Subgroups — C₉×#4981 #13142580

Order 1728.

542 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×Dic₃⋊C₄	C₂	1	2	yes	maximal
4	C₉×C₃⋊#95	C₂	1	2	yes	maximal
5	C₉×C₃⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₃×#4981	C₃	1	1	yes	maximal, characteristic
7	C₉×#398	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
12	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
13	C₃×C₉×C₄⋊C₄	—	2	2	no	—
14	C₉×C₄⋊Dic₃	—	2	2	no	—
15	C₉×Dic₃⋊C₄	—	2	4	no	—
16	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
17	C₂×C₃×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×Dic₃⋊C₄	C₂×C₃	1	2	yes	—
19	C₃×C₃⋊#95	C₂×C₃	1	2	yes	—
20	C₃×C₃⋊#95	C₂×C₃	1	1	yes	characteristic
21	C₂×C₉×C₄⋊C₄	—	2	1	no	—
22	C₂×C₉×C₄⋊C₄	—	3	2	no	—
23	C₂×C₉×C₄⋊C₄	—	3	1	no	—
24	C₂×C₉×C₄⋊C₄	D₃	1	1	yes	characteristic
25	C₉×#95	—	3	1	no	—
26	C₉×#95	—	3	2	no	—
27	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	—	2	1	no	—
29	C₂×C₃×C₄×C₉	—	2	1	no	—
30	C₂×C₃×C₄×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	—	2	1	no	—
32	C₂×C₃×C₄×C₉	—	2	2	no	—
33	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
34	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
35	C₂×C₉×Dic₃	—	2	2	no	—
36	C₂×C₉×Dic₃	—	2	2	no	—
37	C₂×C₉×Dic₃	—	2	2	no	—
38	C₂×C₉×Dic₃	—	2	2	no	—
39	C₂×C₉×Dic₃	—	2	2	no	—
40	C₂×C₉×Dic₃	Q₈	1	4	yes	—
41	C₃×#398	—	3	1	no	—
42	#4981	C₉	1	1	yes	characteristic, direct product factor
43	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
44	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
45	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
46	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
47	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
48	C₃²×C₄⋊C₄	—	2	2	no	—
49	C₃×C₄⋊Dic₃	—	2	2	no	—
50	C₃×Dic₃⋊C₄	—	2	4	no	—
51	C₂²×C₄×C₉	—	2	1	no	—
52	C₂²×C₄×C₉	—	2	1	no	—
53	C₂²×C₄×C₉	—	2	1	no	—
54	C₂²×C₄×C₉	—	3	2	no	—
55	C₂²×C₄×C₉	—	3	2	no	—
56	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
57	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
58	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
59	C₉×C₄⋊C₄	—	2	4	no	—
60	C₉×C₄⋊C₄	—	2	2	no	—
61	C₉×C₄⋊C₄	—	6	4	no	—
62	C₉×C₄⋊C₄	—	6	2	no	—
63	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₃×C₉	C₂×Q₈	1	2	yes	—
66	C₂²×C₃×C₉	#41	1	1	yes	characteristic
67	C₂²×C₃×C₉	#41	1	2	yes	—
68	C₃×C₄×C₉	—	2	2	no	—
69	C₃×C₄×C₉	—	2	2	no	—
70	C₃×C₄×C₉	—	4	1	no	—
71	C₉×Dic₃	—	2	4	no	—
72	C₉×Dic₃	—	4	2	no	—
73	C₂×C₃×C₄⋊C₄	—	2	1	no	—
74	C₂×C₃×C₄⋊C₄	—	3	2	no	—
75	C₂×C₃×C₄⋊C₄	—	3	1	no	—
76	C₂×C₃×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
77	C₂×C₃×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
78	C₂×C₄⋊Dic₃	C₂×C₉	1	1	yes	characteristic
79	C₂×Dic₃⋊C₄	C₂×C₉	1	2	yes	—
80	C₃×#95	—	3	1	no	—
81	C₃×#95	—	3	2	no	—
82	C₃⋊#95	C₂×C₉	1	2	yes	—
83	C₃⋊#95	C₂×C₉	1	1	yes	characteristic
84	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
85	C₂×C₃²×C₄	—	2	1	no	—
86	C₂×C₃²×C₄	—	2	1	no	—
87	C₂×C₃²×C₄	—	2	1	no	—
88	C₂×C₃²×C₄	—	2	2	no	—
89	C₂×C₃²×C₄	—	2	2	no	—
90	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
91	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
92	C₂×C₃×Dic₃	—	2	2	no	—
93	C₂×C₃×Dic₃	—	2	2	no	—
94	C₂×C₃×Dic₃	—	2	2	no	—
95	C₂×C₃×Dic₃	—	2	2	no	—
96	C₂×C₃×Dic₃	—	2	2	no	—
97	C₂×C₃×Dic₃	C₃×Q₈	1	4	yes	—
