1	no	—
109	C₄×D₃×C₉	—	4	1	no	—
110	C₄×D₃×C₉	—	4	1	no	—
111	C₃×#553	—	3	1	no	—
112	#5709	C₉	1	1	yes	characteristic, direct product factor
113	C₂×C₃×#67	C₂²×C₃	1	1	yes	characteristic
114	C₂×C₃×#67	C₂²×C₃	1	1	yes	characteristic
115	C₂×C₃×#67	C₂²×C₃	1	1	yes	characteristic
116	C₂²×C₃×Dic₃	C₂²×C₃	1	1	yes	characteristic
117	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
118	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
119	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
120	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
121	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
122	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
123	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
124	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
125	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
126	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
127	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
128	C₃²×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
129	C₃²×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
130	C₃²×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
131	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
132	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
133	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
134	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
135	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
136	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
137	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
138	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
139	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
140	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
141	C₃×Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
142	C₃×Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
143	C₃×Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
144	C₃×Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
145	C₃×Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
146	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
147	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
148	C₂×D₄×C₉	—	3	1	no	—
149	C₂×D₄×C₉	—	3	1	no	—
150	C₂×D₄×C₉	—	3	1	no	—
151	C₂×D₄×C₉	—	3	1	no	—
152	C₂×D₄×C₉	—	3	1	no	—
153	C₂×D₄×C₉	—	3	1	no	—
154	C₂²×C₄×C₉	—	3	1	no	—
155	C₂²×C₄×C₉	—	3	1	no	—
156	C₂²×C₄×C₉	—	3	1	no	—
157	C₂²×C₄×C₉	—	3	1	no	—
158	C₂²×C₄×C₉	—	3	1	no	—
159	C₂²×C₄×C₉	—	3	1	no	—
160	C₉×C₄⋊C₄	—	2	1	no	—
161	C₉×C₄⋊C₄	—	2	1	no	—
162	C₉×C₄⋊C₄	—	2	1	no	—
163	C₉×C₄⋊C₄	—	3	1	no	—
164	C₉×C₄⋊C₄	—	3	1	no	—
165	C₉×C₄⋊C₄	—	3	1	no	—
166	C₉×C₄⋊C₄	—	3	1	no	—
167	C₉×C₄⋊C₄	—	3	1	no	—
168	C₉×C₄⋊C₄	—	3	1	no	—
