Subgroups — C₉×#4649 #13141293

Order 1728.

658 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×Dic₃⋊C₄	C₂	1	2	yes	maximal
2	C₂×C₄×C₉×Dic₃	C₂	1	2	yes	maximal
3	C₂×C₄×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉×#95	C₂	1	1	yes	maximal, characteristic
5	C₉×C₃⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₃×#4649	C₃	1	1	yes	maximal, characteristic
7	C₄×C₉×C₄⋊C₄	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
10	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
11	C₂²×C₉×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₉×Dic₃	C₂²	1	1	yes	characteristic
13	C₉×Dic₃⋊C₄	C₄	1	8	yes	complemented
14	C₄×C₉×Dic₃	—	2	4	no	—
15	C₄×C₉×Dic₃	C₄	1	4	yes	complemented
16	C₂×C₃×Dic₃⋊C₄	C₂×C₃	1	2	yes	—
17	C₂×C₃×C₄×Dic₃	C₂×C₃	1	2	yes	—
18	C₂×C₃×C₄×Dic₃	C₂×C₃	1	1	yes	characteristic
19	C₃²×#95	C₂×C₃	1	1	yes	characteristic
20	C₃×C₃⋊#95	C₂×C₃	1	1	yes	characteristic
21	C₂×C₉×C₄⋊C₄	—	3	2	no	—
22	C₂×C₄²×C₉	—	3	1	no	—
23	C₂×C₄²×C₉	—	3	2	no	—
24	C₉×#95	—	2	1	no	—
25	C₉×#95	—	3	1	no	—
26	C₉×#95	D₃	1	1	yes	characteristic
27	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	—	2	2	no	—
29	C₂×C₃×C₄×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
33	C₂×C₉×Dic₃	—	2	2	no	—
34	C₂×C₉×Dic₃	—	2	2	no	—
35	C₂×C₉×Dic₃	C₂×C₄	1	4	yes	—
36	C₂×C₉×Dic₃	C₂×C₄	1	2	yes	—
37	C₂×C₉×Dic₃	C₂×C₄	1	4	yes	—
38	C₂×C₉×Dic₃	C₂×C₄	1	2	yes	—
39	C₂×C₉×Dic₃	D₄	1	2	yes	—
40	C₂×C₉×Dic₃	Q₈	1	2	yes	—
41	C₃×C₄×C₄⋊C₄	—	3	1	no	—
42	#4649	C₉	1	1	yes	characteristic, direct product factor
43	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
44	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
45	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
46	C₂²×C₃×Dic₃	C₂²×C₃	1	1	yes	characteristic
47	C₂²×C₃×Dic₃	C₂²×C₃	1	1	yes	characteristic
48	C₃×Dic₃⋊C₄	C₃×C₄	1	8	yes	—
49	C₃×C₄×Dic₃	—	2	4	no	—
50	C₃×C₄×Dic₃	C₃×C₄	1	4	yes	—
51	C₂²×C₄×C₉	—	2	1	no	—
52	C₂²×C₄×C₉	—	2	2	no	—
53	C₂²×C₄×C₉	—	3	1	no	—
54	C₂²×C₄×C₉	—	3	1	no	—
55	C₂²×C₄×C₉	—	3	2	no	—
56	C₂²×C₄×C₉	C₂×D₃	1	2	yes	—
57	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
58	C₉×C₄⋊C₄	—	3	8	no	—
59	C₄²×C₉	—	3	4	no	—
60	C₄²×C₉	—	6	4	no	—
61	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₃×C₉	C₂²×C₄	1	2	yes	—
63	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₃×C₉	#41	1	2	yes	—
66	C₃×C₄×C₉	—	2	2	no	—
67	C₃×C₄×C₉	—	2	4	no	—
68	C₉×Dic₃	—	2	4	no	—
69	C₉×Dic₃	C₄²	1	4	yes	complemented
70	C₉×Dic₃	C₄⋊C₄	1	4	yes	complemented
71	C₂×C₃×C₄⋊C₄	—	3	2	no	—
72	C₂×C₃×C₄²	—	3	1	no	—
73	C₂×C₃×C₄²	—	3	2	no	—
74	C₂×Dic₃⋊C₄	C₂×C₉	1	2	yes	—
