Subgroups — C₉×#5072 #13142977

Order 1728.

528 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#213	C₂	1	2	yes	maximal, complemented
3	C₄×C₉×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#832	C₂	1	2	yes	maximal, complemented
5	C₉×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5072	C₃	1	1	yes	maximal, characteristic
7	C₉×#459	—	3	1	no	maximal
8	C₂×C₃×D₄×C₉	—	2	1	no	—
9	C₂×C₃×D₄×C₉	—	2	1	no	—
10	C₂×C₃×D₄×C₉	C₂²	1	2	yes	—
11	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₃×C₉	—	2	1	no	—
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₉×#213	—	2	4	no	—
15	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
16	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
18	C₃²×#127	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×#213	C₂×C₃	1	2	yes	—
20	C₃×C₄×D₁₂	C₂×C₃	1	1	yes	characteristic
21	C₃×#832	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	—	2	1	no	—
25	C₉×#127	D₃	1	1	yes	characteristic, complemented
26	C₄×D₄×C₉	—	3	1	no	—
27	C₉×C₄⋊C₈	—	3	1	no	—
28	C₉×#102	—	3	2	no	—
29	C₂³×C₃×C₉	—	2	2	no	—
30	C₂²×D₃×C₉	—	2	1	no	—
31	C₃×D₄×C₉	—	2	2	no	—
32	C₃×D₄×C₉	—	2	2	no	—
33	C₃×D₄×C₉	—	4	2	no	—
34	C₃×D₄×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	C₂×C₉×Dic₃	—	2	1	no	—
39	C₉×D₁₂	—	2	1	no	—
40	C₉×D₁₂	D₄	1	2	yes	complemented
41	C₄×D₃×C₉	—	4	2	no	—
42	C₉×C₃⋊C₈	—	2	2	no	—
43	C₃×#459	—	3	1	no	—
44	#5072	C₉	1	1	yes	characteristic, direct product factor
45	C₂×C₃²×D₄	—	2	1	no	—
46	C₂×C₃²×D₄	—	2	1	no	—
47	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
48	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₃×C₄×D₃	—	2	1	no	—
50	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
51	C₃×#213	—	2	4	no	—
52	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
53	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
55	C₂×D₄×C₉	—	2	2	no	—
56	C₂×D₄×C₉	—	2	2	no	—
57	C₂×D₄×C₉	—	2	2	no	—
58	C₂×D₄×C₉	—	2	1	no	—
59	C₂×D₄×C₉	—	2	1	no	—
60	C₂×D₄×C₉	—	3	1	no	—
61	C₂×D₄×C₉	C₂×D₃	1	2	yes	—
62	C₂²×C₄×C₉	—	6	1	no	—
63	C₉×D₈	—	6	4	no	—
64	C₂×C₈×C₉	—	3	2	no	—
65	C₉×C₄⋊C₄	—	3	1	no	—
66	C₉×C₂²⋊C₄	—	6	1	no	—
67	C₄²×C₉	—	2	1	no	—
68	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
69	C₂²×C₃×C₉	—	4	2	no	—
70	C₂²×C₃×C₉	—	4	2	no	—
71	C₂²×C₃×C₉	—	4	2	no	—
72	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
73	C₂×D₃×C₉	—	2	2	no	—
74	C₂×D₃×C₉	—	4	1	no	—
75	C₂×D₃×C₉	—	4	1	no	—
76	C₃×C₄×C₉	—	2	1	no	—
77	C₃×C₄×C₉	D₈	1	2	yes	complemented
78	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
79	C₃×C₄×C₉	#41	1	1	yes	characteristic
80	C₉×Dic₃	—	4	1	no	—
81	C₂×C₃×D₈	—	3	2	no	—
82	C₃×#127	—	2	1	no	—
83	C₃×#127	C₂×C₉	1	1	yes	characteristic, complemented
84	C₃×#127	C₃×D₃	1	1	yes	characteristic
85	C₃×C₄×D₄	—	3	1	no	—
86	C₂×#213	C₂×C₉	1	2	yes	complemented