98	C₂³×C₉	—	2	1	no	—
99	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	2	1	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	1	no	—
105	C₂×C₄×C₉	—	2	2	no	—
106	C₂×C₄×C₉	—	2	2	no	—
107	C₂×C₄×C₉	—	2	1	no	—
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	—	2	2	no	—
110	C₂×C₄×C₉	—	3	4	no	—
111	C₂×C₄×C₉	—	4	2	no	—
112	C₂×C₄×C₉	—	4	2	no	—
113	C₂×C₄×C₉	—	6	2	no	—
114	C₂×C₄×C₉	—	6	2	no	—
115	C₂×C₄×C₉	—	6	2	no	—
116	C₂×C₄×C₉	—	6	2	no	—
117	C₂×C₄×C₉	—	6	2	no	—
118	C₂×C₄×C₉	D₁₂	1	2	yes	—
119	C₂×C₄×C₉	#67	1	2	yes	—
120	#398	—	3	1	no	Sylow
121	C₂×C₃×C₉	#121	1	1	yes	characteristic
122	C₂×C₃×C₉	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₃×C₉	#123	1	1	yes	characteristic
124	C₂×C₃×C₉	#125	1	2	yes	—
125	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
126	C₂²×C₃×C₄	—	2	1	no	—
127	C₂²×C₃×C₄	—	2	1	no	—
128	C₂²×C₃×C₄	—	2	1	no	—
129	C₂²×C₃×C₄	—	3	2	no	—
130	C₂²×C₃×C₄	—	3	2	no	—
131	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
133	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
134	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
135	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
136	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
137	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
138	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
139	C₃×C₄⋊C₄	—	2	4	no	—
140	C₃×C₄⋊C₄	—	2	2	no	—
141	C₃×C₄⋊C₄	—	2	2	no	—
142	C₃×C₄⋊C₄	—	6	4	no	—
143	C₃×C₄⋊C₄	—	6	2	no	—
144	C₄⋊Dic₃	—	2	2	no	—
145	Dic₃⋊C₄	—	2	4	no	—
146	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
147	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
148	C₂²×C₃²	C₂×C₃×Q₈	1	2	yes	—
149	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
150	C₂²×C₃²	C₃×#41	1	2	yes	—
151	C₃²×C₄	—	2	2	no	—
152	C₃²×C₄	—	2	2	no	—
153	C₃²×C₄	—	4	1	no	—
154	C₃×Dic₃	—	2	4	no	—
155	C₃×Dic₃	—	4	2	no	—
156	C₂²×C₉	—	2	1	no	—
157	C₂²×C₉	—	2	1	no	—
158	C₂²×C₉	—	2	2	no	—
159	C₂²×C₉	—	2	2	no	—
160	C₂²×C₉	—	2	1	no	—
161	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
162	C₂²×C₉	#235	1	1	yes	characteristic
163	C₂²×C₉	#237	1	2	yes	—
164	C₂²×C₉	D₃×Q₈	1	2	yes	—
165	C₂²×C₉	C₂×#67	1	1	yes	characteristic
166	C₄×C₉	—	2	4	no	—
167	C₄×C₉	—	2	2	no	—
168	C₄×C₉	—	2	2	no	—
169	C₄×C₉	—	4	2	no	—
170	C₄×C₉	—	4	2	no	—
171	C₄×C₉	—	4	1	no	—
172	C₄×C₉	—	6	4	no	—
173	C₄×C₉	—	12	2	no	—
174	C₂×C₄⋊C₄	—	3	2	no	—
175	C₂×C₄⋊C₄	—	3	1	no	—
176	C₂×C₄⋊C₄	D₃×C₉	1	1	yes	characteristic
177	#95	—	3	1	no	—
178	#95	—	3	2	no	—
179	C₃×C₉	#398	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₃	—	2	1	no	—
181	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
182	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	3	4	no	—
199	C₂×C₃×C₄	—	4	2	no	—
200	C₂×C₃×C₄	—	4	2	no	—
201	C₂×C₃×C₄	—	6	2	no	—
202	C₂×C₃×C₄	—	6	2	no	—
203	C₂×C₃×C₄	—	6	2	no	—
204	C₂×C₃×C₄	—	6	2	no	—
205	C₂×C₃×C₄	—	6	2	no	—
206	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
207	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
208	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
209	C₂×C₃×C₄	C₃×#67	1	2	yes	—
210	C₂×Dic₃	—	2	2	no	—
211	C₂×Dic₃	—	2	2	no	—
212	C₂×Dic₃	—	2	2	no	—
213	C₂×Dic₃	—	2	2	no	—
214	C₂×Dic₃	—	2	2	no	—
215	C₂×Dic₃	Q₈×C₉	1	4	yes	—
216	C₂×C₃²	C₃×#121	1	1	yes	characteristic
217	C₂×C₃²	C₃×C₂²⋊Q₈	1	2	yes	—
