Subgroups — C₉×#4656 #13141299

Order 1728.

622 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊Dic₃	C₂	1	2	yes	maximal
3	C₂×C₉×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉×#95	C₂	1	1	yes	maximal, characteristic
5	C₉×C₃⋊#95	C₂	1	2	yes	maximal
6	C₃×#4656	C₃	1	1	yes	maximal, characteristic
7	C₉×#395	—	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	1	yes	characteristic
11	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
12	C₂²×C₉×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₉×Dic₆	—	2	2	no	—
14	C₂×C₉×Dic₆	—	2	4	no	—
15	C₉×C₄⋊Dic₃	—	2	4	no	—
16	C₉×Dic₃⋊C₄	—	2	2	no	—
17	C₂²×C₃×Dic₆	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×C₄⋊Dic₃	C₂×C₃	1	2	yes	—
19	C₂×C₃×Dic₃⋊C₄	C₂×C₃	1	1	yes	characteristic
20	C₂²×Q₈×C₉	—	3	1	no	—
21	C₃²×#95	C₂×C₃	1	1	yes	characteristic
22	C₃×C₃⋊#95	C₂×C₃	1	2	yes	—
23	C₂×C₉×C₄⋊C₄	—	3	1	no	—
24	C₂×C₉×C₄⋊C₄	—	3	2	no	—
25	C₉×#95	—	2	1	no	—
26	C₉×#95	—	3	2	no	—
27	C₉×#95	D₃	1	1	yes	characteristic
28	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	—	2	2	no	—
32	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
33	C₂×C₃×C₄×C₉	Q₈	1	4	yes	complemented
34	C₂×C₉×Dic₃	—	2	2	no	—
35	C₂×C₉×Dic₃	—	2	2	no	—
36	C₂×C₉×Dic₃	—	2	1	no	—
37	C₂×C₉×Dic₃	—	2	1	no	—
38	C₂×C₉×Dic₃	—	2	1	no	—
39	C₂×C₉×Dic₃	—	2	2	no	—
40	C₂×C₉×Dic₃	D₄	1	4	yes	—
41	C₂×C₉×Dic₃	Q₈	1	2	yes	complemented
42	C₉×Dic₆	—	2	8	no	—
43	C₃×#395	—	3	1	no	—
44	#4656	C₉	1	1	yes	characteristic, direct product factor
45	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
46	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
47	C₂²×C₃×Dic₃	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
49	C₂²×C₃×Dic₃	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₃×Dic₆	—	2	2	no	—
51	C₂×C₃×Dic₆	—	2	4	no	—
52	C₃×C₄⋊Dic₃	—	2	4	no	—
53	C₃×Dic₃⋊C₄	—	2	2	no	—
54	C₂×Q₈×C₉	—	6	4	no	—
55	C₂×Q₈×C₉	—	6	2	no	—
56	C₂²×C₄×C₉	—	2	2	no	—
57	C₂²×C₄×C₉	—	2	1	no	—
58	C₂²×C₄×C₉	—	3	1	no	—
59	C₂²×C₄×C₉	—	3	2	no	—
60	C₂²×C₄×C₉	—	3	1	no	—
61	C₂²×C₄×C₉	C₂×D₃	1	2	yes	—
62	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
63	C₉×C₄⋊C₄	—	6	4	no	—
64	C₉×C₄⋊C₄	—	6	2	no	—
65	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
67	C₂²×C₃×C₉	C₂×Q₈	1	2	yes	—
68	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₃×C₉	#41	1	1	yes	characteristic
70	C₃×C₄×C₉	—	2	2	no	—
71	C₃×C₄×C₉	—	2	4	no	—
72	C₉×Dic₃	—	2	4	no	—
73	C₉×Dic₃	—	2	2	no	—
74	C₉×Dic₃	—	4	1	no	—
75	C₂²×C₃×Q₈	—	3	1	no	—
76	C₂²×Dic₆	C₂×C₉	1	1	yes	characteristic
77	C₂×C₃×C₄⋊C₄	—	3	1	no	—
78	C₂×C₃×C₄⋊C₄	—	3	2	no	—
79	C₂×C₄⋊Dic₃	C₂×C₉	1	2	yes	—
