Subgroups — C₉×#2326 #11767047

Order 1152.

918 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
4	C₉×#391	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#390	C₂	1	1	yes	maximal, characteristic
6	C₉×#388	C₂	1	4	yes	maximal, complemented
7	C₉×#385	C₂	1	4	yes	maximal, complemented
8	C₉×#384	C₂	1	1	yes	maximal, characteristic, complemented
9	C₉×#376	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#2326	C₃	1	1	yes	maximal, characteristic
11	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₉	C₂²	1	2	yes	—
14	C₉×#124	C₄	1	8	yes	complemented
15	C₄×Q₈×C₉	—	2	4	no	—
16	C₄×D₄×C₉	—	2	4	no	—
17	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
21	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
24	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₉	C₂²	1	2	yes	—
26	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
28	C₉×#95	C₂²	1	2	yes	—
29	C₉×#95	C₂²	1	2	yes	—
30	C₉×#95	C₂²	1	4	yes	—
31	C₂×C₃×#124	C₂×C₃	1	1	yes	characteristic
32	C₂×C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
33	C₂×C₃×C₄×D₄	C₂×C₃	1	1	yes	characteristic
34	C₃×#391	C₂×C₃	1	1	yes	characteristic
35	C₃×#390	C₂×C₃	1	1	yes	characteristic
36	C₃×#388	C₂×C₃	1	4	yes	—
37	C₃×#385	C₂×C₃	1	4	yes	—
38	C₃×#384	C₂×C₃	1	1	yes	characteristic
39	C₃×#376	C₂×C₃	1	1	yes	characteristic
40	C₂⁴×C₉	C₂³	1	2	yes	—
41	C₂×Q₈×C₉	—	2	2	no	—
42	C₂×Q₈×C₉	—	2	2	no	—
43	C₂×Q₈×C₉	C₂×C₄	1	4	yes	complemented
44	C₂×D₄×C₉	—	2	2	no	—
45	C₂×D₄×C₉	—	2	2	no	—
46	C₂×D₄×C₉	C₂×C₄	1	4	yes	—
47	C₂²×C₄×C₉	—	2	4	no	—
48	C₂²×C₄×C₉	—	4	4	no	—
49	C₂²×C₄×C₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₉	C₂³	1	2	yes	—
51	C₂²×C₄×C₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₉	C₂³	1	2	yes	—
53	C₂²×C₄×C₉	C₂³	1	2	yes	—
54	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
57	C₉×C₄⋊C₄	—	2	2	no	—
58	C₉×C₄⋊C₄	—	2	4	no	—
59	C₉×C₄⋊C₄	—	2	2	no	—
60	C₉×C₄⋊C₄	—	4	4	no	—
61	C₉×C₂²⋊C₄	—	2	4	no	—
62	C₉×C₂²⋊C₄	—	4	4	no	—
63	C₉×C₂²⋊C₄	C₂×C₄	1	16	yes	complemented
64	C₄²×C₉	—	2	4	no	—
65	C₄²×C₉	—	2	2	no	—
66	C₄²×C₉	C₂×C₄	1	4	yes	complemented
67	C₄²×C₉	D₄	1	4	yes	—
68	#2326	C₉	1	1	yes	characteristic, direct product factor, Sylow
69	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
70	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
71	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
72	C₃×#124	C₃×C₄	1	8	yes	—
73	C₃×C₄×Q₈	—	2	4	no	—
74	C₃×C₄×D₄	—	2	4	no	—
75	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
78	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
79	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
81	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
82	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
84	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
86	C₃×#95	C₂²×C₃	1	2	yes	—
87	C₃×#95	C₂²×C₃	1	2	yes	—
88	C₃×#95	C₂²×C₃	1	4	yes	—
89	C₂³×C₉	—	2	2	no	—
90	C₂³×C₉	—	2	2	no	—
91	C₂³×C₉	—	2	4	no	—
92	C₂³×C₉	—	2	4	no	—
93	C₂³×C₉	C₂²×C₄	1	4	yes	—
94	C₂³×C₉	C₂⁴	1	1	yes	characteristic
95	Q₈×C₉	—	2	8	no	—
96	D₄×C₉	—	2	8	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	2	2	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	2	no	—
105	C₂×C₄×C₉	—	2	2	no	—
106	C₂×C₄×C₉	—	2	2	no	—
107	C₂×C₄×C₉	—	2	4	no	—
108	C₂×C₄×C₉	—	2	4	no	—
109	C₂×C₄×C₉	—	2	2	no	—
110	C₂×C₄×C₉	—	4	8	no	—
111	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
112	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
113	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
114	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₉	#41	1	2	yes	—
118	C₂×#124	C₂×C₉	1	1	yes	characteristic
119	C₂×C₄×Q₈	C₂×C₉	1	1	yes	characteristic, complemented
120	C₂×C₄×D₄	C₂×C₉	1	1	yes	characteristic
