Subgroups — C₉×#4882 #13142202

Order 1728.

378 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#895	C₂	1	1	yes	maximal, characteristic
2	D₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#821	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×(C₂×D₃)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#809	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#807	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4882	C₃	1	1	yes	maximal, characteristic
9	C₉×#480	—	3	1	no	maximal
10	C₂×C₉×D₁₂	—	2	1	no	—
11	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₃×C₉	—	2	1	no	—
13	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
15	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×#895	C₂×C₃	1	1	yes	characteristic
22	C₃×D₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
23	C₃²×#107	C₂×C₃	1	1	yes	characteristic
24	C₃×#821	C₂×C₃	1	1	yes	characteristic
25	C₃×(C₂×D₃)⋊C₈	C₂×C₃	1	1	yes	characteristic
26	C₃×#809	C₂×C₃	1	1	yes	characteristic
27	C₃×#807	C₂×C₃	1	1	yes	characteristic
28	C₉×#121	—	3	1	no	—
29	C₂×C₉×C₄⋊C₄	—	3	1	no	—
30	C₉×#107	—	2	1	no	—
31	C₉×#107	—	3	1	no	—
32	C₉×#107	D₃	1	1	yes	characteristic, complemented
33	C₉×#102	—	3	1	no	—
34	C₉×#102	—	3	1	no	—
35	C₉×C₂²⋊C₈	—	3	1	no	—
36	C₂²×D₃×C₉	—	2	1	no	—
37	C₂²×D₃×C₉	D₄	1	1	yes	characteristic
38	C₂×C₃×C₄×C₉	—	2	1	no	—
39	C₂×C₃×C₄×C₉	—	2	1	no	—
40	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂×C₉×Dic₃	—	2	1	no	—
42	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
43	C₉×D₁₂	—	2	1	no	—
44	C₉×D₁₂	—	2	1	no	—
45	C₉×D₁₂	—	4	2	no	—
46	C₄×D₃×C₉	—	2	1	no	—
47	C₄×D₃×C₉	—	2	1	no	—
48	C₄×D₃×C₉	—	4	2	no	—
49	C₃×C₈×C₉	—	2	1	no	—
50	C₉×C₃⋊C₈	—	2	1	no	—
51	C₃×#480	—	3	1	no	—
52	#4882	C₉	1	1	yes	characteristic, direct product factor
53	C₂×C₃×D₁₂	—	2	1	no	—
54	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₃×C₄×D₃	—	2	1	no	—
56	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
58	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
59	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
60	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
61	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
62	C₃×Dic₃⋊C₄	—	2	1	no	—
63	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
64	C₂×D₄×C₉	—	3	1	no	—
65	C₂×D₄×C₉	—	6	1	no	—
66	C₂²×C₄×C₉	—	3	1	no	—
67	C₂²×C₄×C₉	—	6	1	no	—
68	C₂×C₈×C₉	—	2	1	no	—
69	C₂×C₈×C₉	—	3	1	no	—
70	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
71	C₉×C₄⋊C₄	—	2	1	no	—
72	C₉×C₄⋊C₄	—	2	1	no	—
73	C₉×C₄⋊C₄	—	3	1	no	—
74	C₉×C₄⋊C₄	—	6	1	no	—
75	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
76	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
77	C₉×C₂²⋊C₄	—	6	1	no	—
78	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
79	C₂×D₃×C₉	—	2	1	no	—
80	C₂×D₃×C₉	—	2	1	no	—
81	C₂×D₃×C₉	—	4	1	no	—
82	C₂×D₃×C₉	—	4	1	no	—
83	C₂×D₃×C₉	—	4	1	no	—
84	C₂×D₃×C₉	D₈	1	2	yes	—
85	C₃×C₄×C₉	—	4	1	no	—
86	C₃×C₄×C₉	—	4	1	no	—
87	C₃×C₄×C₉	#41	1	1	yes	characteristic
88	C₃×C₄×C₉	#41	1	1	yes	characteristic
89	C₉×Dic₃	—	2	1	no	—
90	C₉×Dic₃	—	4	1	no	—
91	C₃×#121	—	3	1	no	—
92	C₂×C₃×C₄⋊C₄	—	3	1	no	—
93	#895	C₂×C₉	1	1	yes	characteristic
94	D₃×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
95	C₃×#107	—	2	1	no	—
96	C₃×#107	—	3	1	no	—
97	C₃×#107	C₂×C₉	1	1	yes	characteristic, complemented
98	C₃×#107	C₃×D₃	1	1	yes	characteristic
99	C₃×#102	—	3	1	no	—
100	C₃×#102	—	3	1	no	—
101	C₃×C₂²⋊C₈	—	3	1	no	—
102	#821	C₂×C₉	1	1	yes	characteristic, complemented
103	(C₂×D₃)⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
