Subgroups — C₉×#4757 #13141770

Order 1728.

410 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#214	C₂	1	1	yes	maximal, characteristic
3	C₉×#887	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉×#102	C₂	1	1	yes	maximal, characteristic
5	C₉×#816	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#815	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#807	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4757	C₃	1	1	yes	maximal, characteristic
9	C₉×#440	—	3	1	no	maximal
10	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₉×Dic₃	—	2	1	no	—
12	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₉×#217	—	2	1	no	—
16	C₉×#214	C₄	1	4	yes	complemented
17	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₉×Dic₃⋊C₄	—	2	1	no	—
19	C₄×C₉×Dic₃	—	2	1	no	—
20	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
21	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₃×D₄×Dic₃	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×#214	C₂×C₃	1	1	yes	characteristic
24	C₃×#887	C₂×C₃	1	1	yes	characteristic
25	C₃²×#102	C₂×C₃	1	1	yes	characteristic
26	C₃×#816	C₂×C₃	1	1	yes	characteristic
27	C₃×#815	C₂×C₃	1	1	yes	characteristic
28	C₃×#807	C₂×C₃	1	1	yes	characteristic
29	C₂×C₉×SD₁₆	—	3	1	no	—
30	C₄×Q₈×C₉	—	3	1	no	—
31	C₄×D₄×C₉	—	3	1	no	—
32	C₉×#107	—	3	1	no	—
33	C₉×#103	—	3	1	no	—
34	C₉×#102	—	2	1	no	—
35	C₉×#102	D₃	1	1	yes	characteristic
36	C₉×#97	—	3	1	no	—
37	C₂³×C₃×C₉	—	2	1	no	—
38	C₃×D₄×C₉	—	2	1	no	—
39	C₃×D₄×C₉	C₂×C₄	1	2	yes	—
40	C₂×C₃×C₄×C₉	—	2	1	no	—
41	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂×C₉×Dic₃	—	2	1	no	—
43	C₂×C₉×Dic₃	—	2	1	no	—
44	C₂×C₉×Dic₃	—	4	2	no	—
45	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
46	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
47	C₉×Dic₆	—	2	1	no	—
48	C₉×Dic₆	C₂×C₄	1	2	yes	complemented
49	C₃×C₈×C₉	—	2	1	no	—
50	C₉×C₃⋊C₈	C₂×C₄	1	2	yes	complemented
51	C₃×#440	—	3	1	no	—
52	#4757	C₉	1	1	yes	characteristic, direct product factor
53	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
54	C₂²×C₃×Dic₃	—	2	1	no	—
55	C₂×C₃×Dic₆	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
57	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
58	C₃×#217	—	2	1	no	—
59	C₃×#214	C₃×C₄	1	4	yes	—
60	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
61	C₃×Dic₃⋊C₄	—	2	1	no	—
62	C₃×C₄×Dic₃	—	2	1	no	—
63	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
65	C₂×Q₈×C₉	—	3	1	no	—
66	C₂×D₄×C₉	—	2	1	no	—
67	C₂×D₄×C₉	C₂×D₃	1	1	yes	characteristic
68	C₂²×C₄×C₉	—	6	1	no	—
69	C₉×SD₁₆	—	3	4	no	—
70	C₂×C₈×C₉	—	2	1	no	—
71	C₂×C₈×C₉	—	3	1	no	—
72	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
73	C₉×C₄⋊C₄	—	2	1	no	—
74	C₉×C₄⋊C₄	—	3	1	no	—
75	C₉×C₄⋊C₄	—	6	1	no	—
76	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
77	C₉×C₂²⋊C₄	—	6	1	no	—
78	C₄²×C₉	—	3	1	no	—
79	C₄²×C₉	—	6	1	no	—
80	C₂²×C₃×C₉	—	2	2	no	—
81	C₂²×C₃×C₉	—	4	1	no	—
82	C₂²×C₃×C₉	—	4	1	no	—
83	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
84	C₃×C₄×C₉	—	4	1	no	—
85	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
86	C₃×C₄×C₉	#41	1	1	yes	characteristic
87	C₉×Dic₃	—	2	2	no	—
88	C₉×Dic₃	—	2	1	no	—
89	C₉×Dic₃	—	2	1	no	—
90	C₉×Dic₃	—	4	1	no	—
91	C₂×C₃×SD₁₆	—	3	1	no	—
92	C₃×C₄×Q₈	—	3	1	no	—
93	C₃×C₄×D₄	—	3	1	no	—
94	D₄×Dic₃	C₂×C₉	1	1	yes	characteristic
95	C₂×#214	C₂×C₉	1	1	yes	characteristic
96	#887	C₂×C₉	1	1	yes	characteristic, complemented
97	C₃×#107	—	3	1	no	—
98	C₃×#103	—	3	1	no	—
99	C₃×#102	—	2	1	no	—
100	C₃×#102	C₂×C₉	1	1	yes	characteristic
101	C₃×#102	C₃×D₃	1	1	yes	characteristic
102	C₃×#97	—	3	1	no	—
103	#816	C₂×C₉	1	1	yes	characteristic, complemented
104	#815	C₂×C₉	1	1	yes	characteristic, complemented
