Subgroups — C₉×#4731 #13141591

Order 1728.

614 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₉×#924	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×#204	C₂	1	2	yes	maximal, complemented
4	C₃×C₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#821	C₂	1	2	yes	maximal, complemented
6	C₃×#4731	C₃	1	1	yes	maximal, characteristic
7	C₉×#450	—	3	1	no	maximal
8	C₂³×D₃×C₉	—	2	1	no	—
9	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₉×D₁₂	—	2	1	no	—
11	C₂×C₉×D₁₂	—	2	4	no	—
12	C₂×C₉×D₁₂	C₂²	1	2	yes	—
13	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₃×C₈×C₉	C₂²	1	2	yes	complemented
15	C₉×#217	—	2	1	no	—
16	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
17	C₉×Dic₃⋊C₄	—	2	1	no	—
18	C₉×#204	—	2	4	no	—
19	C₂²×C₃×D₁₂	C₂×C₃	1	1	yes	characteristic
20	C₃×#924	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×#204	C₂×C₃	1	2	yes	—
22	C₂²×D₄×C₉	—	3	1	no	—
23	C₃²×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
24	C₃×#821	C₂×C₃	1	2	yes	—
25	C₂×C₉×SD₁₆	—	3	2	no	—
26	C₉×C₂²⋊Q₈	—	3	1	no	—
27	C₉×#102	—	3	2	no	—
28	C₉×C₂²⋊C₈	—	2	1	no	—
29	C₉×C₂²⋊C₈	D₃	1	1	yes	characteristic, complemented
30	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
31	C₂²×D₃×C₉	—	2	4	no	—
32	C₂²×D₃×C₉	—	2	2	no	—
33	C₂²×D₃×C₉	—	4	2	no	—
34	C₂²×D₃×C₉	—	4	2	no	—
35	C₂×C₃×C₄×C₉	—	2	2	no	—
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	C₂×C₉×Dic₃	—	2	1	no	—
40	C₉×D₁₂	—	2	4	no	—
41	C₉×D₁₂	—	2	2	no	—
42	C₉×D₁₂	D₄	1	4	yes	—
43	C₉×Dic₆	—	2	2	no	—
44	C₃×C₈×C₉	—	2	2	no	—
45	C₃×#450	—	3	1	no	—
46	#4731	C₉	1	1	yes	characteristic, direct product factor
47	C₂³×C₃×D₃	—	2	1	no	—
48	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₃×D₁₂	—	2	1	no	—
50	C₂×C₃×D₁₂	—	2	4	no	—
51	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
52	C₂×C₃×Dic₆	C₂²×C₃	1	1	yes	characteristic
53	C₂⁴×C₉	—	6	1	no	—
54	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
55	C₃×#217	—	2	1	no	—
56	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
57	C₃×Dic₃⋊C₄	—	2	1	no	—
58	C₃×#204	—	2	4	no	—
59	C₂×Q₈×C₉	—	3	1	no	—
60	C₂×D₄×C₉	—	3	2	no	—
61	C₂×D₄×C₉	—	6	4	no	—
62	C₂×D₄×C₉	—	6	1	no	—
63	C₂²×C₄×C₉	—	2	1	no	—
64	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
65	C₉×SD₁₆	—	6	4	no	—
66	C₂×C₈×C₉	—	2	2	no	—
67	C₂×C₈×C₉	C₂×D₃	1	2	yes	complemented
68	C₉×C₄⋊C₄	—	3	1	no	—
69	C₉×C₄⋊C₄	—	6	1	no	—
70	C₉×C₂²⋊C₄	—	6	1	no	—
71	C₂²×C₃×C₉	—	2	1	no	—
72	C₂²×C₃×C₉	—	2	1	no	—
73	C₂²×C₃×C₉	SD₁₆	1	2	yes	complemented
74	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
75	C₂×D₃×C₉	—	2	4	no	—
76	C₂×D₃×C₉	—	4	8	no	—
77	C₂×D₃×C₉	—	4	2	no	—
78	C₂×D₃×C₉	—	4	2	no	—
79	C₃×C₄×C₉	—	2	1	no	—
80	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