169	C₉×C₄⋊C₄	—	3	1	no	—
170	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
171	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
172	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
173	C₉×C₂²⋊C₄	—	2	1	no	—
174	C₉×C₂²⋊C₄	—	2	1	no	—
175	C₉×C₂²⋊C₄	—	2	1	no	—
176	C₉×C₂²⋊C₄	—	3	1	no	—
177	C₉×C₂²⋊C₄	—	3	1	no	—
178	C₉×C₂²⋊C₄	—	3	1	no	—
179	C₉×C₂²⋊C₄	—	3	1	no	—
180	C₉×C₂²⋊C₄	—	3	1	no	—
181	C₉×C₂²⋊C₄	—	3	1	no	—
182	C₉×C₂²⋊C₄	—	3	1	no	—
183	C₉×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
184	C₉×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
185	C₉×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
186	C₄²×C₉	—	2	1	no	—
187	C₄²×C₉	—	3	1	no	—
188	C₄²×C₉	—	3	1	no	—
189	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
190	C₂²×C₃×C₉	—	2	1	no	—
191	C₂²×C₃×C₉	—	2	1	no	—
192	C₂²×C₃×C₉	—	2	1	no	—
193	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
194	C₂×D₃×C₉	—	2	1	no	—
195	C₂×D₃×C₉	—	2	1	no	—
196	C₂×D₃×C₉	—	2	1	no	—
197	C₂×D₃×C₉	—	2	1	no	—
198	C₂×D₃×C₉	—	2	1	no	—
199	C₂×D₃×C₉	—	2	1	no	—
200	C₂×D₃×C₉	—	2	1	no	—
201	C₂×D₃×C₉	—	2	1	no	—
202	C₂×D₃×C₉	#41	1	2	yes	—
203	C₃×C₄×C₉	—	2	1	no	—
204	C₃×C₄×C₉	—	2	1	no	—
205	C₃×C₄×C₉	—	2	1	no	—
206	C₃×C₄×C₉	—	2	1	no	—
207	C₃×C₄×C₉	—	2	1	no	—
208	C₃×C₄×C₉	—	2	1	no	—
209	C₉×Dic₃	—	2	1	no	—
210	C₉×Dic₃	—	2	1	no	—
211	C₉×Dic₃	—	2	1	no	—
212	C₉×Dic₃	—	2	1	no	—
213	C₉×Dic₃	#41	1	2	yes	—
214	C₃×#126	—	2	1	no	—
215	C₃×#126	—	3	1	no	—
216	C₃×#126	C₂×C₉	1	1	yes	characteristic, complemented
217	C₃×#126	C₃×D₃	1	1	yes	characteristic
218	C₃×#125	—	3	1	no	—
219	C₃×#123	—	3	1	no	—
220	C₃×#123	—	3	1	no	—
221	C₃×#121	—	3	1	no	—
222	C₃×#121	—	3	1	no	—
223	C₃×#121	—	3	1	no	—
224	C₃×#121	—	3	1	no	—
225	C₃×C₄×D₄	—	3	1	no	—
226	C₃×C₄×D₄	—	3	1	no	—
227	C₃×C₄×D₄	—	3	1	no	—
228	C₃×C₄×D₄	—	3	1	no	—
229	C₃×#117	—	3	1	no	—
230	C₂×C₃×C₄⋊C₄	—	3	1	no	—
231	#898	C₂×C₉	1	1	yes	characteristic, complemented
232	#895	C₂×C₉	1	1	yes	characteristic, complemented
233	#894	C₂×C₉	1	1	yes	characteristic, complemented
234	#893	C₂×C₉	1	1	yes	characteristic, complemented
235	D₃×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
236	#889	C₂×C₉	1	1	yes	characteristic, complemented
237	#886	C₂×C₉	1	1	yes	characteristic, complemented
238	#884	C₂×C₉	1	1	yes	characteristic, complemented
239	#884	C₂×C₉	1	1	yes	characteristic, complemented
240	#884	C₂×C₉	1	1	yes	characteristic, complemented
241	#67⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
242	#67⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
243	C₄×D₁₂	C₂×C₉	1	1	yes	characteristic, complemented
244	#872	C₂×C₉	1	1	yes	characteristic, complemented
245	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
