Subgroups — C₉×#2970 #11767691

Order 1152.

609 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₉×#41	C₂	1	1	yes	maximal, characteristic
2	C₉×#446	C₂	1	2	yes	maximal, complemented
3	C₉×#435	C₂	1	4	yes	maximal, complemented
4	C₈×D₄×C₉	C₂	1	2	yes	maximal, complemented
5	C₉×#432	C₂	1	2	yes	maximal, complemented
6	C₉×#432	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₉×#97	C₂	1	2	yes	maximal, complemented
9	C₃×#2970	C₃	1	1	yes	maximal, characteristic
10	C₂×C₉×#41	C₄	1	1	yes	characteristic
11	C₂×C₉×M₁₆	C₂²	1	2	yes	—
12	C₂²×C₈×C₉	C₂²	1	4	yes	—
13	C₄×Q₈×C₉	C₄	1	1	yes	characteristic
14	C₄×Q₈×C₉	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₉	C₄	1	2	yes	—
16	C₄×D₄×C₉	C₄	1	1	yes	characteristic
17	C₄×D₄×C₉	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₉	C₂²	1	2	yes	—
19	C₉×#117	C₄	1	1	yes	characteristic
20	C₉×#117	C₄	1	2	yes	—
21	C₂×C₄²×C₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
23	C₉×C₄⋊C₈	C₂²	1	2	yes	—
24	C₉×C₄⋊C₈	C₂²	1	4	yes	—
25	C₉×C₂²⋊C₈	C₂²	1	4	yes	—
26	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
27	C₉×#97	C₂²	1	1	yes	characteristic
28	C₉×#97	C₂²	1	2	yes	—
29	C₉×#97	C₂²	1	4	yes	—
30	C₉×#97	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₉	C₂²	1	2	yes	—
32	C₃×C₄×#41	C₂×C₃	1	1	yes	characteristic
33	C₃×#446	C₂×C₃	1	2	yes	—
34	C₃×#435	C₂×C₃	1	4	yes	—
35	C₃×C₈×D₄	C₂×C₃	1	2	yes	—
36	C₃×#432	C₂×C₃	1	1	yes	characteristic
37	C₃×#432	C₂×C₃	1	2	yes	—
38	C₃×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic
39	C₂×C₃×#97	C₂×C₃	1	2	yes	—
40	C₉×#41	—	2	4	no	—
41	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
45	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
48	C₂²×C₄×C₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
50	C₉×M₁₆	—	2	4	no	—
51	C₂×C₈×C₉	—	2	8	no	—
52	C₂×C₈×C₉	C₂³	1	4	yes	—
53	C₂×C₈×C₉	C₂³	1	2	yes	—
54	C₂×C₈×C₉	C₂³	1	2	yes	—
55	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
56	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
57	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
58	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
59	C₉×C₂²⋊C₄	C₂×C₄	1	2	yes	—
60	C₉×C₂²⋊C₄	C₂×C₄	1	2	yes	—
61	C₉×C₂²⋊C₄	C₂×C₄	1	2	yes	—
62	C₄²×C₉	C₂×C₄	1	2	yes	—
63	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₉	C₂×C₄	1	2	yes	—
65	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
66	C₄²×C₉	C₂³	1	1	yes	characteristic
67	C₄²×C₉	C₂³	1	1	yes	characteristic
68	C₄²×C₉	C₂³	1	2	yes	—
69	#2970	C₉	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×C₃×#41	C₃×C₄	1	1	yes	characteristic
71	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
72	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
73	C₃×C₄×Q₈	C₃×C₄	1	1	yes	characteristic
74	C₃×C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
75	C₃×C₄×D₄	C₃×C₄	1	2	yes	—
76	C₃×C₄×D₄	C₃×C₄	1	1	yes	characteristic
77	C₃×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
78	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
79	C₃×#117	C₃×C₄	1	1	yes	characteristic
80	C₃×#117	C₃×C₄	1	2	yes	—
81	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
82	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
83	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
84	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
85	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
86	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
87	C₃×#97	C₂²×C₃	1	1	yes	characteristic
88	C₃×#97	C₂²×C₃	1	2	yes	—
89	C₃×#97	C₂²×C₃	1	4	yes	—
90	C₃×#97	C₂²×C₃	1	1	yes	characteristic
91	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
92	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
93	C₂³×C₉	C₂²×C₄	1	2	yes	—
94	Q₈×C₉	—	2	2	no	—
95	D₄×C₉	—	2	4	no	—
96	D₄×C₉	M₁₆	1	4	yes	complemented
