Subgroups — C₉×#2951 #11767672

Order 1152.

726 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	C₂×C₉×#135	C₂	1	1	yes	maximal, characteristic
2	C₉×#520	C₂	1	2	yes	maximal, complemented
3	C₉×#440	C₂	1	4	yes	maximal, complemented
4	C₄×C₉×Q₁₆	C₂	1	2	yes	maximal, complemented
5	C₄×C₉×D₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₉×#107	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#417	C₂	1	2	yes	maximal, complemented
8	C₉×#405	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#2951	C₃	1	1	yes	maximal, characteristic
10	C₂×C₉×#41	C₂²	1	2	yes	—
11	C₉×#135	C₄	1	8	yes	complemented
12	C₂×C₉×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
14	C₂×C₉×D₈	C₂²	1	1	yes	characteristic
15	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₉	C₂²	1	4	yes	—
18	C₄×D₄×C₉	—	2	4	no	—
19	C₄×D₄×C₉	C₂²	1	4	yes	—
20	C₉×#117	—	2	2	no	—
21	C₉×#117	C₂²	1	1	yes	characteristic
22	C₂×C₉×C₄⋊C₄	—	2	2	no	—
23	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
24	C₉×#107	C₂²	1	2	yes	—
25	C₉×#107	C₂²	1	2	yes	—
26	C₉×#103	C₂²	1	4	yes	—
27	C₉×#102	C₂²	1	4	yes	—
28	C₉×#97	C₂²	1	2	yes	—
29	C₄×C₈×C₉	C₂²	1	2	yes	—
30	C₂×C₃×#135	C₂×C₃	1	1	yes	characteristic
31	C₃×#520	C₂×C₃	1	2	yes	—
32	C₃×#440	C₂×C₃	1	4	yes	—
33	C₃×C₄×Q₁₆	C₂×C₃	1	2	yes	—
34	C₃×C₄×D₈	C₂×C₃	1	2	yes	—
35	C₂×C₃×#107	C₂×C₃	1	1	yes	characteristic
36	C₃×#417	C₂×C₃	1	2	yes	—
37	C₃×#405	C₂×C₃	1	1	yes	characteristic
38	C₉×#41	—	2	4	no	—
39	C₉×#41	C₂×C₄	1	8	yes	complemented
40	C₂×Q₈×C₉	C₂³	1	2	yes	—
41	C₂×D₄×C₉	—	2	2	no	—
42	C₂×D₄×C₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₉	—	2	2	no	—
44	C₂²×C₄×C₉	—	2	2	no	—
45	C₂²×C₄×C₉	—	2	4	no	—
46	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
47	C₉×Q₁₆	C₂×C₄	1	4	yes	complemented
48	C₉×SD₁₆	C₂×C₄	1	8	yes	complemented
49	C₉×D₈	C₂×C₄	1	4	yes	complemented
50	C₉×M₁₆	—	2	2	no	—
51	C₂×C₈×C₉	C₂×C₄	1	4	yes	complemented
52	C₂×C₈×C₉	C₂³	1	2	yes	—
53	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
55	C₉×C₄⋊C₄	—	2	2	no	—
56	C₉×C₄⋊C₄	—	2	4	no	—
57	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
58	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
59	C₉×C₄⋊C₄	C₂³	1	4	yes	—
60	C₉×C₂²⋊C₄	—	2	4	no	—
61	C₉×C₂²⋊C₄	D₄	1	2	yes	—
62	C₄²×C₉	—	2	4	no	—
63	C₄²×C₉	C₂³	1	2	yes	—
64	#2951	C₉	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
66	C₃×#135	C₃×C₄	1	8	yes	—
67	C₂×C₃×Q₁₆	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
69	C₂×C₃×D₈	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×C₄×Q₈	C₂²×C₃	1	4	yes	—
73	C₃×C₄×D₄	—	2	4	no	—
74	C₃×C₄×D₄	C₂²×C₃	1	4	yes	—
75	C₃×#117	—	2	2	no	—
76	C₃×#117	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₃×C₄⋊C₄	—	2	2	no	—
78	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
79	C₃×#107	C₂²×C₃	1	2	yes	—
80	C₃×#107	C₂²×C₃	1	2	yes	—
81	C₃×#103	C₂²×C₃	1	4	yes	—
82	C₃×#102	C₂²×C₃	1	4	yes	—
83	C₃×#97	C₂²×C₃	1	2	yes	—
84	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
85	C₂³×C₉	—	2	2	no	—
86	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
87	Q₈×C₉	—	2	2	no	—
88	Q₈×C₉	C₂²×C₄	1	4	yes	—
89	D₄×C₉	—	2	8	no	—
90	D₄×C₉	—	2	2	no	—
91	D₄×C₉	C₂²×C₄	1	4	yes	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	8	no	—
94	C₂×C₄×C₉	—	2	4	no	—
95	C₂×C₄×C₉	—	8	4	no	—
96	C₂×C₄×C₉	—	8	2	no	—
97	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₉	#41	1	2	yes	—
102	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
103	C₈×C₉	—	2	2	no	—
104	C₈×C₉	C₂²×C₄	1	2	yes	—
105	C₈×C₉	C₂²×C₄	1	2	yes	—
106	C₂×#135	C₂×C₉	1	1	yes	characteristic