104	#809	C₂×C₉	1	1	yes	characteristic, complemented
105	#807	C₂×C₉	1	1	yes	characteristic, complemented
106	C₂²×C₃×D₃	—	2	1	no	—
107	C₂²×C₃×D₃	C₃×D₄	1	1	yes	characteristic
108	C₂×C₃²×C₄	—	2	1	no	—
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	C₃×D₄	1	1	yes	characteristic
113	C₃×D₁₂	—	2	1	no	—
114	C₃×D₁₂	—	2	1	no	—
115	C₃×D₁₂	—	4	2	no	—
116	C₃×C₄×D₃	—	2	1	no	—
117	C₃×C₄×D₃	—	2	1	no	—
118	C₃×C₄×D₃	—	4	2	no	—
119	C₂³×C₉	—	3	1	no	—
120	C₂³×C₉	—	6	1	no	—
121	C₃²×C₈	—	2	1	no	—
122	C₃×C₃⋊C₈	—	2	1	no	—
123	D₄×C₉	—	6	1	no	—
124	D₄×C₉	—	6	1	no	—
125	D₄×C₉	—	12	2	no	—
126	C₂×C₄×C₉	—	2	1	no	—
127	C₂×C₄×C₉	—	2	1	no	—
128	C₂×C₄×C₉	—	2	1	no	—
129	C₂×C₄×C₉	—	3	1	no	—
130	C₂×C₄×C₉	—	4	1	no	—
131	C₂×C₄×C₉	—	4	1	no	—
132	C₂×C₄×C₉	—	6	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	2	no	—
138	C₈×C₉	—	2	1	no	—
139	C₈×C₉	—	6	1	no	—
140	#480	—	3	1	no	Sylow
141	C₂×C₃×C₉	#123	1	1	yes	characteristic
142	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
143	C₂×C₃×C₉	#137	1	1	yes	characteristic
144	D₃×C₉	—	2	2	no	—
145	D₃×C₉	—	8	1	no	—
146	C₂×C₃×D₄	—	3	1	no	—
147	C₂×C₃×D₄	—	6	1	no	—
148	C₂²×C₃×C₄	—	3	1	no	—
149	C₂²×C₃×C₄	—	6	1	no	—
150	C₂×D₁₂	—	2	1	no	—
151	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
152	C₂×C₄×D₃	—	2	1	no	—
153	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
154	C₂×C₃×C₈	—	2	1	no	—
155	C₂×C₃×C₈	—	3	1	no	—
156	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
157	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
158	C₃×C₄⋊C₄	—	2	1	no	—
159	C₃×C₄⋊C₄	—	2	1	no	—
160	C₃×C₄⋊C₄	—	3	1	no	—
161	C₃×C₄⋊C₄	—	6	1	no	—
162	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
163	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
164	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
165	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
166	C₃×C₂²⋊C₄	—	6	1	no	—
167	(C₂×D₃)⋊C₄	—	2	1	no	—
168	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
169	Dic₃⋊C₄	—	2	1	no	—
170	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
171	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
172	C₂×C₃×D₃	—	2	1	no	—
173	C₂×C₃×D₃	—	2	1	no	—
174	C₂×C₃×D₃	—	4	1	no	—
175	C₂×C₃×D₃	—	4	1	no	—
176	C₂×C₃×D₃	—	4	1	no	—
177	C₂×C₃×D₃	C₃×D₈	1	2	yes	—
178	C₃²×C₄	—	4	1	no	—
179	C₃²×C₄	—	4	1	no	—
180	C₃²×C₄	C₃×#41	1	1	yes	characteristic
181	C₃²×C₄	C₃×#41	1	1	yes	characteristic
182	C₃×Dic₃	—	2	1	no	—
183	C₃×Dic₃	—	4	1	no	—
184	C₂²×C₉	—	2	1	no	—
185	C₂²×C₉	—	3	2	no	—
186	C₂²×C₉	—	6	1	no	—
187	C₂²×C₉	—	6	1	no	—
188	C₂²×C₉	—	12	1	no	—
189	C₂²×C₉	—	12	1	no	—
190	C₂²×C₉	—	12	1	no	—
191	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
192	C₄×C₉	—	2	1	no	—
193	C₄×C₉	—	2	1	no	—
194	C₄×C₉	—	4	1	no	—
195	C₄×C₉	—	4	1	no	—
196	C₄×C₉	—	6	1	no	—
197	C₄×C₉	—	8	1	no	—
198	C₄×C₉	—	8	1	no	—
199	C₄×C₉	—	12	1	no	—
200	C₄×C₉	#235	1	1	yes	characteristic
201	C₄×C₉	#239	1	1	yes	characteristic
202	#121	—	3	1	no	—
203	C₂×C₄⋊C₄	—	3	1	no	—
204	#107	—	3	1	no	—
205	#107	D₃×C₉	1	1	yes	characteristic, complemented
206	#102	—	3	1	no	—
207	#102	—	3	1	no	—
208	C₂²⋊C₈	—	3	1	no	—
209	C₃×C₉	#480	1	1	yes	characteristic, complemented, Sylow
210	C₂³×C₃	—	3	1	no	—
211	C₂³×C₃	—	6	1	no	—
212	C₂²×D₃	—	2	1	no	—
213	C₂²×D₃	D₄×C₉	1	1	yes	characteristic
214	C₃×D₄	—	6	1	no	—
215	C₃×D₄	—	6	1	no	—
216	C₃×D₄	—	12	2	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	2	1	no	—
220	C₂×C₃×C₄	—	2	1	no	—
221	C₂×C₃×C₄	—	2	1	no	—
222	C₂×C₃×C₄	—	3	1	no	—
223	C₂×C₃×C₄	—	4	1	no	—
224	C₂×C₃×C₄	—	4	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	—	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	D₄×C₉	1	1	yes	characteristic
