Subgroups — C₉×#1884 #11766605

Order 1152.

498 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#568	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#532	C₂	1	1	yes	maximal, characteristic
3	C₉×#421	C₂	1	2	yes	maximal, complemented
4	C₉×#410	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#339	C₂	1	2	yes	maximal, complemented
6	C₃×#1884	C₃	1	1	yes	maximal, characteristic
7	C₂×C₉×#41	C₂²	1	1	yes	characteristic
8	C₂×C₉×#41	C₂²	1	1	yes	characteristic
9	C₉×#131	—	2	4	no	—
10	C₂×C₉×M₁₆	—	2	2	no	—
11	C₂×C₉×M₁₆	C₂²	1	2	yes	—
12	C₉×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
13	C₉×#127	C₄	1	1	yes	characteristic, complemented
14	C₉×#124	C₄	1	1	yes	characteristic, complemented
15	C₉×#124	C₄	1	1	yes	characteristic, complemented
16	C₉×#121	—	2	2	no	—
17	C₄×D₄×C₉	—	2	2	no	—
18	C₉×#117	C₂²	1	2	yes	—
19	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
20	C₉×#104	—	2	4	no	—
21	C₉×#99	—	2	2	no	—
22	C₃×#568	C₂×C₃	1	1	yes	characteristic
23	C₃×#532	C₂×C₃	1	1	yes	characteristic
24	C₃×#421	C₂×C₃	1	2	yes	—
25	C₃×#410	C₂×C₃	1	1	yes	characteristic
26	C₃×#339	C₂×C₃	1	2	yes	—
27	C₉×#41	—	2	2	no	—
28	C₉×#41	—	2	2	no	—
29	C₉×#41	—	2	2	no	—
30	C₉×#41	—	2	2	no	—
31	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
33	C₂×D₄×C₉	—	2	2	no	—
34	C₂×D₄×C₉	—	2	1	no	—
35	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
36	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₉	D₄	1	2	yes	—
38	C₂²×C₄×C₉	—	2	1	no	—
39	C₂²×C₄×C₉	—	2	1	no	—
40	C₂²×C₄×C₉	D₄	1	2	yes	—
41	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₉×M₁₆	—	2	4	no	—
43	C₉×M₁₆	—	2	2	no	—
44	C₉×M₁₆	D₄	1	4	yes	complemented
45	C₂×C₈×C₉	—	2	4	no	—
46	C₂×C₈×C₉	—	2	2	no	—
47	C₉×C₄⋊C₄	—	2	2	no	—
48	C₉×C₄⋊C₄	—	2	2	no	—
49	C₉×C₂²⋊C₄	—	2	2	no	—
50	C₉×C₂²⋊C₄	—	2	2	no	—
51	C₉×C₂²⋊C₄	—	2	2	no	—
52	C₄²×C₉	—	2	1	no	—
53	C₄²×C₉	—	2	2	no	—
54	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
56	#1884	C₉	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
59	C₃×#131	—	2	4	no	—
60	C₂×C₃×M₁₆	—	2	2	no	—
61	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
62	C₃×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
63	C₃×#127	C₃×C₄	1	1	yes	characteristic
64	C₃×#124	C₃×C₄	1	1	yes	characteristic
65	C₃×#124	C₃×C₄	1	1	yes	characteristic
66	C₃×#121	—	2	2	no	—
67	C₃×C₄×D₄	—	2	2	no	—
68	C₃×#117	C₂²×C₃	1	2	yes	—
69	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
70	C₃×#104	—	2	4	no	—
71	C₃×#99	—	2	2	no	—
72	C₂³×C₉	—	2	1	no	—
73	C₂³×C₉	C₂²⋊C₄	1	2	yes	—
74	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
75	Q₈×C₉	—	2	2	no	—
76	Q₈×C₉	—	2	2	no	—
77	D₄×C₉	—	2	4	no	—
78	D₄×C₉	—	2	2	no	—
79	D₄×C₉	—	2	2	no	—
80	D₄×C₉	—	4	4	no	—
81	D₄×C₉	—	4	2	no	—
82	C₂×C₄×C₉	—	2	1	no	—
83	C₂×C₄×C₉	—	2	2	no	—
84	C₂×C₄×C₉	—	2	2	no	—
85	C₂×C₄×C₉	—	2	1	no	—
86	C₂×C₄×C₉	—	2	1	no	—
87	C₂×C₄×C₉	—	4	2	no	—
88	C₂×C₄×C₉	—	4	1	no	—
89	C₂×C₄×C₉	—	4	1	no	—
90	C₂×C₄×C₉	—	4	2	no	—
91	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
92	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₉	#41	1	2	yes	—
