Subgroups — C₉×#3043 #11767764

Order 1152.

735 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₉×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#535	C₂	1	1	yes	maximal, characteristic
4	C₉×#519	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#464	C₂	1	4	yes	maximal, complemented
6	C₉×#460	C₂	1	4	yes	maximal, complemented
7	C₉×#424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₉×#418	C₂	1	1	yes	maximal, characteristic, complemented
9	C₉×#416	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#3043	C₃	1	1	yes	maximal, characteristic
11	C₂×C₉×#41	C₂²	1	1	yes	characteristic
12	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
14	C₉×#137	—	2	4	no	—
15	C₂×C₉×Q₁₆	C₂²	1	2	yes	complemented
16	C₂×C₉×SD₁₆	—	2	4	no	—
17	C₂×C₉×SD₁₆	C₂²	1	4	yes	—
18	C₂×C₉×SD₁₆	C₂²	1	2	yes	complemented
19	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
21	C₉×#127	C₂²	1	2	yes	—
22	C₉×#124	C₂²	1	2	yes	—
23	C₉×#121	—	2	2	no	—
24	C₉×#120	—	2	2	no	—
25	C₄×Q₈×C₉	—	2	2	no	—
26	C₄×Q₈×C₉	C₂²	1	4	yes	complemented
27	C₉×#117	—	2	2	no	—
28	C₉×#117	C₂²	1	1	yes	characteristic
29	C₉×#117	C₂²	1	1	yes	characteristic
30	C₉×C₄⋊C₈	C₂²	1	4	yes	complemented
31	C₉×#103	C₂²	1	2	yes	complemented
32	C₉×#102	C₂²	1	4	yes	—
33	C₉×#102	C₂²	1	2	yes	complemented
34	C₂×C₃×#137	C₂×C₃	1	1	yes	characteristic
35	C₂²×C₃×SD₁₆	C₂×C₃	1	1	yes	characteristic
36	C₃×#535	C₂×C₃	1	1	yes	characteristic
37	C₃×#519	C₂×C₃	1	1	yes	characteristic
38	C₃×#464	C₂×C₃	1	4	yes	—
39	C₃×#460	C₂×C₃	1	4	yes	—
40	C₃×#424	C₂×C₃	1	1	yes	characteristic
41	C₃×#418	C₂×C₃	1	1	yes	characteristic
42	C₃×#416	C₂×C₃	1	1	yes	characteristic
43	C₂⁴×C₉	—	2	1	no	—
44	C₉×#41	—	2	2	no	—
45	C₉×#41	—	2	2	no	—
46	C₂×Q₈×C₉	—	2	1	no	—
47	C₂×Q₈×C₉	—	2	2	no	—
48	C₂×Q₈×C₉	D₄	1	4	yes	complemented
49	C₂×Q₈×C₉	C₂³	1	1	yes	characteristic
50	C₂×Q₈×C₉	C₂³	1	2	yes	—
51	C₂×D₄×C₉	—	2	1	no	—
52	C₂×D₄×C₉	—	2	2	no	—
53	C₂×D₄×C₉	—	2	1	no	—
54	C₂×D₄×C₉	—	2	4	no	—
55	C₂×D₄×C₉	—	2	2	no	—
56	C₂×D₄×C₉	C₂³	1	2	yes	—
57	C₂×D₄×C₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₉	—	2	1	no	—
59	C₂²×C₄×C₉	—	2	1	no	—
60	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
61	C₉×Q₁₆	—	2	4	no	—
62	C₉×SD₁₆	—	2	8	no	—
63	C₉×SD₁₆	—	2	4	no	—
64	C₉×M₁₆	—	2	2	no	—
65	C₂×C₈×C₉	—	2	2	no	—
66	C₂×C₈×C₉	C₂³	1	2	yes	—
67	C₂×C₈×C₉	C₂³	1	2	yes	—
68	C₉×C₄⋊C₄	—	2	4	no	—
69	C₉×C₄⋊C₄	D₄	1	2	yes	—
70	C₉×C₄⋊C₄	C₂³	1	2	yes	—
71	C₉×C₂²⋊C₄	—	2	1	no	—
72	C₉×C₂²⋊C₄	—	2	4	no	—
73	C₉×C₂²⋊C₄	D₄	1	2	yes	—
74	C₄²×C₉	—	2	4	no	—
75	C₄²×C₉	—	2	1	no	—
76	C₄²×C₉	C₂³	1	2	yes	—
77	#3043	C₉	1	1	yes	characteristic, direct product factor, Sylow
78	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
79	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
80	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
81	C₃×#137	—	2	4	no	—
82	C₂×C₃×Q₁₆	C₂²×C₃	1	2	yes	—
83	C₂×C₃×SD₁₆	—	2	4	no	—
84	C₂×C₃×SD₁₆	C₂²×C₃	1	4	yes	—
85	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
86	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
