Subgroups — C₉×#3044 #11767765

Order 1152.

687 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×#137	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#536	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×#462	C₂	1	4	yes	maximal, complemented
7	C₉×#424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₉×#419	C₂	1	1	yes	maximal, characteristic, complemented
9	C₉×#416	C₂	1	1	yes	maximal, characteristic
10	C₃×#3044	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×Q₈×C₉	C₂²	1	1	yes	characteristic
14	C₉×#137	—	2	4	no	—
15	C₂×C₉×Q₁₆	—	2	4	no	—
16	C₂×C₉×Q₁₆	C₂²	1	4	yes	—
17	C₂×C₉×Q₁₆	C₂²	1	2	yes	—
18	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
19	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
21	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
22	C₉×#124	C₂²	1	2	yes	—
23	C₉×#123	—	2	2	no	—
24	C₉×C₂²⋊Q₈	—	2	2	no	—
25	C₄×Q₈×C₉	—	2	2	no	—
26	C₄×Q₈×C₉	C₂²	1	4	yes	—
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	—
31	C₉×#103	C₂²	1	2	yes	—
32	C₉×#103	C₂²	1	4	yes	—
33	C₉×#102	C₂²	1	2	yes	—
34	C₂×C₃×#137	C₂×C₃	1	1	yes	characteristic
35	C₂²×C₃×Q₁₆	C₂×C₃	1	1	yes	characteristic
36	C₃×#536	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×#462	C₂×C₃	1	4	yes	—
40	C₃×#424	C₂×C₃	1	1	yes	characteristic
41	C₃×#419	C₂×C₃	1	1	yes	characteristic
42	C₃×#416	C₂×C₃	1	1	yes	characteristic
43	C₉×#41	—	2	2	no	—
44	C₉×#41	—	2	2	no	—
45	C₂×Q₈×C₉	—	2	1	no	—
46	C₂×Q₈×C₉	—	2	1	no	—
47	C₂×Q₈×C₉	—	2	2	no	—
48	C₂×Q₈×C₉	—	2	2	no	—
49	C₂×Q₈×C₉	—	2	2	no	—
50	C₂×Q₈×C₉	D₄	1	4	yes	—
51	C₂×Q₈×C₉	C₂³	1	2	yes	—
52	C₂×Q₈×C₉	C₂³	1	1	yes	characteristic
53	C₂×Q₈×C₉	C₂³	1	2	yes	—
54	C₂×D₄×C₉	—	2	1	no	—
55	C₂×D₄×C₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₉	—	2	1	no	—
57	C₂²×C₄×C₉	—	2	1	no	—
58	C₂²×C₄×C₉	—	2	1	no	—
59	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
60	C₉×Q₁₆	—	2	8	no	—
61	C₉×Q₁₆	—	2	4	no	—
62	C₉×SD₁₆	—	2	4	no	—
63	C₉×M₁₆	—	2	2	no	—
64	C₂×C₈×C₉	—	2	2	no	—
65	C₂×C₈×C₉	C₂³	1	2	yes	—
66	C₂×C₈×C₉	C₂³	1	2	yes	—
67	C₉×C₄⋊C₄	—	2	4	no	—
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	#3044	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×Q₈	C₂²×C₃	1	1	yes	characteristic
81	C₃×#137	—	2	4	no	—
82	C₂×C₃×Q₁₆	—	2	4	no	—
83	C₂×C₃×Q₁₆	C₂²×C₃	1	4	yes	—
84	C₂×C₃×Q₁₆	C₂²×C₃	1	2	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×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
89	C₃×#124	C₂²×C₃	1	2	yes	—
90	C₃×#123	—	2	2	no	—
91	C₃×C₂²⋊Q₈	—	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×#103	C₂²×C₃	1	4	yes	—
100	C₃×#102	C₂²×C₃	1	2	yes	—
101	C₂³×C₉	—	2	1	no	—
102	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
103	Q₈×C₉	—	2	4	no	—
104	Q₈×C₉	—	2	2	no	—
105	Q₈×C₉	—	2	2	no	—
106	Q₈×C₉	—	2	1	no	—
107	Q₈×C₉	—	2	2	no	—
108	Q₈×C₉	—	2	1	no	—
109	Q₈×C₉	—	4	2	no	—
110	Q₈×C₉	C₂×D₄	1	4	yes	—
111	D₄×C₉	—	2	1	no	—
112	D₄×C₉	—	2	1	no	—
113	D₄×C₉	—	4	2	no	—
114	C₂×C₄×C₉	—	2	4	no	—
115	C₂×C₄×C₉	—	2	2	no	—
116	C₂×C₄×C₉	—	2	2	no	—
117	C₂×C₄×C₉	—	2	2	no	—
118	C₂×C₄×C₉	—	2	1	no	—
119	C₂×C₄×C₉	—	4	2	no	—
120	C₂×C₄×C₉	—	4	2	no	—
121	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
122	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
124	C₂×C₄×C₉	#41	1	2	yes	—
125	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
126	C₈×C₉	—	2	2	no	—
127	C₈×C₉	—	2	2	no	—
128	C₂×#137	C₂×C₉	1	1	yes	characteristic
