Subgroups — C₉×#4845 #13142042

Order 1728.

374 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#204	C₂	1	1	yes	maximal, characteristic
3	C₉×#875	C₂	1	1	yes	maximal, characteristic
4	C₄×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#821	C₂	1	1	yes	maximal, characteristic
7	C₉×#816	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4845	C₃	1	1	yes	maximal, characteristic
9	C₉×#464	—	3	1	no	maximal
10	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
11	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
12	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
15	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
17	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
18	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
19	C₉×Dic₃⋊C₄	—	2	1	no	—
20	C₄×C₉×Dic₃	—	2	1	no	—
21	C₉×Dic₁₂	—	2	2	no	—
22	C₉×#204	—	2	2	no	—
23	C₂×C₃×Dic₁₂	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×#204	C₂×C₃	1	1	yes	characteristic
25	C₃×#875	C₂×C₃	1	1	yes	characteristic
26	C₃×C₄×Dic₆	C₂×C₃	1	1	yes	characteristic
27	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
28	C₃×#821	C₂×C₃	1	1	yes	characteristic
29	C₃×#816	C₂×C₃	1	1	yes	characteristic
30	C₂×C₉×Q₁₆	—	3	1	no	—
31	C₂×C₉×SD₁₆	—	3	1	no	—
32	C₉×#124	—	3	1	no	—
33	C₄×Q₈×C₉	—	3	1	no	—
34	C₉×C₄⋊C₈	—	2	1	no	—
35	C₉×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
36	C₉×#103	—	3	1	no	—
37	C₉×#102	—	3	1	no	—
38	C₂²×D₃×C₉	—	2	1	no	—
39	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
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	—	2	1	no	—
45	C₉×D₁₂	—	2	1	no	—
46	C₉×D₁₂	—	2	1	no	—
47	C₉×Dic₆	—	2	1	no	—
48	C₉×Dic₆	—	2	1	no	—
49	C₉×Dic₆	—	2	1	no	—
50	C₉×Dic₆	D₄	1	2	yes	—
51	C₃×C₈×C₉	—	2	1	no	—
52	C₃×C₈×C₉	—	2	1	no	—
53	C₃×#464	—	3	1	no	—
54	#4845	C₉	1	1	yes	characteristic, direct product factor
55	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₃×Dic₆	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃×Dic₆	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
60	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
61	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
62	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
63	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
64	C₃×Dic₃⋊C₄	—	2	1	no	—
65	C₃×C₄×Dic₃	—	2	1	no	—
66	C₃×Dic₁₂	—	2	2	no	—
67	C₃×#204	—	2	2	no	—
68	C₂×Q₈×C₉	—	3	1	no	—
69	C₂×Q₈×C₉	—	3	1	no	—
70	C₂×D₄×C₉	—	3	1	no	—
71	C₉×Q₁₆	—	6	2	no	—
72	C₉×SD₁₆	—	6	2	no	—
73	C₂×C₈×C₉	—	2	1	no	—
74	C₂×C₈×C₉	—	2	1	no	—
75	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
76	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
77	C₉×C₄⋊C₄	—	3	1	no	—
78	C₉×C₄⋊C₄	—	6	1	no	—
79	C₉×C₂²⋊C₄	—	6	1	no	—
80	C₉×C₂²⋊C₄	—	6	1	no	—
81	C₄²×C₉	—	2	1	no	—
82	C₄²×C₉	—	6	1	no	—
83	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
84	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
85	C₂×D₃×C₉	—	4	1	no	—
