Subgroups — C₉×#4800 #13141859

Order 1728.

330 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#947	C₂	1	1	yes	maximal, characteristic
2	C₉×#890	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉×#103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#835	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#821	C₂	1	1	yes	maximal, characteristic
6	C₈×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#809	C₂	1	1	yes	maximal, characteristic
8	C₃×#4800	C₃	1	1	yes	maximal, characteristic
9	C₉×#488	—	3	1	no	maximal
10	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
15	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
16	C₉×C₄⋊Dic₃	—	2	1	no	—
17	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₉×Dic₃⋊C₄	—	2	1	no	—
19	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
20	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×#947	C₂×C₃	1	1	yes	characteristic
22	C₃×#890	C₂×C₃	1	1	yes	characteristic
23	C₃²×#103	C₂×C₃	1	1	yes	characteristic
24	C₃×#835	C₂×C₃	1	1	yes	characteristic
25	C₃×#821	C₂×C₃	1	1	yes	characteristic
26	C₃×C₈×Dic₃	C₂×C₃	1	1	yes	characteristic
27	C₃×#809	C₂×C₃	1	1	yes	characteristic
28	C₉×#125	—	3	1	no	—
29	C₉×#124	—	3	1	no	—
30	C₉×#103	—	2	1	no	—
31	C₉×#103	—	3	1	no	—
32	C₉×#103	D₃	1	1	yes	characteristic, complemented
33	C₉×#102	—	3	1	no	—
34	C₉×#102	—	3	1	no	—
35	C₄×C₈×C₉	—	3	1	no	—
36	C₂²×D₃×C₉	—	2	1	no	—
37	C₃×Q₈×C₉	—	2	1	no	—
38	C₃×Q₈×C₉	—	2	1	no	—
39	C₂×C₃×C₄×C₉	—	2	1	no	—
40	C₂×C₃×C₄×C₉	—	2	1	no	—
41	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂×C₉×Dic₃	—	2	1	no	—
43	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
44	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
45	C₉×D₁₂	—	2	1	no	—
46	C₉×D₁₂	—	2	1	no	—
47	C₃×C₈×C₉	—	2	1	no	—
48	C₉×C₃⋊C₈	—	2	1	no	—
49	C₃×#488	—	3	1	no	—
50	#4800	C₉	1	1	yes	characteristic, direct product factor
51	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
54	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
55	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
56	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
57	C₃×C₄⋊Dic₃	—	2	1	no	—
58	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
59	C₃×Dic₃⋊C₄	—	2	1	no	—
60	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
62	C₂×Q₈×C₉	—	2	1	no	—
63	C₂×Q₈×C₉	C₂×D₃	1	1	yes	characteristic
64	C₂×D₄×C₉	—	3	1	no	—
65	C₂×C₈×C₉	—	2	1	no	—
66	C₂×C₈×C₉	—	3	1	no	—
67	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
68	C₉×C₄⋊C₄	—	2	1	no	—
69	C₉×C₄⋊C₄	—	3	1	no	—
70	C₉×C₄⋊C₄	—	6	1	no	—
71	C₉×C₄⋊C₄	—	6	1	no	—
72	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
73	C₉×C₂²⋊C₄	—	6	1	no	—
74	C₉×C₂²⋊C₄	—	6	1	no	—
75	C₄²×C₉	—	3	1	no	—
76	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
77	C₂×D₃×C₉	—	4	1	no	—
