Subgroups — C₉×#5015 #13142733

Order 1728.

514 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#213	C₂	1	2	yes	maximal, complemented
3	C₉×#874	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#809	C₂	1	2	yes	maximal, complemented
5	C₉×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5015	C₃	1	1	yes	maximal, characteristic
7	C₉×#459	—	3	1	no	maximal
8	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	—	2	1	no	—
10	C₂×C₉×D₁₂	—	2	1	no	—
11	C₂×C₉×D₁₂	—	2	1	no	—
12	C₂×C₉×D₁₂	C₂²	1	2	yes	—
13	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
15	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₉×#213	—	2	4	no	—
17	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
18	C₃²×C₄×D₄	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×#213	C₂×C₃	1	2	yes	—
20	C₃×#874	C₂×C₃	1	1	yes	characteristic
21	C₃×#809	C₂×C₃	1	2	yes	—
22	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
23	C₂×C₉×D₈	—	3	2	no	—
24	C₉×#127	—	3	1	no	—
25	C₄×D₄×C₉	—	2	1	no	—
26	C₄×D₄×C₉	D₃	1	1	yes	characteristic, complemented
27	C₉×C₄⋊C₈	—	3	1	no	—
28	C₉×#102	—	3	2	no	—
29	C₂³×C₃×C₉	—	2	1	no	—
30	C₂²×D₃×C₉	—	2	2	no	—
31	C₃×D₄×C₉	—	2	1	no	—
32	C₃×D₄×C₉	D₄	1	2	yes	complemented
33	C₂×C₃×C₄×C₉	—	2	1	no	—
34	C₂×C₃×C₄×C₉	—	4	2	no	—
35	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₉×D₁₂	—	2	2	no	—
39	C₉×D₁₂	—	2	2	no	—
40	C₉×D₁₂	—	4	2	no	—
41	C₉×D₁₂	—	4	2	no	—
42	C₉×C₃⋊C₈	—	2	2	no	—
43	C₃×#459	—	3	1	no	—
44	#5015	C₉	1	1	yes	characteristic, direct product factor
45	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
46	C₂²×C₃²×C₄	—	2	1	no	—
47	C₂×C₃×D₁₂	—	2	1	no	—
48	C₂×C₃×D₁₂	—	2	1	no	—
49	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
50	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
51	C₃²×C₂²⋊C₄	—	2	1	no	—
52	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
53	C₃×#213	—	2	4	no	—
54	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
55	C₂×D₄×C₉	—	2	1	no	—
56	C₂×D₄×C₉	—	3	2	no	—
57	C₂×D₄×C₉	—	6	1	no	—
58	C₂×D₄×C₉	—	6	1	no	—
59	C₂×D₄×C₉	C₂×D₃	1	1	yes	characteristic
60	C₂²×C₄×C₉	—	2	2	no	—
61	C₂²×C₄×C₉	—	2	1	no	—
62	C₉×D₈	—	6	4	no	—
63	C₂×C₈×C₉	—	3	2	no	—
64	C₉×C₄⋊C₄	—	2	1	no	—
65	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
66	C₉×C₂²⋊C₄	—	2	2	no	—
67	C₉×C₂²⋊C₄	—	2	1	no	—
68	C₄²×C₉	—	2	1	no	—
69	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
70	C₂²×C₃×C₉	—	2	2	no	—
71	C₂²×C₃×C₉	—	4	1	no	—
72	C₂²×C₃×C₉	—	4	1	no	—
73	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
74	C₂×D₃×C₉	—	4	2	no	—
75	C₂×D₃×C₉	—	4	2	no	—
76	C₂×D₃×C₉	—	4	2	no	—
77	C₃×C₄×C₉	—	2	1	no	—
78	C₃×C₄×C₉	—	4	1	no	—
79	C₃×C₄×C₉	D₈	1	2	yes	complemented
80	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
81	C₃×C₄×C₉	#41	1	1	yes	characteristic
82	C₂×C₃×D₈	—	3	2	no	—
83	C₃×#127	—	3	1	no	—
84	C₃×C₄×D₄	—	2	1	no	—
85	C₃×C₄×D₄	C₂×C₉	1	1	yes	characteristic, complemented
86	C₃×C₄×D₄	C₃×D₃	1	1	yes	characteristic