105	#807	C₂×C₉	1	1	yes	characteristic, complemented
106	C₂³×C₃²	—	2	1	no	—
107	C₃²×D₄	—	2	1	no	—
108	C₃²×D₄	C₂×C₃×C₄	1	2	yes	—
109	C₂×C₃²×C₄	—	2	1	no	—
110	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
111	C₂×C₃×Dic₃	—	2	1	no	—
112	C₂×C₃×Dic₃	—	2	1	no	—
113	C₂×C₃×Dic₃	—	4	2	no	—
114	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
115	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
116	C₃×Dic₆	—	2	1	no	—
117	C₃×Dic₆	C₂×C₃×C₄	1	2	yes	—
118	C₂³×C₉	—	2	1	no	—
119	C₂³×C₉	—	4	1	no	—
120	C₃²×C₈	—	2	1	no	—
121	C₃×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
122	Q₈×C₉	—	3	2	no	—
123	Q₈×C₉	—	6	1	no	—
124	D₄×C₉	—	2	1	no	—
125	D₄×C₉	—	2	2	no	—
126	D₄×C₉	—	4	1	no	—
127	D₄×C₉	C₄×D₃	1	2	yes	—
128	C₂×C₄×C₉	—	2	1	no	—
129	C₂×C₄×C₉	—	2	1	no	—
130	C₂×C₄×C₉	—	3	1	no	—
131	C₂×C₄×C₉	—	3	1	no	—
132	C₂×C₄×C₉	—	4	1	no	—
133	C₂×C₄×C₉	—	6	1	no	—
134	C₂×C₄×C₉	—	6	1	no	—
135	C₂×C₄×C₉	—	12	2	no	—
136	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
137	C₈×C₉	—	2	1	no	—
138	C₈×C₉	—	3	2	no	—
139	C₈×C₉	—	4	1	no	—
140	#440	—	3	1	no	Sylow
141	C₂×C₃×C₉	—	4	2	no	—
142	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
143	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₃×C₉	#137	1	1	yes	characteristic
145	C₂×C₃×Q₈	—	3	1	no	—
146	C₂×C₃×D₄	—	2	1	no	—
147	C₂×C₃×D₄	C₂²×C₉	1	1	yes	characteristic
148	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
149	C₂²×C₃×C₄	—	6	1	no	—
150	C₂²×Dic₃	—	2	1	no	—
151	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic
152	C₃×SD₁₆	—	3	4	no	—
153	C₂×C₃×C₈	—	2	1	no	—
154	C₂×C₃×C₈	—	3	1	no	—
155	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
156	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
157	C₃×C₄⋊C₄	—	2	1	no	—
158	C₃×C₄⋊C₄	—	3	1	no	—
159	C₃×C₄⋊C₄	—	6	1	no	—
160	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
161	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
162	C₃×C₂²⋊C₄	—	6	1	no	—
163	C₃×C₄²	—	3	1	no	—
164	C₃×C₄²	—	6	1	no	—
165	#217	—	2	1	no	—
166	#214	C₄×C₉	1	4	yes	complemented
167	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
168	Dic₃⋊C₄	—	2	1	no	—
169	C₄×Dic₃	—	2	1	no	—
170	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
171	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
172	C₂²×C₃²	—	2	2	no	—
173	C₂²×C₃²	—	4	1	no	—
174	C₂²×C₃²	—	4	1	no	—
175	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
176	C₃²×C₄	—	4	1	no	—
177	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₃²×C₄	C₃×#41	1	1	yes	characteristic
179	C₃×Dic₃	—	2	2	no	—
180	C₃×Dic₃	—	2	1	no	—
181	C₃×Dic₃	—	2	1	no	—
182	C₃×Dic₃	—	4	1	no	—
183	C₂²×C₉	—	2	2	no	—
184	C₂²×C₉	—	2	1	no	—
185	C₂²×C₉	—	4	2	no	—
186	C₂²×C₉	—	4	1	no	—
187	C₂²×C₉	—	4	1	no	—
188	C₂²×C₉	—	8	1	no	—
189	C₂²×C₉	—	8	1	no	—
190	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
191	C₄×C₉	—	2	1	no	—
192	C₄×C₉	—	2	1	no	—
193	C₄×C₉	—	4	1	no	—
194	C₄×C₉	—	6	2	no	—
195	C₄×C₉	—	6	1	no	—
196	C₄×C₉	—	6	1	no	—
197	C₄×C₉	—	8	1	no	—
198	C₄×C₉	—	12	1	no	—
199	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
200	C₄×C₉	#237	1	1	yes	characteristic
201	C₂×SD₁₆	—	3	1	no	—
202	C₄×Q₈	—	3	1	no	—
203	C₄×D₄	—	3	1	no	—
204	#107	—	3	1	no	—
205	#103	—	3	1	no	—
206	#102	D₃×C₉	1	1	yes	characteristic
207	#97	—	3	1	no	—
208	C₃×C₉	#440	1	1	yes	characteristic, complemented, Sylow
209	C₂³×C₃	—	2	1	no	—
210	C₂³×C₃	—	2	1	no	—
211	C₂³×C₃	—	4	1	no	—
212	C₃×Q₈	—	3	2	no	—
213	C₃×Q₈	—	6	1	no	—
214	C₃×D₄	—	2	1	no	—
215	C₃×D₄	—	2	1	no	—
216	C₃×D₄	—	2	2	no	—
217	C₃×D₄	—	4	1	no	—
218	C₃×D₄	C₂×C₄×C₉	1	2	yes	—
219	C₃×D₄	C₃×C₄×D₃	1	2	yes	—
220	C₂×C₃×C₄	—	2	1	no	—
221	C₂×C₃×C₄	—	2	1	no	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	3	1	no	—
224	C₂×C₃×C₄	—	3	1	no	—
225	C₂×C₃×C₄	—	4	1	no	—