78	C₂×D₃×C₉	—	4	1	no	—
79	C₂×D₃×C₉	—	4	1	no	—
80	C₃×C₄×C₉	—	4	1	no	—
81	C₃×C₄×C₉	—	4	1	no	—
82	C₃×C₄×C₉	#41	1	1	yes	characteristic
83	C₃×C₄×C₉	#41	1	1	yes	characteristic
84	C₉×Dic₃	—	2	1	no	—
85	C₉×Dic₃	—	2	1	no	—
86	C₉×Dic₃	—	4	1	no	—
87	C₃×#125	—	3	1	no	—
88	C₃×#124	—	3	1	no	—
89	#947	C₂×C₉	1	1	yes	characteristic
90	#890	C₂×C₉	1	1	yes	characteristic, complemented
91	C₃×#103	—	2	1	no	—
92	C₃×#103	—	3	1	no	—
93	C₃×#103	C₂×C₉	1	1	yes	characteristic, complemented
94	C₃×#103	C₃×D₃	1	1	yes	characteristic
95	C₃×#102	—	3	1	no	—
96	C₃×#102	—	3	1	no	—
97	C₃×C₄×C₈	—	3	1	no	—
98	#835	C₂×C₉	1	1	yes	characteristic, complemented
99	#821	C₂×C₉	1	1	yes	characteristic
100	C₈×Dic₃	C₂×C₉	1	1	yes	characteristic, complemented
101	#809	C₂×C₉	1	1	yes	characteristic
102	C₂²×C₃×D₃	—	2	1	no	—
103	C₃²×Q₈	—	2	1	no	—
104	C₃²×Q₈	—	2	1	no	—
105	C₂×C₃²×C₄	—	2	1	no	—
106	C₂×C₃²×C₄	—	2	1	no	—
107	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
108	C₂×C₃×Dic₃	—	2	1	no	—
109	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
110	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
111	C₃×D₁₂	—	2	1	no	—
112	C₃×D₁₂	—	2	1	no	—
113	C₂³×C₉	—	6	1	no	—
114	C₃²×C₈	—	2	1	no	—
115	C₃×C₃⋊C₈	—	2	1	no	—
116	Q₈×C₉	—	2	2	no	—
117	Q₈×C₉	—	2	1	no	—
118	Q₈×C₉	—	2	1	no	—
119	Q₈×C₉	—	4	1	no	—
120	D₄×C₉	—	6	1	no	—
121	D₄×C₉	—	6	1	no	—
122	C₂×C₄×C₉	—	2	1	no	—
123	C₂×C₄×C₉	—	2	1	no	—
124	C₂×C₄×C₉	—	2	1	no	—
125	C₂×C₄×C₉	—	3	1	no	—
126	C₂×C₄×C₉	—	3	1	no	—
127	C₂×C₄×C₉	—	4	1	no	—
128	C₂×C₄×C₉	—	4	1	no	—
129	C₂×C₄×C₉	—	6	1	no	—
130	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
131	C₈×C₉	—	2	1	no	—
132	C₈×C₉	—	4	1	no	—
133	C₈×C₉	—	6	1	no	—
134	#488	—	3	1	no	Sylow
135	C₂×C₃×C₉	#124	1	1	yes	characteristic
136	C₂×C₃×C₉	#135	1	1	yes	characteristic
137	C₂×C₃×C₉	#135	1	1	yes	characteristic
138	D₃×C₉	—	8	1	no	—
139	C₂×C₃×Q₈	—	2	1	no	—
140	C₂×C₃×Q₈	C₂²×C₉	1	1	yes	characteristic
141	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
142	C₂×C₃×D₄	—	3	1	no	—
143	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
144	C₂×C₃×C₈	—	2	1	no	—
145	C₂×C₃×C₈	—	3	1	no	—
146	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
147	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
148	C₃×C₄⋊C₄	—	2	1	no	—
149	C₃×C₄⋊C₄	—	3	1	no	—
150	C₃×C₄⋊C₄	—	6	1	no	—
151	C₃×C₄⋊C₄	—	6	1	no	—
152	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
153	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
154	C₃×C₂²⋊C₄	—	6	1	no	—
155	C₃×C₂²⋊C₄	—	6	1	no	—
156	C₃×C₄²	—	3	1	no	—
157	(C₂×D₃)⋊C₄	—	2	1	no	—
158	(C₂×D₃)⋊C₄	—	2	1	no	—
159	C₄⋊Dic₃	—	2	1	no	—
160	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
161	Dic₃⋊C₄	—	2	1	no	—
162	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
163	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