86	C₂×D₃×C₉	—	4	1	no	—
87	C₂×D₃×C₉	—	4	1	no	—
88	C₃×C₄×C₉	—	2	1	no	—
89	C₃×C₄×C₉	—	2	1	no	—
90	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
91	C₃×C₄×C₉	#41	1	1	yes	characteristic
92	C₉×Dic₃	—	2	2	no	—
93	C₉×Dic₃	—	4	1	no	—
94	C₉×Dic₃	—	4	1	no	—
95	C₂×C₃×Q₁₆	—	3	1	no	—
96	C₂×C₃×SD₁₆	—	3	1	no	—
97	C₃×#124	—	3	1	no	—
98	C₃×C₄×Q₈	—	3	1	no	—
99	C₂×Dic₁₂	C₂×C₉	1	1	yes	characteristic, complemented
100	C₂×#204	C₂×C₉	1	1	yes	characteristic
101	#875	C₂×C₉	1	1	yes	characteristic
102	C₄×Dic₆	C₂×C₉	1	1	yes	characteristic, complemented
103	C₃×C₄⋊C₈	—	2	1	no	—
104	C₃×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
105	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
106	C₃×#103	—	3	1	no	—
107	C₃×#102	—	3	1	no	—
108	#821	C₂×C₉	1	1	yes	characteristic
109	#816	C₂×C₉	1	1	yes	characteristic, complemented
110	C₂²×C₃×D₃	—	2	1	no	—
111	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
112	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
113	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
114	C₂×C₃×Dic₃	—	2	1	no	—
115	C₂×C₃×Dic₃	—	2	1	no	—
116	C₂×C₃×Dic₃	—	2	1	no	—
117	C₃×D₁₂	—	2	1	no	—
118	C₃×D₁₂	—	2	1	no	—
119	C₃×Dic₆	—	2	1	no	—
120	C₃×Dic₆	—	2	1	no	—
121	C₃×Dic₆	—	2	1	no	—
122	C₃×Dic₆	C₃×D₄	1	2	yes	—
123	C₂³×C₉	—	6	1	no	—
124	C₃²×C₈	—	2	1	no	—
125	C₃²×C₈	—	2	1	no	—
126	Q₈×C₉	—	3	2	no	—
127	Q₈×C₉	—	6	1	no	—
128	Q₈×C₉	—	6	1	no	—
129	Q₈×C₉	—	6	1	no	—
130	D₄×C₉	—	6	1	no	—
131	D₄×C₉	—	6	1	no	—
132	C₂×C₄×C₉	—	2	1	no	—
133	C₂×C₄×C₉	—	2	1	no	—
134	C₂×C₄×C₉	—	2	1	no	—
135	C₂×C₄×C₉	—	6	1	no	—
136	C₂×C₄×C₉	—	6	1	no	—
137	C₂×C₄×C₉	—	6	1	no	—
138	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
139	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
140	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
141	C₈×C₉	—	2	1	no	—
142	C₈×C₉	—	2	1	no	—
143	C₈×C₉	—	4	1	no	—
144	C₈×C₉	—	4	1	no	—
145	#464	—	3	1	no	Sylow
146	C₂×C₃×C₉	#121	1	1	yes	characteristic
147	C₂×C₃×C₉	#135	1	1	yes	characteristic
148	C₂×C₃×C₉	#137	1	1	yes	characteristic
149	D₃×C₉	—	8	1	no	—
150	C₂×C₃×Q₈	—	3	1	no	—
151	C₂×C₃×Q₈	—	3	1	no	—
152	C₂×C₃×D₄	—	3	1	no	—
153	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
154	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic
155	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic
156	C₃×Q₁₆	—	6	2	no	—
157	C₃×SD₁₆	—	6	2	no	—
158	C₂×C₃×C₈	—	2	1	no	—
159	C₂×C₃×C₈	—	2	1	no	—
160	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
161	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
162	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
163	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
164	C₃×C₄⋊C₄	—	3	1	no	—
165	C₃×C₄⋊C₄	—	6	1	no	—
166	C₃×C₂²⋊C₄	—	6	1	no	—
167	C₃×C₂²⋊C₄	—	6	1	no	—
168	C₃×C₄²	—	2	1	no	—
169	C₃×C₄²	—	6	1	no	—
170	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
171	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
172	(C₂×D₃)⋊C₄	—	2	1	no	—
