Subgroups — C₉×#2580 #11767301

Order 1152.

846 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
3	C₉×#399	C₂	1	2	yes	maximal, complemented
4	C₉×#393	C₂	1	2	yes	maximal, complemented
5	C₉×#390	C₂	1	2	yes	maximal
6	C₉×#389	C₂	1	2	yes	maximal, complemented
7	C₉×#385	C₂	1	4	yes	maximal, complemented
8	C₉×#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#2580	C₃	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₉	C₂²	1	2	yes	—
13	C₄×Q₈×C₉	—	2	4	no	—
14	C₄×D₄×C₉	—	2	4	no	—
15	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
20	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
21	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₂×C₄²×C₉	C₂²	1	2	yes	—
24	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
25	C₉×#95	C₂²	1	2	yes	—
26	C₉×#95	C₂²	1	2	yes	—
27	C₉×#95	C₂²	1	4	yes	—
28	C₂×C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
29	C₂×C₃×C₄×D₄	C₂×C₃	1	1	yes	characteristic
30	C₃×#399	C₂×C₃	1	2	yes	—
31	C₃×#393	C₂×C₃	1	2	yes	—
32	C₃×#390	C₂×C₃	1	2	yes	—
33	C₃×#389	C₂×C₃	1	2	yes	—
34	C₃×#385	C₂×C₃	1	4	yes	—
35	C₃×#382	C₂×C₃	1	1	yes	characteristic
36	C₂⁴×C₉	C₂³	1	2	yes	—
37	C₂×Q₈×C₉	—	2	2	no	—
38	C₂×Q₈×C₉	—	2	4	no	—
39	C₂×D₄×C₉	—	2	2	no	—
40	C₂×D₄×C₉	—	2	2	no	—
41	C₂×D₄×C₉	Q₈	1	4	yes	complemented
42	C₂²×C₄×C₉	—	2	4	no	—
43	C₂²×C₄×C₉	—	4	4	no	—
44	C₂²×C₄×C₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₉	C₂³	1	4	yes	—
46	C₂²×C₄×C₉	C₂³	1	2	yes	—
47	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
51	C₉×C₄⋊C₄	—	2	2	no	—
52	C₉×C₄⋊C₄	—	2	4	no	—
53	C₉×C₄⋊C₄	—	2	2	no	—
54	C₉×C₄⋊C₄	—	2	4	no	—
55	C₉×C₄⋊C₄	—	4	4	no	—
56	C₉×C₂²⋊C₄	—	2	4	no	—
57	C₉×C₂²⋊C₄	—	2	8	no	—
58	C₉×C₂²⋊C₄	—	4	4	no	—
59	C₄²×C₉	—	2	4	no	—
60	C₄²×C₉	D₄	1	4	yes	—
61	#2580	C₉	1	1	yes	characteristic, direct product factor, Sylow
62	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
63	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
64	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
65	C₃×C₄×Q₈	—	2	4	no	—
66	C₃×C₄×D₄	—	2	4	no	—
67	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
68	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
69	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
72	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
73	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
74	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
75	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
76	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
77	C₃×#95	C₂²×C₃	1	2	yes	—
78	C₃×#95	C₂²×C₃	1	2	yes	—
79	C₃×#95	C₂²×C₃	1	4	yes	—
80	C₂³×C₉	—	2	2	no	—
81	C₂³×C₉	—	2	4	no	—
82	C₂³×C₉	—	2	2	no	—
83	C₂³×C₉	—	2	4	no	—
84	C₂³×C₉	C₂×Q₈	1	4	yes	—
85	C₂³×C₉	C₂⁴	1	1	yes	characteristic
86	Q₈×C₉	—	2	8	no	—
87	D₄×C₉	—	2	8	no	—
88	C₂×C₄×C₉	—	2	2	no	—
89	C₂×C₄×C₉	—	2	2	no	—
90	C₂×C₄×C₉	—	2	2	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	2	no	—
94	C₂×C₄×C₉	—	2	4	no	—
95	C₂×C₄×C₉	—	2	4	no	—
96	C₂×C₄×C₉	—	2	2	no	—
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	—	4	8	no	—
101	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
105	C₂×C₄×C₉	#41	1	4	yes	—
106	C₂×C₄×C₉	#41	1	2	yes	—
107	C₂×C₄×Q₈	C₂×C₉	1	1	yes	characteristic, complemented
108	C₂×C₄×D₄	C₂×C₉	1	1	yes	characteristic
109	#399	C₂×C₉	1	2	yes	complemented
110	#393	C₂×C₉	1	2	yes	complemented
111	#390	C₂×C₉	1	2	yes	—
112	#389	C₂×C₉	1	2	yes	complemented
