Subgroups — C₉×#5567 #13152673

Order 1728.

670 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₉×#944	C₂	1	2	yes	maximal
3	C₉×#889	C₂	1	4	yes	maximal
4	C₉×#887	C₂	1	2	yes	maximal
5	C₉×#870	C₂	1	2	yes	maximal
6	C₉×#869	C₂	1	1	yes	maximal, characteristic
7	C₄×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic
8	C₄×C₉×Dic₆	C₂	1	2	yes	maximal
9	C₃×#5567	C₃	1	1	yes	maximal, characteristic
10	C₉×#557	—	3	1	no	maximal
11	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×Dic₆	C₂²	1	2	yes	—
13	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
14	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
17	C₃×C₄²×C₉	C₂²	1	2	yes	—
18	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
19	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
20	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
21	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
22	C₉×Dic₃⋊C₄	C₂²	1	2	yes	—
23	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
24	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
25	C₄×C₉×Dic₃	C₂²	1	2	yes	—
26	C₄×C₉×Dic₃	C₂²	1	4	yes	—
27	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
28	C₃×#944	C₂×C₃	1	2	yes	—
29	C₃×#889	C₂×C₃	1	4	yes	—
30	C₃×#887	C₂×C₃	1	2	yes	—
31	C₃×#870	C₂×C₃	1	2	yes	—
32	C₃×#869	C₂×C₃	1	1	yes	characteristic
33	C₃×C₄×Dic₆	C₂×C₃	1	2	yes	—
34	C₃×C₄×Dic₆	C₂×C₃	1	1	yes	characteristic
35	C₉×C₄⋊Q₈	—	3	2	no	—
36	C₉×C₄⋊Q₈	—	3	1	no	—
37	C₉×#125	—	3	4	no	—
38	C₉×#125	—	3	2	no	—
39	C₄×Q₈×C₉	—	2	1	no	—
40	C₄×Q₈×C₉	—	3	2	no	—
41	C₄×Q₈×C₉	—	3	2	no	—
42	C₄×Q₈×C₉	—	3	1	no	—
43	C₄×Q₈×C₉	D₃	1	1	yes	characteristic
44	C₃×Q₈×C₉	—	2	2	no	—
45	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
46	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
47	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
48	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
50	C₂×C₉×Dic₃	C₂³	1	4	yes	—
51	C₂×C₉×Dic₃	C₂³	1	2	yes	—
52	C₂×C₉×Dic₃	C₂³	1	2	yes	—
53	C₉×Dic₆	—	2	4	no	—
54	C₉×Dic₆	Q₈	1	4	yes	complemented
55	C₃×#557	—	3	1	no	—
56	#5567	C₉	1	1	yes	characteristic, direct product factor
57	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
59	C₂×C₃×Dic₆	C₂²×C₃	1	1	yes	characteristic
60	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
61	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
62	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
63	C₃²×C₄²	C₂²×C₃	1	2	yes	—
64	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
65	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
66	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
67	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
68	C₃×Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
69	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
70	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
71	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
72	C₃×C₄×Dic₃	C₂²×C₃	1	2	yes	—
73	C₂×Q₈×C₉	—	2	1	no	—
74	C₂×Q₈×C₉	—	3	1	no	—
75	C₂×Q₈×C₉	—	3	2	no	—
76	C₂×Q₈×C₉	C₂×D₃	1	1	yes	characteristic
77	C₉×C₄⋊C₄	—	2	1	no	—