246	C₂²×C₃×D₃	C₂³×C₃	1	1	yes	characteristic
247	C₂²×C₃×D₃	C₂³×C₃	1	1	yes	characteristic
248	C₂²×C₃×D₃	C₂³×C₃	1	1	yes	characteristic
249	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
250	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
251	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
252	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
253	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
254	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
255	C₃×#67	—	2	2	no	—
256	C₃×#67	—	2	2	no	—
257	C₃×#67	—	4	1	no	—
258	C₂×C₃×Dic₃	—	4	1	no	—
259	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
260	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
261	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
262	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
263	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
264	C₃×D₁₂	—	2	2	no	—
265	C₃×D₁₂	—	2	2	no	—
266	C₃×D₁₂	—	4	1	no	—
267	C₃×C₄×D₃	—	2	2	no	—
268	C₃×C₄×D₃	—	2	2	no	—
269	C₃×C₄×D₃	—	4	1	no	—
270	C₃×C₄×D₃	—	4	1	no	—
271	C₃×C₄×D₃	—	4	1	no	—
272	C₂³×C₉	—	2	1	no	—
273	C₂³×C₉	—	3	1	no	—
274	C₂³×C₉	—	3	1	no	—
275	C₂³×C₉	—	3	1	no	—
276	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
277	D₄×C₉	—	6	2	no	—
278	D₄×C₉	—	6	2	no	—
279	D₄×C₉	—	6	2	no	—
280	D₄×C₉	—	6	2	no	—
281	D₄×C₉	—	12	1	no	—
282	D₄×C₉	—	12	1	no	—
283	C₂×C₄×C₉	—	2	1	no	—
284	C₂×C₄×C₉	—	2	1	no	—
285	C₂×C₄×C₉	—	2	1	no	—
286	C₂×C₄×C₉	—	2	1	no	—
287	C₂×C₄×C₉	—	2	1	no	—
288	C₂×C₄×C₉	—	2	1	no	—
289	C₂×C₄×C₉	—	3	1	no	—
290	C₂×C₄×C₉	—	3	1	no	—
291	C₂×C₄×C₉	—	3	1	no	—
292	C₂×C₄×C₉	—	3	1	no	—
293	C₂×C₄×C₉	—	3	1	no	—
294	C₂×C₄×C₉	—	6	2	no	—
295	C₂×C₄×C₉	—	6	2	no	—
296	C₂×C₄×C₉	—	12	1	no	—
297	C₂×C₄×C₉	—	12	1	no	—
298	C₂×C₄×C₉	—	12	1	no	—
299	C₂×C₄×C₉	—	12	1	no	—
300	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
301	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
302	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
303	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
304	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
305	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
306	#553	—	3	1	no	Sylow
307	C₂×C₃×C₉	—	4	1	no	—
308	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
309	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
310	C₂×C₃×C₉	H(5,𝔽₂)	1	1	yes	characteristic
311	D₃×C₉	—	2	2	no	—
312	D₃×C₉	—	4	1	no	—
313	D₃×C₉	—	4	1	no	—
314	C₂×C₃×D₄	—	3	1	no	—
315	C₂×C₃×D₄	—	3	1	no	—
316	C₂×C₃×D₄	—	3	1	no	—
317	C₂×C₃×D₄	—	3	1	no	—
318	C₂×C₃×D₄	—	3	1	no	—
319	C₂×C₃×D₄	—	3	1	no	—
320	C₂²×C₃×C₄	—	3	1	no	—
321	C₂²×C₃×C₄	—	3	1	no	—
322	C₂²×C₃×C₄	—	3	1	no	—
323	C₂²×C₃×C₄	—	3	1	no	—
324	C₂²×C₃×C₄	—	3	1	no	—
325	C₂²×C₃×C₄	—	3	1	no	—
326	C₂×#67	C₂²×C₉	1	1	yes	characteristic