75	C₂×C₄×Dic₃	C₂×C₉	1	2	yes	—
76	C₂×C₄×Dic₃	C₂×C₉	1	1	yes	characteristic
77	C₃×#95	—	2	1	no	—
78	C₃×#95	—	3	1	no	—
79	C₃×#95	C₂×C₉	1	1	yes	characteristic
80	C₃×#95	C₃×D₃	1	1	yes	characteristic
81	C₃⋊#95	C₂×C₉	1	1	yes	characteristic
82	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
83	C₂×C₃²×C₄	—	2	2	no	—
84	C₂×C₃²×C₄	—	2	2	no	—
85	C₂×C₃²×C₄	—	2	2	no	—
86	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
87	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
88	C₂×C₃×Dic₃	—	2	2	no	—
89	C₂×C₃×Dic₃	—	2	2	no	—
90	C₂×C₃×Dic₃	C₂×C₃×C₄	1	4	yes	—
91	C₂×C₃×Dic₃	C₂×C₃×C₄	1	2	yes	—
92	C₂×C₃×Dic₃	C₂×C₃×C₄	1	4	yes	—
93	C₂×C₃×Dic₃	C₂×C₃×C₄	1	2	yes	—
94	C₂×C₃×Dic₃	C₃×D₄	1	2	yes	—
95	C₂×C₃×Dic₃	C₃×Q₈	1	2	yes	—
96	C₂³×C₉	—	2	1	no	—
97	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	2	4	no	—
103	C₂×C₄×C₉	—	3	2	no	—
104	C₂×C₄×C₉	—	3	2	no	—
105	C₂×C₄×C₉	—	3	4	no	—
106	C₂×C₄×C₉	—	3	2	no	—
107	C₂×C₄×C₉	—	3	2	no	—
108	C₂×C₄×C₉	—	3	4	no	—
109	C₂×C₄×C₉	—	4	2	no	—
110	C₂×C₄×C₉	—	4	2	no	—
111	C₂×C₄×C₉	—	4	2	no	—
112	C₂×C₄×C₉	—	6	2	no	—
113	C₂×C₄×C₉	—	6	2	no	—
114	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
115	C₂×C₄×C₉	C₄×D₃	1	4	yes	—
116	C₄×C₄⋊C₄	—	3	1	no	Sylow
117	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₃×C₉	#117	1	1	yes	characteristic
120	C₂×C₃×C₉	C₄×D₄	1	2	yes	—
121	C₂×C₃×C₉	C₄×Q₈	1	2	yes	—
122	C₂²×C₃×C₄	—	2	1	no	—
123	C₂²×C₃×C₄	—	2	2	no	—
124	C₂²×C₃×C₄	—	3	1	no	—
125	C₂²×C₃×C₄	—	3	1	no	—
126	C₂²×C₃×C₄	—	3	2	no	—
127	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₂²×C₉	1	2	yes	—
129	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
131	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
132	C₂²×Dic₃	C₂²×C₉	1	1	yes	characteristic
133	C₂²×Dic₃	C₂²×C₉	1	1	yes	characteristic
134	C₃×C₄⋊C₄	—	3	8	no	—
135	C₃×C₄²	—	3	4	no	—
136	C₃×C₄²	—	6	4	no	—
137	Dic₃⋊C₄	C₄×C₉	1	8	yes	complemented
138	C₄×Dic₃	—	2	4	no	—
139	C₄×Dic₃	C₄×C₉	1	4	yes	complemented
140	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
141	C₂²×C₃²	C₂²×C₃×C₄	1	2	yes	—
142	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
143	C₂²×C₃²	C₂×C₃×Q₈	1	1	yes	characteristic
144	C₂²×C₃²	C₃×#41	1	2	yes	—
145	C₃²×C₄	—	2	2	no	—
146	C₃²×C₄	—	2	4	no	—
147	C₃×Dic₃	—	2	4	no	—
148	C₃×Dic₃	C₃×C₄²	1	4	yes	—
149	C₃×Dic₃	C₃×C₄⋊C₄	1	4	yes	—
150	C₂²×C₉	—	2	1	no	—
151	C₂²×C₉	—	2	2	no	—
152	C₂²×C₉	—	2	1	no	—
153	C₂²×C₉	—	2	2	no	—
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
156	C₂²×C₉	C₂×C₄×D₃	1	2	yes	—
157	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
158	C₂²×C₉	#237	1	2	yes	—
159	C₂²×C₉	D₃×Q₈	1	1	yes	characteristic