78	C₉×C₄⋊C₄	—	2	2	no	—
79	C₉×C₄⋊C₄	—	3	2	no	—
80	C₉×C₄⋊C₄	—	3	4	no	—
81	C₉×C₄⋊C₄	—	3	1	no	—
82	C₉×C₄⋊C₄	—	3	4	no	—
83	C₉×C₄⋊C₄	—	3	2	no	—
84	C₉×C₄⋊C₄	—	3	4	no	—
85	C₉×C₄⋊C₄	—	3	2	no	—
86	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
87	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
88	C₄²×C₉	—	2	2	no	—
89	C₄²×C₉	—	2	1	no	—
90	C₄²×C₉	—	3	4	no	—
91	C₄²×C₉	—	3	2	no	—
92	C₄²×C₉	C₂×D₃	1	2	yes	—
93	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
94	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
95	C₃×C₄×C₉	—	2	2	no	—
96	C₃×C₄×C₉	—	2	2	no	—
97	C₃×C₄×C₉	—	2	1	no	—
98	C₃×C₄×C₉	C₂×Q₈	1	2	yes	—
99	C₃×C₄×C₉	#41	1	2	yes	—
100	C₉×Dic₃	—	2	4	no	—
101	C₉×Dic₃	—	2	2	no	—
102	C₉×Dic₃	C₂×Q₈	1	4	yes	—
103	C₃×C₄⋊Q₈	—	3	2	no	—
104	C₃×C₄⋊Q₈	—	3	1	no	—
105	C₃×#125	—	3	4	no	—
106	C₃×#125	—	3	2	no	—
107	C₃×C₄×Q₈	—	2	1	no	—
108	C₃×C₄×Q₈	—	3	2	no	—
109	C₃×C₄×Q₈	—	3	2	no	—
110	C₃×C₄×Q₈	—	3	1	no	—
111	C₃×C₄×Q₈	C₂×C₉	1	1	yes	characteristic
112	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
113	#944	C₂×C₉	1	2	yes	—
114	#889	C₂×C₉	1	4	yes	—
115	#887	C₂×C₉	1	2	yes	—
116	#870	C₂×C₉	1	2	yes	—
117	#869	C₂×C₉	1	1	yes	characteristic
118	C₄×Dic₆	C₂×C₉	1	2	yes	—
119	C₄×Dic₆	C₂×C₉	1	1	yes	characteristic
120	C₃²×Q₈	—	2	2	no	—
121	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
122	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
123	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
124	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
125	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
126	C₂×C₃×Dic₃	C₂³×C₃	1	2	yes	—
127	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
128	C₂×C₃×Dic₃	C₂³×C₃	1	2	yes	—
129	C₃×Dic₆	—	2	4	no	—
130	C₃×Dic₆	C₃×Q₈	1	4	yes	—
131	Q₈×C₉	—	2	4	no	—
132	Q₈×C₉	—	2	2	no	—
133	Q₈×C₉	—	3	4	no	—
134	Q₈×C₉	—	6	4	no	—
135	C₂×C₄×C₉	—	2	1	no	—
136	C₂×C₄×C₉	—	2	2	no	—
137	C₂×C₄×C₉	—	2	1	no	—
138	C₂×C₄×C₉	—	2	2	no	—
139	C₂×C₄×C₉	—	2	1	no	—
140	C₂×C₄×C₉	—	3	2	no	—
141	C₂×C₄×C₉	—	3	2	no	—
142	C₂×C₄×C₉	—	3	4	no	—
143	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
144	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
145	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
146	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
147	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
148	#557	—	3	1	no	Sylow
149	C₂×C₃×C₉	C₂²×Q₈	1	1	yes	characteristic
150	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
151	C₂×C₃×C₉	2₋¹⁺⁴	1	1	yes	characteristic
152	C₂×C₃×Q₈	—	2	1	no	—
153	C₂×C₃×Q₈	—	3	1	no	—
154	C₂×C₃×Q₈	—	3	2	no	—
155	C₂×C₃×Q₈	C₂²×C₉	1	1	yes	characteristic
156	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
157	C₂×Dic₆	C₂²×C₉	1	2	yes	—
158	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic
159	C₃×C₄⋊C₄	—	2	1	no	—
160	C₃×C₄⋊C₄	—	2	2	no	—
161	C₃×C₄⋊C₄	—	3	2	no	—
162	C₃×C₄⋊C₄	—	3	4	no	—
163	C₃×C₄⋊C₄	—	3	4	no	—
164	C₃×C₄⋊C₄	—	3	4	no	—
165	C₃×C₄⋊C₄	—	3	1	no	—
166	C₃×C₄⋊C₄	—	3	2	no	—
167	C₃×C₄⋊C₄	—	3	2	no	—
168	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