87	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
88	C₃×#127	C₂²×C₃	1	2	yes	—
89	C₃×#124	C₂²×C₃	1	2	yes	—
90	C₃×#121	—	2	2	no	—
91	C₃×#120	—	2	2	no	—
92	C₃×C₄×Q₈	—	2	2	no	—
93	C₃×C₄×Q₈	C₂²×C₃	1	4	yes	—
94	C₃×#117	—	2	2	no	—
95	C₃×#117	C₂²×C₃	1	1	yes	characteristic
96	C₃×#117	C₂²×C₃	1	1	yes	characteristic
97	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
98	C₃×#103	C₂²×C₃	1	2	yes	—
99	C₃×#102	C₂²×C₃	1	4	yes	—
100	C₃×#102	C₂²×C₃	1	2	yes	—
101	C₂³×C₉	—	2	2	no	—
102	C₂³×C₉	—	2	1	no	—
103	C₂³×C₉	—	4	2	no	—
104	C₂³×C₉	—	8	2	no	—
105	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
106	Q₈×C₉	—	2	4	no	—
107	Q₈×C₉	—	2	2	no	—
108	Q₈×C₉	—	2	1	no	—
109	Q₈×C₉	—	2	1	no	—
110	Q₈×C₉	C₂×D₄	1	4	yes	—
111	D₄×C₉	—	2	2	no	—
112	D₄×C₉	—	2	1	no	—
113	D₄×C₉	—	2	2	no	—
114	D₄×C₉	—	2	1	no	—
115	D₄×C₉	—	4	2	no	—
116	D₄×C₉	—	4	2	no	—
117	D₄×C₉	—	8	4	no	—
118	C₂×C₄×C₉	—	2	4	no	—
119	C₂×C₄×C₉	—	2	2	no	—
120	C₂×C₄×C₉	—	2	2	no	—
121	C₂×C₄×C₉	—	2	1	no	—
122	C₂×C₄×C₉	—	4	2	no	—
123	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
124	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
126	C₂×C₄×C₉	#41	1	2	yes	—
127	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
128	C₈×C₉	—	2	2	no	—
129	C₈×C₉	—	2	2	no	—
130	C₂×#137	C₂×C₉	1	1	yes	characteristic, complemented
131	C₂²×SD₁₆	C₂×C₉	1	1	yes	characteristic, complemented
132	#535	C₂×C₉	1	1	yes	characteristic
133	#519	C₂×C₉	1	1	yes	characteristic, complemented
134	#464	C₂×C₉	1	4	yes	complemented
135	#460	C₂×C₉	1	4	yes	complemented
136	#424	C₂×C₉	1	1	yes	characteristic, complemented
137	#418	C₂×C₉	1	1	yes	characteristic, complemented
138	#416	C₂×C₉	1	1	yes	characteristic, complemented
139	C₂⁴×C₃	—	2	1	no	—
140	C₃×#41	—	2	2	no	—
141	C₃×#41	—	2	2	no	—
142	C₂×C₃×Q₈	—	2	1	no	—
143	C₂×C₃×Q₈	—	2	2	no	—
144	C₂×C₃×Q₈	C₃×D₄	1	4	yes	—
145	C₂×C₃×Q₈	C₂³×C₃	1	1	yes	characteristic
146	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
147	C₂×C₃×D₄	—	2	1	no	—
148	C₂×C₃×D₄	—	2	2	no	—
149	C₂×C₃×D₄	—	2	1	no	—
150	C₂×C₃×D₄	—	2	4	no	—
151	C₂×C₃×D₄	—	2	2	no	—
152	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
153	C₂×C₃×D₄	C₂³×C₃	1	1	yes	characteristic
154	C₂²×C₃×C₄	—	2	1	no	—
155	C₂²×C₃×C₄	—	2	1	no	—
156	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
157	C₃×Q₁₆	—	2	4	no	—
158	C₃×SD₁₆	—	2	8	no	—
159	C₃×SD₁₆	—	2	4	no	—
160	C₃×M₁₆	—	2	2	no	—
161	C₂×C₃×C₈	—	2	2	no	—
162	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
163	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
164	C₃×C₄⋊C₄	—	2	4	no	—
165	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
166	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
167	C₃×C₂²⋊C₄	—	2	1	no	—
168	C₃×C₂²⋊C₄	—	2	4	no	—
169	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
170	C₃×C₄²	—	2	4	no	—
171	C₃×C₄²	—	2	1	no	—
172	C₃×C₄²	C₂³×C₃	1	2	yes	—
173	C₂²×C₉	—	2	2	no	—
174	C₂²×C₉	—	4	1	no	—
175	C₂²×C₉	—	4	2	no	—
176	C₂²×C₉	—	4	2	no	—
177	C₂²×C₉	—	4	4	no	—
178	C₂²×C₉	—	8	4	no	—
179	C₂²×C₉	#121	1	2	yes	—