113	#385	C₂×C₉	1	4	yes	complemented
114	#382	C₂×C₉	1	1	yes	characteristic, complemented
115	C₂⁴×C₃	C₂³×C₃	1	2	yes	—
116	C₂×C₃×Q₈	—	2	2	no	—
117	C₂×C₃×Q₈	—	2	4	no	—
118	C₂×C₃×D₄	—	2	2	no	—
119	C₂×C₃×D₄	—	2	2	no	—
120	C₂×C₃×D₄	C₃×Q₈	1	4	yes	—
121	C₂²×C₃×C₄	—	2	4	no	—
122	C₂²×C₃×C₄	—	4	4	no	—
123	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
124	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
125	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
126	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
127	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
129	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
130	C₃×C₄⋊C₄	—	2	2	no	—
131	C₃×C₄⋊C₄	—	2	4	no	—
132	C₃×C₄⋊C₄	—	2	2	no	—
133	C₃×C₄⋊C₄	—	2	4	no	—
134	C₃×C₄⋊C₄	—	4	4	no	—
135	C₃×C₂²⋊C₄	—	2	4	no	—
136	C₃×C₂²⋊C₄	—	2	8	no	—
137	C₃×C₂²⋊C₄	—	4	4	no	—
138	C₃×C₄²	—	2	4	no	—
139	C₃×C₄²	C₃×D₄	1	4	yes	—
140	C₂²×C₉	—	2	4	no	—
141	C₂²×C₉	—	2	8	no	—
142	C₂²×C₉	—	4	2	no	—
143	C₂²×C₉	—	4	2	no	—
144	C₂²×C₉	—	4	4	no	—
145	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
146	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
147	C₂²×C₉	C₂×#41	1	1	yes	characteristic
148	C₂²×C₉	C₂×#41	1	2	yes	—
149	C₂²×C₉	H(5,𝔽₂)	1	2	yes	—
150	C₄×C₉	—	2	2	no	—
151	C₄×C₉	—	2	4	no	—
152	C₄×C₉	—	2	2	no	—
153	C₄×C₉	—	4	2	no	—
154	C₄×C₉	—	4	2	no	—
155	C₄×C₉	—	4	4	no	—
156	C₄×C₉	C₂²⋊Q₈	1	4	yes	—
157	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
158	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
159	C₂³×C₄	C₂²×C₉	1	2	yes	—
160	C₄×Q₈	—	2	4	no	—
161	C₄×D₄	—	2	4	no	—
162	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
163	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
164	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
166	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
167	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
168	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
169	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
170	C₂×C₄²	C₂²×C₉	1	2	yes	—
171	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
172	#95	C₂²×C₉	1	2	yes	—
173	#95	C₂²×C₉	1	2	yes	—
174	#95	C₂²×C₉	1	4	yes	—
175	C₂³×C₃	—	2	2	no	—
176	C₂³×C₃	—	2	4	no	—
177	C₂³×C₃	—	2	2	no	—
178	C₂³×C₃	—	2	4	no	—
179	C₂³×C₃	C₂×C₃×Q₈	1	4	yes	—
180	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
181	C₃×Q₈	—	2	8	no	—
182	C₃×D₄	—	2	8	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	2	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	4	no	—
190	C₂×C₃×C₄	—	2	4	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	4	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	4	8	no	—
196	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
197	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
198	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
199	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
200	C₂×C₃×C₄	C₃×#41	1	4	yes	—
201	C₂×C₃×C₄	C₃×#41	1	2	yes	—
202	C₂×C₉	—	4	4	no	—
203	C₂×C₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
204	C₂×C₉	#525	1	1	yes	characteristic
205	C₂×C₉	#535	1	1	yes	characteristic
206	C₂×C₉	#554	1	2	yes	—
207	C₂×C₉	#559	1	2	yes	—
208	C₂⁴	C₂³×C₉	1	2	yes	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	4	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	Q₈×C₉	1	4	yes	complemented