226	C₂×C₃×C₄	—	6	1	no	—
227	C₂×C₃×C₄	—	6	1	no	—
228	C₂×C₃×C₄	—	12	2	no	—
229	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
231	C₂×Dic₃	—	2	1	no	—
232	C₂×Dic₃	—	2	1	no	—
233	C₂×Dic₃	—	4	2	no	—
234	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
235	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
236	Dic₆	—	2	1	no	—
237	Dic₆	C₂×C₄×C₉	1	2	yes	complemented
238	C₃×C₈	—	2	1	no	—
239	C₃×C₈	—	2	1	no	—
240	C₃×C₈	—	3	2	no	—
241	C₃×C₈	—	4	1	no	—
242	C₃⋊C₈	C₂×C₄×C₉	1	2	yes	complemented
243	C₂×C₃²	—	4	2	no	—
244	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
245	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
246	C₂×C₃²	C₃×#137	1	1	yes	characteristic
247	C₂×C₉	—	2	1	no	—
248	C₂×C₉	—	2	1	no	—
249	C₂×C₉	—	2	1	no	—
250	C₂×C₉	—	4	2	no	—
251	C₂×C₉	—	8	2	no	—
252	C₂×C₉	#67⋊C₄	1	1	yes	characteristic
253	C₂×C₉	#911	1	1	yes	characteristic
254	C₂×C₉	#915	1	1	yes	characteristic
255	C₂×Q₈	—	3	1	no	—
256	C₂×D₄	C₂×D₃×C₉	1	1	yes	characteristic
257	C₂²×C₄	—	6	1	no	—
258	SD₁₆	—	3	4	no	—
259	C₂×C₈	—	3	1	no	—
260	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
261	C₄⋊C₄	—	3	1	no	—
262	C₄⋊C₄	—	6	1	no	—
263	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
264	C₂²⋊C₄	—	6	1	no	—
265	C₄²	—	3	1	no	—
266	C₄²	—	6	1	no	—
267	C₂²×C₃	—	2	2	no	—
268	C₂²×C₃	—	2	2	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	4	2	no	—
271	C₂²×C₃	—	4	1	no	—
272	C₂²×C₃	—	4	1	no	—
273	C₂²×C₃	—	4	1	no	—
274	C₂²×C₃	—	4	1	no	—
275	C₂²×C₃	—	8	1	no	—
276	C₂²×C₃	—	8	1	no	—
277	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
278	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	4	1	no	—
282	C₃×C₄	—	4	1	no	—
283	C₃×C₄	—	6	2	no	—
284	C₃×C₄	—	6	1	no	—
285	C₃×C₄	—	6	1	no	—
286	C₃×C₄	—	8	1	no	—
287	C₃×C₄	—	12	1	no	—
288	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
289	C₃×C₄	C₉×#41	1	1	yes	characteristic
290	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
291	C₃×C₄	C₃×#237	1	1	yes	characteristic
292	Dic₃	—	2	2	no	—
293	Dic₃	—	2	1	no	—
294	Dic₃	—	2	1	no	—
295	Dic₃	—	4	1	no	—
296	C₃²	C₃×#440	1	1	yes	characteristic
297	C₉	—	2	1	no	—
298	C₉	#4757	1	1	yes	characteristic, direct product factor
299	C₂³	—	2	1	no	—
300	Q₈	—	3	2	no	—
301	Q₈	—	6	1	no	—
302	D₄	—	2	1	no	—
303	D₄	C₄×D₃×C₉	1	2	yes	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	3	1	no	—
306	C₂×C₄	—	3	1	no	—
307	C₂×C₄	—	6	1	no	—
308	C₂×C₄	—	6	1	no	—
309	C₂×C₄	—	12	2	no	—
310	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	3	2	no	—
313	C₂×C₃	—	2	1	no	—
314	C₂×C₃	—	2	1	no	—
315	C₂×C₃	—	2	1	no	—
316	C₂×C₃	—	4	2	no	—
317	C₂×C₃	—	4	2	no	—
318	C₂×C₃	—	8	2	no	—
319	C₂×C₃	C₄×D₄×C₉	1	1	yes	characteristic
320	C₂×C₃	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
321	C₂×C₃	C₉×#137	1	1	yes	characteristic
322	C₂×C₃	C₃×#67⋊C₄	1	1	yes	characteristic
323	C₂×C₃	C₃×#911	1	1	yes	characteristic
324	C₂×C₃	C₃×#915	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
329	C₄	—	4	1	no	—
330	C₄	—	6	1	no	—
331	C₄	—	6	1	no	—
332	C₄	—	6	2	no	—
333	C₄	—	12	1	no	—
334	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
335	C₄	C₉×#237	1	1	yes	characteristic
336	C₃	—	2	1	no	—
337	C₃	C₉×#440	1	1	yes	characteristic, complemented
338	C₃	C₃×#4757	1	1	yes	characteristic
339	C₂	—	4	2	no	—
340	C₂	C₉×#67⋊C₄	1	1	yes	characteristic
341	C₂	C₉×#911	1	1	yes	characteristic
342	C₂	C₉×#915	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	D₃×D₄
commutator	C₃×C₄	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#102	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃²×D₄	C₂²×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#102	D₃
hypocenter	C₃	C₉×#440
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#102