180	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
181	C₄×C₉	—	2	4	no	—
182	C₄×C₉	—	2	4	no	—
183	C₄×C₉	—	4	1	no	—
184	C₄×C₉	—	4	2	no	—
185	C₄×C₉	#121	1	2	yes	—
186	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
187	C₄×C₉	C₂×#41	1	1	yes	characteristic
188	C₂×#41	C₂²×C₉	1	1	yes	characteristic
189	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
190	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
191	#137	—	2	4	no	—
192	C₂×Q₁₆	C₂²×C₉	1	2	yes	complemented
193	C₂×SD₁₆	—	2	4	no	—
194	C₂×SD₁₆	C₂²×C₉	1	4	yes	—
195	C₂×SD₁₆	C₂²×C₉	1	2	yes	complemented
196	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
197	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
198	#127	C₂²×C₉	1	2	yes	—
199	#124	C₂²×C₉	1	2	yes	—
200	#121	—	2	2	no	—
201	#120	—	2	2	no	—
202	C₄×Q₈	—	2	2	no	—
203	C₄×Q₈	C₂²×C₉	1	4	yes	complemented
204	#117	—	2	2	no	—
205	#117	C₂²×C₉	1	1	yes	characteristic
206	#117	C₂²×C₉	1	1	yes	characteristic
207	C₄⋊C₈	C₂²×C₉	1	4	yes	complemented
208	#103	C₂²×C₉	1	2	yes	complemented
209	#102	C₂²×C₉	1	4	yes	—
210	#102	C₂²×C₉	1	2	yes	complemented
211	C₂³×C₃	—	2	2	no	—
212	C₂³×C₃	—	2	1	no	—
213	C₂³×C₃	—	4	2	no	—
214	C₂³×C₃	—	8	2	no	—
215	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
216	C₃×Q₈	—	2	4	no	—
217	C₃×Q₈	—	2	2	no	—
218	C₃×Q₈	—	2	1	no	—
219	C₃×Q₈	—	2	1	no	—
220	C₃×Q₈	C₂×C₃×D₄	1	4	yes	—
221	C₃×D₄	—	2	2	no	—
222	C₃×D₄	—	2	1	no	—
223	C₃×D₄	—	2	2	no	—
224	C₃×D₄	—	2	1	no	—
225	C₃×D₄	—	4	2	no	—
226	C₃×D₄	—	4	2	no	—
227	C₃×D₄	—	8	4	no	—
228	C₂×C₃×C₄	—	2	4	no	—
229	C₂×C₃×C₄	—	2	2	no	—
230	C₂×C₃×C₄	—	2	2	no	—
231	C₂×C₃×C₄	—	2	1	no	—
232	C₂×C₃×C₄	—	4	2	no	—
233	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
234	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
236	C₂×C₃×C₄	C₃×#41	1	2	yes	—
237	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
238	C₃×C₈	—	2	2	no	—
239	C₃×C₈	—	2	2	no	—
240	C₂×C₉	—	2	2	no	—
241	C₂×C₉	—	8	1	no	—
242	C₂×C₉	—	8	2	no	—
243	C₂×C₉	C₂×#121	1	1	yes	characteristic
244	C₂×C₉	#577	1	2	yes	—
245	C₂⁴	—	2	1	no	—
246	#41	—	2	2	no	—
247	#41	—	2	2	no	—
248	C₂×Q₈	—	2	1	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	D₄×C₉	1	4	yes	complemented
251	C₂×Q₈	C₂³×C₉	1	1	yes	characteristic
252	C₂×Q₈	C₂³×C₉	1	2	yes	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	C₂³×C₉	1	2	yes	—
259	C₂×D₄	C₂³×C₉	1	1	yes	characteristic
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	2	1	no	—
262	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
263	Q₁₆	—	2	4	no	—
264	SD₁₆	—	2	8	no	—
265	SD₁₆	—	2	4	no	—
266	M₁₆	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	C₂³×C₉	1	2	yes	—
269	C₂×C₈	C₂³×C₉	1	2	yes	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	D₄×C₉	1	2	yes	—
272	C₄⋊C₄	C₂³×C₉	1	2	yes	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	2	4	no	—
275	C₂²⋊C₄	D₄×C₉	1	2	yes	—
276	C₄²	—	2	4	no	—
277	C₄²	—	2	1	no	—
278	C₄²	C₂³×C₉	1	2	yes	—
279	C₂²×C₃	—	2	2	no	—
280	C₂²×C₃	—	4	2	no	—
281	C₂²×C₃	—	4	4	no	—
282	C₂²×C₃	—	4	1	no	—
