Subgroups — C₉×#2092 #11766813

Order 1152.

411 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#107	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#106	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×#102	C₂	1	1	yes	maximal, characteristic
4	C₂×C₉×#102	C₂	1	1	yes	maximal, characteristic
5	C₉×#390	C₂	1	1	yes	maximal, characteristic
6	C₉×#382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2092	C₃	1	1	yes	maximal, characteristic
9	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₉×C₂²⋊C₄	—	2	2	no	—
16	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
17	C₉×#107	—	2	2	no	—
18	C₉×#106	—	2	2	no	—
19	C₉×#102	—	2	2	no	—
20	C₉×#102	—	2	2	no	—
21	C₉×#95	—	2	2	no	—
22	C₂×C₃×#107	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×#106	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×#102	C₂×C₃	1	1	yes	characteristic
25	C₂×C₃×#102	C₂×C₃	1	1	yes	characteristic
26	C₃×#390	C₂×C₃	1	1	yes	characteristic
27	C₃×#382	C₂×C₃	1	1	yes	characteristic
28	C₃×#336	C₂×C₃	1	1	yes	characteristic
29	C₂⁴×C₉	—	2	1	no	—
30	C₂×D₄×C₉	—	2	1	no	—
31	C₂×D₄×C₉	—	2	1	no	—
32	C₂×D₄×C₉	—	2	1	no	—
33	C₂×D₄×C₉	—	2	1	no	—
34	C₂×D₄×C₉	—	2	1	no	—
35	C₂×D₄×C₉	Q₈	1	2	yes	—
36	C₂²×C₄×C₉	—	2	1	no	—
37	C₂²×C₄×C₉	—	2	1	no	—
38	C₂²×C₄×C₉	—	2	1	no	—
39	C₂²×C₄×C₉	D₄	1	2	yes	—
40	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₉	—	2	1	no	—
42	C₂×C₈×C₉	—	2	1	no	—
43	C₂×C₈×C₉	—	2	1	no	—
44	C₂×C₈×C₉	—	2	1	no	—
45	C₂×C₈×C₉	D₄	1	2	yes	—
46	C₂×C₈×C₉	D₄	1	2	yes	—
47	C₉×C₄⋊C₄	—	2	1	no	—
48	C₉×C₄⋊C₄	—	2	1	no	—
49	C₉×C₄⋊C₄	—	2	1	no	—
50	C₉×C₄⋊C₄	—	2	1	no	—
51	C₉×C₄⋊C₄	—	2	1	no	—
52	C₉×C₄⋊C₄	—	2	1	no	—
53	C₉×C₂²⋊C₄	—	4	4	no	—
54	C₄²×C₉	—	2	1	no	—
55	C₄²×C₉	—	2	1	no	—
56	#2092	C₉	1	1	yes	characteristic, direct product factor, Sylow
57	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
58	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
64	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
65	C₃×#107	—	2	2	no	—
66	C₃×#106	—	2	2	no	—
67	C₃×#102	—	2	2	no	—
68	C₃×#102	—	2	2	no	—
69	C₃×#95	—	2	2	no	—
70	C₂³×C₉	—	2	2	no	—
71	C₂³×C₉	—	4	1	no	—
72	C₂³×C₉	—	4	1	no	—
73	C₂³×C₉	—	4	2	no	—
74	C₂³×C₉	—	4	2	no	—
75	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
76	D₄×C₉	—	2	2	no	—
77	D₄×C₉	—	2	2	no	—
78	D₄×C₉	—	4	1	no	—
79	D₄×C₉	—	4	1	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	2	2	no	—
82	C₂×C₄×C₉	—	2	2	no	—
83	C₂×C₄×C₉	—	4	1	no	—
84	C₂×C₄×C₉	—	4	1	no	—
85	C₂×C₄×C₉	—	4	1	no	—
86	C₂×C₄×C₉	—	4	2	no	—
87	C₂×C₄×C₉	—	4	2	no	—
88	C₂×C₄×C₉	—	4	2	no	—
89	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₉	#41	1	1	yes	characteristic
93	C₂×C₄×C₉	#41	1	1	yes	characteristic
94	C₂×C₄×C₉	#41	1	1	yes	characteristic
95	C₈×C₉	—	2	2	no	—
96	C₈×C₉	—	2	2	no	—
97	C₂×#107	C₂×C₉	1	1	yes	characteristic, complemented
98	C₂×#106	C₂×C₉	1	1	yes	characteristic, complemented
99	C₂×#102	C₂×C₉	1	1	yes	characteristic
100	C₂×#102	C₂×C₉	1	1	yes	characteristic
101	#390	C₂×C₉	1	1	yes	characteristic
102	#382	C₂×C₉	1	1	yes	characteristic, complemented