87	C₄×D₁₂	C₂×C₉	1	1	yes	characteristic, complemented
88	C₃×C₄⋊C₈	—	3	1	no	—
89	C₃×#102	—	3	2	no	—
90	#832	C₂×C₉	1	2	yes	complemented
91	C₄⋊(C₃⋊C₈)	C₂×C₉	1	1	yes	characteristic, complemented
92	C₂³×C₃²	—	2	2	no	—
93	C₂²×C₃×D₃	—	2	1	no	—
94	C₃²×D₄	—	2	2	no	—
95	C₃²×D₄	—	2	2	no	—
96	C₃²×D₄	—	4	2	no	—
97	C₃²×D₄	—	4	2	no	—
98	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
99	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
100	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
101	C₂×C₃×Dic₃	—	2	1	no	—
102	C₃×D₁₂	—	2	1	no	—
103	C₃×D₁₂	C₃×D₄	1	2	yes	—
104	C₃×C₄×D₃	—	4	2	no	—
105	C₂³×C₉	—	2	4	no	—
106	C₂³×C₉	—	2	2	no	—
107	C₂³×C₉	—	6	1	no	—
108	C₃×C₃⋊C₈	—	2	2	no	—
109	D₄×C₉	—	2	2	no	—
110	D₄×C₉	—	2	2	no	—
111	D₄×C₉	—	3	2	no	—
112	D₄×C₉	—	4	2	no	—
113	D₄×C₉	—	4	2	no	—
114	D₄×C₉	—	4	4	no	—
115	D₄×C₉	—	4	2	no	—
116	D₄×C₉	—	4	4	no	—
117	D₄×C₉	—	4	2	no	—
118	D₄×C₉	—	6	1	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	—	6	1	no	—
123	C₂×C₄×C₉	—	12	2	no	—
124	C₂×C₄×C₉	#67	1	1	yes	characteristic
125	C₂×C₄×C₉	#67	1	1	yes	characteristic
126	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
127	C₈×C₉	—	6	2	no	—
128	#459	—	3	1	no	Sylow
129	C₂×C₃×C₉	—	8	2	no	—
130	C₂×C₃×C₉	#121	1	1	yes	characteristic
131	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
132	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
133	D₃×C₉	—	4	2	no	—
134	C₂×C₃×D₄	—	2	2	no	—
135	C₂×C₃×D₄	—	2	2	no	—
136	C₂×C₃×D₄	—	2	1	no	—
137	C₂×C₃×D₄	—	2	1	no	—
138	C₂×C₃×D₄	—	2	1	no	—
139	C₂×C₃×D₄	—	2	1	no	—
140	C₂×C₃×D₄	—	2	2	no	—
141	C₂×C₃×D₄	—	3	1	no	—
142	C₂×C₃×D₄	C₂²×C₉	1	2	yes	—
143	C₂×C₃×D₄	C₂×C₃×D₃	1	2	yes	—
144	C₂²×C₃×C₄	—	6	1	no	—
145	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
146	C₂×C₄×D₃	—	2	1	no	—
147	C₃×D₈	—	6	4	no	—
148	C₂×C₃×C₈	—	3	2	no	—
149	C₃×C₄⋊C₄	—	3	1	no	—
150	C₃×C₂²⋊C₄	—	6	1	no	—
151	C₃×C₄²	—	2	1	no	—
152	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
153	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
154	#213	—	2	4	no	—
155	(C₂×D₃)⋊C₄	—	2	1	no	—
156	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
157	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
158	C₂²×C₃²	—	4	2	no	—
159	C₂²×C₃²	—	4	2	no	—
160	C₂²×C₃²	—	4	2	no	—
161	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
162	C₂×C₃×D₃	—	2	2	no	—
163	C₂×C₃×D₃	—	4	1	no	—
164	C₂×C₃×D₃	—	4	1	no	—
165	C₃²×C₄	—	2	1	no	—
166	C₃²×C₄	C₃×D₈	1	2	yes	—
167	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
168	C₃²×C₄	C₃×#41	1	1	yes	characteristic
169	C₃×Dic₃	—	4	1	no	—
170	C₂²×C₉	—	2	1	no	—
171	C₂²×C₉	—	4	4	no	—
172	C₂²×C₉	—	4	4	no	—
173	C₂²×C₉	—	4	4	no	—
174	C₂²×C₉	—	4	2	no	—
175	C₂²×C₉	—	4	2	no	—
176	C₂²×C₉	—	4	2	no	—
177	C₂²×C₉	—	6	2	no	—
178	C₂²×C₉	—	12	1	no	—
