Subgroups — C₉×#4884 #13142204

Order 1728.

348 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#897	C₂	1	1	yes	maximal, characteristic
2	D₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×(C₂×D₃)⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₉×#816	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#810	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#807	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4884	C₃	1	1	yes	maximal, characteristic
9	C₉×#484	—	3	1	no	maximal
10	C₂×C₄×D₃×C₉	—	2	1	no	—
11	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
17	C₉×C₄⋊Dic₃	—	2	1	no	—
18	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
19	C₉×Dic₃⋊C₄	—	2	1	no	—
20	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×#897	C₂×C₃	1	1	yes	characteristic
22	C₃×D₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
23	C₃²×#107	C₂×C₃	1	1	yes	characteristic
24	C₃×(C₂×D₃)⋊C₈	C₂×C₃	1	1	yes	characteristic
25	C₃×#816	C₂×C₃	1	1	yes	characteristic
26	C₃×#810	C₂×C₃	1	1	yes	characteristic
27	C₃×#807	C₂×C₃	1	1	yes	characteristic
28	C₉×C₂²⋊Q₈	—	3	1	no	—
29	C₂×C₉×C₄⋊C₄	—	3	1	no	—
30	C₉×#107	—	2	1	no	—
31	C₉×#107	—	3	1	no	—
32	C₉×#107	D₃	1	1	yes	characteristic, complemented
33	C₉×#103	—	3	1	no	—
34	C₉×#103	—	3	1	no	—
35	C₉×C₂²⋊C₈	—	3	1	no	—
36	C₂²×D₃×C₉	D₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₉	—	2	1	no	—
38	C₂×C₃×C₄×C₉	—	2	1	no	—
39	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂×C₉×Dic₃	—	2	1	no	—
41	C₂×C₉×Dic₃	—	2	1	no	—
42	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
43	C₄×D₃×C₉	—	2	1	no	—
44	C₄×D₃×C₉	—	2	1	no	—
45	C₄×D₃×C₉	—	4	2	no	—
46	C₉×Dic₆	—	2	1	no	—
47	C₉×Dic₆	—	2	1	no	—
48	C₃×C₈×C₉	—	2	1	no	—
49	C₉×C₃⋊C₈	—	2	1	no	—
50	C₃×#484	—	3	1	no	—
51	#4884	C₉	1	1	yes	characteristic, direct product factor
52	C₂×C₃×C₄×D₃	—	2	1	no	—
53	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₃×Dic₆	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
56	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
57	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
58	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
59	C₃×C₄⋊Dic₃	—	2	1	no	—
60	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
61	C₃×Dic₃⋊C₄	—	2	1	no	—
62	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
63	C₂×Q₈×C₉	—	3	1	no	—
64	C₂²×C₄×C₉	—	3	1	no	—
65	C₂²×C₄×C₉	—	6	1	no	—
66	C₂×C₈×C₉	—	2	1	no	—
67	C₂×C₈×C₉	—	3	1	no	—
68	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
69	C₉×C₄⋊C₄	—	2	1	no	—
70	C₉×C₄⋊C₄	—	2	1	no	—
71	C₉×C₄⋊C₄	—	3	1	no	—
72	C₉×C₄⋊C₄	—	6	1	no	—
73	C₉×C₄⋊C₄	—	6	1	no	—
74	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
75	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
76	C₉×C₂²⋊C₄	—	6	1	no	—
77	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