81	C₉×Dic₃	—	4	1	no	—
82	C₉×Dic₃	—	4	1	no	—
83	C₂²×C₃×D₄	—	3	1	no	—
84	C₂²×D₁₂	C₂×C₉	1	1	yes	characteristic
85	C₂×C₃×SD₁₆	—	3	2	no	—
86	C₃×C₂²⋊Q₈	—	3	1	no	—
87	#924	C₂×C₉	1	1	yes	characteristic, complemented
88	C₂×#204	C₂×C₉	1	2	yes	complemented
89	C₃×#102	—	3	2	no	—
90	C₃×C₂²⋊C₈	—	2	1	no	—
91	C₃×C₂²⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
92	C₃×C₂²⋊C₈	C₃×D₃	1	1	yes	characteristic
93	#821	C₂×C₉	1	2	yes	complemented
94	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
95	C₂²×C₃×D₃	—	2	4	no	—
96	C₂²×C₃×D₃	—	2	2	no	—
97	C₂²×C₃×D₃	—	4	2	no	—
98	C₂²×C₃×D₃	—	4	2	no	—
99	C₂×C₃²×C₄	—	2	2	no	—
100	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
101	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
102	C₂×C₃×Dic₃	—	2	1	no	—
103	C₂×C₃×Dic₃	—	2	1	no	—
104	C₃×D₁₂	—	2	4	no	—
105	C₃×D₁₂	—	2	2	no	—
106	C₃×D₁₂	C₃×D₄	1	4	yes	—
107	C₃×Dic₆	—	2	2	no	—
108	C₂³×C₉	—	2	1	no	—
109	C₂³×C₉	—	6	4	no	—
110	C₂³×C₉	—	6	2	no	—
111	C₂³×C₉	—	12	2	no	—
112	C₂³×C₉	—	12	2	no	—
113	C₂³×C₉	D₁₂	1	1	yes	characteristic
114	C₃²×C₈	—	2	2	no	—
115	Q₈×C₉	—	6	2	no	—
116	D₄×C₉	—	3	4	no	—
117	D₄×C₉	—	6	4	no	—
118	D₄×C₉	—	6	2	no	—
119	C₂×C₄×C₉	—	2	2	no	—
120	C₂×C₄×C₉	—	2	1	no	—
121	C₂×C₄×C₉	—	2	1	no	—
122	C₂×C₄×C₉	—	4	2	no	—
123	C₂×C₄×C₉	—	6	1	no	—
124	C₂×C₄×C₉	—	6	1	no	—
125	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
126	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
127	C₈×C₉	—	2	2	no	—
128	C₈×C₉	—	4	2	no	—
129	#450	—	3	1	no	Sylow
130	C₂×C₃×C₉	—	2	2	no	—
131	C₂×C₃×C₉	#120	1	1	yes	characteristic
132	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
134	D₃×C₉	—	4	4	no	—
135	C₂⁴×C₃	—	6	1	no	—
136	C₂³×D₃	—	2	1	no	—
137	C₂×C₃×Q₈	—	3	1	no	—
138	C₂×C₃×D₄	—	3	2	no	—
139	C₂×C₃×D₄	—	6	4	no	—
140	C₂×C₃×D₄	—	6	1	no	—
141	C₂²×C₃×C₄	—	2	1	no	—
142	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
143	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
144	C₂×D₁₂	—	2	1	no	—
145	C₂×D₁₂	—	2	4	no	—
146	C₂×D₁₂	C₂²×C₉	1	2	yes	—
147	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic, complemented
148	C₃×SD₁₆	—	6	4	no	—
149	C₂×C₃×C₈	—	2	2	no	—
150	C₂×C₃×C₈	C₂²×C₉	1	2	yes	complemented
151	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
152	C₃×C₄⋊C₄	—	3	1	no	—
153	C₃×C₄⋊C₄	—	6	1	no	—
154	C₃×C₂²⋊C₄	—	6	1	no	—
155	#217	—	2	1	no	—
156	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic, complemented
157	Dic₃⋊C₄	—	2	1	no	—
158	#204	—	2	4	no	—
159	C₂²×C₃²	—	2	1	no	—
160	C₂²×C₃²	—	2	1	no	—
161	C₂²×C₃²	C₃×SD₁₆	1	2	yes	—
162	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
163	C₂×C₃×D₃	—	2	4	no	—
164	C₂×C₃×D₃	—	4	8	no	—
165	C₂×C₃×D₃	—	4	2	no	—
166	C₂×C₃×D₃	—	4	2	no	—
167	C₃²×C₄	—	2	1	no	—