103	#336	C₂×C₉	1	1	yes	characteristic, complemented
104	C₂⁴×C₃	—	2	1	no	—
105	C₂×C₃×D₄	—	2	1	no	—
106	C₂×C₃×D₄	—	2	1	no	—
107	C₂×C₃×D₄	—	2	1	no	—
108	C₂×C₃×D₄	—	2	1	no	—
109	C₂×C₃×D₄	—	2	1	no	—
110	C₂×C₃×D₄	C₃×Q₈	1	2	yes	—
111	C₂²×C₃×C₄	—	2	1	no	—
112	C₂²×C₃×C₄	—	2	1	no	—
113	C₂²×C₃×C₄	—	2	1	no	—
114	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
115	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
116	C₂×C₃×C₈	—	2	1	no	—
117	C₂×C₃×C₈	—	2	1	no	—
118	C₂×C₃×C₈	—	2	1	no	—
119	C₂×C₃×C₈	—	2	1	no	—
120	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
121	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
122	C₃×C₄⋊C₄	—	2	1	no	—
123	C₃×C₄⋊C₄	—	2	1	no	—
124	C₃×C₄⋊C₄	—	2	1	no	—
125	C₃×C₄⋊C₄	—	2	1	no	—
126	C₃×C₄⋊C₄	—	2	1	no	—
127	C₃×C₄⋊C₄	—	2	1	no	—
128	C₃×C₂²⋊C₄	—	4	4	no	—
129	C₃×C₄²	—	2	1	no	—
130	C₃×C₄²	—	2	1	no	—
131	C₂²×C₉	—	4	2	no	—
132	C₂²×C₉	—	4	4	no	—
133	C₂²×C₉	—	8	2	no	—
134	C₂²×C₉	—	8	1	no	—
135	C₂²×C₉	—	8	1	no	—
136	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂²×C₉	#123	1	1	yes	characteristic
138	C₂²×C₉	#127	1	1	yes	characteristic
139	C₂²×C₉	#135	1	2	yes	—
140	C₂²×C₉	AGL(1,ℤ/8)	1	2	yes	—
141	C₄×C₉	—	4	2	no	—
142	C₄×C₉	—	8	1	no	—
143	C₄×C₉	—	8	1	no	—
144	C₄×C₉	—	8	1	no	—
145	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
146	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
147	C₄×C₉	#123	1	1	yes	characteristic
148	C₄×C₉	#123	1	1	yes	characteristic
149	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
150	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
151	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
152	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	—	2	2	no	—
156	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
157	#107	—	2	2	no	—
158	#106	—	2	2	no	—
159	#102	—	2	2	no	—
160	#102	—	2	2	no	—
161	#95	—	2	2	no	—
162	C₂³×C₃	—	2	2	no	—
163	C₂³×C₃	—	4	1	no	—
164	C₂³×C₃	—	4	1	no	—
165	C₂³×C₃	—	4	2	no	—
166	C₂³×C₃	—	4	2	no	—
167	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
168	C₃×D₄	—	2	2	no	—
169	C₃×D₄	—	2	2	no	—
170	C₃×D₄	—	4	1	no	—
171	C₃×D₄	—	4	1	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	4	1	no	—
176	C₂×C₃×C₄	—	4	1	no	—
177	C₂×C₃×C₄	—	4	1	no	—
178	C₂×C₃×C₄	—	4	2	no	—
179	C₂×C₃×C₄	—	4	2	no	—
180	C₂×C₃×C₄	—	4	2	no	—
181	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
184	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
185	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
186	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
187	C₃×C₈	—	2	2	no	—
188	C₃×C₈	—	2	2	no	—
189	C₂×C₉	—	8	2	no	—
190	C₂×C₉	#399	1	1	yes	characteristic
191	C₂×C₉	#478	1	2	yes	—
192	C₂×C₉	#482	1	2	yes	—
193	C₂×C₉	#495	1	1	yes	characteristic
194	C₂×C₉	#496	1	1	yes	characteristic
195	C₂⁴	—	2	1	no	—
196	C₂×D₄	—	2	1	no	—
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	Q₈×C₉	1	2	yes	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	D₄×C₉	1	2	yes	—