80	C₂×Dic₃⋊C₄	C₂×C₉	1	1	yes	characteristic
81	C₃×#95	—	2	1	no	—
82	C₃×#95	—	3	2	no	—
83	C₃×#95	C₂×C₉	1	1	yes	characteristic
84	C₃×#95	C₃×D₃	1	1	yes	characteristic
85	C₃⋊#95	C₂×C₉	1	2	yes	—
86	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
87	C₂×C₃²×C₄	—	2	2	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×Q₈	1	4	yes	—
92	C₂×C₃×Dic₃	—	2	1	no	—
93	C₂×C₃×Dic₃	—	2	2	no	—
94	C₂×C₃×Dic₃	—	2	2	no	—
95	C₂×C₃×Dic₃	—	2	1	no	—
96	C₂×C₃×Dic₃	—	2	1	no	—
97	C₂×C₃×Dic₃	—	2	2	no	—
98	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
99	C₂×C₃×Dic₃	C₃×Q₈	1	2	yes	—
100	C₃×Dic₆	—	2	8	no	—
101	C₂³×C₉	—	2	1	no	—
102	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
103	Q₈×C₉	—	6	8	no	—
104	C₂×C₄×C₉	—	2	2	no	—
105	C₂×C₄×C₉	—	2	2	no	—
106	C₂×C₄×C₉	—	2	2	no	—
107	C₂×C₄×C₉	—	2	4	no	—
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	—	3	4	no	—
110	C₂×C₄×C₉	—	3	2	no	—
111	C₂×C₄×C₉	—	4	2	no	—
112	C₂×C₄×C₉	—	4	2	no	—
113	C₂×C₄×C₉	—	4	2	no	—
114	C₂×C₄×C₉	—	6	1	no	—
115	C₂×C₄×C₉	—	6	2	no	—
116	C₂×C₄×C₉	—	6	2	no	—
117	C₂×C₄×C₉	—	6	1	no	—
118	C₂×C₄×C₉	—	6	1	no	—
119	C₂×C₄×C₉	—	6	2	no	—
120	C₂×C₄×C₉	Dic₆	1	4	yes	complemented
121	C₂×C₄×C₉	D₁₂	1	2	yes	—
122	#395	—	3	1	no	Sylow
123	C₂×C₃×C₉	#120	1	1	yes	characteristic
124	C₂×C₃×C₉	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
127	C₂×C₃×C₉	C₄⋊Q₈	1	2	yes	—
128	C₂×C₃×Q₈	—	6	4	no	—
129	C₂×C₃×Q₈	—	6	2	no	—
130	C₂²×C₃×C₄	—	2	2	no	—
131	C₂²×C₃×C₄	—	2	1	no	—
132	C₂²×C₃×C₄	—	3	1	no	—
133	C₂²×C₃×C₄	—	3	2	no	—
134	C₂²×C₃×C₄	—	3	1	no	—
135	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
136	C₂²×C₃×C₄	C₂²×C₉	1	2	yes	—
137	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
138	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
139	C₂²×Dic₃	C₂²×C₉	1	1	yes	characteristic
140	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
141	C₂²×Dic₃	C₂²×C₉	1	1	yes	characteristic
142	C₂×Dic₆	—	2	2	no	—
143	C₂×Dic₆	—	2	4	no	—
144	C₃×C₄⋊C₄	—	6	4	no	—
145	C₃×C₄⋊C₄	—	6	2	no	—
146	C₄⋊Dic₃	—	2	4	no	—
147	Dic₃⋊C₄	—	2	2	no	—
148	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
149	C₂²×C₃²	C₂×C₃×D₄	1	2	yes	—
150	C₂²×C₃²	C₂×C₃×Q₈	1	2	yes	—
151	C₂²×C₃²	C₂×C₃×Q₈	1	1	yes	characteristic
152	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
153	C₃²×C₄	—	2	2	no	—
154	C₃²×C₄	—	2	4	no	—
155	C₃×Dic₃	—	2	4	no	—
156	C₃×Dic₃	—	2	2	no	—
157	C₃×Dic₃	—	4	1	no	—
158	C₂²×C₉	—	2	2	no	—
159	C₂²×C₉	—	2	1	no	—
160	C₂²×C₉	—	2	2	no	—
161	C₂²×C₉	—	2	1	no	—
162	C₂²×C₉	—	2	1	no	—
163	C₂²×C₉	C₂×Dic₆	1	2	yes	—
164	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