87	C₂×#213	C₂×C₉	1	2	yes	complemented
88	#874	C₂×C₉	1	1	yes	characteristic, complemented
89	C₃×C₄⋊C₈	—	3	1	no	—
90	C₃×#102	—	3	2	no	—
91	#809	C₂×C₉	1	2	yes	complemented
92	C₄⋊(C₃⋊C₈)	C₂×C₉	1	1	yes	characteristic, complemented
93	C₂³×C₃²	—	2	1	no	—
94	C₂²×C₃×D₃	—	2	2	no	—
95	C₃²×D₄	—	2	1	no	—
96	C₃²×D₄	C₃×D₄	1	2	yes	—
97	C₂×C₃²×C₄	—	2	1	no	—
98	C₂×C₃²×C₄	—	4	2	no	—
99	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
100	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
101	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
102	C₃×D₁₂	—	2	2	no	—
103	C₃×D₁₂	—	2	2	no	—
104	C₃×D₁₂	—	4	2	no	—
105	C₃×D₁₂	—	4	2	no	—
106	C₂³×C₉	—	2	2	no	—
107	C₂³×C₉	—	2	1	no	—
108	C₂³×C₉	—	6	2	no	—
109	C₃×C₃⋊C₈	—	2	2	no	—
110	D₄×C₉	—	2	2	no	—
111	D₄×C₉	—	2	1	no	—
112	D₄×C₉	—	2	2	no	—
113	D₄×C₉	—	6	2	no	—
114	D₄×C₉	—	6	2	no	—
115	D₄×C₉	—	12	2	no	—
116	D₄×C₉	—	12	2	no	—
117	D₄×C₉	D₁₂	1	2	yes	complemented
118	C₂×C₄×C₉	—	2	2	no	—
119	C₂×C₄×C₉	—	2	1	no	—
120	C₂×C₄×C₉	—	2	1	no	—
121	C₂×C₄×C₉	—	2	1	no	—
122	C₂×C₄×C₉	—	2	1	no	—
123	C₂×C₄×C₉	—	4	4	no	—
124	C₂×C₄×C₉	—	4	2	no	—
125	C₂×C₄×C₉	#67	1	1	yes	characteristic
126	C₂×C₄×C₉	#67	1	1	yes	characteristic
127	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
128	C₈×C₉	—	6	2	no	—
129	#459	—	3	1	no	Sylow
130	C₂×C₃×C₉	—	4	2	no	—
131	C₂×C₃×C₉	#121	1	1	yes	characteristic
132	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
134	D₃×C₉	—	8	2	no	—
135	C₂×C₃×D₄	—	2	1	no	—
136	C₂×C₃×D₄	—	3	2	no	—
137	C₂×C₃×D₄	—	6	1	no	—
138	C₂×C₃×D₄	—	6	1	no	—
139	C₂×C₃×D₄	C₂²×C₉	1	1	yes	characteristic
140	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
141	C₂²×C₃×C₄	—	2	2	no	—
142	C₂²×C₃×C₄	—	2	1	no	—
143	C₂²×C₃×C₄	—	2	1	no	—
144	C₂×D₁₂	—	2	1	no	—
145	C₂×D₁₂	—	2	1	no	—
146	C₂×D₁₂	C₂²×C₉	1	2	yes	—
147	C₃×D₈	—	6	4	no	—
148	C₂×C₃×C₈	—	3	2	no	—
149	C₃×C₄⋊C₄	—	2	1	no	—
150	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
151	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
152	C₃×C₂²⋊C₄	—	2	2	no	—
153	C₃×C₂²⋊C₄	—	2	1	no	—
154	C₃×C₂²⋊C₄	—	2	1	no	—
155	C₃×C₄²	—	2	1	no	—
156	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
157	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
158	#213	—	2	4	no	—
159	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
160	C₂²×C₃²	—	2	2	no	—
161	C₂²×C₃²	—	4	1	no	—
162	C₂²×C₃²	—	4	1	no	—
163	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
164	C₂×C₃×D₃	—	4	2	no	—
165	C₂×C₃×D₃	—	4	2	no	—
166	C₂×C₃×D₃	—	4	2	no	—
167	C₃²×C₄	—	2	1	no	—
168	C₃²×C₄	—	4	1	no	—
169	C₃²×C₄	C₃×D₈	1	2	yes	—
170	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
171	C₃²×C₄	C₃×#41	1	1	yes	characteristic
172	C₂²×C₉	—	2	4	no	—
173	C₂²×C₉	—	2	2	no	—
174	C₂²×C₉	—	2	1	no	—
175	C₂²×C₉	—	4	2	no	—
176	C₂²×C₉	—	4	2	no	—
177	C₂²×C₉	—	4	1	no	—
178	C₂²×C₉	—	4	1	no	—
179	C₂²×C₉	—	12	2	no	—
180	C₂²×C₉	—	12	2	no	—
181	C₂²×C₉	—	12	2	no	—
