Subgroups — C₉×#2046 #11766767

Order 1152.

414 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×SD₁₆	C₂	1	1	yes	maximal, characteristic
2	C₉×#424	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#419	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#418	C₂	1	1	yes	maximal, characteristic
5	C₂×C₉×#101	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₉×#100	C₂	1	1	yes	maximal, characteristic
7	C₉×#341	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2046	C₃	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₉×SD₁₆	—	2	2	no	—
12	C₂×C₉×SD₁₆	—	2	2	no	—
13	C₂×C₉×SD₁₆	—	2	2	no	—
14	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
18	C₉×#117	C₂²	1	1	yes	characteristic
19	C₉×C₄⋊C₈	—	2	2	no	—
20	C₉×#103	—	2	2	no	—
21	C₉×#102	—	2	2	no	—
22	C₉×#101	—	2	2	no	—
23	C₉×#100	—	2	2	no	—
24	C₂²×C₃×SD₁₆	C₂×C₃	1	1	yes	characteristic
25	C₃×#424	C₂×C₃	1	1	yes	characteristic
26	C₃×#419	C₂×C₃	1	1	yes	characteristic
27	C₃×#418	C₂×C₃	1	1	yes	characteristic
28	C₂×C₃×#101	C₂×C₃	1	1	yes	characteristic
29	C₂×C₃×#100	C₂×C₃	1	1	yes	characteristic
30	C₃×#341	C₂×C₃	1	1	yes	characteristic
31	C₂⁴×C₉	—	2	1	no	—
32	C₂×Q₈×C₉	—	2	1	no	—
33	C₂×Q₈×C₉	—	2	1	no	—
34	C₂×Q₈×C₉	—	2	1	no	—
35	C₂×Q₈×C₉	—	2	1	no	—
36	C₂×Q₈×C₉	—	2	1	no	—
37	C₂×Q₈×C₉	D₄	1	2	yes	—
38	C₂×D₄×C₉	—	2	1	no	—
39	C₂×D₄×C₉	—	2	1	no	—
40	C₂×D₄×C₉	—	2	1	no	—
41	C₂×D₄×C₉	—	2	1	no	—
42	C₂×D₄×C₉	—	2	1	no	—
43	C₂×D₄×C₉	D₄	1	2	yes	—
44	C₂²×C₄×C₉	—	2	1	no	—
45	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
46	C₉×SD₁₆	—	2	8	no	—
47	C₉×M₁₆	—	2	1	no	—
48	C₉×M₁₆	—	2	1	no	—
49	C₉×M₁₆	—	2	1	no	—
50	C₉×M₁₆	—	2	1	no	—
51	C₉×M₁₆	—	2	1	no	—
52	C₉×M₁₆	—	2	1	no	—
53	C₂×C₈×C₉	—	2	1	no	—
54	C₂×C₈×C₉	—	2	1	no	—
55	C₂×C₈×C₉	—	2	1	no	—
56	C₂×C₈×C₉	—	2	1	no	—
57	C₂×C₈×C₉	—	2	1	no	—
58	C₂×C₈×C₉	D₄	1	2	yes	—
59	C₉×C₄⋊C₄	D₄	1	2	yes	—
60	C₉×C₂²⋊C₄	—	2	1	no	—
61	C₄²×C₉	—	2	1	no	—
62	#2046	C₉	1	1	yes	characteristic, direct product factor, Sylow
63	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
64	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₃×SD₁₆	—	2	2	no	—
66	C₂×C₃×SD₁₆	—	2	2	no	—
67	C₂×C₃×SD₁₆	—	2	2	no	—
68	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
71	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
72	C₃×#117	C₂²×C₃	1	1	yes	characteristic
73	C₃×C₄⋊C₈	—	2	2	no	—
74	C₃×#103	—	2	2	no	—
75	C₃×#102	—	2	2	no	—
76	C₃×#101	—	2	2	no	—
77	C₃×#100	—	2	2	no	—
78	C₂³×C₉	—	2	2	no	—
79	C₂³×C₉	—	4	1	no	—
80	C₂³×C₉	—	4	1	no	—
81	C₂³×C₉	—	8	2	no	—
82	C₂³×C₉	#41	1	1	yes	characteristic
83	Q₈×C₉	—	2	2	no	—
84	Q₈×C₉	—	2	2	no	—
85	Q₈×C₉	—	4	1	no	—
86	Q₈×C₉	—	4	1	no	—
87	D₄×C₉	—	2	2	no	—
88	D₄×C₉	—	2	2	no	—
89	D₄×C₉	—	4	1	no	—
90	D₄×C₉	—	4	1	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	4	1	no	—
94	C₂×C₄×C₉	—	4	1	no	—
95	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₉	#41	1	1	yes	characteristic
100	C₂×C₄×C₉	#41	1	1	yes	characteristic
101	C₈×C₉	—	2	2	no	—
102	C₈×C₉	—	4	1	no	—
103	C₈×C₉	—	4	1	no	—
104	C₈×C₉	—	4	1	no	—
