Subgroups — C₉×#2001 #11766722

Order 1152.

588 conjugacy classes of subgroups.

The subgroup lattice has 279 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×C₂²⋊Q₈	C₂	1	2	yes	maximal
3	C₂×C₉×#103	C₂	1	2	yes	maximal, complemented
4	C₂×C₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₉×#340	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2001	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	C₂²	1	2	yes	—
8	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×Q₁₆	—	2	4	no	—
10	C₂×C₉×Q₁₆	—	2	2	no	—
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₉×C₂²⋊Q₈	—	2	4	no	—
14	C₉×C₂²⋊Q₈	—	2	4	no	—
15	C₂×C₉×C₄⋊C₄	—	2	2	no	—
16	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₉×C₂²⋊C₄	—	2	2	no	—
18	C₉×#103	—	2	4	no	—
19	C₉×C₂²⋊C₈	—	2	2	no	—
20	C₂²×C₃×Q₁₆	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×C₂²⋊Q₈	C₂×C₃	1	2	yes	—
22	C₂×C₃×#103	C₂×C₃	1	2	yes	—
23	C₂×C₃×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
24	C₃×#340	C₂×C₃	1	1	yes	characteristic
25	C₂⁴×C₉	D₄	1	1	yes	characteristic
26	C₂×Q₈×C₉	—	2	2	no	—
27	C₂×Q₈×C₉	—	2	2	no	—
28	C₂×Q₈×C₉	—	2	2	no	—
29	C₂×Q₈×C₉	—	2	2	no	—
30	C₂×Q₈×C₉	—	2	2	no	—
31	C₂×Q₈×C₉	D₄	1	4	yes	—
32	C₂²×C₄×C₉	—	2	2	no	—
33	C₂²×C₄×C₉	—	2	2	no	—
34	C₂²×C₄×C₉	—	2	2	no	—
35	C₂²×C₄×C₉	—	2	2	no	—
36	C₂²×C₄×C₉	—	2	1	no	—
37	C₂²×C₄×C₉	—	2	1	no	—
38	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₉×Q₁₆	—	2	8	no	—
41	C₂×C₈×C₉	—	2	2	no	—
42	C₂×C₈×C₉	—	2	1	no	—
43	C₂×C₈×C₉	—	2	2	no	—
44	C₂×C₈×C₉	D₄	1	2	yes	—
45	C₉×C₄⋊C₄	—	2	8	no	—
46	C₉×C₄⋊C₄	—	2	2	no	—
47	C₉×C₄⋊C₄	D₄	1	4	yes	—
48	C₉×C₂²⋊C₄	—	2	8	no	—
49	#2001	C₉	1	1	yes	characteristic, direct product factor, Sylow
50	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
51	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₃×Q₁₆	—	2	4	no	—
53	C₂×C₃×Q₁₆	—	2	2	no	—
54	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
55	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
56	C₃×C₂²⋊Q₈	—	2	4	no	—
57	C₃×C₂²⋊Q₈	—	2	4	no	—
58	C₂×C₃×C₄⋊C₄	—	2	2	no	—
59	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
60	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
61	C₃×#103	—	2	4	no	—
62	C₃×C₂²⋊C₈	—	2	2	no	—
63	C₂³×C₉	—	2	2	no	—
64	C₂³×C₉	—	2	1	no	—
65	C₂³×C₉	—	2	1	no	—
66	C₂³×C₉	—	2	2	no	—
67	C₂³×C₉	Q₁₆	1	2	yes	complemented
68	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₉	—	2	4	no	—
70	Q₈×C₉	—	2	4	no	—
71	Q₈×C₉	—	4	2	no	—
72	Q₈×C₉	—	4	2	no	—
73	C₂×C₄×C₉	—	2	4	no	—
74	C₂×C₄×C₉	—	2	4	no	—
75	C₂×C₄×C₉	—	2	2	no	—
76	C₂×C₄×C₉	—	2	1	no	—
77	C₂×C₄×C₉	—	4	2	no	—
78	C₂×C₄×C₉	—	4	2	no	—
79	C₂×C₄×C₉	—	4	2	no	—
80	C₂×C₄×C₉	—	4	2	no	—
81	C₂×C₄×C₉	—	4	1	no	—
82	C₂×C₄×C₉	—	4	2	no	—
83	C₂×C₄×C₉	—	4	1	no	—
84	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₉	#41	1	1	yes	characteristic