78	C₂×D₃×C₉	—	2	1	no	—
79	C₂×D₃×C₉	—	2	1	no	—
80	C₂×D₃×C₉	Q₁₆	1	2	yes	—
81	C₃×C₄×C₉	—	4	1	no	—
82	C₃×C₄×C₉	—	4	1	no	—
83	C₃×C₄×C₉	#41	1	1	yes	characteristic
84	C₃×C₄×C₉	#41	1	1	yes	characteristic
85	C₉×Dic₃	—	2	1	no	—
86	C₉×Dic₃	—	4	1	no	—
87	C₉×Dic₃	—	4	1	no	—
88	C₃×C₂²⋊Q₈	—	3	1	no	—
89	C₂×C₃×C₄⋊C₄	—	3	1	no	—
90	#897	C₂×C₉	1	1	yes	characteristic
91	D₃×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
92	C₃×#107	—	2	1	no	—
93	C₃×#107	—	3	1	no	—
94	C₃×#107	C₂×C₉	1	1	yes	characteristic, complemented
95	C₃×#107	C₃×D₃	1	1	yes	characteristic
96	C₃×#103	—	3	1	no	—
97	C₃×#103	—	3	1	no	—
98	C₃×C₂²⋊C₈	—	3	1	no	—
99	(C₂×D₃)⋊C₈	C₂×C₉	1	1	yes	characteristic
100	#816	C₂×C₉	1	1	yes	characteristic, complemented
101	#810	C₂×C₉	1	1	yes	characteristic, complemented
102	#807	C₂×C₉	1	1	yes	characteristic, complemented
103	C₂²×C₃×D₃	C₃×D₄	1	1	yes	characteristic
104	C₂×C₃²×C₄	—	2	1	no	—
105	C₂×C₃²×C₄	—	2	1	no	—
106	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
107	C₂×C₃×Dic₃	—	2	1	no	—
108	C₂×C₃×Dic₃	—	2	1	no	—
109	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
110	C₃×C₄×D₃	—	2	1	no	—
111	C₃×C₄×D₃	—	2	1	no	—
112	C₃×C₄×D₃	—	4	2	no	—
113	C₃×Dic₆	—	2	1	no	—
114	C₃×Dic₆	—	2	1	no	—
115	C₂³×C₉	—	3	1	no	—
116	C₃²×C₈	—	2	1	no	—
117	C₃×C₃⋊C₈	—	2	1	no	—
118	Q₈×C₉	—	6	1	no	—
119	Q₈×C₉	—	6	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	—	3	1	no	—
124	C₂×C₄×C₉	—	4	1	no	—
125	C₂×C₄×C₉	—	4	1	no	—
126	C₂×C₄×C₉	—	6	1	no	—
127	C₂×C₄×C₉	—	6	1	no	—
128	C₂×C₄×C₉	—	6	1	no	—
129	C₂×C₄×C₉	—	6	1	no	—
130	C₂×C₄×C₉	—	12	2	no	—
131	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
132	C₈×C₉	—	2	2	no	—
133	C₈×C₉	—	2	1	no	—
134	C₈×C₉	—	6	1	no	—
135	#484	—	3	1	no	Sylow
136	C₂×C₃×C₉	#123	1	1	yes	characteristic
137	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
138	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
139	D₃×C₉	—	2	2	no	—
140	C₂×C₃×Q₈	—	3	1	no	—
141	C₂²×C₃×C₄	—	3	1	no	—
142	C₂²×C₃×C₄	—	6	1	no	—
143	C₂×C₄×D₃	—	2	1	no	—
144	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
145	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic
146	C₂×C₃×C₈	—	2	1	no	—
147	C₂×C₃×C₈	—	3	1	no	—
148	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
149	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
150	C₃×C₄⋊C₄	—	2	1	no	—
151	C₃×C₄⋊C₄	—	2	1	no	—
152	C₃×C₄⋊C₄	—	3	1	no	—
153	C₃×C₄⋊C₄	—	6	1	no	—
154	C₃×C₄⋊C₄	—	6	1	no	—
155	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
156	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
157	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
158	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
159	C₃×C₂²⋊C₄	—	6	1	no	—
160	(C₂×D₃)⋊C₄	—	2	1	no	—
161	C₄⋊Dic₃	—	2	1	no	—
162	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