327	C₂×#67	C₂²×C₉	1	1	yes	characteristic
328	C₂×#67	C₂²×C₉	1	1	yes	characteristic
329	C₂²×Dic₃	C₂²×C₉	1	1	yes	characteristic
330	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
331	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
332	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
333	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
334	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
335	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
336	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
337	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
338	C₃×C₄⋊C₄	—	2	1	no	—
339	C₃×C₄⋊C₄	—	2	1	no	—
340	C₃×C₄⋊C₄	—	2	1	no	—
341	C₃×C₄⋊C₄	—	3	1	no	—
342	C₃×C₄⋊C₄	—	3	1	no	—
343	C₃×C₄⋊C₄	—	3	1	no	—
344	C₃×C₄⋊C₄	—	3	1	no	—
345	C₃×C₄⋊C₄	—	3	1	no	—
346	C₃×C₄⋊C₄	—	3	1	no	—
347	C₃×C₄⋊C₄	—	3	1	no	—
348	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
349	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
350	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
351	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
352	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
353	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
354	C₃×C₂²⋊C₄	—	2	1	no	—
355	C₃×C₂²⋊C₄	—	2	1	no	—
356	C₃×C₂²⋊C₄	—	2	1	no	—
357	C₃×C₂²⋊C₄	—	3	1	no	—
358	C₃×C₂²⋊C₄	—	3	1	no	—
359	C₃×C₂²⋊C₄	—	3	1	no	—
360	C₃×C₂²⋊C₄	—	3	1	no	—
361	C₃×C₂²⋊C₄	—	3	1	no	—
362	C₃×C₂²⋊C₄	—	3	1	no	—
363	C₃×C₂²⋊C₄	—	3	1	no	—
364	C₃×C₂²⋊C₄	C₂²×C₉	1	1	yes	characteristic
365	C₃×C₂²⋊C₄	C₂²×C₉	1	1	yes	characteristic
366	C₃×C₂²⋊C₄	C₂²×C₉	1	1	yes	characteristic
367	C₃×C₂²⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
368	C₃×C₂²⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
369	C₃×C₂²⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
370	C₃×C₄²	—	2	1	no	—
371	C₃×C₄²	—	3	1	no	—
372	C₃×C₄²	—	3	1	no	—
373	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
374	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
375	(C₂×D₃)⋊C₄	C₂²×C₉	1	1	yes	characteristic
376	(C₂×D₃)⋊C₄	C₂²×C₉	1	1	yes	characteristic
377	(C₂×D₃)⋊C₄	C₂²×C₉	1	1	yes	characteristic
378	(C₂×D₃)⋊C₄	C₂²×C₉	1	1	yes	characteristic
379	(C₂×D₃)⋊C₄	C₂²×C₉	1	1	yes	characteristic
380	(C₂×D₃)⋊C₄	C₂²×C₉	1	1	yes	characteristic
381	(C₂×D₃)⋊C₄	C₂²×C₉	1	1	yes	characteristic
382	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
383	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
384	Dic₃⋊C₄	C₂²×C₉	1	1	yes	characteristic
385	Dic₃⋊C₄	C₂²×C₉	1	1	yes	characteristic
386	Dic₃⋊C₄	C₂²×C₉	1	1	yes	characteristic
387	Dic₃⋊C₄	C₂²×C₉	1	1	yes	characteristic
388	Dic₃⋊C₄	C₂²×C₉	1	1	yes	characteristic
389	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