96	C₈×C₉	—	2	4	no	—
97	#568	C₂×C₉	1	1	yes	characteristic, complemented
98	#532	C₂×C₉	1	1	yes	characteristic
99	#421	C₂×C₉	1	2	yes	complemented
100	#410	C₂×C₉	1	1	yes	characteristic, complemented
101	#339	C₂×C₉	1	2	yes	complemented
102	C₃×#41	—	2	2	no	—
103	C₃×#41	—	2	2	no	—
104	C₃×#41	—	2	2	no	—
105	C₃×#41	—	2	2	no	—
106	C₂×C₃×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
107	C₂×C₃×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂×C₃×D₄	—	2	2	no	—
109	C₂×C₃×D₄	—	2	1	no	—
110	C₂×C₃×D₄	C₂×C₃×C₄	1	1	yes	characteristic
111	C₂×C₃×D₄	C₂×C₃×C₄	1	1	yes	characteristic
112	C₂×C₃×D₄	C₃×D₄	1	2	yes	—
113	C₂²×C₃×C₄	—	2	1	no	—
114	C₂²×C₃×C₄	—	2	1	no	—
115	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
116	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
117	C₃×M₁₆	—	2	4	no	—
118	C₃×M₁₆	—	2	2	no	—
119	C₃×M₁₆	C₃×D₄	1	4	yes	—
120	C₂×C₃×C₈	—	2	4	no	—
121	C₂×C₃×C₈	—	2	2	no	—
122	C₃×C₄⋊C₄	—	2	2	no	—
123	C₃×C₄⋊C₄	—	2	2	no	—
124	C₃×C₂²⋊C₄	—	2	2	no	—
125	C₃×C₂²⋊C₄	—	2	2	no	—
126	C₃×C₂²⋊C₄	—	2	2	no	—
127	C₃×C₄²	—	2	1	no	—
128	C₃×C₄²	—	2	2	no	—
129	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
130	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
131	C₂²×C₉	—	2	2	no	—
132	C₂²×C₉	—	4	1	no	—
133	C₂²×C₉	—	4	2	no	—
134	C₂²×C₉	—	4	2	no	—
135	C₂²×C₉	—	4	1	no	—
136	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₉	#121	1	2	yes	—
138	C₄×C₉	—	2	2	no	—
139	C₄×C₉	—	2	2	no	—
140	C₄×C₉	—	4	1	no	—
141	C₄×C₉	—	4	1	no	—
142	C₄×C₉	—	8	2	no	—
143	C₄×C₉	C₄×D₄	1	2	yes	—
144	C₄×C₉	#120	1	1	yes	characteristic
145	C₄×C₉	#123	1	1	yes	characteristic
146	C₂×#41	C₂²×C₉	1	1	yes	characteristic
147	C₂×#41	C₂²×C₉	1	1	yes	characteristic
148	#131	—	2	4	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	C₂²×C₉	1	2	yes	—
151	C₄⋊Q₈	C₄×C₉	1	1	yes	characteristic, complemented
152	#127	C₄×C₉	1	1	yes	characteristic, complemented
153	#124	C₄×C₉	1	1	yes	characteristic, complemented
154	#124	C₄×C₉	1	1	yes	characteristic, complemented
155	#121	—	2	2	no	—
156	C₄×D₄	—	2	2	no	—
157	#117	C₂²×C₉	1	2	yes	—
158	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
159	#104	—	2	4	no	—
160	#99	—	2	2	no	—
161	C₂³×C₃	—	2	1	no	—
162	C₂³×C₃	C₃×C₂²⋊C₄	1	2	yes	—
163	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
164	C₃×Q₈	—	2	2	no	—
165	C₃×Q₈	—	2	2	no	—
166	C₃×D₄	—	2	4	no	—
167	C₃×D₄	—	2	2	no	—
168	C₃×D₄	—	2	2	no	—
169	C₃×D₄	—	4	4	no	—
170	C₃×D₄	—	4	2	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	4	2	no	—
177	C₂×C₃×C₄	—	4	1	no	—
178	C₂×C₃×C₄	—	4	1	no	—
179	C₂×C₃×C₄	—	4	2	no	—
180	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
181	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
183	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
184	C₂×C₃×C₄	C₃×#41	1	2	yes	—
185	C₃×C₈	—	2	4	no	—
186	C₂×C₉	—	2	1	no	—
187	C₂×C₉	—	2	2	no	—
188	C₂×C₉	—	4	2	no	—
189	C₂×C₉	—	8	1	no	—
190	C₂×C₉	#386	1	1	yes	characteristic
191	#41	—	2	2	no	—
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	C₂×Q₈	C₂×C₄×C₉	1	1	yes	characteristic
