Subgroups — C₉×#3250 #11767971

Order 1152.

540 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#529	C₂	1	2	yes	maximal, complemented
2	C₉×#479	C₂	1	2	yes	maximal, complemented
3	C₉×#478	C₂	1	2	yes	maximal, complemented
4	C₉×#467	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#466	C₂	1	1	yes	maximal, characteristic
6	C₉×#465	C₂	1	2	yes	maximal, complemented
7	C₉×#464	C₂	1	2	yes	maximal, complemented
8	C₉×#463	C₂	1	2	yes	maximal, complemented
9	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#3250	C₃	1	1	yes	maximal, characteristic
11	C₂×C₉×Q₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
13	C₂×C₉×D₈	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
16	C₉×#126	—	2	2	no	—
17	C₉×#125	C₂²	1	2	yes	—
18	C₉×#124	C₂²	1	2	yes	—
19	C₉×#123	—	2	2	no	—
20	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₉×#121	C₂²	1	2	yes	—
22	C₄×Q₈×C₉	C₂²	1	2	yes	—
23	C₄×D₄×C₉	—	2	2	no	—
24	C₄×D₄×C₉	C₂²	1	2	yes	—
25	C₉×#117	—	2	2	no	—
26	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
27	C₉×#107	C₂²	1	1	yes	characteristic
28	C₉×#107	C₂²	1	1	yes	characteristic
29	C₉×#106	C₂²	1	2	yes	—
30	C₉×C₄⋊C₈	C₂²	1	2	yes	—
31	C₉×C₄⋊C₈	C₂²	1	2	yes	—
32	C₉×#103	C₂²	1	2	yes	—
33	C₉×#103	C₂²	1	2	yes	—
34	C₉×#102	C₂²	1	2	yes	—
35	C₉×#102	C₂²	1	2	yes	—
36	C₃×#529	C₂×C₃	1	2	yes	—
37	C₃×#479	C₂×C₃	1	2	yes	—
38	C₃×#478	C₂×C₃	1	2	yes	—
39	C₃×#467	C₂×C₃	1	1	yes	characteristic
40	C₃×#466	C₂×C₃	1	1	yes	characteristic
41	C₃×#465	C₂×C₃	1	2	yes	—
42	C₃×#464	C₂×C₃	1	2	yes	—
43	C₃×#463	C₂×C₃	1	2	yes	—
44	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
45	C₂×Q₈×C₉	C₂³	1	2	yes	—
46	C₂×D₄×C₉	—	2	2	no	—
47	C₂×D₄×C₉	C₂³	1	2	yes	—
48	C₂²×C₄×C₉	—	2	2	no	—
49	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
52	C₉×Q₁₆	—	4	1	no	—
53	C₉×SD₁₆	—	4	2	no	—
54	C₉×D₈	—	4	1	no	—
55	C₂×C₈×C₉	—	4	1	no	—
56	C₂×C₈×C₉	—	4	1	no	—
57	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₉	C₂³	1	2	yes	—
59	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
60	C₉×C₄⋊C₄	—	2	2	no	—
61	C₉×C₄⋊C₄	—	2	2	no	—
62	C₉×C₄⋊C₄	—	2	2	no	—
63	C₉×C₄⋊C₄	—	2	2	no	—
64	C₉×C₄⋊C₄	C₂³	1	2	yes	—
65	C₉×C₄⋊C₄	C₂³	1	2	yes	—
66	C₉×C₂²⋊C₄	—	2	2	no	—
67	C₉×C₂²⋊C₄	—	2	2	no	—
68	C₉×C₂²⋊C₄	—	2	2	no	—
69	C₉×C₂²⋊C₄	—	2	2	no	—
70	C₉×C₂²⋊C₄	—	2	2	no	—
71	C₄²×C₉	—	2	2	no	—
72	C₄²×C₉	D₄	1	2	yes	—
73	C₄²×C₉	C₂³	1	2	yes	—
74	#3250	C₉	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×C₃×Q₁₆	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
77	C₂×C₃×D₈	C₂²×C₃	1	1	yes	characteristic
78	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
79	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
80	C₃×#126	—	2	2	no	—
81	C₃×#125	C₂²×C₃	1	2	yes	—
82	C₃×#124	C₂²×C₃	1	2	yes	—
83	C₃×#123	—	2	2	no	—
84	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
85	C₃×#121	C₂²×C₃	1	2	yes	—
86	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
87	C₃×C₄×D₄	—	2	2	no	—
88	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
89	C₃×#117	—	2	2	no	—
90	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
91	C₃×#107	C₂²×C₃	1	1	yes	characteristic
92	C₃×#107	C₂²×C₃	1	1	yes	characteristic
93	C₃×#106	C₂²×C₃	1	2	yes	—
94	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
95	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
96	C₃×#103	C₂²×C₃	1	2	yes	—
97	C₃×#103	C₂²×C₃	1	2	yes	—
98	C₃×#102	C₂²×C₃	1	2	yes	—
99	C₃×#102	C₂²×C₃	1	2	yes	—
100	C₂³×C₉	—	2	2	no	—
101	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