Complements

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

N	H ≅ G/N	type
3. C₉×#887	C₂	semidirect
5. C₉×#816	C₂	semidirect
6. C₉×#815	C₂	semidirect
7. C₉×#807	C₂	semidirect
16. C₉×#214	C₄	semidirect
48. C₉×Dic₆	C₂×C₄	semidirect
50. C₉×C₃⋊C₈	C₂×C₄	semidirect
52. #4757	C₉	direct (N × H)
96. #887	C₂×C₉	semidirect
103. #816	C₂×C₉	semidirect
104. #815	C₂×C₉	semidirect
105. #807	C₂×C₉	semidirect
166. #214	C₄×C₉	semidirect
208. C₃×C₉	#440	semidirect
237. Dic₆	C₂×C₄×C₉	semidirect
242. C₃⋊C₈	C₂×C₄×C₉	semidirect
298. C₉	#4757	direct (N × H)
337. C₃	C₉×#440	semidirect

Maximal subgroups

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

subgroup	index
D₄×C₉×Dic₃	2
C₂×C₉×#214	2
C₉×#887	2
C₃×C₉×#102	2
C₉×#816	2
C₉×#815	2
C₉×#807	2
C₃×#4757	3
C₉×#440	3

Subgroups — C9×#4757 #13141770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4757 #13141770