163	Dic₃⋊C₄	—	2	1	no	—
164	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
165	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
166	C₂×C₃×D₃	—	2	1	no	—
167	C₂×C₃×D₃	—	2	1	no	—
168	C₂×C₃×D₃	C₃×Q₁₆	1	2	yes	—
169	C₃²×C₄	—	4	1	no	—
170	C₃²×C₄	—	4	1	no	—
171	C₃²×C₄	C₃×#41	1	1	yes	characteristic
172	C₃²×C₄	C₃×#41	1	1	yes	characteristic
173	C₃×Dic₃	—	2	1	no	—
174	C₃×Dic₃	—	4	1	no	—
175	C₃×Dic₃	—	4	1	no	—
176	C₂²×C₉	—	2	1	no	—
177	C₂²×C₉	—	3	2	no	—
178	C₂²×C₉	—	6	1	no	—
179	C₂²×C₉	—	6	1	no	—
180	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
181	C₄×C₉	—	2	1	no	—
182	C₄×C₉	—	2	1	no	—
183	C₄×C₉	—	4	1	no	—
184	C₄×C₉	—	4	1	no	—
185	C₄×C₉	—	6	1	no	—
186	C₄×C₉	—	8	1	no	—
187	C₄×C₉	—	8	1	no	—
188	C₄×C₉	—	12	1	no	—
189	C₄×C₉	—	12	1	no	—
190	C₄×C₉	#235	1	1	yes	characteristic
191	C₄×C₉	#239	1	1	yes	characteristic
192	C₂²⋊Q₈	—	3	1	no	—
193	C₂×C₄⋊C₄	—	3	1	no	—
194	#107	—	3	1	no	—
195	#107	D₃×C₉	1	1	yes	characteristic, complemented
196	#103	—	3	1	no	—
197	#103	—	3	1	no	—
198	C₂²⋊C₈	—	3	1	no	—
199	C₃×C₉	#484	1	1	yes	characteristic, complemented, Sylow
200	C₂³×C₃	—	3	1	no	—
201	C₂²×D₃	D₄×C₉	1	1	yes	characteristic
202	C₃×Q₈	—	6	1	no	—
203	C₃×Q₈	—	6	1	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	3	1	no	—
210	C₂×C₃×C₄	—	4	1	no	—
211	C₂×C₃×C₄	—	4	1	no	—
212	C₂×C₃×C₄	—	6	1	no	—
213	C₂×C₃×C₄	—	6	1	no	—
214	C₂×C₃×C₄	—	6	1	no	—
215	C₂×C₃×C₄	—	6	1	no	—
216	C₂×C₃×C₄	—	12	2	no	—
217	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
219	C₂×Dic₃	—	2	1	no	—
220	C₂×Dic₃	—	2	1	no	—
221	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
222	C₄×D₃	—	2	1	no	—
223	C₄×D₃	—	2	1	no	—
224	C₄×D₃	—	4	2	no	—
225	Dic₆	—	2	1	no	—
226	Dic₆	—	2	1	no	—
227	C₃×C₈	—	2	2	no	—
228	C₃×C₈	—	2	1	no	—
229	C₃×C₈	—	2	1	no	—
230	C₃×C₈	—	6	1	no	—
231	C₃⋊C₈	—	2	1	no	—
232	C₂×C₃²	C₃×#123	1	1	yes	characteristic
233	C₂×C₃²	C₂×C₃×Q₁₆	1	1	yes	characteristic
234	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
235	C₃×D₃	—	2	2	no	—
236	C₂×C₉	—	2	1	no	—
237	C₂×C₉	—	2	1	no	—
238	C₂×C₉	—	2	1	no	—
239	C₂×C₉	—	6	2	no	—
240	C₂×C₉	#894	1	1	yes	characteristic
241	C₂×C₉	#911	1	1	yes	characteristic
242	C₂×C₉	D₃×Q₁₆	1	1	yes	characteristic
243	C₂×Q₈	—	3	1	no	—
244	C₂²×C₄	—	3	1	no	—
245	C₂²×C₄	—	6	1	no	—
246	C₂×C₈	—	3	1	no	—
247	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
248	C₄⋊C₄	—	3	1	no	—
249	C₄⋊C₄	—	6	1	no	—
250	C₄⋊C₄	—	6	1	no	—
251	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
252	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
253	C₂²⋊C₄	—	6	1	no	—
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	3	2	no	—
256	C₂²×C₃	—	6	1	no	—
257	C₂²×C₃	—	6	1	no	—
258	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
259	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