88	C₈×C₉	—	2	2	no	—
89	C₈×C₉	—	4	1	no	—
90	C₂²×Q₁₆	C₂×C₉	1	1	yes	characteristic, complemented
91	C₂×C₂²⋊Q₈	C₂×C₉	1	2	yes	—
92	C₂×#103	C₂×C₉	1	2	yes	complemented
93	C₂×C₂²⋊C₈	C₂×C₉	1	1	yes	characteristic
94	#340	C₂×C₉	1	1	yes	characteristic, complemented
95	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
96	C₂×C₃×Q₈	—	2	2	no	—
97	C₂×C₃×Q₈	—	2	2	no	—
98	C₂×C₃×Q₈	—	2	2	no	—
99	C₂×C₃×Q₈	—	2	2	no	—
100	C₂×C₃×Q₈	—	2	2	no	—
101	C₂×C₃×Q₈	C₃×D₄	1	4	yes	—
102	C₂²×C₃×C₄	—	2	2	no	—
103	C₂²×C₃×C₄	—	2	2	no	—
104	C₂²×C₃×C₄	—	2	1	no	—
105	C₂²×C₃×C₄	—	2	2	no	—
106	C₂²×C₃×C₄	—	2	2	no	—
107	C₂²×C₃×C₄	—	2	1	no	—
108	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
110	C₃×Q₁₆	—	2	8	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	—	2	1	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
115	C₃×C₄⋊C₄	—	2	8	no	—
116	C₃×C₄⋊C₄	—	2	2	no	—
117	C₃×C₄⋊C₄	C₃×D₄	1	4	yes	—
118	C₃×C₂²⋊C₄	—	2	8	no	—
119	C₂²×C₉	—	2	4	no	—
120	C₂²×C₉	—	2	2	no	—
121	C₂²×C₉	—	4	1	no	—
122	C₂²×C₉	—	4	2	no	—
123	C₂²×C₉	—	4	1	no	—
124	C₂²×C₉	#120	1	2	yes	—
125	C₂²×C₉	#121	1	1	yes	characteristic
126	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
127	C₂²×C₉	#135	1	1	yes	characteristic
128	C₂²×C₉	#137	1	2	yes	—
129	C₄×C₉	—	4	1	no	—
130	C₄×C₉	—	4	4	no	—
131	C₄×C₉	—	4	4	no	—
132	C₄×C₉	#120	1	1	yes	characteristic
133	C₄×C₉	#121	1	2	yes	—
134	C₄×C₉	#127	1	1	yes	characteristic
135	C₂²×Q₈	C₂²×C₉	1	2	yes	—
136	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
137	C₂×Q₁₆	—	2	4	no	—
138	C₂×Q₁₆	—	2	2	no	—
139	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
140	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
141	C₂²⋊Q₈	—	2	4	no	—
142	C₂²⋊Q₈	—	2	4	no	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
145	C₂×C₂²⋊C₄	—	2	2	no	—
146	#103	—	2	4	no	—
147	C₂²⋊C₈	—	2	2	no	—
148	C₂³×C₃	—	2	2	no	—
149	C₂³×C₃	—	2	1	no	—
150	C₂³×C₃	—	2	1	no	—
151	C₂³×C₃	—	2	2	no	—
152	C₂³×C₃	C₃×Q₁₆	1	2	yes	—
153	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
154	C₃×Q₈	—	2	4	no	—
155	C₃×Q₈	—	2	4	no	—
156	C₃×Q₈	—	4	2	no	—
157	C₃×Q₈	—	4	2	no	—
158	C₂×C₃×C₄	—	2	4	no	—
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	4	2	no	—
163	C₂×C₃×C₄	—	4	2	no	—
164	C₂×C₃×C₄	—	4	2	no	—
165	C₂×C₃×C₄	—	4	2	no	—
166	C₂×C₃×C₄	—	4	1	no	—
167	C₂×C₃×C₄	—	4	2	no	—
168	C₂×C₃×C₄	—	4	1	no	—
169	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
170	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
172	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
173	C₃×C₈	—	2	2	no	—
174	C₃×C₈	—	4	1	no	—
175	C₂×C₉	—	4	2	no	—
176	C₂×C₉	#392	1	1	yes	characteristic
177	C₂×C₉	#451	1	2	yes	—
178	C₂×C₉	#452	1	2	yes	—
179	C₂×C₉	#467	1	1	yes	characteristic
180	C₂×C₉	#470	1	1	yes	characteristic
181	C₂⁴	D₄×C₉	1	1	yes	characteristic