173	(C₂×D₃)⋊C₄	—	2	1	no	—
174	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
175	Dic₃⋊C₄	—	2	1	no	—
176	C₄×Dic₃	—	2	1	no	—
177	Dic₁₂	—	2	2	no	—
178	#204	—	2	2	no	—
179	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂×C₃×D₃	—	4	1	no	—
181	C₂×C₃×D₃	—	4	1	no	—
182	C₂×C₃×D₃	—	4	1	no	—
183	C₃²×C₄	—	2	1	no	—
184	C₃²×C₄	—	2	1	no	—
185	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
186	C₃²×C₄	C₃×#41	1	1	yes	characteristic
187	C₃×Dic₃	—	2	2	no	—
188	C₃×Dic₃	—	4	1	no	—
189	C₃×Dic₃	—	4	1	no	—
190	C₂²×C₉	—	2	1	no	—
191	C₂²×C₉	—	12	1	no	—
192	C₂²×C₉	—	12	1	no	—
193	C₂²×C₉	—	12	1	no	—
194	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
195	C₄×C₉	—	2	2	no	—
196	C₄×C₉	—	2	1	no	—
197	C₄×C₉	—	2	1	no	—
198	C₄×C₉	—	2	1	no	—
199	C₄×C₉	—	2	1	no	—
200	C₄×C₉	—	4	1	no	—
201	C₄×C₉	—	6	2	no	—
202	C₄×C₉	—	12	1	no	—
203	C₄×C₉	—	12	1	no	—
204	C₄×C₉	D₃×D₄	1	1	yes	characteristic
205	C₄×C₉	#239	1	1	yes	characteristic
206	C₂×Q₁₆	—	3	1	no	—
207	C₂×SD₁₆	—	3	1	no	—
208	#124	—	3	1	no	—
209	C₄×Q₈	—	3	1	no	—
210	C₄⋊C₈	D₃×C₉	1	1	yes	characteristic, complemented
211	#103	—	3	1	no	—
212	#102	—	3	1	no	—
213	C₃×C₉	#464	1	1	yes	characteristic, complemented, Sylow
214	C₂³×C₃	—	6	1	no	—
215	C₂²×D₃	—	2	1	no	—
216	C₃×Q₈	—	3	2	no	—
217	C₃×Q₈	—	6	1	no	—
218	C₃×Q₈	—	6	1	no	—
219	C₃×Q₈	—	6	1	no	—
220	C₃×D₄	—	6	1	no	—
221	C₃×D₄	—	6	1	no	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	2	1	no	—
224	C₂×C₃×C₄	—	2	1	no	—
225	C₂×C₃×C₄	—	6	1	no	—
226	C₂×C₃×C₄	—	6	1	no	—
227	C₂×C₃×C₄	—	6	1	no	—
228	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
229	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
231	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
232	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
234	C₂×Dic₃	—	2	1	no	—
235	C₂×Dic₃	—	2	1	no	—
236	C₂×Dic₃	—	2	1	no	—
237	D₁₂	—	2	1	no	—
238	D₁₂	—	2	1	no	—
239	Dic₆	—	2	1	no	—
240	Dic₆	—	2	1	no	—
241	Dic₆	—	2	1	no	—
242	Dic₆	D₄×C₉	1	2	yes	—
243	C₃×C₈	—	2	1	no	—
244	C₃×C₈	—	2	1	no	—
245	C₃×C₈	—	2	1	no	—
246	C₃×C₈	—	2	1	no	—
247	C₃×C₈	—	4	1	no	—
248	C₃×C₈	—	4	1	no	—
249	C₂×C₃²	C₃×#121	1	1	yes	characteristic
250	C₂×C₃²	C₃×#135	1	1	yes	characteristic
251	C₂×C₃²	C₃×#137	1	1	yes	characteristic
252	C₃×D₃	—	8	1	no	—
253	C₂×C₉	—	2	1	no	—
254	C₂×C₉	—	2	1	no	—
255	C₂×C₉	—	2	1	no	—
256	C₂×C₉	—	24	1	no	—
257	C₂×C₉	#895	1	1	yes	characteristic
258	C₂×C₉	#904	1	1	yes	characteristic
259	C₂×C₉	#909	1	1	yes	characteristic
260	C₂×Q₈	—	3	1	no	—
261	C₂×Q₈	—	3	1	no	—
262	C₂×D₄	—	3	1	no	—
263	Q₁₆	—	6	2	no	—
264	SD₁₆	—	6	2	no	—
265	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
266	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
267	C₄⋊C₄	—	3	1	no	—
268	C₄⋊C₄	—	6	1	no	—
269	C₂²⋊C₄	—	6	1	no	—
270	C₂²⋊C₄	—	6	1	no	—
271	C₄²	—	6	1	no	—
272	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	—	12	1	no	—
275	C₂²×C₃	—	12	1	no	—
276	C₂²×C₃	—	12	1	no	—