169	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
170	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
171	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
172	C₃×C₄²	—	2	2	no	—
173	C₃×C₄²	—	2	1	no	—
174	C₃×C₄²	—	3	4	no	—
175	C₃×C₄²	—	3	2	no	—
176	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
177	C₃×C₄²	C₂²×C₉	1	2	yes	—
178	C₃×C₄²	C₂×C₃×D₃	1	2	yes	—
179	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
180	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
181	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
182	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
183	Dic₃⋊C₄	C₂²×C₉	1	2	yes	—
184	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
185	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
186	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
187	C₄×Dic₃	C₂²×C₉	1	4	yes	—
188	C₄×Dic₃	C₂²×C₉	1	2	yes	—
189	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
190	C₃²×C₄	—	2	2	no	—
191	C₃²×C₄	—	2	2	no	—
192	C₃²×C₄	—	2	1	no	—
193	C₃²×C₄	C₂×C₃×Q₈	1	2	yes	—
194	C₃²×C₄	C₃×#41	1	2	yes	—
195	C₃×Dic₃	—	2	4	no	—
196	C₃×Dic₃	—	2	2	no	—
197	C₃×Dic₃	C₂×C₃×Q₈	1	4	yes	—
198	C₂²×C₉	—	2	1	no	—
199	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
200	C₄×C₉	—	2	4	no	—
201	C₄×C₉	—	2	2	no	—
202	C₄×C₉	—	2	2	no	—
203	C₄×C₉	—	2	1	no	—
204	C₄×C₉	—	2	2	no	—
205	C₄×C₉	—	2	2	no	—
206	C₄×C₉	—	3	4	no	—
207	C₄×C₉	—	4	2	no	—
208	C₄×C₉	—	4	1	no	—
209	C₄×C₉	—	6	4	no	—
210	C₄×C₉	—	6	2	no	—
211	C₄×C₉	#235	1	2	yes	—
212	C₄×C₉	D₃×Q₈	1	2	yes	—
213	C₄⋊Q₈	—	3	2	no	—
214	C₄⋊Q₈	—	3	1	no	—
215	#125	—	3	4	no	—
216	#125	—	3	2	no	—
217	C₄×Q₈	—	3	2	no	—
218	C₄×Q₈	—	3	2	no	—
219	C₄×Q₈	—	3	1	no	—
220	C₄×Q₈	D₃×C₉	1	1	yes	characteristic
221	C₃×C₉	#557	1	1	yes	characteristic, complemented, Sylow
222	C₃×Q₈	—	2	4	no	—
223	C₃×Q₈	—	2	2	no	—
224	C₃×Q₈	—	2	2	no	—
225	C₃×Q₈	—	3	4	no	—
226	C₃×Q₈	—	6	4	no	—
227	C₂×C₃×C₄	—	2	1	no	—
228	C₂×C₃×C₄	—	2	2	no	—
229	C₂×C₃×C₄	—	2	1	no	—
230	C₂×C₃×C₄	—	2	2	no	—
231	C₂×C₃×C₄	—	2	1	no	—
232	C₂×C₃×C₄	—	3	2	no	—
233	C₂×C₃×C₄	—	3	2	no	—
234	C₂×C₃×C₄	—	3	4	no	—
235	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
236	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
237	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
241	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
242	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
243	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
245	C₂×Dic₃	C₂³×C₉	1	2	yes	—
246	C₂×Dic₃	C₂³×C₉	1	4	yes	—
247	C₂×Dic₃	C₂³×C₉	1	2	yes	—
248	Dic₆	—	2	4	no	—
249	Dic₆	Q₈×C₉	1	4	yes	complemented
250	C₂×C₃²	C₂²×C₃×Q₈	1	1	yes	characteristic
251	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
252	C₂×C₃²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
253	C₂×C₉	—	2	1	no	—
254	C₂×C₉	—	2	1	no	—
255	C₂×C₉	—	2	1	no	—
256	C₂×C₉	C₂×#235	1	1	yes	characteristic
257	C₂×C₉	C₂×D₃×Q₈	1	1	yes	characteristic
258	C₂×C₉	#1010	1	1	yes	characteristic
259	C₂×Q₈	—	3	1	no	—
260	C₂×Q₈	—	3	2	no	—
261	C₂×Q₈	C₂×D₃×C₉	1	1	yes	characteristic