218	C₂×C₃²	C₃×#123	1	1	yes	characteristic
219	C₂×C₃²	C₃×#125	1	2	yes	—
220	C₂×C₃²	C₃×C₄⋊Q₈	1	1	yes	characteristic
221	C₂×C₉	—	2	2	no	—
222	C₂×C₉	—	2	2	no	—
223	C₂×C₉	—	2	1	no	—
224	C₂×C₉	—	2	1	no	—
225	C₂×C₉	—	2	1	no	—
226	C₂×C₉	#889	1	2	yes	—
227	C₂×C₉	#897	1	2	yes	—
228	C₂×C₉	#930	1	1	yes	characteristic
229	C₂×C₉	#935	1	1	yes	characteristic
230	C₂×C₉	#944	1	1	yes	characteristic
231	C₂²×C₄	—	3	2	no	—
232	C₂²×C₄	—	3	2	no	—
233	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	6	4	no	—
238	C₄⋊C₄	—	6	2	no	—
239	C₂²×C₃	—	2	1	no	—
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
245	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
246	C₂²×C₃	C₂×Q₈×C₉	1	2	yes	—
247	C₂²×C₃	C₉×#41	1	1	yes	characteristic
248	C₂²×C₃	C₉×#41	1	2	yes	—
249	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
250	C₂²×C₃	C₃×#235	1	1	yes	characteristic
251	C₂²×C₃	C₃×#237	1	2	yes	—
252	C₂²×C₃	C₃×D₃×Q₈	1	2	yes	—
253	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
254	C₃×C₄	—	2	4	no	—
255	C₃×C₄	—	2	2	no	—
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	2	2	no	—
259	C₃×C₄	—	4	2	no	—
260	C₃×C₄	—	4	2	no	—
261	C₃×C₄	—	4	1	no	—
262	C₃×C₄	—	4	1	no	—
263	C₃×C₄	—	6	4	no	—
264	C₃×C₄	—	12	2	no	—
265	Dic₃	—	2	4	no	—
266	Dic₃	—	4	2	no	—
267	C₃²	C₃×#398	1	1	yes	characteristic
268	C₉	—	2	1	no	—
269	C₉	#4981	1	1	yes	characteristic, direct product factor
270	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	3	4	no	—
277	C₂×C₄	—	6	2	no	—
278	C₂×C₄	—	6	2	no	—
279	C₂×C₄	—	6	2	no	—
280	C₂×C₄	—	6	2	no	—
281	C₂×C₄	—	6	2	no	—
282	C₂×C₄	C₉×D₁₂	1	2	yes	—
283	C₂×C₄	C₉×#67	1	2	yes	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	2	2	no	—
288	C₂×C₃	—	2	2	no	—
289	C₂×C₃	C₉×#121	1	1	yes	characteristic
290	C₂×C₃	C₉×C₂²⋊Q₈	1	2	yes	—
291	C₂×C₃	C₉×#123	1	1	yes	characteristic
292	C₂×C₃	C₉×#125	1	2	yes	—
293	C₂×C₃	C₉×C₄⋊Q₈	1	1	yes	characteristic
294	C₂×C₃	C₃×#889	1	2	yes	—
295	C₂×C₃	C₃×#897	1	2	yes	—
296	C₂×C₃	C₃×#930	1	1	yes	characteristic
297	C₂×C₃	C₃×#935	1	1	yes	characteristic
298	C₂×C₃	C₃×#944	1	1	yes	characteristic
299	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
300	C₂²	C₉×#235	1	1	yes	characteristic
301	C₂²	C₉×#237	1	2	yes	—
302	C₂²	D₃×Q₈×C₉	1	2	yes	—
303	C₂²	C₂×C₉×#67	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	4	1	no	—
307	C₄	—	6	4	no	—
308	C₄	—	12	2	no	—
309	C₃	—	2	1	no	—
310	C₃	C₉×#398	1	1	yes	characteristic, complemented
311	C₃	C₃×#4981	1	1	yes	characteristic
312	C₂	C₉×#889	1	2	yes	—
313	C₂	C₉×#897	1	2	yes	—
314	C₂	C₉×#930	1	1	yes	characteristic
315	C₂	C₉×#935	1	1	yes	characteristic
316	C₂	C₉×#944	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂²×D₃
commutator	C₂³×C₃	C₂³×C₉
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₃×C₉×C₄⋊C₄	C₂
socle, prime radical, cosocle	C₂³×C₃²	C₂³×C₃
hypercenter	C₂×C₉×C₄⋊C₄	D₃
hypocenter	C₃	C₉×#398
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₉×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
42. #4981	C₉	direct (N × H)
179. C₃×C₉	#398	semidirect
269. C₉	#4981	direct (N × H)
310. C₃	C₉×#398	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×C₄⋊C₄	2
C₂×C₉×C₄⋊Dic₃	2
C₂×C₉×Dic₃⋊C₄	2
C₉×C₃⋊#95	2
C₃×#4981	3
C₉×#398	3

Subgroups — C9×#4981 #13142580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4981 #13142580