182	C₂²×C₉	C₂×#67	1	1	yes	characteristic
183	C₄×C₉	—	2	1	no	—
184	C₄×C₉	—	2	2	no	—
185	C₄×C₉	—	2	1	no	—
186	C₄×C₉	—	2	1	no	—
187	C₄×C₉	—	4	1	no	—
188	C₄×C₉	—	4	2	no	—
189	C₄×C₉	—	4	1	no	—
190	C₄×C₉	#213	1	2	yes	complemented
191	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
192	C₄×C₉	#235	1	1	yes	characteristic
193	C₂×D₈	—	3	2	no	—
194	#127	—	3	1	no	—
195	C₄×D₄	D₃×C₉	1	1	yes	characteristic, complemented
196	C₄⋊C₈	—	3	1	no	—
197	#102	—	3	2	no	—
198	C₃×C₉	#459	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₃	—	2	2	no	—
200	C₂³×C₃	—	2	1	no	—
201	C₂³×C₃	—	2	1	no	—
202	C₂³×C₃	—	6	2	no	—
203	C₂²×D₃	—	2	2	no	—
204	C₃×D₄	—	2	2	no	—
205	C₃×D₄	—	2	1	no	—
206	C₃×D₄	—	2	1	no	—
207	C₃×D₄	—	2	2	no	—
208	C₃×D₄	—	6	2	no	—
209	C₃×D₄	—	6	2	no	—
210	C₃×D₄	—	12	2	no	—
211	C₃×D₄	—	12	2	no	—
212	C₃×D₄	D₄×C₉	1	2	yes	complemented
213	C₃×D₄	C₃×D₁₂	1	2	yes	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	—	2	1	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	2	1	no	—
220	C₂×C₃×C₄	—	4	4	no	—
221	C₂×C₃×C₄	—	4	2	no	—
222	C₂×C₃×C₄	—	4	2	no	—
223	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
224	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
226	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
227	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
229	D₁₂	—	2	2	no	—
230	D₁₂	—	2	2	no	—
231	D₁₂	—	4	2	no	—
232	D₁₂	—	4	2	no	—
233	C₃×C₈	—	6	2	no	—
234	C₃⋊C₈	—	2	2	no	—
235	C₂×C₃²	—	4	2	no	—
236	C₂×C₃²	C₃×#121	1	1	yes	characteristic
237	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
238	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
239	C₃×D₃	—	8	2	no	—
240	C₂×C₉	—	2	1	no	—
241	C₂×C₉	—	2	1	no	—
242	C₂×C₉	—	2	1	no	—
243	C₂×C₉	—	4	4	no	—
244	C₂×C₉	—	4	2	no	—
245	C₂×C₉	—	24	2	no	—
246	C₂×C₉	#930	1	1	yes	characteristic
247	C₂×C₉	C₂×#213	1	1	yes	characteristic
248	C₂×C₉	#949	1	1	yes	characteristic
249	C₂×D₄	—	3	2	no	—
250	C₂×D₄	—	6	1	no	—
251	C₂×D₄	—	6	1	no	—
252	C₂×D₄	C₂×D₃×C₉	1	1	yes	characteristic
253	C₂²×C₄	—	2	1	no	—
254	D₈	—	6	4	no	—
255	C₂×C₈	—	3	2	no	—
256	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
259	C₂²×C₃	—	2	4	no	—
260	C₂²×C₃	—	2	2	no	—
261	C₂²×C₃	—	2	2	no	—
262	C₂²×C₃	—	2	1	no	—
263	C₂²×C₃	—	4	2	no	—
264	C₂²×C₃	—	4	2	no	—
265	C₂²×C₃	—	4	1	no	—
266	C₂²×C₃	—	4	1	no	—
267	C₂²×C₃	—	4	1	no	—
268	C₂²×C₃	—	4	1	no	—
269	C₂²×C₃	—	12	2	no	—
270	C₂²×C₃	—	12	2	no	—
271	C₂²×C₃	—	12	2	no	—
272	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
273	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
274	C₂×D₃	—	4	2	no	—
275	C₂×D₃	—	4	2	no	—
276	C₂×D₃	—	4	2	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	2	no	—
282	C₃×C₄	—	4	1	no	—
283	C₃×C₄	—	4	2	no	—
284	C₃×C₄	—	4	1	no	—
285	C₃×C₄	—	4	1	no	—
286	C₃×C₄	C₉×D₈	1	2	yes	complemented
287	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
288	C₃×C₄	C₉×#41	1	1	yes	characteristic
289	C₃×C₄	C₃×#213	1	2	yes	—