129	C₂²×Q₁₆	C₂×C₉	1	1	yes	characteristic, complemented
130	#536	C₂×C₉	1	1	yes	characteristic
131	#519	C₂×C₉	1	1	yes	characteristic, complemented
132	#464	C₂×C₉	1	4	yes	complemented
133	#462	C₂×C₉	1	4	yes	complemented
134	#424	C₂×C₉	1	1	yes	characteristic, complemented
135	#419	C₂×C₉	1	1	yes	characteristic, complemented
136	#416	C₂×C₉	1	1	yes	characteristic
137	C₃×#41	—	2	2	no	—
138	C₃×#41	—	2	2	no	—
139	C₂×C₃×Q₈	—	2	1	no	—
140	C₂×C₃×Q₈	—	2	1	no	—
141	C₂×C₃×Q₈	—	2	2	no	—
142	C₂×C₃×Q₈	—	2	2	no	—
143	C₂×C₃×Q₈	—	2	2	no	—
144	C₂×C₃×Q₈	C₃×D₄	1	4	yes	—
145	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
146	C₂×C₃×Q₈	C₂³×C₃	1	1	yes	characteristic
147	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
148	C₂×C₃×D₄	—	2	1	no	—
149	C₂×C₃×D₄	C₂³×C₃	1	1	yes	characteristic
150	C₂²×C₃×C₄	—	2	1	no	—
151	C₂²×C₃×C₄	—	2	1	no	—
152	C₂²×C₃×C₄	—	2	1	no	—
153	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
154	C₃×Q₁₆	—	2	8	no	—
155	C₃×Q₁₆	—	2	4	no	—
156	C₃×SD₁₆	—	2	4	no	—
157	C₃×M₁₆	—	2	2	no	—
158	C₂×C₃×C₈	—	2	2	no	—
159	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
160	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
161	C₃×C₄⋊C₄	—	2	4	no	—
162	C₃×C₄⋊C₄	—	2	4	no	—
163	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
164	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
165	C₃×C₂²⋊C₄	—	2	1	no	—
166	C₃×C₂²⋊C₄	—	2	4	no	—
167	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
168	C₃×C₄²	—	2	4	no	—
169	C₃×C₄²	—	2	1	no	—
170	C₃×C₄²	C₂³×C₃	1	2	yes	—
171	C₂²×C₉	—	2	2	no	—
172	C₂²×C₉	—	4	1	no	—
173	C₂²×C₉	—	4	2	no	—
174	C₂²×C₉	#121	1	2	yes	—
175	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
176	C₄×C₉	—	2	4	no	—
177	C₄×C₉	—	2	4	no	—
178	C₄×C₉	—	4	1	no	—
179	C₄×C₉	—	4	2	no	—
180	C₄×C₉	—	4	2	no	—
181	C₄×C₉	#121	1	2	yes	—
182	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
183	C₄×C₉	C₂×#41	1	1	yes	characteristic
184	C₂×#41	C₂²×C₉	1	1	yes	characteristic
185	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
186	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
187	#137	—	2	4	no	—
188	C₂×Q₁₆	—	2	4	no	—
189	C₂×Q₁₆	C₂²×C₉	1	4	yes	—
190	C₂×Q₁₆	C₂²×C₉	1	2	yes	—
191	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
192	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
193	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
194	C₄⋊Q₈	C₂²×C₉	1	2	yes	—
195	#124	C₂²×C₉	1	2	yes	—
196	#123	—	2	2	no	—
197	C₂²⋊Q₈	—	2	2	no	—
198	C₄×Q₈	—	2	2	no	—
199	C₄×Q₈	C₂²×C₉	1	4	yes	—
200	#117	—	2	2	no	—
201	#117	C₂²×C₉	1	1	yes	characteristic
202	#117	C₂²×C₉	1	1	yes	characteristic
203	C₄⋊C₈	C₂²×C₉	1	4	yes	—
204	#103	C₂²×C₉	1	2	yes	—
205	#103	C₂²×C₉	1	4	yes	—
206	#102	C₂²×C₉	1	2	yes	—
207	C₂³×C₃	—	2	1	no	—
208	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
209	C₃×Q₈	—	2	4	no	—
210	C₃×Q₈	—	2	2	no	—
211	C₃×Q₈	—	2	2	no	—
212	C₃×Q₈	—	2	1	no	—
213	C₃×Q₈	—	2	2	no	—
214	C₃×Q₈	—	2	1	no	—
215	C₃×Q₈	—	4	2	no	—
216	C₃×Q₈	C₂×C₃×D₄	1	4	yes	—
217	C₃×D₄	—	2	1	no	—
218	C₃×D₄	—	2	1	no	—
219	C₃×D₄	—	4	2	no	—
220	C₂×C₃×C₄	—	2	4	no	—
221	C₂×C₃×C₄	—	2	2	no	—
222	C₂×C₃×C₄	—	2	2	no	—
223	C₂×C₃×C₄	—	2	2	no	—
224	C₂×C₃×C₄	—	2	1	no	—
225	C₂×C₃×C₄	—	4	2	no	—
226	C₂×C₃×C₄	—	4	2	no	—
227	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
228	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
230	C₂×C₃×C₄	C₃×#41	1	2	yes	—
231	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