121	#391	C₂×C₉	1	1	yes	characteristic, complemented
122	#390	C₂×C₉	1	1	yes	characteristic
123	#388	C₂×C₉	1	4	yes	complemented
124	#385	C₂×C₉	1	4	yes	complemented
125	#384	C₂×C₉	1	1	yes	characteristic, complemented
126	#376	C₂×C₉	1	1	yes	characteristic, complemented
127	C₂⁴×C₃	C₂³×C₃	1	2	yes	—
128	C₂×C₃×Q₈	—	2	2	no	—
129	C₂×C₃×Q₈	—	2	2	no	—
130	C₂×C₃×Q₈	C₂×C₃×C₄	1	4	yes	—
131	C₂×C₃×D₄	—	2	2	no	—
132	C₂×C₃×D₄	—	2	2	no	—
133	C₂×C₃×D₄	C₂×C₃×C₄	1	4	yes	—
134	C₂²×C₃×C₄	—	2	4	no	—
135	C₂²×C₃×C₄	—	4	4	no	—
136	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
137	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
138	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
139	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
140	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
141	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
142	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
143	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
144	C₃×C₄⋊C₄	—	2	2	no	—
145	C₃×C₄⋊C₄	—	2	4	no	—
146	C₃×C₄⋊C₄	—	2	2	no	—
147	C₃×C₄⋊C₄	—	4	4	no	—
148	C₃×C₂²⋊C₄	—	2	4	no	—
149	C₃×C₂²⋊C₄	—	4	4	no	—
150	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	16	yes	—
151	C₃×C₄²	—	2	4	no	—
152	C₃×C₄²	—	2	2	no	—
153	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
154	C₃×C₄²	C₃×D₄	1	4	yes	—
155	C₂²×C₉	—	2	4	no	—
156	C₂²×C₉	—	2	8	no	—
157	C₂²×C₉	—	4	2	no	—
158	C₂²×C₉	—	4	2	no	—
159	C₂²×C₉	—	4	4	no	—
160	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
161	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
162	C₂²×C₉	C₂×#41	1	1	yes	characteristic
163	C₂²×C₉	H(5,𝔽₂)	1	2	yes	—
164	C₂²×C₉	2₋¹⁺⁴	1	2	yes	—
165	C₄×C₉	—	2	4	no	—
166	C₄×C₉	—	2	4	no	—
167	C₄×C₉	—	2	2	no	—
168	C₄×C₉	—	2	2	no	—
169	C₄×C₉	—	4	2	no	—
170	C₄×C₉	—	4	2	no	—
171	C₄×C₉	—	4	2	no	—
172	C₄×C₉	C₄×D₄	1	4	yes	—
173	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
174	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
175	C₂³×C₄	C₂²×C₉	1	2	yes	—
176	#124	C₄×C₉	1	8	yes	complemented
177	C₄×Q₈	—	2	4	no	—
178	C₄×D₄	—	2	4	no	—
179	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
180	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
182	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
183	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
184	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
185	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
186	C₂×C₄²	C₂²×C₉	1	2	yes	—
187	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
188	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
189	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
190	#95	C₂²×C₉	1	2	yes	—
191	#95	C₂²×C₉	1	2	yes	—
192	#95	C₂²×C₉	1	4	yes	—
193	C₂³×C₃	—	2	2	no	—
194	C₂³×C₃	—	2	2	no	—
195	C₂³×C₃	—	2	4	no	—
196	C₂³×C₃	—	2	4	no	—
197	C₂³×C₃	C₂²×C₃×C₄	1	4	yes	—
198	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
199	C₃×Q₈	—	2	8	no	—
200	C₃×D₄	—	2	8	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	2	4	no	—
212	C₂×C₃×C₄	—	2	4	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	4	8	no	—
215	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
216	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
217	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
218	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
219	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
220	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
221	C₂×C₃×C₄	C₃×#41	1	2	yes	—
222	C₂×C₉	—	4	4	no	—
223	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
224	C₂×C₉	#520	1	2	yes	—
225	C₂×C₉	#535	1	1	yes	characteristic
226	C₂×C₉	#536	1	1	yes	characteristic
227	C₂×C₉	#542	1	2	yes	—
228	C₂⁴	C₂³×C₉	1	2	yes	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	C₂×C₄×C₉	1	4	yes	complemented