168	C₃²×C₄	C₂×C₃×D₄	1	2	yes	—
169	C₃×Dic₃	—	4	1	no	—
170	C₃×Dic₃	—	4	1	no	—
171	C₂²×C₉	—	2	1	no	—
172	C₂²×C₉	—	2	1	no	—
173	C₂²×C₉	—	2	2	no	—
174	C₂²×C₉	—	2	1	no	—
175	C₂²×C₉	—	4	1	no	—
176	C₂²×C₉	—	4	1	no	—
177	C₂²×C₉	—	6	4	no	—
178	C₂²×C₉	—	12	8	no	—
179	C₂²×C₉	—	12	2	no	—
180	C₂²×C₉	—	12	2	no	—
181	C₂²×C₉	#204	1	2	yes	complemented
182	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
183	C₄×C₉	—	2	1	no	—
184	C₄×C₉	—	2	2	no	—
185	C₄×C₉	—	4	1	no	—
186	C₄×C₉	—	12	1	no	—
187	C₄×C₉	—	12	1	no	—
188	C₄×C₉	D₃×D₄	1	2	yes	—
189	C₂²×D₄	—	3	1	no	—
190	C₂×SD₁₆	—	3	2	no	—
191	C₂²⋊Q₈	—	3	1	no	—
192	#102	—	3	2	no	—
193	C₂²⋊C₈	D₃×C₉	1	1	yes	characteristic, complemented
194	C₃×C₉	#450	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₃	—	2	1	no	—
196	C₂³×C₃	—	6	4	no	—
197	C₂³×C₃	—	6	2	no	—
198	C₂³×C₃	—	12	2	no	—
199	C₂³×C₃	—	12	2	no	—
200	C₂³×C₃	D₄×C₉	1	1	yes	characteristic
201	C₂³×C₃	C₃×D₁₂	1	1	yes	characteristic
202	C₂²×D₃	—	2	4	no	—
203	C₂²×D₃	—	2	2	no	—
204	C₂²×D₃	—	4	2	no	—
205	C₂²×D₃	—	4	2	no	—
206	C₃×Q₈	—	6	2	no	—
207	C₃×D₄	—	3	4	no	—
208	C₃×D₄	—	6	4	no	—
209	C₃×D₄	—	6	2	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	1	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	4	2	no	—
215	C₂×C₃×C₄	—	6	1	no	—
216	C₂×C₃×C₄	—	6	1	no	—
217	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
219	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
221	C₂×Dic₃	—	2	1	no	—
222	C₂×Dic₃	—	2	1	no	—
223	D₁₂	—	2	4	no	—
224	D₁₂	—	2	2	no	—
225	D₁₂	D₄×C₉	1	4	yes	—
226	Dic₆	—	2	2	no	—
227	C₃×C₈	—	2	2	no	—
228	C₃×C₈	—	2	2	no	—
229	C₃×C₈	—	4	2	no	—
230	C₂×C₃²	—	2	2	no	—
231	C₂×C₃²	C₃×#120	1	1	yes	characteristic
232	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
233	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
234	C₃×D₃	—	4	4	no	—
235	C₂×C₉	—	2	2	no	—
236	C₂×C₉	—	2	1	no	—
237	C₂×C₉	—	2	1	no	—
238	C₂×C₉	—	2	1	no	—
239	C₂×C₉	—	4	2	no	—
240	C₂×C₉	—	12	4	no	—
241	C₂×C₉	#882	1	1	yes	characteristic
242	C₂×C₉	C₂×#204	1	1	yes	characteristic
243	C₂×C₉	#908	1	1	yes	characteristic
244	C₂⁴	—	6	1	no	—
245	C₂×Q₈	—	3	1	no	—
246	C₂×D₄	—	3	2	no	—
247	C₂×D₄	—	6	4	no	—
248	C₂×D₄	—	6	1	no	—
249	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
250	SD₁₆	—	6	4	no	—
251	C₂×C₈	C₂×D₃×C₉	1	2	yes	complemented
252	C₄⋊C₄	—	3	1	no	—
253	C₄⋊C₄	—	6	1	no	—
254	C₂²⋊C₄	—	6	1	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	2	no	—
261	C₂²×C₃	—	4	1	no	—
262	C₂²×C₃	—	4	1	no	—
263	C₂²×C₃	—	6	4	no	—
264	C₂²×C₃	—	12	8	no	—
265	C₂²×C₃	—	12	2	no	—
266	C₂²×C₃	—	12	2	no	—
267	C₂²×C₃	C₉×SD₁₆	1	2	yes	complemented
268	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
269	C₂²×C₃	C₃×#204	1	2	yes	—
270	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