232	C₃×C₈	—	2	2	no	—
233	C₃×C₈	—	2	2	no	—
234	C₂×C₉	—	2	2	no	—
235	C₂×C₉	—	8	1	no	—
236	C₂×C₉	C₂×#121	1	1	yes	characteristic
237	C₂×C₉	#578	1	2	yes	—
238	#41	—	2	2	no	—
239	#41	—	2	2	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	D₄×C₉	1	4	yes	—
246	C₂×Q₈	C₂³×C₉	1	2	yes	—
247	C₂×Q₈	C₂³×C₉	1	1	yes	characteristic
248	C₂×Q₈	C₂³×C₉	1	2	yes	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	C₂³×C₉	1	1	yes	characteristic
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
255	Q₁₆	—	2	8	no	—
256	Q₁₆	—	2	4	no	—
257	SD₁₆	—	2	4	no	—
258	M₁₆	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	C₂³×C₉	1	2	yes	—
261	C₂×C₈	C₂³×C₉	1	2	yes	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	D₄×C₉	1	2	yes	—
265	C₄⋊C₄	C₂³×C₉	1	2	yes	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	4	no	—
268	C₂²⋊C₄	D₄×C₉	1	2	yes	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	1	no	—
271	C₄²	C₂³×C₉	1	2	yes	—
272	C₂²×C₃	—	2	2	no	—
273	C₂²×C₃	—	4	2	no	—
274	C₂²×C₃	—	4	1	no	—
275	C₂²×C₃	C₃×#121	1	2	yes	—
276	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
277	C₃×C₄	—	2	4	no	—
278	C₃×C₄	—	2	4	no	—
279	C₃×C₄	—	4	2	no	—
280	C₃×C₄	—	4	1	no	—
281	C₃×C₄	—	4	2	no	—
282	C₃×C₄	C₃×#121	1	2	yes	—
283	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
284	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
285	C₉	#3044	1	1	yes	characteristic, direct product factor, Sylow
286	C₂³	—	2	1	no	—
287	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
288	Q₈	—	2	4	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	2	2	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	2	2	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	4	2	no	—
295	Q₈	C₂×D₄×C₉	1	4	yes	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	4	2	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
307	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
309	C₂×C₄	C₉×#41	1	2	yes	—
310	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	2	2	no	—
313	C₂×C₃	—	2	2	no	—
314	C₂×C₃	—	8	1	no	—
315	C₂×C₃	C₂×C₃×#121	1	1	yes	characteristic
316	C₂×C₃	C₃×#578	1	2	yes	—
317	C₂²	—	2	2	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	2	no	—
320	C₂²	C₉×#121	1	2	yes	—
321	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
322	C₄	—	2	4	no	—
323	C₄	—	2	4	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	C₉×#121	1	2	yes	—
328	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
329	C₄	C₂×C₉×#41	1	1	yes	characteristic
330	C₃	C₃×#3044	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	—	8	1	no	—
333	C₂	C₂×C₉×#121	1	1	yes	characteristic
334	C₂	C₉×#578	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	C₂×C₃×#41	C₂²×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
2. C₂²×C₉×Q₁₆	C₂	semidirect
4. C₉×#519	C₂	semidirect
5. C₉×#464	C₂	semidirect
6. C₉×#462	C₂	semidirect
7. C₉×#424	C₂	semidirect
8. C₉×#419	C₂	semidirect
77. #3044	C₉	direct (N × H)
129. C₂²×Q₁₆	C₂×C₉	semidirect
131. #519	C₂×C₉	semidirect
132. #464	C₂×C₉	semidirect
133. #462	C₂×C₉	semidirect
134. #424	C₂×C₉	semidirect
135. #419	C₂×C₉	semidirect
285. C₉	#3044	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×Q₁₆	2
C₉×#536	2
C₉×#519	2
C₉×#464	2
C₉×#462	2
C₉×#424	2
C₉×#419	2
C₉×#416	2
C₃×#3044	3

Subgroups — C9×#3044 #11767765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3044 #11767765