277	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
278	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
279	C₂×D₃	—	4	1	no	—
280	C₂×D₃	—	4	1	no	—
281	C₂×D₃	—	4	1	no	—
282	C₃×C₄	—	2	2	no	—
283	C₃×C₄	—	2	1	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	2	1	no	—
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	—	4	1	no	—
290	C₃×C₄	—	6	2	no	—
291	C₃×C₄	—	12	1	no	—
292	C₃×C₄	—	12	1	no	—
293	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
294	C₃×C₄	C₉×#41	1	1	yes	characteristic
295	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
296	C₃×C₄	C₃×#239	1	1	yes	characteristic
297	Dic₃	—	2	2	no	—
298	Dic₃	—	4	1	no	—
299	Dic₃	—	4	1	no	—
300	C₃²	C₃×#464	1	1	yes	characteristic
301	C₉	—	2	1	no	—
302	C₉	#4845	1	1	yes	characteristic, direct product factor
303	C₂³	—	6	1	no	—
304	Q₈	—	3	2	no	—
305	Q₈	—	6	1	no	—
306	Q₈	—	6	1	no	—
307	Q₈	—	6	1	no	—
308	D₄	—	6	1	no	—
309	D₄	—	6	1	no	—
310	C₂×C₄	—	6	1	no	—
311	C₂×C₄	—	6	1	no	—
312	C₂×C₄	—	6	1	no	—
313	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
314	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
315	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	2	1	no	—
320	C₂×C₃	—	2	1	no	—
321	C₂×C₃	—	24	1	no	—
322	C₂×C₃	C₉×#121	1	1	yes	characteristic
323	C₂×C₃	C₉×#135	1	1	yes	characteristic
324	C₂×C₃	C₉×#137	1	1	yes	characteristic
325	C₂×C₃	C₃×#895	1	1	yes	characteristic
326	C₂×C₃	C₃×#904	1	1	yes	characteristic
327	C₂×C₃	C₃×#909	1	1	yes	characteristic
328	D₃	—	8	1	no	—
329	C₂²	—	12	1	no	—
330	C₂²	—	12	1	no	—
331	C₂²	—	12	1	no	—
332	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	6	2	no	—
336	C₄	—	12	1	no	—
337	C₄	—	12	1	no	—
338	C₄	D₃×D₄×C₉	1	1	yes	characteristic
339	C₄	C₉×#239	1	1	yes	characteristic
340	C₃	—	2	1	no	—
341	C₃	C₉×#464	1	1	yes	characteristic, complemented
342	C₃	C₃×#4845	1	1	yes	characteristic
343	C₂	—	24	1	no	—
344	C₂	C₉×#895	1	1	yes	characteristic
345	C₂	C₉×#904	1	1	yes	characteristic
346	C₂	C₉×#909	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×C₄⋊C₈	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃×D₁₂	C₂²×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×C₄⋊C₈	D₃
hypocenter	C₃	C₉×#464
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₉×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×Dic₁₂	C₂	semidirect
4. C₄×C₉×Dic₆	C₂	semidirect
5. C₃×C₉×C₄⋊C₈	C₂	semidirect
7. C₉×#816	C₂	semidirect
35. C₉×C₄⋊C₈	D₃	semidirect
54. #4845	C₉	direct (N × H)
99. C₂×Dic₁₂	C₂×C₉	semidirect
102. C₄×Dic₆	C₂×C₉	semidirect
104. C₃×C₄⋊C₈	C₂×C₉	semidirect
109. #816	C₂×C₉	semidirect
210. C₄⋊C₈	D₃×C₉	semidirect
213. C₃×C₉	#464	semidirect
302. C₉	#4845	direct (N × H)
341. C₃	C₉×#464	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×Dic₁₂	2
C₂×C₉×#204	2
C₉×#875	2
C₄×C₉×Dic₆	2
C₃×C₉×C₄⋊C₈	2
C₉×#821	2
C₉×#816	2
C₃×#4845	3
C₉×#464	3

Subgroups — C9×#4845 #13142042

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4845 #13142042