165	C₂²×C₉	D₃×D₄	1	2	yes	—
166	C₂²×C₉	#237	1	1	yes	characteristic
167	C₂²×C₉	D₃×Q₈	1	1	yes	characteristic
168	C₄×C₉	—	2	4	no	—
169	C₄×C₉	—	2	2	no	—
170	C₄×C₉	—	4	4	no	—
171	C₄×C₉	—	4	2	no	—
172	C₄×C₉	—	6	4	no	—
173	C₄×C₉	—	6	2	no	—
174	C₄×C₉	—	12	1	no	—
175	C₂²×Q₈	—	3	1	no	—
176	C₂×C₄⋊C₄	—	3	1	no	—
177	C₂×C₄⋊C₄	—	3	2	no	—
178	#95	—	3	2	no	—
179	#95	D₃×C₉	1	1	yes	characteristic
180	C₃×C₉	#395	1	1	yes	characteristic, complemented, Sylow
181	C₂³×C₃	—	2	1	no	—
182	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
183	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
184	C₃×Q₈	—	6	8	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	4	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	3	4	no	—
194	C₂×C₃×C₄	—	3	2	no	—
195	C₂×C₃×C₄	—	4	2	no	—
196	C₂×C₃×C₄	—	4	2	no	—
197	C₂×C₃×C₄	—	4	2	no	—
198	C₂×C₃×C₄	—	6	1	no	—
199	C₂×C₃×C₄	—	6	2	no	—
200	C₂×C₃×C₄	—	6	2	no	—
201	C₂×C₃×C₄	—	6	1	no	—
202	C₂×C₃×C₄	—	6	1	no	—
203	C₂×C₃×C₄	—	6	2	no	—
204	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
205	C₂×C₃×C₄	Q₈×C₉	1	4	yes	complemented
206	C₂×C₃×C₄	C₃×Dic₆	1	4	yes	—
207	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
208	C₂×Dic₃	—	2	1	no	—
209	C₂×Dic₃	—	2	2	no	—
210	C₂×Dic₃	—	2	2	no	—
211	C₂×Dic₃	—	2	1	no	—
212	C₂×Dic₃	—	2	1	no	—
213	C₂×Dic₃	—	2	2	no	—
214	C₂×Dic₃	D₄×C₉	1	4	yes	—
215	C₂×Dic₃	Q₈×C₉	1	2	yes	complemented
216	Dic₆	—	2	8	no	—
217	C₂×C₃²	C₃×#120	1	1	yes	characteristic
218	C₂×C₃²	C₃×C₂²⋊Q₈	1	2	yes	—
219	C₂×C₃²	C₃×C₂²⋊Q₈	1	1	yes	characteristic
220	C₂×C₃²	C₃×C₄⋊Q₈	1	2	yes	—
221	C₂×C₃²	C₃×C₄⋊Q₈	1	1	yes	characteristic
222	C₂×C₉	—	2	2	no	—
223	C₂×C₉	—	2	1	no	—
224	C₂×C₉	—	2	2	no	—
225	C₂×C₉	—	2	1	no	—
226	C₂×C₉	—	2	1	no	—
227	C₂×C₉	#869	1	1	yes	characteristic
228	C₂×C₉	#878	1	2	yes	—
229	C₂×C₉	#882	1	1	yes	characteristic
230	C₂×C₉	#888	1	2	yes	—
231	C₂×C₉	#897	1	1	yes	characteristic
232	C₂×Q₈	—	6	4	no	—
233	C₂×Q₈	—	6	2	no	—
234	C₂²×C₄	—	3	1	no	—
235	C₂²×C₄	—	3	2	no	—
236	C₂²×C₄	—	3	1	no	—
237	C₂²×C₄	C₂×D₃×C₉	1	2	yes	—
238	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
239	C₄⋊C₄	—	6	4	no	—
240	C₄⋊C₄	—	6	2	no	—
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
247	C₂²×C₃	C₂×D₄×C₉	1	2	yes	—
248	C₂²×C₃	C₂×Q₈×C₉	1	2	yes	—
249	C₂²×C₃	C₂×Q₈×C₉	1	1	yes	characteristic
250	C₂²×C₃	C₉×#41	1	1	yes	characteristic
251	C₂²×C₃	C₂×C₃×Dic₆	1	2	yes	—
252	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
253	C₂²×C₃	C₃×D₃×D₄	1	2	yes	—
254	C₂²×C₃	C₃×#237	1	1	yes	characteristic
255	C₂²×C₃	C₃×D₃×Q₈	1	1	yes	characteristic
256	C₃×C₄	—	2	4	no	—
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	2	2	no	—