283	C₂²×C₃	—	4	2	no	—
284	C₂²×C₃	—	8	4	no	—
285	C₂²×C₃	C₃×#121	1	2	yes	—
286	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
287	C₃×C₄	—	2	4	no	—
288	C₃×C₄	—	2	4	no	—
289	C₃×C₄	—	4	2	no	—
290	C₃×C₄	—	4	1	no	—
291	C₃×C₄	C₃×#121	1	2	yes	—
292	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
293	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
294	C₉	#3043	1	1	yes	characteristic, direct product factor, Sylow
295	C₂³	—	2	2	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	8	2	no	—
299	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
300	Q₈	—	2	4	no	—
301	Q₈	—	2	2	no	—
302	Q₈	—	2	1	no	—
303	Q₈	—	2	1	no	—
304	Q₈	C₂×D₄×C₉	1	4	yes	—
305	D₄	—	2	2	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	2	no	—
308	D₄	—	2	1	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	8	4	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
318	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
320	C₂×C₄	C₉×#41	1	2	yes	—
321	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₂×C₃	—	2	2	no	—
325	C₂×C₃	—	8	2	no	—
326	C₂×C₃	—	8	1	no	—
327	C₂×C₃	C₂×C₃×#121	1	1	yes	characteristic
328	C₂×C₃	C₃×#577	1	2	yes	—
329	C₂²	—	2	2	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	4	no	—
334	C₂²	—	8	4	no	—
335	C₂²	C₉×#121	1	2	yes	—
336	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
337	C₄	—	2	4	no	—
338	C₄	—	2	4	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	2	no	—
341	C₄	C₉×#121	1	2	yes	—
342	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
343	C₄	C₂×C₉×#41	1	1	yes	characteristic
344	C₃	C₃×#3043	1	1	yes	characteristic
345	C₂	—	2	2	no	—
346	C₂	—	8	2	no	—
347	C₂	—	8	1	no	—
348	C₂	C₂×C₉×#121	1	1	yes	characteristic
349	C₂	C₉×#577	1	2	yes	—

	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×#535	C₂×C₃
LCS	G ▷ C₂×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
1. C₂×C₉×#137	C₂	semidirect
2. C₂²×C₉×SD₁₆	C₂	semidirect
4. C₉×#519	C₂	semidirect
5. C₉×#464	C₂	semidirect
6. C₉×#460	C₂	semidirect
7. C₉×#424	C₂	semidirect
8. C₉×#418	C₂	semidirect
9. C₉×#416	C₂	semidirect
15. C₂×C₉×Q₁₆	C₂²	semidirect
18. C₂×C₉×SD₁₆	C₂²	semidirect
26. C₄×Q₈×C₉	C₂²	semidirect
30. C₉×C₄⋊C₈	C₂²	semidirect
31. C₉×#103	C₂²	semidirect
33. C₉×#102	C₂²	semidirect
48. C₂×Q₈×C₉	D₄	semidirect
77. #3043	C₉	direct (N × H)
130. C₂×#137	C₂×C₉	semidirect
131. C₂²×SD₁₆	C₂×C₉	semidirect
133. #519	C₂×C₉	semidirect
134. #464	C₂×C₉	semidirect
135. #460	C₂×C₉	semidirect
136. #424	C₂×C₉	semidirect
137. #418	C₂×C₉	semidirect
138. #416	C₂×C₉	semidirect
192. C₂×Q₁₆	C₂²×C₉	semidirect
195. C₂×SD₁₆	C₂²×C₉	semidirect
203. C₄×Q₈	C₂²×C₉	semidirect
207. C₄⋊C₈	C₂²×C₉	semidirect
208. #103	C₂²×C₉	semidirect
210. #102	C₂²×C₉	semidirect
250. C₂×Q₈	D₄×C₉	semidirect
294. C₉	#3043	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×#137	2
C₂²×C₉×SD₁₆	2
C₉×#535	2
C₉×#519	2
C₉×#464	2
C₉×#460	2
C₉×#424	2
C₉×#418	2
C₉×#416	2
C₃×#3043	3

Subgroups — C9×#3043 #11767764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3043 #11767764