97	C₂×C₄×C₉	—	2	4	no	—
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	2	4	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	4	2	no	—
102	C₂×C₄×C₉	—	4	1	no	—
103	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
112	C₈×C₉	—	2	4	no	—
113	C₈×C₉	—	2	2	no	—
114	C₈×C₉	#41	1	4	yes	complemented
115	C₄×#41	C₂×C₉	1	1	yes	characteristic
116	#446	C₂×C₉	1	2	yes	complemented
117	#435	C₂×C₉	1	4	yes	complemented
118	C₈×D₄	C₂×C₉	1	2	yes	complemented
119	#432	C₂×C₉	1	1	yes	characteristic, complemented
120	#432	C₂×C₉	1	2	yes	complemented
121	C₄×M₁₆	C₂×C₉	1	1	yes	characteristic, complemented
122	C₂×#97	C₂×C₉	1	2	yes	complemented
123	C₃×#41	—	2	4	no	—
124	C₂×C₃×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
125	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
126	C₂×C₃×D₄	C₂×C₃×C₄	1	1	yes	characteristic
127	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
128	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
131	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
132	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
133	C₃×M₁₆	—	2	4	no	—
134	C₂×C₃×C₈	—	2	8	no	—
135	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
136	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
137	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
138	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
139	C₃×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
140	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
141	C₃×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
142	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
143	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
144	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
145	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
146	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
147	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
148	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
149	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
150	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
151	C₃×C₄²	C₂³×C₃	1	2	yes	—
152	C₂²×C₉	—	2	2	no	—
153	C₂²×C₉	—	2	2	no	—
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	—	2	1	no	—
156	C₂²×C₉	—	2	1	no	—
157	C₂²×C₉	C₂×M₁₆	1	4	yes	—
158	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
159	C₄×C₉	—	2	2	no	—
160	C₄×C₉	—	2	1	no	—
161	C₄×C₉	—	2	1	no	—
162	C₄×C₉	—	2	2	no	—
163	C₄×C₉	—	2	1	no	—
164	C₄×C₉	—	2	2	no	—
165	C₄×C₉	—	2	1	no	—
166	C₄×C₉	C₂×M₁₆	1	2	yes	—
167	C₄×C₉	C₂×#41	1	2	yes	—
168	C₂×#41	C₄×C₉	1	1	yes	characteristic
169	C₂×M₁₆	C₂²×C₉	1	2	yes	—
170	C₂²×C₈	C₂²×C₉	1	4	yes	—
171	C₄×Q₈	C₄×C₉	1	1	yes	characteristic
172	C₄×Q₈	C₂²×C₉	1	1	yes	characteristic
173	C₄×D₄	C₄×C₉	1	2	yes	—
174	C₄×D₄	C₄×C₉	1	1	yes	characteristic
175	C₄×D₄	C₂²×C₉	1	1	yes	characteristic
176	C₄×D₄	C₂²×C₉	1	2	yes	—
177	#117	C₄×C₉	1	1	yes	characteristic
178	#117	C₄×C₉	1	2	yes	—
179	C₂×C₄²	C₂²×C₉	1	2	yes	—
180	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
181	C₄⋊C₈	C₂²×C₉	1	2	yes	—
182	C₄⋊C₈	C₂²×C₉	1	4	yes	—
183	C₂²⋊C₈	C₂²×C₉	1	4	yes	—
184	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
185	#97	C₂²×C₉	1	1	yes	characteristic
186	#97	C₂²×C₉	1	2	yes	—
187	#97	C₂²×C₉	1	4	yes	—
188	#97	C₂²×C₉	1	1	yes	characteristic
189	C₄×C₈	C₂²×C₉	1	2	yes	—
190	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
191	C₂³×C₃	C₂²×C₃×C₄	1	2	yes	—
192	C₃×Q₈	—	2	2	no	—
193	C₃×D₄	—	2	4	no	—
194	C₃×D₄	C₃×M₁₆	1	4	yes	—
195	C₂×C₃×C₄	—	2	4	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	2	4	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	4	2	no	—
200	C₂×C₃×C₄	—	4	1	no	—
201	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
204	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