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	D₄×C₉	1	4	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
196	Q₁₆	—	2	8	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	D₄×C₉	1	2	yes	—
201	C₄⋊C₄	—	2	8	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	D₄×C₉	1	4	yes	—
204	C₂²⋊C₄	—	2	8	no	—
205	C₂²×C₃	—	2	4	no	—
206	C₂²×C₃	—	2	2	no	—
207	C₂²×C₃	—	4	1	no	—
208	C₂²×C₃	—	4	2	no	—
209	C₂²×C₃	—	4	1	no	—
210	C₂²×C₃	C₃×#120	1	2	yes	—
211	C₂²×C₃	C₃×#121	1	1	yes	characteristic
212	C₂²×C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic
213	C₂²×C₃	C₃×#135	1	1	yes	characteristic
214	C₂²×C₃	C₃×#137	1	2	yes	—
215	C₃×C₄	—	4	1	no	—
216	C₃×C₄	—	4	4	no	—
217	C₃×C₄	—	4	4	no	—
218	C₃×C₄	C₃×#120	1	1	yes	characteristic
219	C₃×C₄	C₃×#121	1	2	yes	—
220	C₃×C₄	C₃×#127	1	1	yes	characteristic
221	C₉	#2001	1	1	yes	characteristic, direct product factor, Sylow
222	C₂³	—	2	2	no	—
223	C₂³	—	2	1	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	2	no	—
226	C₂³	C₉×Q₁₆	1	2	yes	complemented
227	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
228	Q₈	—	2	4	no	—
229	Q₈	—	2	4	no	—
230	Q₈	—	4	2	no	—
231	Q₈	—	4	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
244	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
246	C₂×C₄	C₉×#41	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	4	1	no	—
249	C₂×C₃	—	4	2	no	—
250	C₂×C₃	C₃×#392	1	1	yes	characteristic
251	C₂×C₃	C₃×#451	1	2	yes	—
252	C₂×C₃	C₃×#452	1	2	yes	—
253	C₂×C₃	C₃×#467	1	1	yes	characteristic
254	C₂×C₃	C₃×#470	1	1	yes	characteristic
255	C₂²	—	2	4	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	C₉×#120	1	2	yes	—
261	C₂²	C₉×#121	1	1	yes	characteristic
262	C₂²	C₂×C₉×Q₁₆	1	1	yes	characteristic
263	C₂²	C₉×#135	1	1	yes	characteristic
264	C₂²	C₉×#137	1	2	yes	—
265	C₄	—	4	1	no	—
266	C₄	—	4	4	no	—
267	C₄	—	4	4	no	—
268	C₄	C₉×#120	1	1	yes	characteristic
269	C₄	C₉×#121	1	2	yes	—
270	C₄	C₉×#127	1	1	yes	characteristic
271	C₃	C₃×#2001	1	1	yes	characteristic
272	C₂	—	4	2	no	—
273	C₂	C₉×#392	1	1	yes	characteristic
274	C₂	C₉×#451	1	2	yes	—
275	C₂	C₉×#452	1	2	yes	—
276	C₂	C₉×#467	1	1	yes	characteristic
277	C₂	C₉×#470	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂⁴×C₃	C₃×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂³×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₉×Q₁₆	C₂	semidirect
3. C₂×C₉×#103	C₂	semidirect
5. C₉×#340	C₂	semidirect
49. #2001	C₉	direct (N × H)
67. C₂³×C₉	Q₁₆	semidirect
90. C₂²×Q₁₆	C₂×C₉	semidirect
92. C₂×#103	C₂×C₉	semidirect
94. #340	C₂×C₉	semidirect
221. C₉	#2001	direct (N × H)
226. C₂³	C₉×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×Q₁₆	2
C₂×C₉×C₂²⋊Q₈	2
C₂×C₉×#103	2
C₂×C₉×C₂²⋊C₈	2
C₉×#340	2
C₃×#2001	3

Subgroups — C9×#2001 #11766722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2001 #11766722