196	C₂×Q₈	C₂×C₄×C₉	1	1	yes	characteristic
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	C₂×C₄×C₉	1	1	yes	characteristic
200	C₂×D₄	C₂×C₄×C₉	1	1	yes	characteristic
201	C₂×D₄	D₄×C₉	1	2	yes	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	D₄×C₉	1	2	yes	—
205	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
206	M₁₆	—	2	4	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	D₄×C₉	1	4	yes	complemented
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	2	2	no	—
218	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
219	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
220	C₂²×C₃	—	2	2	no	—
221	C₂²×C₃	—	4	1	no	—
222	C₂²×C₃	—	4	2	no	—
223	C₂²×C₃	—	4	2	no	—
224	C₂²×C₃	—	4	1	no	—
225	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
226	C₂²×C₃	C₃×#121	1	2	yes	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	—	4	1	no	—
230	C₃×C₄	—	4	1	no	—
231	C₃×C₄	—	8	2	no	—
232	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
233	C₃×C₄	C₃×#120	1	1	yes	characteristic
234	C₃×C₄	C₃×#123	1	1	yes	characteristic
235	C₉	#1884	1	1	yes	characteristic, direct product factor, Sylow
236	C₂³	—	2	1	no	—
237	C₂³	C₉×C₂²⋊C₄	1	2	yes	—
238	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
239	Q₈	—	2	2	no	—
240	Q₈	—	2	2	no	—
241	D₄	—	2	4	no	—
242	D₄	—	2	2	no	—
243	D₄	—	2	2	no	—
244	D₄	—	4	4	no	—
245	D₄	—	4	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
256	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
257	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
258	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
259	C₂×C₄	C₉×#41	1	2	yes	—
260	C₈	—	2	4	no	—
261	C₂×C₃	—	2	1	no	—
262	C₂×C₃	—	2	2	no	—
263	C₂×C₃	—	4	2	no	—
264	C₂×C₃	—	8	1	no	—
265	C₂×C₃	C₃×#386	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
272	C₂²	C₉×#121	1	2	yes	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	8	2	no	—
278	C₄	C₄×D₄×C₉	1	2	yes	—
279	C₄	C₉×#120	1	1	yes	characteristic
280	C₄	C₉×#123	1	1	yes	characteristic
281	C₃	C₃×#1884	1	1	yes	characteristic
282	C₂	—	2	1	no	—
283	C₂	—	2	2	no	—
284	C₂	—	4	2	no	—
285	C₂	—	8	1	no	—
286	C₂	C₉×#386	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₉	#120
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#386
prime radical	C₃×#532	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×#568	C₂	semidirect
3. C₉×#421	C₂	semidirect
4. C₉×#410	C₂	semidirect
5. C₉×#339	C₂	semidirect
12. C₉×C₄⋊Q₈	C₄	semidirect
13. C₉×#127	C₄	semidirect
14. C₉×#124	C₄	semidirect
15. C₉×#124	C₄	semidirect
44. C₉×M₁₆	D₄	semidirect
56. #1884	C₉	direct (N × H)
97. #568	C₂×C₉	semidirect
99. #421	C₂×C₉	semidirect
100. #410	C₂×C₉	semidirect
101. #339	C₂×C₉	semidirect
151. C₄⋊Q₈	C₄×C₉	semidirect
152. #127	C₄×C₉	semidirect
153. #124	C₄×C₉	semidirect
154. #124	C₄×C₉	semidirect
208. M₁₆	D₄×C₉	semidirect
235. C₉	#1884	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#568	2
C₉×#532	2
C₉×#421	2
C₉×#410	2
C₉×#339	2
C₃×#1884	3

Subgroups — C9×#1884 #11766605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1884 #11766605