179	C₂²×C₉	—	12	1	no	—
180	C₂²×C₉	C₂×#67	1	1	yes	characteristic
181	C₄×C₉	—	2	2	no	—
182	C₄×C₉	—	2	1	no	—
183	C₄×C₉	—	2	2	no	—
184	C₄×C₉	—	2	1	no	—
185	C₄×C₉	—	2	1	no	—
186	C₄×C₉	—	12	1	no	—
187	C₄×C₉	#213	1	2	yes	complemented
188	C₄×C₉	D₃×D₄	1	1	yes	characteristic
189	C₄×C₉	#237	1	1	yes	characteristic
190	C₂×D₈	—	3	2	no	—
191	#127	D₃×C₉	1	1	yes	characteristic, complemented
192	C₄×D₄	—	3	1	no	—
193	C₄⋊C₈	—	3	1	no	—
194	#102	—	3	2	no	—
195	C₃×C₉	#459	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₃	—	2	4	no	—
197	C₂³×C₃	—	2	2	no	—
198	C₂³×C₃	—	2	2	no	—
199	C₂³×C₃	—	6	1	no	—
200	C₂²×D₃	—	2	1	no	—
201	C₃×D₄	—	2	2	no	—
202	C₃×D₄	—	2	2	no	—
203	C₃×D₄	—	2	2	no	—
204	C₃×D₄	—	2	2	no	—
205	C₃×D₄	—	3	2	no	—
206	C₃×D₄	—	4	2	no	—
207	C₃×D₄	—	4	2	no	—
208	C₃×D₄	—	4	2	no	—
209	C₃×D₄	—	4	4	no	—
210	C₃×D₄	—	4	2	no	—
211	C₃×D₄	—	4	4	no	—
212	C₃×D₄	—	4	2	no	—
213	C₃×D₄	—	4	2	no	—
214	C₃×D₄	—	6	1	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	—	6	1	no	—
219	C₂×C₃×C₄	—	12	2	no	—
220	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
221	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
223	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
224	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
226	C₂×Dic₃	—	2	1	no	—
227	D₁₂	—	2	1	no	—
228	D₁₂	D₄×C₉	1	2	yes	complemented
229	C₄×D₃	—	4	2	no	—
230	C₃×C₈	—	6	2	no	—
231	C₃⋊C₈	—	2	2	no	—
232	C₂×C₃²	—	8	2	no	—
233	C₂×C₃²	C₃×#121	1	1	yes	characteristic
234	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
235	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
236	C₃×D₃	—	4	2	no	—
237	C₂×C₉	—	2	1	no	—
238	C₂×C₉	—	2	1	no	—
239	C₂×C₉	—	2	1	no	—
240	C₂×C₉	—	8	4	no	—
241	C₂×C₉	—	8	2	no	—
242	C₂×C₉	—	12	2	no	—
243	C₂×C₉	C₂×#213	1	1	yes	characteristic
244	C₂×C₉	#932	1	1	yes	characteristic
245	C₂×C₉	#938	1	1	yes	characteristic
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	3	1	no	—
249	C₂×D₄	C₂×D₃×C₉	1	2	yes	—
250	C₂²×C₄	—	6	1	no	—
251	D₈	—	6	4	no	—
252	C₂×C₈	—	3	2	no	—
253	C₄⋊C₄	—	3	1	no	—
254	C₂²⋊C₄	—	6	1	no	—
255	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	4	4	no	—
258	C₂²×C₃	—	4	4	no	—
259	C₂²×C₃	—	4	4	no	—
260	C₂²×C₃	—	4	2	no	—
261	C₂²×C₃	—	4	2	no	—
262	C₂²×C₃	—	4	2	no	—
263	C₂²×C₃	—	4	2	no	—
264	C₂²×C₃	—	4	2	no	—
265	C₂²×C₃	—	4	2	no	—
266	C₂²×C₃	—	6	2	no	—
267	C₂²×C₃	—	12	1	no	—
268	C₂²×C₃	—	12	1	no	—
269	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
270	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
271	C₂×D₃	—	2	2	no	—
272	C₂×D₃	—	4	1	no	—
273	C₂×D₃	—	4	1	no	—
274	C₃×C₄	—	2	2	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	2	no	—
280	C₃×C₄	—	12	1	no	—
281	C₃×C₄	C₉×D₈	1	2	yes	complemented
282	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
283	C₃×C₄	C₉×#41	1	1	yes	characteristic