207	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
208	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
210	C₃×C₈	—	2	4	no	—
211	C₃×C₈	—	2	2	no	—
212	C₃×C₈	C₃×#41	1	4	yes	—
213	C₂×C₉	—	2	4	no	—
214	C₂×C₉	—	4	1	no	—
215	C₂×C₉	C₄×#41	1	1	yes	characteristic
216	C₂×C₉	C₂²×M₁₆	1	1	yes	characteristic
217	C₂×C₉	#568	1	1	yes	characteristic
218	#41	—	2	4	no	—
219	C₂×Q₈	C₂×C₄×C₉	1	1	yes	characteristic
220	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
221	C₂×D₄	C₂×C₄×C₉	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
223	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
226	C₂²×C₄	C₂³×C₉	1	2	yes	—
227	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
228	M₁₆	—	2	4	no	—
229	C₂×C₈	—	2	8	no	—
230	C₂×C₈	C₂³×C₉	1	2	yes	—
231	C₂×C₈	C₂³×C₉	1	4	yes	—
232	C₂×C₈	C₂³×C₉	1	2	yes	—
233	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
234	C₄⋊C₄	C₂×C₄×C₉	1	1	yes	characteristic
235	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
236	C₄⋊C₄	C₂×C₄×C₉	1	1	yes	characteristic
237	C₂²⋊C₄	C₂×C₄×C₉	1	2	yes	—
238	C₂²⋊C₄	C₂×C₄×C₉	1	2	yes	—
239	C₂²⋊C₄	C₂×C₄×C₉	1	2	yes	—
240	C₄²	C₂×C₄×C₉	1	2	yes	—
241	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
242	C₄²	C₂×C₄×C₉	1	2	yes	—
243	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
244	C₄²	C₂³×C₉	1	1	yes	characteristic
245	C₄²	C₂³×C₉	1	1	yes	characteristic
246	C₄²	C₂³×C₉	1	2	yes	—
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	2	2	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	C₂×C₃×M₁₆	1	4	yes	—
253	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	2	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	2	2	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	C₂×C₃×M₁₆	1	2	yes	—
262	C₃×C₄	C₂×C₃×#41	1	2	yes	—
263	C₉	#2970	1	1	yes	characteristic, direct product factor, Sylow
264	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
265	C₂³	C₂²×C₄×C₉	1	2	yes	—
266	Q₈	—	2	2	no	—
267	D₄	—	2	4	no	—
268	D₄	C₉×M₁₆	1	4	yes	complemented
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
278	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
281	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
282	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
283	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	—	2	2	no	—
286	C₈	C₉×#41	1	4	yes	complemented
287	C₂×C₃	—	2	4	no	—
288	C₂×C₃	—	4	1	no	—
289	C₂×C₃	C₃×C₄×#41	1	1	yes	characteristic
290	C₂×C₃	C₂²×C₃×M₁₆	1	1	yes	characteristic
291	C₂×C₃	C₃×#568	1	1	yes	characteristic
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	C₂×C₉×M₁₆	1	4	yes	—
298	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	C₂×C₉×M₁₆	1	2	yes	—
307	C₄	C₂×C₉×#41	1	2	yes	—
308	C₃	C₃×#2970	1	1	yes	characteristic
309	C₂	—	2	4	no	—
310	C₂	—	4	1	no	—
311	C₂	C₄×C₉×#41	1	1	yes	characteristic
312	C₂	C₂²×C₉×M₁₆	1	1	yes	characteristic
313	C₂	C₉×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂³×C₃×C₄
prime radical	C₂×C₃×#41	C₃×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#446	C₂	semidirect
3. C₉×#435	C₂	semidirect
4. C₈×D₄×C₉	C₂	semidirect
5. C₉×#432	C₂	semidirect
6. C₉×#432	C₂	semidirect
7. C₄×C₉×M₁₆	C₂	semidirect
8. C₂×C₉×#97	C₂	semidirect
69. #2970	C₉	direct (N × H)
96. D₄×C₉	M₁₆	semidirect
114. C₈×C₉	#41	semidirect
116. #446	C₂×C₉	semidirect
117. #435	C₂×C₉	semidirect
118. C₈×D₄	C₂×C₉	semidirect
119. #432	C₂×C₉	semidirect
120. #432	C₂×C₉	semidirect
121. C₄×M₁₆	C₂×C₉	semidirect
122. C₂×#97	C₂×C₉	semidirect
263. C₉	#2970	direct (N × H)
268. D₄	C₉×M₁₆	semidirect
286. C₈	C₉×#41	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₉×#41	2
C₉×#446	2
C₉×#435	2
C₈×D₄×C₉	2
C₉×#432	2
C₄×C₉×M₁₆	2
C₂×C₉×#97	2
C₃×#2970	3

Subgroups — C9×#2970 #11767691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2970 #11767691