164	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂×C₃×D₃	—	4	1	no	—
166	C₂×C₃×D₃	—	4	1	no	—
167	C₂×C₃×D₃	—	4	1	no	—
168	C₃²×C₄	—	4	1	no	—
169	C₃²×C₄	—	4	1	no	—
170	C₃²×C₄	C₃×#41	1	1	yes	characteristic
171	C₃²×C₄	C₃×#41	1	1	yes	characteristic
172	C₃×Dic₃	—	2	1	no	—
173	C₃×Dic₃	—	2	1	no	—
174	C₃×Dic₃	—	4	1	no	—
175	C₂²×C₉	—	2	1	no	—
176	C₂²×C₉	—	12	1	no	—
177	C₂²×C₉	—	12	1	no	—
178	C₂²×C₉	—	12	1	no	—
179	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
180	C₄×C₉	—	2	1	no	—
181	C₄×C₉	—	2	1	no	—
182	C₄×C₉	—	4	2	no	—
183	C₄×C₉	—	4	1	no	—
184	C₄×C₉	—	4	1	no	—
185	C₄×C₉	—	6	1	no	—
186	C₄×C₉	—	6	1	no	—
187	C₄×C₉	—	8	1	no	—
188	C₄×C₉	—	12	1	no	—
189	C₄×C₉	#235	1	1	yes	characteristic
190	C₄×C₉	#237	1	1	yes	characteristic
191	#125	—	3	1	no	—
192	#124	—	3	1	no	—
193	#103	—	3	1	no	—
194	#103	D₃×C₉	1	1	yes	characteristic, complemented
195	#102	—	3	1	no	—
196	#102	—	3	1	no	—
197	C₄×C₈	—	3	1	no	—
198	C₃×C₉	#488	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₃	—	6	1	no	—
200	C₂²×D₃	—	2	1	no	—
201	C₃×Q₈	—	2	2	no	—
202	C₃×Q₈	—	2	1	no	—
203	C₃×Q₈	—	2	1	no	—
204	C₃×Q₈	—	2	1	no	—
205	C₃×Q₈	—	2	1	no	—
206	C₃×Q₈	—	4	1	no	—
207	C₃×D₄	—	6	1	no	—
208	C₃×D₄	—	6	1	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	1	no	—
212	C₂×C₃×C₄	—	2	1	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	3	1	no	—
215	C₂×C₃×C₄	—	3	1	no	—
216	C₂×C₃×C₄	—	4	1	no	—
217	C₂×C₃×C₄	—	4	1	no	—
218	C₂×C₃×C₄	—	6	1	no	—
219	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
221	C₂×Dic₃	—	2	1	no	—
222	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
223	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
224	D₁₂	—	2	1	no	—
225	D₁₂	—	2	1	no	—
226	C₃×C₈	—	2	1	no	—
227	C₃×C₈	—	2	1	no	—
228	C₃×C₈	—	4	1	no	—
229	C₃×C₈	—	6	1	no	—
230	C₃⋊C₈	—	2	1	no	—
231	C₂×C₃²	C₃×#124	1	1	yes	characteristic
232	C₂×C₃²	C₃×#135	1	1	yes	characteristic
233	C₂×C₃²	C₃×#135	1	1	yes	characteristic
234	C₃×D₃	—	8	1	no	—
235	C₂×C₉	—	2	1	no	—
236	C₂×C₉	—	2	1	no	—
237	C₂×C₉	—	2	1	no	—
238	C₂×C₉	—	24	1	no	—
239	C₂×C₉	#885	1	1	yes	characteristic
240	C₂×C₉	#916	1	1	yes	characteristic
241	C₂×C₉	#919	1	1	yes	characteristic
242	C₂×Q₈	C₂×D₃×C₉	1	1	yes	characteristic
243	C₂×D₄	—	3	1	no	—
244	C₂×C₈	—	3	1	no	—
245	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
246	C₄⋊C₄	—	3	1	no	—
247	C₄⋊C₄	—	6	1	no	—
248	C₄⋊C₄	—	6	1	no	—
249	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
250	C₂²⋊C₄	—	6	1	no	—
251	C₂²⋊C₄	—	6	1	no	—
252	C₄²	—	3	1	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	12	1	no	—
255	C₂²×C₃	—	12	1	no	—
256	C₂²×C₃	—	12	1	no	—
257	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
258	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
259	C₂×D₃	—	4	1	no	—
260	C₂×D₃	—	4	1	no	—