102	Q₈×C₉	—	2	2	no	—
103	Q₈×C₉	—	2	2	no	—
104	D₄×C₉	—	2	2	no	—
105	D₄×C₉	—	2	2	no	—
106	D₄×C₉	—	4	4	no	—
107	C₂×C₄×C₉	—	2	2	no	—
108	C₂×C₄×C₉	—	2	1	no	—
109	C₂×C₄×C₉	—	2	1	no	—
110	C₂×C₄×C₉	—	2	2	no	—
111	C₂×C₄×C₉	—	2	2	no	—
112	C₂×C₄×C₉	—	2	2	no	—
113	C₂×C₄×C₉	—	4	1	no	—
114	C₂×C₄×C₉	—	4	4	no	—
115	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
119	C₈×C₉	—	2	1	no	—
120	C₈×C₉	—	2	2	no	—
121	C₈×C₉	—	2	1	no	—
122	#529	C₂×C₉	1	2	yes	complemented
123	#479	C₂×C₉	1	2	yes	complemented
124	#478	C₂×C₉	1	2	yes	complemented
125	#467	C₂×C₉	1	1	yes	characteristic, complemented
126	#466	C₂×C₉	1	1	yes	characteristic
127	#465	C₂×C₉	1	2	yes	complemented
128	#464	C₂×C₉	1	2	yes	complemented
129	#463	C₂×C₉	1	2	yes	complemented
130	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
131	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
132	C₂×C₃×D₄	—	2	2	no	—
133	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
134	C₂²×C₃×C₄	—	2	2	no	—
135	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
136	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
137	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
138	C₃×Q₁₆	—	4	1	no	—
139	C₃×SD₁₆	—	4	2	no	—
140	C₃×D₈	—	4	1	no	—
141	C₂×C₃×C₈	—	4	1	no	—
142	C₂×C₃×C₈	—	4	1	no	—
143	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
144	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
145	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
146	C₃×C₄⋊C₄	—	2	2	no	—
147	C₃×C₄⋊C₄	—	2	2	no	—
148	C₃×C₄⋊C₄	—	2	2	no	—
149	C₃×C₄⋊C₄	—	2	2	no	—
150	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
151	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
152	C₃×C₂²⋊C₄	—	2	2	no	—
153	C₃×C₂²⋊C₄	—	2	2	no	—
154	C₃×C₂²⋊C₄	—	2	2	no	—
155	C₃×C₂²⋊C₄	—	2	2	no	—
156	C₃×C₂²⋊C₄	—	2	2	no	—
157	C₃×C₄²	—	2	2	no	—
158	C₃×C₄²	C₃×D₄	1	2	yes	—
159	C₃×C₄²	C₂³×C₃	1	2	yes	—
160	C₂²×C₉	—	2	1	no	—
161	C₂²×C₉	—	2	1	no	—
162	C₂²×C₉	—	4	2	no	—
163	C₂²×C₉	—	4	2	no	—
164	C₂²×C₉	—	4	2	no	—
165	C₂²×C₉	#135	1	2	yes	—
166	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
167	C₄×C₉	—	2	1	no	—
168	C₄×C₉	—	2	2	no	—
169	C₄×C₉	—	2	2	no	—
170	C₄×C₉	—	4	2	no	—
171	C₄×C₉	—	4	2	no	—
172	C₄×C₉	—	4	2	no	—
173	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
174	C₄×C₉	2₋¹⁺⁴	1	1	yes	characteristic
175	C₂×Q₁₆	C₂²×C₉	1	1	yes	characteristic
176	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
177	C₂×D₈	C₂²×C₉	1	1	yes	characteristic
178	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
179	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
180	#126	—	2	2	no	—
181	#125	C₂²×C₉	1	2	yes	—
182	#124	C₂²×C₉	1	2	yes	—
183	#123	—	2	2	no	—
184	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
185	#121	C₂²×C₉	1	2	yes	—
186	C₄×Q₈	C₂²×C₉	1	2	yes	—
187	C₄×D₄	—	2	2	no	—
188	C₄×D₄	C₂²×C₉	1	2	yes	—
189	#117	—	2	2	no	—
190	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
191	#107	C₂²×C₉	1	1	yes	characteristic
192	#107	C₂²×C₉	1	1	yes	characteristic
193	#106	C₂²×C₉	1	2	yes	—
194	C₄⋊C₈	C₂²×C₉	1	2	yes	—
195	C₄⋊C₈	C₂²×C₉	1	2	yes	—
196	#103	C₂²×C₉	1	2	yes	—
197	#103	C₂²×C₉	1	2	yes	—
198	#102	C₂²×C₉	1	2	yes	—
199	#102	C₂²×C₉	1	2	yes	—
200	C₂³×C₃	—	2	2	no	—
201	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
202	C₃×Q₈	—	2	2	no	—
203	C₃×Q₈	—	2	2	no	—
204	C₃×D₄	—	2	2	no	—
205	C₃×D₄	—	2	2	no	—
206	C₃×D₄	—	4	4	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	4	1	no	—