390	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
391	C₂²×C₃²	—	2	1	no	—
392	C₂²×C₃²	—	2	1	no	—
393	C₂²×C₃²	—	2	1	no	—
394	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
395	C₂×C₃×D₃	—	2	1	no	—
396	C₂×C₃×D₃	—	2	1	no	—
397	C₂×C₃×D₃	—	2	1	no	—
398	C₂×C₃×D₃	—	2	1	no	—
399	C₂×C₃×D₃	—	2	1	no	—
400	C₂×C₃×D₃	—	2	1	no	—
401	C₂×C₃×D₃	—	2	1	no	—
402	C₂×C₃×D₃	—	2	1	no	—
403	C₂×C₃×D₃	C₃×#41	1	2	yes	—
404	C₃²×C₄	—	2	1	no	—
405	C₃²×C₄	—	2	1	no	—
406	C₃²×C₄	—	2	1	no	—
407	C₃²×C₄	—	2	1	no	—
408	C₃²×C₄	—	2	1	no	—
409	C₃²×C₄	—	2	1	no	—
410	C₃×Dic₃	—	2	1	no	—
411	C₃×Dic₃	—	2	1	no	—
412	C₃×Dic₃	—	2	1	no	—
413	C₃×Dic₃	—	2	1	no	—
414	C₃×Dic₃	C₃×#41	1	2	yes	—
415	C₂²×C₉	—	2	1	no	—
416	C₂²×C₉	—	2	1	no	—
417	C₂²×C₉	—	2	1	no	—
418	C₂²×C₉	—	2	1	no	—
419	C₂²×C₉	—	3	2	no	—
420	C₂²×C₉	—	4	1	no	—
421	C₂²×C₉	—	4	1	no	—
422	C₂²×C₉	—	4	1	no	—
423	C₂²×C₉	—	6	1	no	—
424	C₂²×C₉	—	6	1	no	—
425	C₂²×C₉	—	6	1	no	—
426	C₂²×C₉	—	6	1	no	—
427	C₂²×C₉	—	6	1	no	—
428	C₂²×C₉	—	6	1	no	—
429	C₂²×C₉	—	6	1	no	—
430	C₂²×C₉	—	6	1	no	—
431	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
432	C₄×C₉	—	2	1	no	—
433	C₄×C₉	—	2	1	no	—
434	C₄×C₉	—	2	1	no	—
435	C₄×C₉	—	2	1	no	—
436	C₄×C₉	—	2	1	no	—
437	C₄×C₉	—	2	1	no	—
438	C₄×C₉	—	3	2	no	—
439	C₄×C₉	—	4	1	no	—
440	C₄×C₉	—	4	1	no	—
441	C₄×C₉	—	4	1	no	—
442	C₄×C₉	—	4	1	no	—
443	C₄×C₉	—	4	1	no	—
444	C₄×C₉	—	4	1	no	—
445	C₄×C₉	—	6	1	no	—
446	C₄×C₉	—	6	1	no	—
447	C₄×C₉	—	6	1	no	—
448	C₄×C₉	—	6	1	no	—
449	#126	—	3	1	no	—
450	#126	D₃×C₉	1	1	yes	characteristic, complemented
451	#125	—	3	1	no	—
452	#123	—	3	1	no	—
453	#123	—	3	1	no	—
454	#121	—	3	1	no	—
455	#121	—	3	1	no	—
456	#121	—	3	1	no	—
457	#121	—	3	1	no	—
458	C₄×D₄	—	3	1	no	—
459	C₄×D₄	—	3	1	no	—
460	C₄×D₄	—	3	1	no	—
461	C₄×D₄	—	3	1	no	—
462	#117	—	3	1	no	—
463	C₂×C₄⋊C₄	—	3	1	no	—
464	C₃×C₉	#553	1	1	yes	characteristic, complemented, Sylow
465	C₂³×C₃	—	2	1	no	—
466	C₂³×C₃	—	3	1	no	—
467	C₂³×C₃	—	3	1	no	—
468	C₂³×C₃	—	3	1	no	—
469	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
470	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
471	C₂²×D₃	C₂³×C₉	1	1	yes	characteristic
472	C₂²×D₃	C₂³×C₉	1	1	yes	characteristic
473	C₂²×D₃	C₂³×C₉	1	1	yes	characteristic
474	C₃×D₄	—	6	2	no	—
475	C₃×D₄	—	6	2	no	—
476	C₃×D₄	—	6	2	no	—
477	C₃×D₄	—	6	2	no	—
478	C₃×D₄	—	12	1	no	—
479	C₃×D₄	—	12	1	no	—
480	C₂×C₃×C₄	—	2	1	no	—
481	C₂×C₃×C₄	—	2	1	no	—
482	C₂×C₃×C₄	—	2	1	no	—
483	C₂×C₃×C₄	—	2	1	no	—
484	C₂×C₃×C₄	—	2	1	no	—
485	C₂×C₃×C₄	—	2	1	no	—
486	C₂×C₃×C₄	—	3	1	no	—
487	C₂×C₃×C₄	—	3	1	no	—
488	C₂×C₃×C₄	—	3	1	no	—
489	C₂×C₃×C₄	—	3	1	no	—
490	C₂×C₃×C₄	—	3	1	no	—
491	C₂×C₃×C₄	—	6	2	no	—
492	C₂×C₃×C₄	—	6	2	no	—
493	C₂×C₃×C₄	—	12	1	no	—
494	C₂×C₃×C₄	—	12	1	no	—