233	D₁₂	—	2	1	no	—
234	D₁₂	—	2	1	no	—
235	D₁₂	—	4	2	no	—
236	C₄×D₃	—	2	1	no	—
237	C₄×D₃	—	2	1	no	—
238	C₄×D₃	—	4	2	no	—
239	C₃×C₈	—	2	2	no	—
240	C₃×C₈	—	2	1	no	—
241	C₃×C₈	—	2	1	no	—
242	C₃×C₈	—	6	1	no	—
243	C₃⋊C₈	—	2	1	no	—
244	C₂×C₃²	C₃×#123	1	1	yes	characteristic
245	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
246	C₂×C₃²	C₃×#137	1	1	yes	characteristic
247	C₃×D₃	—	2	2	no	—
248	C₃×D₃	—	8	1	no	—
249	C₂×C₉	—	2	1	no	—
250	C₂×C₉	—	2	1	no	—
251	C₂×C₉	—	2	1	no	—
252	C₂×C₉	—	6	2	no	—
253	C₂×C₉	—	24	1	no	—
254	C₂×C₉	#894	1	1	yes	characteristic
255	C₂×C₉	D₃×D₈	1	1	yes	characteristic
256	C₂×C₉	#918	1	1	yes	characteristic
257	C₂×D₄	—	3	1	no	—
258	C₂×D₄	—	6	1	no	—
259	C₂²×C₄	—	3	1	no	—
260	C₂²×C₄	—	6	1	no	—
261	C₂×C₈	—	3	1	no	—
262	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
263	C₄⋊C₄	—	3	1	no	—
264	C₄⋊C₄	—	6	1	no	—
265	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
266	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
267	C₂²⋊C₄	—	6	1	no	—
268	C₂²×C₃	—	2	1	no	—
269	C₂²×C₃	—	3	2	no	—
270	C₂²×C₃	—	6	1	no	—
271	C₂²×C₃	—	6	1	no	—
272	C₂²×C₃	—	12	1	no	—
273	C₂²×C₃	—	12	1	no	—
274	C₂²×C₃	—	12	1	no	—
275	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
276	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
277	C₂×D₃	—	2	1	no	—
278	C₂×D₃	—	2	1	no	—
279	C₂×D₃	—	4	1	no	—
280	C₂×D₃	—	4	1	no	—
281	C₂×D₃	—	4	1	no	—
282	C₂×D₃	C₉×D₈	1	2	yes	—
283	C₃×C₄	—	2	1	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	4	1	no	—
286	C₃×C₄	—	4	1	no	—
287	C₃×C₄	—	4	1	no	—
288	C₃×C₄	—	4	1	no	—
289	C₃×C₄	—	6	1	no	—
290	C₃×C₄	—	8	1	no	—
291	C₃×C₄	—	8	1	no	—
292	C₃×C₄	—	12	1	no	—
293	C₃×C₄	C₉×#41	1	1	yes	characteristic
294	C₃×C₄	C₉×#41	1	1	yes	characteristic
295	C₃×C₄	C₃×#235	1	1	yes	characteristic
296	C₃×C₄	C₃×#239	1	1	yes	characteristic
297	Dic₃	—	2	1	no	—
298	Dic₃	—	4	1	no	—
299	C₃²	C₃×#480	1	1	yes	characteristic
300	C₉	—	2	1	no	—
301	C₉	#4882	1	1	yes	characteristic, direct product factor
302	C₂³	—	3	1	no	—
303	C₂³	—	6	1	no	—
304	D₄	—	6	1	no	—
305	D₄	—	6	1	no	—
306	D₄	—	12	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	3	1	no	—
310	C₂×C₄	—	6	1	no	—
311	C₂×C₄	—	6	1	no	—
312	C₂×C₄	—	6	1	no	—
313	C₂×C₄	—	12	2	no	—
314	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	6	1	no	—
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	2	1	no	—
320	C₂×C₃	—	6	2	no	—
321	C₂×C₃	—	24	1	no	—
322	C₂×C₃	C₉×#123	1	1	yes	characteristic
323	C₂×C₃	C₂×C₉×D₈	1	1	yes	characteristic
324	C₂×C₃	C₉×#137	1	1	yes	characteristic
325	C₂×C₃	C₃×#894	1	1	yes	characteristic
326	C₂×C₃	C₃×D₃×D₈	1	1	yes	characteristic
327	C₂×C₃	C₃×#918	1	1	yes	characteristic
328	D₃	—	2	2	no	—
329	D₃	—	8	1	no	—
330	C₂²	—	3	2	no	—
331	C₂²	—	6	1	no	—
332	C₂²	—	6	1	no	—
333	C₂²	—	12	1	no	—
334	C₂²	—	12	1	no	—
335	C₂²	—	12	1	no	—
336	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	6	1	no	—
340	C₄	—	12	1	no	—
341	C₄	C₉×#235	1	1	yes	characteristic
342	C₄	C₉×#239	1	1	yes	characteristic
343	C₃	—	2	1	no	—
344	C₃	C₉×#480	1	1	yes	characteristic, complemented
345	C₃	C₃×#4882	1	1	yes	characteristic
346	C₂	—	6	2	no	—
347	C₂	—	24	1	no	—
348	C₂	C₉×#894	1	1	yes	characteristic
349	C₂	D₃×C₉×D₈	1	1	yes	characteristic
350	C₂	C₉×#918	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	D₃×D₄
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#107	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃×#895	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#107	D₃
hypocenter	C₃	C₉×#480
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#107