105	C₂²×SD₁₆	C₂×C₉	1	1	yes	characteristic
106	#424	C₂×C₉	1	1	yes	characteristic, complemented
107	#419	C₂×C₉	1	1	yes	characteristic, complemented
108	#418	C₂×C₉	1	1	yes	characteristic
109	C₂×#101	C₂×C₉	1	1	yes	characteristic, complemented
110	C₂×#100	C₂×C₉	1	1	yes	characteristic
111	#341	C₂×C₉	1	1	yes	characteristic, complemented
112	C₂⁴×C₃	—	2	1	no	—
113	C₂×C₃×Q₈	—	2	1	no	—
114	C₂×C₃×Q₈	—	2	1	no	—
115	C₂×C₃×Q₈	—	2	1	no	—
116	C₂×C₃×Q₈	—	2	1	no	—
117	C₂×C₃×Q₈	—	2	1	no	—
118	C₂×C₃×Q₈	C₃×D₄	1	2	yes	—
119	C₂×C₃×D₄	—	2	1	no	—
120	C₂×C₃×D₄	—	2	1	no	—
121	C₂×C₃×D₄	—	2	1	no	—
122	C₂×C₃×D₄	—	2	1	no	—
123	C₂×C₃×D₄	—	2	1	no	—
124	C₂×C₃×D₄	C₃×D₄	1	2	yes	—
125	C₂²×C₃×C₄	—	2	1	no	—
126	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
127	C₃×SD₁₆	—	2	8	no	—
128	C₃×M₁₆	—	2	1	no	—
129	C₃×M₁₆	—	2	1	no	—
130	C₃×M₁₆	—	2	1	no	—
131	C₃×M₁₆	—	2	1	no	—
132	C₃×M₁₆	—	2	1	no	—
133	C₃×M₁₆	—	2	1	no	—
134	C₂×C₃×C₈	—	2	1	no	—
135	C₂×C₃×C₈	—	2	1	no	—
136	C₂×C₃×C₈	—	2	1	no	—
137	C₂×C₃×C₈	—	2	1	no	—
138	C₂×C₃×C₈	—	2	1	no	—
139	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
140	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
141	C₃×C₂²⋊C₄	—	2	1	no	—
142	C₃×C₄²	—	2	1	no	—
143	C₂²×C₉	—	2	2	no	—
144	C₂²×C₉	—	4	2	no	—
145	C₂²×C₉	—	8	2	no	—
146	C₂²×C₉	—	8	2	no	—
147	C₂²×C₉	—	8	2	no	—
148	C₂²×C₉	#121	1	1	yes	characteristic
149	C₂²×C₉	#123	1	1	yes	characteristic
150	C₂²×C₉	#123	1	1	yes	characteristic
151	C₄×C₉	—	4	2	no	—
152	C₄×C₉	—	4	2	no	—
153	C₄×C₉	#120	1	1	yes	characteristic
154	C₄×C₉	#121	1	1	yes	characteristic
155	C₄×C₉	#121	1	1	yes	characteristic
156	C₄×C₉	#124	1	1	yes	characteristic
157	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
158	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×SD₁₆	—	2	2	no	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
163	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
164	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
165	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
166	#117	C₂²×C₉	1	1	yes	characteristic
167	C₄⋊C₈	—	2	2	no	—
168	#103	—	2	2	no	—
169	#102	—	2	2	no	—
170	#101	—	2	2	no	—
171	#100	—	2	2	no	—
172	C₂³×C₃	—	2	2	no	—
173	C₂³×C₃	—	4	1	no	—
174	C₂³×C₃	—	4	1	no	—
175	C₂³×C₃	—	8	2	no	—
176	C₂³×C₃	C₃×#41	1	1	yes	characteristic
177	C₃×Q₈	—	2	2	no	—
178	C₃×Q₈	—	2	2	no	—
179	C₃×Q₈	—	4	1	no	—
180	C₃×Q₈	—	4	1	no	—
181	C₃×D₄	—	2	2	no	—
182	C₃×D₄	—	2	2	no	—
183	C₃×D₄	—	4	1	no	—
184	C₃×D₄	—	4	1	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	4	1	no	—
188	C₂×C₃×C₄	—	4	1	no	—
189	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
193	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
194	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
195	C₃×C₈	—	2	2	no	—
196	C₃×C₈	—	4	1	no	—
197	C₃×C₈	—	4	1	no	—
198	C₃×C₈	—	4	1	no	—
199	C₂×C₉	—	2	1	no	—
200	C₂×C₉	—	2	1	no	—
201	C₂×C₉	—	8	2	no	—
202	C₂×C₉	#394	1	1	yes	characteristic
203	C₂×C₉	#471	1	2	yes	—
204	C₂⁴	—	2	1	no	—
205	C₂×Q₈	—	2	1	no	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	2	1	no	—