160	C₄×C₉	—	2	2	no	—
161	C₄×C₉	—	2	4	no	—
162	C₄×C₉	—	3	4	no	—
163	C₄×C₉	—	3	4	no	—
164	C₄×C₉	—	4	2	no	—
165	C₄×C₉	—	4	4	no	—
166	C₄×C₉	—	6	4	no	—
167	C₂×C₄⋊C₄	—	3	2	no	—
168	C₂×C₄²	—	3	1	no	—
169	C₂×C₄²	—	3	2	no	—
170	#95	—	3	1	no	—
171	#95	D₃×C₉	1	1	yes	characteristic
172	C₃×C₉	C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₃	—	2	1	no	—
174	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
175	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	—	3	2	no	—
185	C₂×C₃×C₄	—	3	2	no	—
186	C₂×C₃×C₄	—	3	4	no	—
187	C₂×C₃×C₄	—	3	2	no	—
188	C₂×C₃×C₄	—	3	2	no	—
189	C₂×C₃×C₄	—	3	4	no	—
190	C₂×C₃×C₄	—	4	2	no	—
191	C₂×C₃×C₄	—	4	2	no	—
192	C₂×C₃×C₄	—	4	2	no	—
193	C₂×C₃×C₄	—	6	2	no	—
194	C₂×C₃×C₄	—	6	2	no	—
195	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
196	C₂×C₃×C₄	C₂×C₄×C₉	1	4	yes	—
197	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
198	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
199	C₂×Dic₃	—	2	2	no	—
200	C₂×Dic₃	—	2	2	no	—
201	C₂×Dic₃	C₂×C₄×C₉	1	4	yes	—
202	C₂×Dic₃	C₂×C₄×C₉	1	2	yes	—
203	C₂×Dic₃	C₂×C₄×C₉	1	4	yes	—
204	C₂×Dic₃	C₂×C₄×C₉	1	2	yes	—
205	C₂×Dic₃	D₄×C₉	1	2	yes	—
206	C₂×Dic₃	Q₈×C₉	1	2	yes	—
207	C₂×C₃²	C₂×C₃×C₄²	1	1	yes	characteristic
208	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
209	C₂×C₃²	C₃×#117	1	1	yes	characteristic
210	C₂×C₃²	C₃×C₄×D₄	1	2	yes	—
211	C₂×C₃²	C₃×C₄×Q₈	1	2	yes	—
212	C₂×C₉	—	2	1	no	—
213	C₂×C₉	—	2	2	no	—
214	C₂×C₉	—	2	2	no	—
215	C₂×C₉	—	2	1	no	—
216	C₂×C₉	—	2	1	no	—
217	C₂×C₉	C₄²×D₃	1	1	yes	characteristic
218	C₂×C₉	#877	1	1	yes	characteristic
219	C₂×C₉	#67⋊C₄	1	2	yes	—
220	C₂×C₉	#887	1	2	yes	—
221	C₂×C₉	D₃×C₄⋊C₄	1	1	yes	characteristic
222	C₂²×C₄	—	3	1	no	—
223	C₂²×C₄	—	3	1	no	—
224	C₂²×C₄	—	3	2	no	—
225	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₃×C₉	1	2	yes	—
227	C₄⋊C₄	—	3	8	no	—
228	C₄²	—	3	4	no	—
229	C₄²	—	6	4	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
236	C₂²×C₃	C₂²×C₄×C₉	1	2	yes	—
237	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
238	C₂²×C₃	C₂×Q₈×C₉	1	1	yes	characteristic
239	C₂²×C₃	C₉×#41	1	2	yes	—
240	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
241	C₂²×C₃	C₂×C₃×C₄×D₃	1	2	yes	—
242	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
243	C₂²×C₃	C₃×#237	1	2	yes	—
244	C₂²×C₃	C₃×D₃×Q₈	1	1	yes	characteristic
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	—	2	4	no	—
247	C₃×C₄	—	2	2	no	—
248	C₃×C₄	—	2	4	no	—
249	C₃×C₄	—	3	4	no	—
250	C₃×C₄	—	3	4	no	—
251	C₃×C₄	—	4	2	no	—
252	C₃×C₄	—	4	4	no	—
253	C₃×C₄	—	6	4	no	—
254	Dic₃	—	2	4	no	—