259	C₃×C₄	—	2	4	no	—
260	C₃×C₄	—	4	4	no	—
261	C₃×C₄	—	4	2	no	—
262	C₃×C₄	—	6	4	no	—
263	C₃×C₄	—	6	2	no	—
264	C₃×C₄	—	12	1	no	—
265	Dic₃	—	2	4	no	—
266	Dic₃	—	2	2	no	—
267	Dic₃	—	4	1	no	—
268	C₃²	C₃×#395	1	1	yes	characteristic
269	C₉	—	2	1	no	—
270	C₉	#4656	1	1	yes	characteristic, direct product factor
271	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
272	Q₈	—	6	8	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	3	4	no	—
277	C₂×C₄	—	3	2	no	—
278	C₂×C₄	—	6	1	no	—
279	C₂×C₄	—	6	2	no	—
280	C₂×C₄	—	6	2	no	—
281	C₂×C₄	—	6	1	no	—
282	C₂×C₄	—	6	1	no	—
283	C₂×C₄	—	6	2	no	—
284	C₂×C₄	C₉×Dic₆	1	4	yes	complemented
285	C₂×C₄	C₉×D₁₂	1	2	yes	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	2	2	no	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	—	2	2	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	C₉×#120	1	1	yes	characteristic
292	C₂×C₃	C₉×C₂²⋊Q₈	1	2	yes	—
293	C₂×C₃	C₉×C₂²⋊Q₈	1	1	yes	characteristic
294	C₂×C₃	C₉×C₄⋊Q₈	1	1	yes	characteristic
295	C₂×C₃	C₉×C₄⋊Q₈	1	2	yes	—
296	C₂×C₃	C₃×#869	1	1	yes	characteristic
297	C₂×C₃	C₃×#878	1	2	yes	—
298	C₂×C₃	C₃×#882	1	1	yes	characteristic
299	C₂×C₃	C₃×#888	1	2	yes	—
300	C₂×C₃	C₃×#897	1	1	yes	characteristic
301	C₂²	C₂×C₉×Dic₆	1	2	yes	—
302	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
303	C₂²	D₃×D₄×C₉	1	2	yes	—
304	C₂²	C₉×#237	1	1	yes	characteristic
305	C₂²	D₃×Q₈×C₉	1	1	yes	characteristic
306	C₄	—	2	4	no	—
307	C₄	—	2	2	no	—
308	C₄	—	6	4	no	—
309	C₄	—	6	2	no	—
310	C₄	—	12	1	no	—
311	C₃	—	2	1	no	—
312	C₃	C₉×#395	1	1	yes	characteristic, complemented
313	C₃	C₃×#4656	1	1	yes	characteristic
314	C₂	C₉×#869	1	1	yes	characteristic
315	C₂	C₉×#878	1	2	yes	—
316	C₂	C₉×#882	1	1	yes	characteristic
317	C₂	C₉×#888	1	2	yes	—
318	C₂	C₉×#897	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂²×D₃
commutator	C₂³×C₃	C₂³×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	C₉×#395
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₉×#95

Complements

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

N	H ≅ G/N	type
33. C₂×C₃×C₄×C₉	Q₈	semidirect
41. C₂×C₉×Dic₃	Q₈	semidirect
44. #4656	C₉	direct (N × H)
120. C₂×C₄×C₉	Dic₆	semidirect
180. C₃×C₉	#395	semidirect
205. C₂×C₃×C₄	Q₈×C₉	semidirect
215. C₂×Dic₃	Q₈×C₉	semidirect
270. C₉	#4656	direct (N × H)
284. C₂×C₄	C₉×Dic₆	semidirect
312. C₃	C₉×#395	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×Dic₆	2
C₂×C₉×C₄⋊Dic₃	2
C₂×C₉×Dic₃⋊C₄	2
C₃×C₉×#95	2
C₉×C₃⋊#95	2
C₃×#4656	3
C₉×#395	3

Subgroups — C9×#4656 #13141299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4656 #13141299