261	C₂×D₃	—	4	1	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	4	2	no	—
265	C₃×C₄	—	4	1	no	—
266	C₃×C₄	—	4	1	no	—
267	C₃×C₄	—	4	1	no	—
268	C₃×C₄	—	4	1	no	—
269	C₃×C₄	—	6	1	no	—
270	C₃×C₄	—	6	1	no	—
271	C₃×C₄	—	8	1	no	—
272	C₃×C₄	—	12	1	no	—
273	C₃×C₄	C₉×#41	1	1	yes	characteristic
274	C₃×C₄	C₉×#41	1	1	yes	characteristic
275	C₃×C₄	C₃×#235	1	1	yes	characteristic
276	C₃×C₄	C₃×#237	1	1	yes	characteristic
277	Dic₃	—	2	1	no	—
278	Dic₃	—	2	1	no	—
279	Dic₃	—	4	1	no	—
280	C₃²	C₃×#488	1	1	yes	characteristic
281	C₉	—	2	1	no	—
282	C₉	#4800	1	1	yes	characteristic, direct product factor
283	C₂³	—	6	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	D₄	—	6	1	no	—
287	D₄	—	6	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	3	1	no	—
291	C₂×C₄	—	3	1	no	—
292	C₂×C₄	—	6	1	no	—
293	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	6	1	no	—
296	C₂×C₃	—	2	1	no	—
297	C₂×C₃	—	2	1	no	—
298	C₂×C₃	—	2	1	no	—
299	C₂×C₃	—	24	1	no	—
300	C₂×C₃	C₉×#124	1	1	yes	characteristic
301	C₂×C₃	C₉×#135	1	1	yes	characteristic
302	C₂×C₃	C₉×#135	1	1	yes	characteristic
303	C₂×C₃	C₃×#885	1	1	yes	characteristic
304	C₂×C₃	C₃×#916	1	1	yes	characteristic
305	C₂×C₃	C₃×#919	1	1	yes	characteristic
306	D₃	—	8	1	no	—
307	C₂²	—	12	1	no	—
308	C₂²	—	12	1	no	—
309	C₂²	—	12	1	no	—
310	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	6	1	no	—
314	C₄	—	6	1	no	—
315	C₄	—	12	1	no	—
316	C₄	C₉×#235	1	1	yes	characteristic
317	C₄	C₉×#237	1	1	yes	characteristic
318	C₃	—	2	1	no	—
319	C₃	C₉×#488	1	1	yes	characteristic, complemented
320	C₃	C₃×#4800	1	1	yes	characteristic
321	C₂	—	24	1	no	—
322	C₂	C₉×#885	1	1	yes	characteristic
323	C₂	C₉×#916	1	1	yes	characteristic
324	C₂	C₉×#919	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	D₃×D₄
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#103	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃×D₁₂	C₂²×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#103	D₃
hypocenter	C₃	C₉×#488
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#103

Complements

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

N	H ≅ G/N	type
2. C₉×#890	C₂	semidirect
3. C₃×C₉×#103	C₂	semidirect
4. C₉×#835	C₂	semidirect
6. C₈×C₉×Dic₃	C₂	semidirect
32. C₉×#103	D₃	semidirect
50. #4800	C₉	direct (N × H)
90. #890	C₂×C₉	semidirect
93. C₃×#103	C₂×C₉	semidirect
98. #835	C₂×C₉	semidirect
100. C₈×Dic₃	C₂×C₉	semidirect
194. #103	D₃×C₉	semidirect
198. C₃×C₉	#488	semidirect
282. C₉	#4800	direct (N × H)
319. C₃	C₉×#488	semidirect

Maximal subgroups

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

subgroup	index
C₉×#947	2
C₉×#890	2
C₃×C₉×#103	2
C₉×#835	2
C₉×#821	2
C₈×C₉×Dic₃	2
C₉×#809	2
C₃×#4800	3
C₉×#488	3

Subgroups — C9×#4800 #13141859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4800 #13141859