255	Dic₃	C₄²×C₉	1	4	yes	complemented
256	Dic₃	C₉×C₄⋊C₄	1	4	yes	complemented
257	C₃²	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
258	C₉	—	2	1	no	—
259	C₉	#4649	1	1	yes	characteristic, direct product factor
260	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	3	2	no	—
265	C₂×C₄	—	3	2	no	—
266	C₂×C₄	—	3	4	no	—
267	C₂×C₄	—	3	2	no	—
268	C₂×C₄	—	3	2	no	—
269	C₂×C₄	—	3	4	no	—
270	C₂×C₄	—	6	2	no	—
271	C₂×C₄	—	6	2	no	—
272	C₂×C₄	C₄×D₃×C₉	1	2	yes	—
273	C₂×C₄	C₄×D₃×C₉	1	4	yes	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	—	2	1	no	—
276	C₂×C₃	—	2	2	no	—
277	C₂×C₃	—	2	2	no	—
278	C₂×C₃	—	2	1	no	—
279	C₂×C₃	C₂×C₄²×C₉	1	1	yes	characteristic
280	C₂×C₃	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
281	C₂×C₃	C₉×#117	1	1	yes	characteristic
282	C₂×C₃	C₄×D₄×C₉	1	2	yes	—
283	C₂×C₃	C₄×Q₈×C₉	1	2	yes	—
284	C₂×C₃	C₃×C₄²×D₃	1	1	yes	characteristic
285	C₂×C₃	C₃×#877	1	1	yes	characteristic
286	C₂×C₃	C₃×#67⋊C₄	1	2	yes	—
287	C₂×C₃	C₃×#887	1	2	yes	—
288	C₂×C₃	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
289	C₂²	C₂×C₄×D₃×C₉	1	1	yes	characteristic
290	C₂²	C₂×C₄×D₃×C₉	1	2	yes	—
291	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
292	C₂²	C₉×#237	1	2	yes	—
293	C₂²	D₃×Q₈×C₉	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	4	no	—
296	C₄	—	3	4	no	—
297	C₄	—	3	4	no	—
298	C₄	—	6	4	no	—
299	C₃	—	2	1	no	—
300	C₃	C₄×C₉×C₄⋊C₄	1	1	yes	characteristic, complemented
301	C₃	C₃×#4649	1	1	yes	characteristic
302	C₂	C₄²×D₃×C₉	1	1	yes	characteristic
303	C₂	C₉×#877	1	1	yes	characteristic
304	C₂	C₉×#67⋊C₄	1	2	yes	—
305	C₂	C₉×#887	1	2	yes	—
306	C₂	D₃×C₉×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂²×D₃
commutator	C₂×C₃	C₂×C₄²×C₉
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₃×C₉×#95	C₂
socle, prime radical, cosocle	C₂³×C₃²	C₂³×C₃
hypercenter	C₉×#95	D₃
hypocenter	C₃	C₄×C₉×C₄⋊C₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₉×#95

Complements

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

N	H ≅ G/N	type
13. C₉×Dic₃⋊C₄	C₄	semidirect
15. C₄×C₉×Dic₃	C₄	semidirect
42. #4649	C₉	direct (N × H)
69. C₉×Dic₃	C₄²	semidirect
70. C₉×Dic₃	C₄⋊C₄	semidirect
137. Dic₃⋊C₄	C₄×C₉	semidirect
139. C₄×Dic₃	C₄×C₉	semidirect
172. C₃×C₉	C₄×C₄⋊C₄	semidirect
255. Dic₃	C₄²×C₉	semidirect
256. Dic₃	C₉×C₄⋊C₄	semidirect
259. C₉	#4649	direct (N × H)
300. C₃	C₄×C₉×C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×Dic₃⋊C₄	2
C₂×C₄×C₉×Dic₃	2
C₃×C₉×#95	2
C₉×C₃⋊#95	2
C₃×#4649	3
C₄×C₉×C₄⋊C₄	3

Subgroups — C9×#4649 #13141293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4649 #13141293