Subgroups — C₉×#2989 #11767710

Order 1152.

582 conjugacy classes of subgroups.

The subgroup lattice has 324 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×#436	C₂	1	2	yes	maximal, complemented
4	C₉×#435	C₂	1	4	yes	maximal, complemented
5	C₉×#432	C₂	1	2	yes	maximal, complemented
6	C₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#423	C₂	1	2	yes	maximal, complemented
8	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#2989	C₃	1	1	yes	maximal, characteristic
10	C₂×C₉×#41	C₄	1	1	yes	characteristic
11	C₂×C₉×M₁₆	C₂²	1	4	yes	—
12	C₂²×C₈×C₉	C₂²	1	2	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	2	yes	—
18	C₄×D₄×C₉	C₂²	1	1	yes	characteristic
19	C₉×#117	C₄	1	2	yes	—
20	C₉×#117	C₄	1	1	yes	characteristic
21	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₉	C₂²	1	2	yes	—
23	C₉×C₄⋊C₈	C₂²	1	2	yes	—
24	C₉×C₄⋊C₈	C₂²	1	2	yes	—
25	C₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₉×C₄⋊C₈	C₂²	1	4	yes	—
27	C₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₉×C₂²⋊C₈	C₂²	1	4	yes	—
30	C₉×#97	C₂²	1	4	yes	—
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×#436	C₂×C₃	1	2	yes	—
35	C₃×#435	C₂×C₃	1	4	yes	—
36	C₃×#432	C₂×C₃	1	2	yes	—
37	C₃×#431	C₂×C₃	1	1	yes	characteristic
38	C₃×#423	C₂×C₃	1	2	yes	—
39	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
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	2	yes	—
46	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₉	C₂³	1	2	yes	—
50	C₉×M₁₆	—	4	4	no	—
51	C₂×C₈×C₉	—	4	2	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	1	yes	characteristic
63	C₄²×C₉	C₂×C₄	1	2	yes	—
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	#2989	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	4	yes	—
72	C₂²×C₃×C₈	C₂²×C₃	1	2	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	2	yes	—
78	C₃×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
79	C₃×#117	C₃×C₄	1	2	yes	—
80	C₃×#117	C₃×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
83	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
84	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
85	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
86	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
87	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
88	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
89	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
90	C₃×#97	C₂²×C₃	1	4	yes	—
91	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
92	C₂³×C₉	C₂²×C₄	1	2	yes	—
93	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
94	Q₈×C₉	—	2	2	no	—
95	D₄×C₉	—	2	4	no	—
96	D₄×C₉	—	2	2	no	—
97	C₂×C₄×C₉	—	2	4	no	—
98	C₂×C₄×C₉	—	2	1	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	1	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	4	1	no	—
103	C₂×C₄×C₉	—	4	2	no	—
104	C₂×C₄×C₉	M₁₆	1	4	yes	complemented
105	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
111	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
114	C₈×C₉	—	2	4	no	—
115	C₈×C₉	—	2	2	no	—
116	C₈×C₉	—	2	2	no	—
117	C₄×#41	C₂×C₉	1	1	yes	characteristic
118	#446	C₂×C₉	1	2	yes	complemented
119	#436	C₂×C₉	1	2	yes	complemented
120	#435	C₂×C₉	1	4	yes	complemented
121	#432	C₂×C₉	1	2	yes	complemented
122	#431	C₂×C₉	1	1	yes	characteristic, complemented
123	#423	C₂×C₉	1	2	yes	complemented
124	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
125	C₃×#41	—	2	4	no	—
126	C₂×C₃×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
127	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
128	C₂×C₃×D₄	C₂×C₃×C₄	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
130	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
131	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
133	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
134	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
135	C₃×M₁₆	—	4	4	no	—
136	C₂×C₃×C₈	—	4	2	no	—
137	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
138	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
139	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
140	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
141	C₃×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
142	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
143	C₃×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
144	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
145	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
146	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
147	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
148	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
149	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
150	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
151	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
152	C₃×C₄²	C₂³×C₃	1	2	yes	—
153	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	—	2	1	no	—
156	C₂²×C₉	—	2	2	no	—
157	C₂²×C₉	—	2	2	no	—
158	C₂²×C₉	—	2	2	no	—
159	C₂²×C₉	C₂×M₁₆	1	2	yes	—
160	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
161	C₄×C₉	—	2	1	no	—
162	C₄×C₉	—	2	2	no	—
163	C₄×C₉	—	2	2	no	—
164	C₄×C₉	—	2	1	no	—
165	C₄×C₉	—	2	1	no	—
166	C₄×C₉	—	2	2	no	—
167	C₄×C₉	C₂×M₁₆	1	2	yes	—
168	C₄×C₉	C₂×M₁₆	1	2	yes	—
169	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
170	C₄×C₉	2₋¹⁺⁴	1	1	yes	characteristic