495	C₂×C₃×C₄	—	12	1	no	—
496	C₂×C₃×C₄	—	12	1	no	—
497	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
498	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
499	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
500	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
501	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
502	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
503	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
504	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
505	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
506	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
507	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
508	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
509	#67	—	2	2	no	—
510	#67	—	2	2	no	—
511	#67	—	4	1	no	—
512	C₂×Dic₃	—	4	1	no	—
513	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
514	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
515	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
516	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
517	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
518	D₁₂	—	2	2	no	—
519	D₁₂	—	2	2	no	—
520	D₁₂	—	4	1	no	—
521	C₄×D₃	—	2	2	no	—
522	C₄×D₃	—	2	2	no	—
523	C₄×D₃	—	4	1	no	—
524	C₄×D₃	—	4	1	no	—
525	C₄×D₃	—	4	1	no	—
526	C₂×C₃²	—	4	1	no	—
527	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
528	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
529	C₂×C₃²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
530	C₃×D₃	—	2	2	no	—
531	C₃×D₃	—	4	1	no	—
532	C₃×D₃	—	4	1	no	—
533	C₂×C₉	—	2	1	no	—
534	C₂×C₉	—	2	1	no	—
535	C₂×C₉	—	2	1	no	—
536	C₂×C₉	—	4	1	no	—
537	C₂×C₉	—	6	2	no	—
538	C₂×C₉	—	8	1	no	—
539	C₂×C₉	—	12	1	no	—
540	C₂×C₉	—	12	1	no	—
541	C₂×C₉	D₃×#41	1	1	yes	characteristic
542	C₂×C₉	D₃×#41	1	1	yes	characteristic
543	C₂×C₉	#1009	1	1	yes	characteristic
544	C₂×D₄	—	3	1	no	—
545	C₂×D₄	—	3	1	no	—
546	C₂×D₄	—	3	1	no	—
547	C₂×D₄	—	3	1	no	—
548	C₂×D₄	—	3	1	no	—
549	C₂×D₄	—	3	1	no	—
550	C₂²×C₄	—	3	1	no	—
551	C₂²×C₄	—	3	1	no	—
552	C₂²×C₄	—	3	1	no	—
553	C₂²×C₄	—	3	1	no	—
554	C₂²×C₄	—	3	1	no	—
555	C₂²×C₄	—	3	1	no	—
556	C₄⋊C₄	—	3	1	no	—
557	C₄⋊C₄	—	3	1	no	—
558	C₄⋊C₄	—	3	1	no	—
559	C₄⋊C₄	—	3	1	no	—
560	C₄⋊C₄	—	3	1	no	—
561	C₄⋊C₄	—	3	1	no	—
562	C₄⋊C₄	—	3	1	no	—
563	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
564	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
565	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
566	C₂²⋊C₄	—	3	1	no	—
567	C₂²⋊C₄	—	3	1	no	—
568	C₂²⋊C₄	—	3	1	no	—
569	C₂²⋊C₄	—	3	1	no	—
570	C₂²⋊C₄	—	3	1	no	—
571	C₂²⋊C₄	—	3	1	no	—
572	C₂²⋊C₄	—	3	1	no	—
573	C₂²⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
574	C₂²⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