262	C₄⋊C₄	—	3	2	no	—
263	C₄⋊C₄	—	3	4	no	—
264	C₄⋊C₄	—	3	1	no	—
265	C₄⋊C₄	—	3	4	no	—
266	C₄⋊C₄	—	3	2	no	—
267	C₄⋊C₄	—	3	4	no	—
268	C₄⋊C₄	—	3	2	no	—
269	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
270	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
271	C₄²	—	3	4	no	—
272	C₄²	—	3	2	no	—
273	C₄²	C₂×D₃×C₉	1	2	yes	—
274	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
275	C₂²×C₃	—	2	1	no	—
276	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
277	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
278	C₃×C₄	—	2	4	no	—
279	C₃×C₄	—	2	2	no	—
280	C₃×C₄	—	2	2	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	2	no	—
283	C₃×C₄	—	2	2	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	2	2	no	—
286	C₃×C₄	—	2	2	no	—
287	C₃×C₄	—	3	4	no	—
288	C₃×C₄	—	4	2	no	—
289	C₃×C₄	—	4	1	no	—
290	C₃×C₄	—	6	4	no	—
291	C₃×C₄	—	6	2	no	—
292	C₃×C₄	C₂×Q₈×C₉	1	2	yes	—
293	C₃×C₄	C₉×#41	1	2	yes	—
294	C₃×C₄	C₃×#235	1	2	yes	—
295	C₃×C₄	C₃×D₃×Q₈	1	2	yes	—
296	Dic₃	—	2	4	no	—
297	Dic₃	—	2	2	no	—
298	Dic₃	C₂×Q₈×C₉	1	4	yes	—
299	C₃²	C₃×#557	1	1	yes	characteristic
300	C₉	—	2	1	no	—
301	C₉	#5567	1	1	yes	characteristic, direct product factor
302	Q₈	—	2	2	no	—
303	Q₈	—	3	4	no	—
304	Q₈	—	6	4	no	—
305	C₂×C₄	—	3	2	no	—
306	C₂×C₄	—	3	2	no	—
307	C₂×C₄	—	3	4	no	—
308	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
309	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
310	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
311	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
313	C₂×C₃	—	2	1	no	—
314	C₂×C₃	—	2	1	no	—
315	C₂×C₃	—	2	1	no	—
316	C₂×C₃	C₂²×Q₈×C₉	1	1	yes	characteristic
317	C₂×C₃	C₂×C₉×#41	1	1	yes	characteristic
318	C₂×C₃	C₉×2₋¹⁺⁴	1	1	yes	characteristic
319	C₂×C₃	C₂×C₃×#235	1	1	yes	characteristic
320	C₂×C₃	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
321	C₂×C₃	C₃×#1010	1	1	yes	characteristic
322	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	1	no	—
326	C₄	—	3	4	no	—
327	C₄	—	6	4	no	—
328	C₄	—	6	2	no	—
329	C₄	C₉×#235	1	2	yes	—
330	C₄	D₃×Q₈×C₉	1	2	yes	—
331	C₃	—	2	1	no	—
332	C₃	C₉×#557	1	1	yes	characteristic, complemented
333	C₃	C₃×#5567	1	1	yes	characteristic
334	C₂	C₂×C₉×#235	1	1	yes	characteristic
335	C₂	C₂×D₃×Q₈×C₉	1	1	yes	characteristic
336	C₂	C₉×#1010	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂³×D₃
commutator	C₂²×C₃	C₂⁴×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle, prime radical, cosocle	C₂²×C₃²	C₂⁴×C₃
hypercenter	C₄×Q₈×C₉	D₃
hypocenter	C₃	C₉×#557
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄×Q₈×C₉

Complements

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

N	H ≅ G/N	type
54. C₉×Dic₆	Q₈	semidirect
56. #5567	C₉	direct (N × H)
221. C₃×C₉	#557	semidirect
249. Dic₆	Q₈×C₉	semidirect
301. C₉	#5567	direct (N × H)
332. C₃	C₉×#557	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×Q₈×C₉	2
C₉×#944	2
C₉×#889	2
C₉×#887	2
C₉×#870	2
C₉×#869	2
C₄×C₉×Dic₆	2
C₃×#5567	3
C₉×#557	3

Subgroups — C9×#5567 #13152673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5567 #13152673