214	C₂×C₃×C₄	—	4	4	no	—
215	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
216	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
218	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
219	C₃×C₈	—	2	1	no	—
220	C₃×C₈	—	2	2	no	—
221	C₃×C₈	—	2	1	no	—
222	C₂×C₉	—	2	2	no	—
223	C₂×C₉	—	8	2	no	—
224	C₂×C₉	#537	1	1	yes	characteristic
225	C₂×C₉	C₂×#135	1	1	yes	characteristic
226	C₂×C₉	#575	1	1	yes	characteristic
227	C₂×Q₈	C₂³×C₉	1	2	yes	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂³×C₉	1	2	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
232	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
233	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
234	Q₁₆	—	4	1	no	—
235	SD₁₆	—	4	2	no	—
236	D₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
240	C₂×C₈	C₂³×C₉	1	2	yes	—
241	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	C₂³×C₉	1	2	yes	—
247	C₄⋊C₄	C₂³×C₉	1	2	yes	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	2	2	no	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	D₄×C₉	1	2	yes	—
255	C₄²	C₂³×C₉	1	2	yes	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	4	2	no	—
259	C₂²×C₃	—	4	2	no	—
260	C₂²×C₃	—	4	2	no	—
261	C₂²×C₃	C₃×#135	1	2	yes	—
262	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
263	C₃×C₄	—	2	2	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	4	2	no	—
267	C₃×C₄	—	4	2	no	—
268	C₃×C₄	—	4	2	no	—
269	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
270	C₃×C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
271	C₉	#3250	1	1	yes	characteristic, direct product factor, Sylow
272	C₂³	—	2	2	no	—
273	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
274	Q₈	—	2	2	no	—
275	Q₈	—	2	2	no	—
276	D₄	—	2	2	no	—
277	D₄	—	2	2	no	—
278	D₄	—	4	4	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
288	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
289	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
290	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	2	no	—
293	C₈	—	2	1	no	—
294	C₂×C₃	—	2	2	no	—
295	C₂×C₃	—	8	2	no	—
296	C₂×C₃	C₃×#537	1	1	yes	characteristic
297	C₂×C₃	C₂×C₃×#135	1	1	yes	characteristic
298	C₂×C₃	C₃×#575	1	1	yes	characteristic
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	C₉×#135	1	2	yes	—
305	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	2	no	—
312	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
313	C₄	C₉×2₋¹⁺⁴	1	1	yes	characteristic
314	C₃	C₃×#3250	1	1	yes	characteristic
315	C₂	—	2	2	no	—
316	C₂	—	8	2	no	—
317	C₂	C₉×#537	1	1	yes	characteristic
318	C₂	C₂×C₉×#135	1	1	yes	characteristic
319	C₂	C₉×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₃×#466	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×#529	C₂	semidirect
2. C₉×#479	C₂	semidirect
3. C₉×#478	C₂	semidirect
4. C₉×#467	C₂	semidirect
6. C₉×#465	C₂	semidirect
7. C₉×#464	C₂	semidirect
8. C₉×#463	C₂	semidirect
9. C₂×C₉×C₄⋊C₈	C₂	semidirect
74. #3250	C₉	direct (N × H)
122. #529	C₂×C₉	semidirect
123. #479	C₂×C₉	semidirect
124. #478	C₂×C₉	semidirect
125. #467	C₂×C₉	semidirect
127. #465	C₂×C₉	semidirect
128. #464	C₂×C₉	semidirect
129. #463	C₂×C₉	semidirect
130. C₂×C₄⋊C₈	C₂×C₉	semidirect
271. C₉	#3250	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×#529	2
C₉×#479	2
C₉×#478	2
C₉×#467	2
C₉×#466	2
C₉×#465	2
C₉×#464	2
C₉×#463	2
C₂×C₉×C₄⋊C₈	2
C₃×#3250	3

Subgroups — C9×#3250 #11767971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3250 #11767971