171	C₂×#41	C₄×C₉	1	1	yes	characteristic
172	C₂×M₁₆	C₂²×C₉	1	4	yes	—
173	C₂²×C₈	C₂²×C₉	1	2	yes	—
174	C₄×Q₈	C₄×C₉	1	1	yes	characteristic
175	C₄×Q₈	C₂²×C₉	1	1	yes	characteristic
176	C₄×D₄	C₄×C₉	1	2	yes	—
177	C₄×D₄	C₄×C₉	1	1	yes	characteristic
178	C₄×D₄	C₂²×C₉	1	2	yes	—
179	C₄×D₄	C₂²×C₉	1	1	yes	characteristic
180	#117	C₄×C₉	1	2	yes	—
181	#117	C₄×C₉	1	1	yes	characteristic
182	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
183	C₂×C₄²	C₂²×C₉	1	2	yes	—
184	C₄⋊C₈	C₂²×C₉	1	2	yes	—
185	C₄⋊C₈	C₂²×C₉	1	2	yes	—
186	C₄⋊C₈	C₂²×C₉	1	1	yes	characteristic
187	C₄⋊C₈	C₂²×C₉	1	4	yes	—
188	C₄⋊C₈	C₂²×C₉	1	1	yes	characteristic
189	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
190	C₂²⋊C₈	C₂²×C₉	1	4	yes	—
191	#97	C₂²×C₉	1	4	yes	—
192	C₄×C₈	C₂²×C₉	1	2	yes	—
193	C₂³×C₃	C₂²×C₃×C₄	1	2	yes	—
194	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
195	C₃×Q₈	—	2	2	no	—
196	C₃×D₄	—	2	4	no	—
197	C₃×D₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	4	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	4	1	no	—
204	C₂×C₃×C₄	—	4	2	no	—
205	C₂×C₃×C₄	C₃×M₁₆	1	4	yes	—
206	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
209	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
211	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
212	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
215	C₃×C₈	—	2	4	no	—
216	C₃×C₈	—	2	2	no	—
217	C₃×C₈	—	2	2	no	—
218	C₂×C₉	—	2	2	no	—
219	C₂×C₉	—	4	2	no	—
220	C₂×C₉	#520	1	1	yes	characteristic
221	C₂×C₉	C₂²×M₁₆	1	1	yes	characteristic
222	C₂×C₉	#568	1	1	yes	characteristic
223	#41	—	2	4	no	—
224	C₂×Q₈	C₂×C₄×C₉	1	1	yes	characteristic
225	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
226	C₂×D₄	C₂×C₄×C₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
228	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
229	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
231	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
232	C₂²×C₄	C₂³×C₉	1	2	yes	—
233	M₁₆	—	4	4	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	C₂³×C₉	1	2	yes	—
236	C₂×C₈	C₂³×C₉	1	2	yes	—
237	C₂×C₈	C₂³×C₉	1	4	yes	—
238	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
239	C₄⋊C₄	C₂×C₄×C₉	1	1	yes	characteristic
240	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
241	C₄⋊C₄	C₂×C₄×C₉	1	1	yes	characteristic
242	C₂²⋊C₄	C₂×C₄×C₉	1	2	yes	—
243	C₂²⋊C₄	C₂×C₄×C₉	1	2	yes	—
244	C₂²⋊C₄	C₂×C₄×C₉	1	2	yes	—
245	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
246	C₄²	C₂×C₄×C₉	1	2	yes	—
247	C₄²	C₂×C₄×C₉	1	2	yes	—
248	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
249	C₄²	C₂³×C₉	1	1	yes	characteristic
250	C₄²	C₂³×C₉	1	2	yes	—
251	C₄²	C₂³×C₉	1	1	yes	characteristic
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	2	2	no	—
255	C₂²×C₃	—	2	2	no	—
256	C₂²×C₃	—	2	2	no	—
257	C₂²×C₃	C₂×C₃×M₁₆	1	2	yes	—
258	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
259	C₃×C₄	—	2	2	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	2	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	C₂×C₃×M₁₆	1	2	yes	—
266	C₃×C₄	C₂×C₃×M₁₆	1	2	yes	—
267	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
268	C₃×C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
269	C₉	#2989	1	1	yes	characteristic, direct product factor, Sylow
270	C₂³	C₂²×C₄×C₉	1	2	yes	—
271	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
272	Q₈	—	2	2	no	—
273	D₄	—	2	4	no	—
274	D₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	C₉×M₁₆	1	4	yes	complemented
283	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
284	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
286	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
287	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
288	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
289	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
291	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
292	C₈	—	2	4	no	—
293	C₈	—	2	2	no	—
294	C₈	—	2	2	no	—
295	C₂×C₃	—	2	2	no	—
296	C₂×C₃	—	4	2	no	—
297	C₂×C₃	C₃×#520	1	1	yes	characteristic
298	C₂×C₃	C₂²×C₃×M₁₆	1	1	yes	characteristic
299	C₂×C₃	C₃×#568	1	1	yes	characteristic
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	C₂×C₉×M₁₆	1	2	yes	—
306	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	2	no	—
313	C₄	C₂×C₉×M₁₆	1	2	yes	—
314	C₄	C₂×C₉×M₁₆	1	2	yes	—
315	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
316	C₄	C₉×2₋¹⁺⁴	1	1	yes	characteristic
317	C₃	C₃×#2989	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	2	no	—
320	C₂	C₉×#520	1	1	yes	characteristic
321	C₂	C₂²×C₉×M₁₆	1	1	yes	characteristic
322	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×#436	C₂	semidirect
4. C₉×#435	C₂	semidirect
5. C₉×#432	C₂	semidirect
6. C₉×#431	C₂	semidirect
7. C₉×#423	C₂	semidirect
8. C₂×C₉×C₄⋊C₈	C₂	semidirect
69. #2989	C₉	direct (N × H)
104. C₂×C₄×C₉	M₁₆	semidirect
118. #446	C₂×C₉	semidirect
119. #436	C₂×C₉	semidirect
120. #435	C₂×C₉	semidirect
121. #432	C₂×C₉	semidirect
122. #431	C₂×C₉	semidirect
123. #423	C₂×C₉	semidirect
124. C₂×C₄⋊C₈	C₂×C₉	semidirect
269. C₉	#2989	direct (N × H)
282. C₂×C₄	C₉×M₁₆	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×#436	2
C₉×#435	2
C₉×#432	2
C₉×#431	2
C₉×#423	2
C₂×C₉×C₄⋊C₈	2
C₃×#2989	3

Subgroups — C9×#2989 #11767710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2989 #11767710