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	4	4	no	—
216	C₂²×C₄	C₂³×C₉	1	4	yes	—
217	C₂²×C₄	C₂³×C₉	1	2	yes	—
218	C₂²×C₄	C₂³×C₉	1	2	yes	—
219	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
220	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
221	C₂²×C₄	C₂³×C₉	1	2	yes	—
222	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₂²⋊C₄	—	2	4	no	—
229	C₂²⋊C₄	—	2	8	no	—
230	C₂²⋊C₄	—	4	4	no	—
231	C₄²	—	2	4	no	—
232	C₄²	D₄×C₉	1	4	yes	—
233	C₂²×C₃	—	2	4	no	—
234	C₂²×C₃	—	2	8	no	—
235	C₂²×C₃	—	4	4	no	—
236	C₂²×C₃	—	4	2	no	—
237	C₂²×C₃	—	4	2	no	—
238	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
239	C₂²×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
240	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
241	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
242	C₂²×C₃	C₃×H(5,𝔽₂)	1	2	yes	—
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	—	2	4	no	—
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	—	4	4	no	—
247	C₃×C₄	—	4	2	no	—
248	C₃×C₄	—	4	2	no	—
249	C₃×C₄	C₃×C₂²⋊Q₈	1	4	yes	—
250	C₉	#2580	1	1	yes	characteristic, direct product factor, Sylow
251	C₂³	—	2	2	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	4	no	—
255	C₂³	C₂×Q₈×C₉	1	4	yes	—
256	C₂³	C₂⁴×C₉	1	1	yes	characteristic
257	Q₈	—	2	8	no	—
258	D₄	—	2	8	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	8	no	—
272	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
273	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
274	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
275	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
276	C₂×C₄	C₉×#41	1	4	yes	—
277	C₂×C₄	C₉×#41	1	2	yes	—
278	C₂×C₃	—	4	4	no	—
279	C₂×C₃	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
280	C₂×C₃	C₃×#525	1	1	yes	characteristic
281	C₂×C₃	C₃×#535	1	1	yes	characteristic
282	C₂×C₃	C₃×#554	1	2	yes	—
283	C₂×C₃	C₃×#559	1	2	yes	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	8	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	4	no	—
289	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
290	C₂²	C₂²×Q₈×C₉	1	1	yes	characteristic
291	C₂²	C₂×C₉×#41	1	1	yes	characteristic
292	C₂²	C₂×C₉×#41	1	2	yes	—
293	C₂²	C₉×H(5,𝔽₂)	1	2	yes	—
294	C₄	—	2	2	no	—
295	C₄	—	2	4	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	4	no	—
300	C₄	C₉×C₂²⋊Q₈	1	4	yes	—
301	C₃	C₃×#2580	1	1	yes	characteristic
302	C₂	—	4	4	no	—
303	C₂	C₂×C₉×C₂²⋊Q₈	1	1	yes	characteristic
304	C₂	C₉×#525	1	1	yes	characteristic
305	C₂	C₉×#535	1	1	yes	characteristic
306	C₂	C₉×#554	1	2	yes	—
307	C₂	C₉×#559	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂³	C₂⁴×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	C₂²×C₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×Q₈×C₉	C₂	semidirect
3. C₉×#399	C₂	semidirect
4. C₉×#393	C₂	semidirect
6. C₉×#389	C₂	semidirect
7. C₉×#385	C₂	semidirect
8. C₉×#382	C₂	semidirect
41. C₂×D₄×C₉	Q₈	semidirect
61. #2580	C₉	direct (N × H)
107. C₂×C₄×Q₈	C₂×C₉	semidirect
109. #399	C₂×C₉	semidirect
110. #393	C₂×C₉	semidirect
112. #389	C₂×C₉	semidirect
113. #385	C₂×C₉	semidirect
114. #382	C₂×C₉	semidirect
213. C₂×D₄	Q₈×C₉	semidirect
250. C₉	#2580	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₉	2
C₂×C₄×D₄×C₉	2
C₉×#399	2
C₉×#393	2
C₉×#390	2
C₉×#389	2
C₉×#385	2
C₉×#382	2
C₃×#2580	3

Subgroups — C9×#2580 #11767301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2580 #11767301