208	C₂×Q₈	—	2	1	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×Q₈	D₄×C₉	1	2	yes	—
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	—	2	1	no	—
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	D₄×C₉	1	2	yes	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
219	SD₁₆	—	2	8	no	—
220	M₁₆	—	2	1	no	—
221	M₁₆	—	2	1	no	—
222	M₁₆	—	2	1	no	—
223	M₁₆	—	2	1	no	—
224	M₁₆	—	2	1	no	—
225	M₁₆	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	D₄×C₉	1	2	yes	—
232	C₄⋊C₄	D₄×C₉	1	2	yes	—
233	C₂²⋊C₄	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	4	2	no	—
237	C₂²×C₃	—	8	2	no	—
238	C₂²×C₃	—	8	2	no	—
239	C₂²×C₃	—	8	2	no	—
240	C₂²×C₃	C₃×#121	1	1	yes	characteristic
241	C₂²×C₃	C₃×#123	1	1	yes	characteristic
242	C₂²×C₃	C₃×#123	1	1	yes	characteristic
243	C₃×C₄	—	4	2	no	—
244	C₃×C₄	—	4	2	no	—
245	C₃×C₄	C₃×#120	1	1	yes	characteristic
246	C₃×C₄	C₃×#121	1	1	yes	characteristic
247	C₃×C₄	C₃×#121	1	1	yes	characteristic
248	C₃×C₄	C₃×#124	1	1	yes	characteristic
249	C₉	#2046	1	1	yes	characteristic, direct product factor, Sylow
250	C₂³	—	2	2	no	—
251	C₂³	—	4	1	no	—
252	C₂³	—	4	1	no	—
253	C₂³	—	8	2	no	—
254	C₂³	C₉×#41	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	Q₈	—	2	2	no	—
257	Q₈	—	4	1	no	—
258	Q₈	—	4	1	no	—
259	D₄	—	2	2	no	—
260	D₄	—	2	2	no	—
261	D₄	—	4	1	no	—
262	D₄	—	4	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
269	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
270	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
271	C₂×C₄	C₉×#41	1	1	yes	characteristic
272	C₂×C₄	C₉×#41	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	—	4	1	no	—
275	C₈	—	4	1	no	—
276	C₈	—	4	1	no	—
277	C₂×C₃	—	2	1	no	—
278	C₂×C₃	—	2	1	no	—
279	C₂×C₃	—	8	2	no	—
280	C₂×C₃	C₃×#394	1	1	yes	characteristic
281	C₂×C₃	C₃×#471	1	2	yes	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	C₉×#121	1	1	yes	characteristic
288	C₂²	C₉×#123	1	1	yes	characteristic
289	C₂²	C₉×#123	1	1	yes	characteristic
290	C₄	—	4	2	no	—
291	C₄	—	4	2	no	—
292	C₄	C₉×#120	1	1	yes	characteristic
293	C₄	C₉×#121	1	1	yes	characteristic
294	C₄	C₉×#121	1	1	yes	characteristic
295	C₄	C₉×#124	1	1	yes	characteristic
296	C₃	C₃×#2046	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	8	2	no	—
300	C₂	C₉×#394	1	1	yes	characteristic
301	C₂	C₉×#471	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	#121
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₃×#121
prime radical	C₂²×C₃×D₄	C₂²×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#424	C₂	semidirect
3. C₉×#419	C₂	semidirect
5. C₂×C₉×#101	C₂	semidirect
7. C₉×#341	C₂	semidirect
62. #2046	C₉	direct (N × H)
106. #424	C₂×C₉	semidirect
107. #419	C₂×C₉	semidirect
109. C₂×#101	C₂×C₉	semidirect
111. #341	C₂×C₉	semidirect
249. C₉	#2046	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×SD₁₆	2
C₉×#424	2
C₉×#419	2
C₉×#418	2
C₂×C₉×#101	2
C₂×C₉×#100	2
C₉×#341	2
C₃×#2046	3

Subgroups — C9×#2046 #11766767

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2046 #11766767