290	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
291	C₃×C₄	C₃×#235	1	1	yes	characteristic
292	C₃²	C₃×#459	1	1	yes	characteristic
293	C₉	—	2	1	no	—
294	C₉	#5015	1	1	yes	characteristic, direct product factor
295	C₂³	—	2	1	no	—
296	C₂³	—	6	2	no	—
297	D₄	—	2	1	no	—
298	D₄	—	6	2	no	—
299	D₄	—	6	2	no	—
300	D₄	—	12	2	no	—
301	D₄	—	12	2	no	—
302	D₄	C₉×D₁₂	1	2	yes	complemented
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	C₉×#67	1	1	yes	characteristic
306	C₂×C₄	C₉×#67	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
308	C₈	—	6	2	no	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	2	1	no	—
311	C₂×C₃	—	2	1	no	—
312	C₂×C₃	—	4	4	no	—
313	C₂×C₃	—	4	2	no	—
314	C₂×C₃	—	4	2	no	—
315	C₂×C₃	—	24	2	no	—
316	C₂×C₃	C₉×#121	1	1	yes	characteristic
317	C₂×C₃	C₂×C₉×D₈	1	1	yes	characteristic
318	C₂×C₃	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
319	C₂×C₃	C₃×#930	1	1	yes	characteristic
320	C₂×C₃	C₂×C₃×#213	1	1	yes	characteristic
321	C₂×C₃	C₃×#949	1	1	yes	characteristic
322	D₃	—	8	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	12	2	no	—
327	C₂²	—	12	2	no	—
328	C₂²	—	12	2	no	—
329	C₂²	C₂×C₉×#67	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	4	1	no	—
332	C₄	C₉×#213	1	2	yes	complemented
333	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
334	C₄	C₉×#235	1	1	yes	characteristic
335	C₃	—	2	1	no	—
336	C₃	C₉×#459	1	1	yes	characteristic, complemented
337	C₃	C₃×#5015	1	1	yes	characteristic
338	C₂	—	4	2	no	—
339	C₂	—	24	2	no	—
340	C₂	C₉×#930	1	1	yes	characteristic
341	C₂	C₂×C₉×#213	1	1	yes	characteristic
342	C₂	C₉×#949	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×#67
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×D₄×C₉	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃×#5015	C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₄×D₄×C₉	D₃
hypocenter	C₃	C₉×#459
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₄×D₄×C₉

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×D₄×C₉	C₂	semidirect
2. C₂×C₉×#213	C₂	semidirect
3. C₉×#874	C₂	semidirect
4. C₉×#809	C₂	semidirect
5. C₉×C₄⋊(C₃⋊C₈)	C₂	semidirect
26. C₄×D₄×C₉	D₃	semidirect
32. C₃×D₄×C₉	D₄	semidirect
44. #5015	C₉	direct (N × H)
79. C₃×C₄×C₉	D₈	semidirect
85. C₃×C₄×D₄	C₂×C₉	semidirect
87. C₂×#213	C₂×C₉	semidirect
88. #874	C₂×C₉	semidirect
91. #809	C₂×C₉	semidirect
92. C₄⋊(C₃⋊C₈)	C₂×C₉	semidirect
117. D₄×C₉	D₁₂	semidirect
190. C₄×C₉	#213	semidirect
195. C₄×D₄	D₃×C₉	semidirect
198. C₃×C₉	#459	semidirect
212. C₃×D₄	D₄×C₉	semidirect
286. C₃×C₄	C₉×D₈	semidirect
294. C₉	#5015	direct (N × H)
302. D₄	C₉×D₁₂	semidirect
332. C₄	C₉×#213	semidirect
336. C₃	C₉×#459	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₄×D₄×C₉	2
C₂×C₉×#213	2
C₉×#874	2
C₉×#809	2
C₉×C₄⋊(C₃⋊C₈)	2
C₃×#5015	3
C₉×#459	3

Subgroups — C9×#5015 #13142733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5015 #13142733