206	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	D₄×C₉	1	2	yes	—
212	C₂×C₈	D₄×C₉	1	2	yes	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	—	2	1	no	—
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	4	4	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	2	1	no	—
222	C₂²×C₃	—	4	2	no	—
223	C₂²×C₃	—	4	4	no	—
224	C₂²×C₃	—	8	2	no	—
225	C₂²×C₃	—	8	1	no	—
226	C₂²×C₃	—	8	1	no	—
227	C₂²×C₃	C₃×C₂²⋊Q₈	1	1	yes	characteristic
228	C₂²×C₃	C₃×#123	1	1	yes	characteristic
229	C₂²×C₃	C₃×#127	1	1	yes	characteristic
230	C₂²×C₃	C₃×#135	1	2	yes	—
231	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	2	yes	—
232	C₃×C₄	—	4	2	no	—
233	C₃×C₄	—	8	1	no	—
234	C₃×C₄	—	8	1	no	—
235	C₃×C₄	—	8	1	no	—
236	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
237	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
238	C₃×C₄	C₃×#123	1	1	yes	characteristic
239	C₃×C₄	C₃×#123	1	1	yes	characteristic
240	C₉	#2092	1	1	yes	characteristic, direct product factor, Sylow
241	C₂³	—	2	2	no	—
242	C₂³	—	4	1	no	—
243	C₂³	—	4	1	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
247	D₄	—	2	2	no	—
248	D₄	—	2	2	no	—
249	D₄	—	4	1	no	—
250	D₄	—	4	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
261	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
262	C₂×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
263	C₂×C₄	C₉×#41	1	1	yes	characteristic
264	C₂×C₄	C₉×#41	1	1	yes	characteristic
265	C₂×C₄	C₉×#41	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	2	2	no	—
268	C₂×C₃	—	8	2	no	—
269	C₂×C₃	C₃×#399	1	1	yes	characteristic
270	C₂×C₃	C₃×#478	1	2	yes	—
271	C₂×C₃	C₃×#482	1	2	yes	—
272	C₂×C₃	C₃×#495	1	1	yes	characteristic
273	C₂×C₃	C₃×#496	1	1	yes	characteristic
274	C₂²	—	4	2	no	—
275	C₂²	—	4	4	no	—
276	C₂²	—	8	2	no	—
277	C₂²	—	8	1	no	—
278	C₂²	—	8	1	no	—
279	C₂²	C₉×C₂²⋊Q₈	1	1	yes	characteristic
280	C₂²	C₉×#123	1	1	yes	characteristic
281	C₂²	C₉×#127	1	1	yes	characteristic
282	C₂²	C₉×#135	1	2	yes	—
283	C₂²	C₉×AGL(1,ℤ/8)	1	2	yes	—
284	C₄	—	4	2	no	—
285	C₄	—	8	1	no	—
286	C₄	—	8	1	no	—
287	C₄	—	8	1	no	—
288	C₄	C₉×C₂²⋊Q₈	1	1	yes	characteristic
289	C₄	C₉×C₂²⋊Q₈	1	1	yes	characteristic
290	C₄	C₉×#123	1	1	yes	characteristic
291	C₄	C₉×#123	1	1	yes	characteristic
292	C₃	C₃×#2092	1	1	yes	characteristic
293	C₂	—	8	2	no	—
294	C₂	C₉×#399	1	1	yes	characteristic
295	C₂	C₉×#478	1	2	yes	—
296	C₂	C₉×#482	1	2	yes	—
297	C₂	C₉×#495	1	1	yes	characteristic
298	C₂	C₉×#496	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	C₂²×C₃×D₄	C₂²×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×#107	C₂	semidirect
2. C₂×C₉×#106	C₂	semidirect
6. C₉×#382	C₂	semidirect
7. C₉×#336	C₂	semidirect
56. #2092	C₉	direct (N × H)
97. C₂×#107	C₂×C₉	semidirect
98. C₂×#106	C₂×C₉	semidirect
102. #382	C₂×C₉	semidirect
103. #336	C₂×C₉	semidirect
240. C₉	#2092	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×C₉×#107	2
C₂×C₉×#106	2
C₂×C₉×#102	2
C₉×#390	2
C₉×#382	2
C₉×#336	2
C₃×#2092	3

Subgroups — C9×#2092 #11766813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2092 #11766813