Complements

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

N	H ≅ G/N	type
2. D₃×C₉×C₄⋊C₄	C₂	semidirect
3. C₃×C₉×#107	C₂	semidirect
4. C₉×#821	C₂	semidirect
5. C₉×(C₂×D₃)⋊C₈	C₂	semidirect
6. C₉×#809	C₂	semidirect
7. C₉×#807	C₂	semidirect
32. C₉×#107	D₃	semidirect
52. #4882	C₉	direct (N × H)
94. D₃×C₄⋊C₄	C₂×C₉	semidirect
97. C₃×#107	C₂×C₉	semidirect
102. #821	C₂×C₉	semidirect
103. (C₂×D₃)⋊C₈	C₂×C₉	semidirect
104. #809	C₂×C₉	semidirect
105. #807	C₂×C₉	semidirect
205. #107	D₃×C₉	semidirect
209. C₃×C₉	#480	semidirect
301. C₉	#4882	direct (N × H)
344. C₃	C₉×#480	semidirect

Maximal subgroups

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

subgroup	index
C₉×#895	2
D₃×C₉×C₄⋊C₄	2
C₃×C₉×#107	2
C₉×#821	2
C₉×(C₂×D₃)⋊C₈	2
C₉×#809	2
C₉×#807	2
C₃×#4882	3
C₉×#480	3

Subgroups — C9×#4882 #13142202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4882 #13142202