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	C₂×C₄×C₉	1	4	yes	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	4	4	no	—
237	C₂²×C₄	C₂³×C₉	1	2	yes	—
238	C₂²×C₄	C₂³×C₉	1	2	yes	—
239	C₂²×C₄	C₂³×C₉	1	2	yes	—
240	C₂²×C₄	C₂³×C₉	1	2	yes	—
241	C₂²×C₄	C₂³×C₉	1	2	yes	—
242	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
243	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
244	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₂²⋊C₄	—	4	4	no	—
251	C₂²⋊C₄	C₂×C₄×C₉	1	16	yes	complemented
252	C₄²	—	2	4	no	—
253	C₄²	—	2	2	no	—
254	C₄²	C₂×C₄×C₉	1	4	yes	complemented
255	C₄²	D₄×C₉	1	4	yes	—
256	C₂²×C₃	—	2	4	no	—
257	C₂²×C₃	—	2	8	no	—
258	C₂²×C₃	—	4	2	no	—
259	C₂²×C₃	—	4	2	no	—
260	C₂²×C₃	—	4	4	no	—
261	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
262	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
263	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
264	C₂²×C₃	C₃×H(5,𝔽₂)	1	2	yes	—
265	C₂²×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
266	C₃×C₄	—	2	4	no	—
267	C₃×C₄	—	2	4	no	—
268	C₃×C₄	—	2	2	no	—
269	C₃×C₄	—	2	2	no	—
270	C₃×C₄	—	4	2	no	—
271	C₃×C₄	—	4	2	no	—
272	C₃×C₄	—	4	2	no	—
273	C₃×C₄	C₃×C₄×D₄	1	4	yes	—
274	C₉	#2326	1	1	yes	characteristic, direct product factor, Sylow
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	2	4	no	—
278	C₂³	—	2	4	no	—
279	C₂³	C₂²×C₄×C₉	1	4	yes	—
280	C₂³	C₂⁴×C₉	1	1	yes	characteristic
281	Q₈	—	2	8	no	—
282	D₄	—	2	8	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	8	no	—
297	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
298	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
299	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
300	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
301	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
302	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
303	C₂×C₄	C₉×#41	1	2	yes	—
304	C₂×C₃	—	4	4	no	—
305	C₂×C₃	C₂×C₃×C₄×D₄	1	1	yes	characteristic
306	C₂×C₃	C₃×#520	1	2	yes	—
307	C₂×C₃	C₃×#535	1	1	yes	characteristic
308	C₂×C₃	C₃×#536	1	1	yes	characteristic
309	C₂×C₃	C₃×#542	1	2	yes	—
310	C₂²	—	2	8	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	4	no	—
315	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
316	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
317	C₂²	C₂×C₉×#41	1	1	yes	characteristic
318	C₂²	C₉×H(5,𝔽₂)	1	2	yes	—
319	C₂²	C₉×2₋¹⁺⁴	1	2	yes	—
320	C₄	—	2	4	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	4	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	C₄×D₄×C₉	1	4	yes	—
328	C₃	C₃×#2326	1	1	yes	characteristic
329	C₂	—	4	4	no	—
330	C₂	C₂×C₄×D₄×C₉	1	1	yes	characteristic
331	C₂	C₉×#520	1	2	yes	—
332	C₂	C₉×#535	1	1	yes	characteristic
333	C₂	C₉×#536	1	1	yes	characteristic
334	C₂	C₉×#542	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₈×C₉	C₂	semidirect
4. C₉×#391	C₂	semidirect
6. C₉×#388	C₂	semidirect
7. C₉×#385	C₂	semidirect
8. C₉×#384	C₂	semidirect
9. C₉×#376	C₂	semidirect
14. C₉×#124	C₄	semidirect
43. C₂×Q₈×C₉	C₂×C₄	semidirect
63. C₉×C₂²⋊C₄	C₂×C₄	semidirect
66. C₄²×C₉	C₂×C₄	semidirect
68. #2326	C₉	direct (N × H)
119. C₂×C₄×Q₈	C₂×C₉	semidirect
121. #391	C₂×C₉	semidirect
123. #388	C₂×C₉	semidirect
124. #385	C₂×C₉	semidirect
125. #384	C₂×C₉	semidirect
126. #376	C₂×C₉	semidirect
176. #124	C₄×C₉	semidirect
231. C₂×Q₈	C₂×C₄×C₉	semidirect
251. C₂²⋊C₄	C₂×C₄×C₉	semidirect
254. C₄²	C₂×C₄×C₉	semidirect
274. C₉	#2326	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×#124	2
C₂×C₄×Q₈×C₉	2
C₂×C₄×D₄×C₉	2
C₉×#391	2
C₉×#390	2
C₉×#388	2
C₉×#385	2
C₉×#384	2
C₉×#376	2
C₃×#2326	3

Subgroups — C9×#2326 #11767047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2326 #11767047