260	C₂×D₃	—	2	1	no	—
261	C₂×D₃	—	2	1	no	—
262	C₂×D₃	C₉×Q₁₆	1	2	yes	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	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	—	8	1	no	—
271	C₃×C₄	—	8	1	no	—
272	C₃×C₄	—	12	1	no	—
273	C₃×C₄	—	12	1	no	—
274	C₃×C₄	C₉×#41	1	1	yes	characteristic
275	C₃×C₄	C₉×#41	1	1	yes	characteristic
276	C₃×C₄	C₃×#235	1	1	yes	characteristic
277	C₃×C₄	C₃×#239	1	1	yes	characteristic
278	Dic₃	—	2	1	no	—
279	Dic₃	—	4	1	no	—
280	Dic₃	—	4	1	no	—
281	C₃²	C₃×#484	1	1	yes	characteristic
282	C₉	—	2	1	no	—
283	C₉	#4884	1	1	yes	characteristic, direct product factor
284	C₂³	—	3	1	no	—
285	Q₈	—	6	1	no	—
286	Q₈	—	6	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	3	1	no	—
290	C₂×C₄	—	6	1	no	—
291	C₂×C₄	—	6	1	no	—
292	C₂×C₄	—	6	1	no	—
293	C₂×C₄	—	6	1	no	—
294	C₂×C₄	—	12	2	no	—
295	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	6	1	no	—
298	C₂×C₃	—	2	1	no	—
299	C₂×C₃	—	2	1	no	—
300	C₂×C₃	—	2	1	no	—
301	C₂×C₃	—	6	2	no	—
302	C₂×C₃	C₉×#123	1	1	yes	characteristic
303	C₂×C₃	C₂×C₉×Q₁₆	1	1	yes	characteristic
304	C₂×C₃	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
305	C₂×C₃	C₃×#894	1	1	yes	characteristic
306	C₂×C₃	C₃×#911	1	1	yes	characteristic
307	C₂×C₃	C₃×D₃×Q₁₆	1	1	yes	characteristic
308	D₃	—	2	2	no	—
309	C₂²	—	3	2	no	—
310	C₂²	—	6	1	no	—
311	C₂²	—	6	1	no	—
312	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	6	1	no	—
316	C₄	—	12	1	no	—
317	C₄	—	12	1	no	—
318	C₄	C₉×#235	1	1	yes	characteristic
319	C₄	C₉×#239	1	1	yes	characteristic
320	C₃	—	2	1	no	—
321	C₃	C₉×#484	1	1	yes	characteristic, complemented
322	C₃	C₃×#4884	1	1	yes	characteristic
323	C₂	—	6	2	no	—
324	C₂	C₉×#894	1	1	yes	characteristic
325	C₂	C₉×#911	1	1	yes	characteristic
326	C₂	D₃×C₉×Q₁₆	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×#107	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂²×C₃×D₃	C₃×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#107	D₃
hypocenter	C₃	C₉×#484
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#107

Complements

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

N	H ≅ G/N	type
3. C₃×C₉×#107	C₂	semidirect
5. C₉×#816	C₂	semidirect
6. C₉×#810	C₂	semidirect
7. C₉×#807	C₂	semidirect
32. C₉×#107	D₃	semidirect
51. #4884	C₉	direct (N × H)
94. C₃×#107	C₂×C₉	semidirect
100. #816	C₂×C₉	semidirect
101. #810	C₂×C₉	semidirect
102. #807	C₂×C₉	semidirect
195. #107	D₃×C₉	semidirect
199. C₃×C₉	#484	semidirect
283. C₉	#4884	direct (N × H)
321. C₃	C₉×#484	semidirect

Maximal subgroups

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

subgroup	index
C₉×#897	2
D₃×C₉×C₄⋊C₄	2
C₃×C₉×#107	2
C₉×(C₂×D₃)⋊C₈	2
C₉×#816	2
C₉×#810	2
C₉×#807	2
C₃×#4884	3
C₉×#484	3

Subgroups — C9×#4884 #13142204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4884 #13142204