575	C₂²⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
576	C₄²	—	3	1	no	—
577	C₄²	—	3	1	no	—
578	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
579	C₂²×C₃	—	2	1	no	—
580	C₂²×C₃	—	2	1	no	—
581	C₂²×C₃	—	2	1	no	—
582	C₂²×C₃	—	2	1	no	—
583	C₂²×C₃	—	2	1	no	—
584	C₂²×C₃	—	2	1	no	—
585	C₂²×C₃	—	2	1	no	—
586	C₂²×C₃	—	3	2	no	—
587	C₂²×C₃	—	4	1	no	—
588	C₂²×C₃	—	4	1	no	—
589	C₂²×C₃	—	4	1	no	—
590	C₂²×C₃	—	6	1	no	—
591	C₂²×C₃	—	6	1	no	—
592	C₂²×C₃	—	6	1	no	—
593	C₂²×C₃	—	6	1	no	—
594	C₂²×C₃	—	6	1	no	—
595	C₂²×C₃	—	6	1	no	—
596	C₂²×C₃	—	6	1	no	—
597	C₂²×C₃	—	6	1	no	—
598	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
599	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
600	C₂×D₃	—	2	1	no	—
601	C₂×D₃	—	2	1	no	—
602	C₂×D₃	—	2	1	no	—
603	C₂×D₃	—	2	1	no	—
604	C₂×D₃	—	2	1	no	—
605	C₂×D₃	—	2	1	no	—
606	C₂×D₃	—	2	1	no	—
607	C₂×D₃	—	2	1	no	—
608	C₂×D₃	C₉×#41	1	2	yes	—
609	C₃×C₄	—	2	1	no	—
610	C₃×C₄	—	2	1	no	—
611	C₃×C₄	—	2	1	no	—
612	C₃×C₄	—	2	1	no	—
613	C₃×C₄	—	2	1	no	—
614	C₃×C₄	—	2	1	no	—
615	C₃×C₄	—	2	1	no	—
616	C₃×C₄	—	2	1	no	—
617	C₃×C₄	—	2	1	no	—
618	C₃×C₄	—	2	1	no	—
619	C₃×C₄	—	2	1	no	—
620	C₃×C₄	—	2	1	no	—
621	C₃×C₄	—	3	2	no	—
622	C₃×C₄	—	4	1	no	—
623	C₃×C₄	—	4	1	no	—
624	C₃×C₄	—	4	1	no	—
625	C₃×C₄	—	4	1	no	—
626	C₃×C₄	—	4	1	no	—
627	C₃×C₄	—	4	1	no	—
628	C₃×C₄	—	6	1	no	—
629	C₃×C₄	—	6	1	no	—
630	C₃×C₄	—	6	1	no	—
631	C₃×C₄	—	6	1	no	—
632	Dic₃	—	2	1	no	—
633	Dic₃	—	2	1	no	—
634	Dic₃	—	2	1	no	—
635	Dic₃	—	2	1	no	—
636	Dic₃	C₉×#41	1	2	yes	—
637	C₃²	C₃×#553	1	1	yes	characteristic
638	C₉	—	2	1	no	—
639	C₉	#5709	1	1	yes	characteristic, direct product factor
640	C₂³	—	3	1	no	—
641	C₂³	—	3	1	no	—
642	C₂³	—	3	1	no	—
643	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
644	D₄	—	6	2	no	—
645	D₄	—	6	2	no	—
646	D₄	—	6	2	no	—
647	D₄	—	6	2	no	—
648	D₄	—	12	1	no	—
649	D₄	—	12	1	no	—
650	C₂×C₄	—	3	1	no	—
651	C₂×C₄	—	3	1	no	—
652	C₂×C₄	—	3	1	no	—
653	C₂×C₄	—	3	1	no	—
654	C₂×C₄	—	3	1	no	—
655	C₂×C₄	—	6	2	no	—
656	C₂×C₄	—	6	2	no	—
657	C₂×C₄	—	12	1	no	—
658	C₂×C₄	—	12	1	no	—
659	C₂×C₄	—	12	1	no	—
660	C₂×C₄	—	12	1	no	—
661	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
662	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
663	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
664	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
665	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
666	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
667	C₂×C₃	—	2	1	no	—
668	C₂×C₃	—	2	1	no	—
669	C₂×C₃	—	2	1	no	—
670	C₂×C₃	—	4	1	no	—
671	C₂×C₃	—	4	1	no	—
672	C₂×C₃	—	6	2	no	—
673	C₂×C₃	—	8	1	no	—
674	C₂×C₃	—	12	1	no	—
675	C₂×C₃	—	12	1	no	—
676	C₂×C₃	C₂×C₉×#41	1	1	yes	characteristic