284	C₃×C₄	C₃×#213	1	2	yes	—
285	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
286	C₃×C₄	C₃×#237	1	1	yes	characteristic
287	Dic₃	—	4	1	no	—
288	C₃²	C₃×#459	1	1	yes	characteristic
289	C₉	—	2	1	no	—
290	C₉	#5072	1	1	yes	characteristic, direct product factor
291	C₂³	—	2	2	no	—
292	C₂³	—	6	1	no	—
293	D₄	—	2	2	no	—
294	D₄	—	2	2	no	—
295	D₄	—	3	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	6	1	no	—
299	C₂×C₄	—	6	1	no	—
300	C₂×C₄	—	12	2	no	—
301	C₂×C₄	C₉×#67	1	1	yes	characteristic
302	C₂×C₄	C₉×#67	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
304	C₈	—	6	2	no	—
305	C₂×C₃	—	2	1	no	—
306	C₂×C₃	—	2	1	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	8	4	no	—
309	C₂×C₃	—	8	2	no	—
310	C₂×C₃	—	8	2	no	—
311	C₂×C₃	—	12	2	no	—
312	C₂×C₃	C₉×#121	1	1	yes	characteristic
313	C₂×C₃	C₂×C₉×D₈	1	1	yes	characteristic
314	C₂×C₃	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
315	C₂×C₃	C₂×C₃×#213	1	1	yes	characteristic
316	C₂×C₃	C₃×#932	1	1	yes	characteristic
317	C₂×C₃	C₃×#938	1	1	yes	characteristic
318	D₃	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	6	2	no	—
323	C₂²	—	12	1	no	—
324	C₂²	—	12	1	no	—
325	C₂²	C₂×C₉×#67	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	12	1	no	—
328	C₄	C₉×#213	1	2	yes	complemented
329	C₄	D₃×D₄×C₉	1	1	yes	characteristic
330	C₄	C₉×#237	1	1	yes	characteristic
331	C₃	—	2	1	no	—
332	C₃	C₉×#459	1	1	yes	characteristic, complemented
333	C₃	C₃×#5072	1	1	yes	characteristic
334	C₂	—	8	2	no	—
335	C₂	—	12	2	no	—
336	C₂	C₂×C₉×#213	1	1	yes	characteristic
337	C₂	C₉×#932	1	1	yes	characteristic
338	C₂	C₉×#938	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×C₉×#127	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃×#5072	C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#127	D₃
hypocenter	C₃	C₉×#459
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₉×#127

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×#127	C₂	semidirect
2. C₂×C₉×#213	C₂	semidirect
3. C₄×C₉×D₁₂	C₂	semidirect
4. C₉×#832	C₂	semidirect
5. C₉×C₄⋊(C₃⋊C₈)	C₂	semidirect
25. C₉×#127	D₃	semidirect
40. C₉×D₁₂	D₄	semidirect
44. #5072	C₉	direct (N × H)
77. C₃×C₄×C₉	D₈	semidirect
83. C₃×#127	C₂×C₉	semidirect
86. C₂×#213	C₂×C₉	semidirect
87. C₄×D₁₂	C₂×C₉	semidirect
90. #832	C₂×C₉	semidirect
91. C₄⋊(C₃⋊C₈)	C₂×C₉	semidirect
187. C₄×C₉	#213	semidirect
191. #127	D₃×C₉	semidirect
195. C₃×C₉	#459	semidirect
228. D₁₂	D₄×C₉	semidirect
281. C₃×C₄	C₉×D₈	semidirect
290. C₉	#5072	direct (N × H)
328. C₄	C₉×#213	semidirect
332. 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×C₉×#127	2
C₂×C₉×#213	2
C₄×C₉×D₁₂	2
C₉×#832	2
C₉×C₄⋊(C₃⋊C₈)	2
C₃×#5072	3
C₉×#459	3

Subgroups — C9×#5072 #13142977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5072 #13142977