107	#520	C₂×C₉	1	2	yes	complemented
108	#440	C₂×C₉	1	4	yes	complemented
109	C₄×Q₁₆	C₂×C₉	1	2	yes	complemented
110	C₄×D₈	C₂×C₉	1	2	yes	complemented
111	C₂×#107	C₂×C₉	1	1	yes	characteristic, complemented
112	#417	C₂×C₉	1	2	yes	complemented
113	#405	C₂×C₉	1	1	yes	characteristic, complemented
114	C₃×#41	—	2	4	no	—
115	C₃×#41	C₂×C₃×C₄	1	8	yes	—
116	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
117	C₂×C₃×D₄	—	2	2	no	—
118	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
119	C₂²×C₃×C₄	—	2	2	no	—
120	C₂²×C₃×C₄	—	2	2	no	—
121	C₂²×C₃×C₄	—	2	4	no	—
122	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
123	C₃×Q₁₆	C₂×C₃×C₄	1	4	yes	—
124	C₃×SD₁₆	C₂×C₃×C₄	1	8	yes	—
125	C₃×D₈	C₂×C₃×C₄	1	4	yes	—
126	C₃×M₁₆	—	2	2	no	—
127	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
128	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
129	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
130	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
131	C₃×C₄⋊C₄	—	2	2	no	—
132	C₃×C₄⋊C₄	—	2	4	no	—
133	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
134	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
135	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
136	C₃×C₂²⋊C₄	—	2	4	no	—
137	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
138	C₃×C₄²	—	2	4	no	—
139	C₃×C₄²	C₂³×C₃	1	2	yes	—
140	C₂²×C₉	—	2	1	no	—
141	C₂²×C₉	—	2	1	no	—
142	C₂²×C₉	—	2	4	no	—
143	C₂²×C₉	—	4	2	no	—
144	C₂²×C₉	—	4	2	no	—
145	C₂²×C₉	C₄×D₄	1	2	yes	—
146	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
147	C₄×C₉	—	2	4	no	—
148	C₄×C₉	—	2	2	no	—
149	C₄×C₉	—	2	2	no	—
150	C₄×C₉	—	4	4	no	—
151	C₄×C₉	C₄×D₄	1	2	yes	—
152	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
153	C₄×C₉	C₂×#41	1	1	yes	characteristic
154	C₂×#41	C₂²×C₉	1	2	yes	—
155	#135	C₄×C₉	1	8	yes	complemented
156	C₂×Q₁₆	C₂²×C₉	1	1	yes	characteristic
157	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
158	C₂×D₈	C₂²×C₉	1	1	yes	characteristic
159	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
160	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
161	C₄×Q₈	C₂²×C₉	1	4	yes	—
162	C₄×D₄	—	2	4	no	—
163	C₄×D₄	C₂²×C₉	1	4	yes	—
164	#117	—	2	2	no	—
165	#117	C₂²×C₉	1	1	yes	characteristic
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
168	#107	C₂²×C₉	1	2	yes	—
169	#107	C₂²×C₉	1	2	yes	—
170	#103	C₂²×C₉	1	4	yes	—
171	#102	C₂²×C₉	1	4	yes	—
172	#97	C₂²×C₉	1	2	yes	—
173	C₄×C₈	C₂²×C₉	1	2	yes	—
174	C₂³×C₃	—	2	2	no	—
175	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
176	C₃×Q₈	—	2	2	no	—
177	C₃×Q₈	C₂²×C₃×C₄	1	4	yes	—
178	C₃×D₄	—	2	8	no	—
179	C₃×D₄	—	2	2	no	—
180	C₃×D₄	C₂²×C₃×C₄	1	4	yes	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	8	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	—	8	4	no	—
185	C₂×C₃×C₄	—	8	2	no	—
186	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
187	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
188	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
189	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
190	C₂×C₃×C₄	C₃×#41	1	2	yes	—
191	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
192	C₃×C₈	—	2	2	no	—
193	C₃×C₈	C₂²×C₃×C₄	1	2	yes	—
194	C₃×C₈	C₂²×C₃×C₄	1	2	yes	—
195	C₂×C₉	—	2	2	no	—
196	C₂×C₉	—	4	4	no	—
197	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
198	C₂×C₉	#576	1	1	yes	characteristic
199	C₂×C₉	#578	1	1	yes	characteristic
200	#41	—	2	4	no	—
201	#41	C₂×C₄×C₉	1	8	yes	complemented
202	C₂×Q₈	C₂³×C₉	1	2	yes	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	C₂³×C₉	1	2	yes	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