677	C₂×C₃	C₂×C₉×#41	1	1	yes	characteristic
678	C₂×C₃	C₉×H(5,𝔽₂)	1	1	yes	characteristic
679	C₂×C₃	C₃×D₃×#41	1	1	yes	characteristic
680	C₂×C₃	C₃×D₃×#41	1	1	yes	characteristic
681	C₂×C₃	C₃×#1009	1	1	yes	characteristic
682	D₃	—	2	2	no	—
683	D₃	—	4	1	no	—
684	D₃	—	4	1	no	—
685	C₂²	—	2	1	no	—
686	C₂²	—	2	1	no	—
687	C₂²	—	2	1	no	—
688	C₂²	—	3	2	no	—
689	C₂²	—	6	1	no	—
690	C₂²	—	6	1	no	—
691	C₂²	—	6	1	no	—
692	C₂²	—	6	1	no	—
693	C₂²	—	6	1	no	—
694	C₂²	—	6	1	no	—
695	C₂²	—	6	1	no	—
696	C₂²	—	6	1	no	—
697	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
698	C₄	—	2	1	no	—
699	C₄	—	2	1	no	—
700	C₄	—	2	1	no	—
701	C₄	—	2	1	no	—
702	C₄	—	2	1	no	—
703	C₄	—	2	1	no	—
704	C₄	—	3	2	no	—
705	C₄	—	6	1	no	—
706	C₄	—	6	1	no	—
707	C₄	—	6	1	no	—
708	C₄	—	6	1	no	—
709	C₃	—	2	1	no	—
710	C₃	C₉×#553	1	1	yes	characteristic, complemented
711	C₃	C₃×#5709	1	1	yes	characteristic
712	C₂	—	4	1	no	—
713	C₂	—	6	2	no	—
714	C₂	—	12	1	no	—
715	C₂	—	12	1	no	—
716	C₂	D₃×C₉×#41	1	1	yes	characteristic
717	C₂	D₃×C₉×#41	1	1	yes	characteristic
718	C₂	C₉×#1009	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂³×D₃
commutator	C₂²×C₃	C₂⁴×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₉×#126	C₂
socle, cosocle	C₂²×C₃²	C₂⁴×C₃
prime radical	C₃×#5709	C₃
hypercenter	C₉×#126	D₃
hypocenter	C₃	C₉×#553
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#126

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×#126	C₂	semidirect
2. C₉×#898	C₂	semidirect
3. C₉×#895	C₂	semidirect
4. C₉×#894	C₂	semidirect
5. C₉×#893	C₂	semidirect
6. D₃×C₉×C₄⋊C₄	C₂	semidirect
7. C₉×#889	C₂	semidirect
8. C₉×#886	C₂	semidirect
9. C₉×#884	C₂	semidirect
10. C₉×#884	C₂	semidirect
11. C₉×#884	C₂	semidirect
12. C₉×#67⋊C₄	C₂	semidirect
13. C₉×#67⋊C₄	C₂	semidirect
14. C₄×C₉×D₁₂	C₂	semidirect
15. C₉×#872	C₂	semidirect
70. C₉×#126	D₃	semidirect
112. #5709	C₉	direct (N × H)
216. C₃×#126	C₂×C₉	semidirect
231. #898	C₂×C₉	semidirect
232. #895	C₂×C₉	semidirect
233. #894	C₂×C₉	semidirect
234. #893	C₂×C₉	semidirect
235. D₃×C₄⋊C₄	C₂×C₉	semidirect
236. #889	C₂×C₉	semidirect
237. #886	C₂×C₉	semidirect
238. #884	C₂×C₉	semidirect
239. #884	C₂×C₉	semidirect
240. #884	C₂×C₉	semidirect
241. #67⋊C₄	C₂×C₉	semidirect
242. #67⋊C₄	C₂×C₉	semidirect
243. C₄×D₁₂	C₂×C₉	semidirect
244. #872	C₂×C₉	semidirect
450. #126	D₃×C₉	semidirect
464. C₃×C₉	#553	semidirect
639. C₉	#5709	direct (N × H)
710. C₃	C₉×#553	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#126	2
C₉×#898	2
C₉×#895	2
C₉×#894	2
C₉×#893	2
D₃×C₉×C₄⋊C₄	2
C₉×#889	2
C₉×#886	2
C₉×#884	2
C₉×#67⋊C₄	2
C₄×C₉×D₁₂	2
C₉×#872	2
C₃×#5709	3
C₉×#553	3

Subgroups — C9×#5709 #13153228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5709 #13153228