209	Q₁₆	C₂×C₄×C₉	1	4	yes	complemented
210	SD₁₆	C₂×C₄×C₉	1	8	yes	complemented
211	D₈	C₂×C₄×C₉	1	4	yes	complemented
212	M₁₆	—	2	2	no	—
213	C₂×C₈	C₂×C₄×C₉	1	4	yes	complemented
214	C₂×C₈	C₂³×C₉	1	2	yes	—
215	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
216	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	D₄×C₉	1	1	yes	characteristic
220	C₄⋊C₄	D₄×C₉	1	1	yes	characteristic
221	C₄⋊C₄	C₂³×C₉	1	4	yes	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₂²⋊C₄	D₄×C₉	1	2	yes	—
224	C₄²	—	2	4	no	—
225	C₄²	C₂³×C₉	1	2	yes	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	2	4	no	—
229	C₂²×C₃	—	4	2	no	—
230	C₂²×C₃	—	4	2	no	—
231	C₂²×C₃	C₃×C₄×D₄	1	2	yes	—
232	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	2	2	no	—
235	C₃×C₄	—	2	4	no	—
236	C₃×C₄	—	4	4	no	—
237	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
238	C₃×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
239	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
240	C₉	#2951	1	1	yes	characteristic, direct product factor, Sylow
241	C₂³	—	2	2	no	—
242	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
243	Q₈	—	2	2	no	—
244	Q₈	C₂²×C₄×C₉	1	4	yes	—
245	D₄	—	2	8	no	—
246	D₄	—	2	2	no	—
247	D₄	C₂²×C₄×C₉	1	4	yes	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	8	4	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
254	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
255	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
256	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
257	C₂×C₄	C₉×#41	1	2	yes	—
258	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	C₂²×C₄×C₉	1	2	yes	—
261	C₈	C₂²×C₄×C₉	1	2	yes	—
262	C₂×C₃	—	2	2	no	—
263	C₂×C₃	—	4	4	no	—
264	C₂×C₃	C₂×C₃×C₄×D₄	1	1	yes	characteristic
265	C₂×C₃	C₃×#576	1	1	yes	characteristic
266	C₂×C₃	C₃×#578	1	1	yes	characteristic
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	C₄×D₄×C₉	1	2	yes	—
273	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	4	no	—
277	C₄	—	4	4	no	—
278	C₄	C₄×D₄×C₉	1	2	yes	—
279	C₄	C₂³×C₄×C₉	1	1	yes	characteristic
280	C₄	C₂×C₉×#41	1	1	yes	characteristic
281	C₃	C₃×#2951	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	4	4	no	—
284	C₂	C₂×C₄×D₄×C₉	1	1	yes	characteristic
285	C₂	C₉×#576	1	1	yes	characteristic
286	C₂	C₉×#578	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₄	C₂³×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₂×C₃×#135	C₂×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#520	C₂	semidirect
3. C₉×#440	C₂	semidirect
4. C₄×C₉×Q₁₆	C₂	semidirect
5. C₄×C₉×D₈	C₂	semidirect
6. C₂×C₉×#107	C₂	semidirect
7. C₉×#417	C₂	semidirect
8. C₉×#405	C₂	semidirect
11. C₉×#135	C₄	semidirect
39. C₉×#41	C₂×C₄	semidirect
47. C₉×Q₁₆	C₂×C₄	semidirect
48. C₉×SD₁₆	C₂×C₄	semidirect
49. C₉×D₈	C₂×C₄	semidirect
51. C₂×C₈×C₉	C₂×C₄	semidirect
64. #2951	C₉	direct (N × H)
107. #520	C₂×C₉	semidirect
108. #440	C₂×C₉	semidirect
109. C₄×Q₁₆	C₂×C₉	semidirect
110. C₄×D₈	C₂×C₉	semidirect
111. C₂×#107	C₂×C₉	semidirect
112. #417	C₂×C₉	semidirect
113. #405	C₂×C₉	semidirect
155. #135	C₄×C₉	semidirect
201. #41	C₂×C₄×C₉	semidirect
209. Q₁₆	C₂×C₄×C₉	semidirect
210. SD₁₆	C₂×C₄×C₉	semidirect
211. D₈	C₂×C₄×C₉	semidirect
213. C₂×C₈	C₂×C₄×C₉	semidirect
240. C₉	#2951	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×#135	2
C₉×#520	2
C₉×#440	2
C₄×C₉×Q₁₆	2
C₄×C₉×D₈	2
C₂×C₉×#107	2
C₉×#417	2
C₉×#405	2
C₃×#2951	3

Subgroups — C9×#2951 #11767672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2951 #11767672