271	C₂×D₃	—	2	4	no	—
272	C₂×D₃	—	4	8	no	—
273	C₂×D₃	—	4	2	no	—
274	C₂×D₃	—	4	2	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	2	no	—
278	C₃×C₄	—	4	1	no	—
279	C₃×C₄	—	12	1	no	—
280	C₃×C₄	—	12	1	no	—
281	C₃×C₄	C₂×D₄×C₉	1	2	yes	—
282	C₃×C₄	C₃×D₃×D₄	1	2	yes	—
283	Dic₃	—	4	1	no	—
284	Dic₃	—	4	1	no	—
285	C₃²	C₃×#450	1	1	yes	characteristic
286	C₉	—	2	1	no	—
287	C₉	#4731	1	1	yes	characteristic, direct product factor
288	C₂³	—	6	4	no	—
289	C₂³	—	6	2	no	—
290	C₂³	—	12	2	no	—
291	C₂³	—	12	2	no	—
292	C₂³	C₉×D₁₂	1	1	yes	characteristic
293	Q₈	—	6	2	no	—
294	D₄	—	3	4	no	—
295	D₄	—	6	4	no	—
296	D₄	—	6	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	6	1	no	—
299	C₂×C₄	—	6	1	no	—
300	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₂×C₃	—	2	2	no	—
304	C₂×C₃	—	2	2	no	—
305	C₂×C₃	—	2	1	no	—
306	C₂×C₃	—	2	1	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	4	2	no	—
309	C₂×C₃	—	12	4	no	—
310	C₂×C₃	C₉×#120	1	1	yes	characteristic
311	C₂×C₃	C₂×C₉×SD₁₆	1	1	yes	characteristic
312	C₂×C₃	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
313	C₂×C₃	C₃×#882	1	1	yes	characteristic
314	C₂×C₃	C₂×C₃×#204	1	1	yes	characteristic
315	C₂×C₃	C₃×#908	1	1	yes	characteristic
316	D₃	—	4	4	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	6	4	no	—
320	C₂²	—	12	8	no	—
321	C₂²	—	12	2	no	—
322	C₂²	—	12	2	no	—
323	C₂²	C₉×#204	1	2	yes	complemented
324	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	12	1	no	—
327	C₄	—	12	1	no	—
328	C₄	D₃×D₄×C₉	1	2	yes	—
329	C₃	—	2	1	no	—
330	C₃	C₉×#450	1	1	yes	characteristic, complemented
331	C₃	C₃×#4731	1	1	yes	characteristic
332	C₂	—	2	2	no	—
333	C₂	—	12	4	no	—
334	C₂	C₉×#882	1	1	yes	characteristic
335	C₂	C₂×C₉×#204	1	1	yes	characteristic
336	C₂	C₉×#908	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×C₂²⋊C₈	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂²×C₃×D₁₂	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×C₂²⋊C₈	D₃
hypocenter	C₃	C₉×#450
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ C₉×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₉×#924	C₂	semidirect
3. C₂×C₉×#204	C₂	semidirect
4. C₃×C₉×C₂²⋊C₈	C₂	semidirect
5. C₉×#821	C₂	semidirect
13. C₂×C₉×Dic₆	C₂²	semidirect
14. C₂×C₃×C₈×C₉	C₂²	semidirect
16. C₉×C₄⋊Dic₃	C₂²	semidirect
29. C₉×C₂²⋊C₈	D₃	semidirect
46. #4731	C₉	direct (N × H)
67. C₂×C₈×C₉	C₂×D₃	semidirect
73. C₂²×C₃×C₉	SD₁₆	semidirect
87. #924	C₂×C₉	semidirect
88. C₂×#204	C₂×C₉	semidirect
91. C₃×C₂²⋊C₈	C₂×C₉	semidirect
93. #821	C₂×C₉	semidirect
147. C₂×Dic₆	C₂²×C₉	semidirect
150. C₂×C₃×C₈	C₂²×C₉	semidirect
156. C₄⋊Dic₃	C₂²×C₉	semidirect
181. C₂²×C₉	#204	semidirect
193. C₂²⋊C₈	D₃×C₉	semidirect
194. C₃×C₉	#450	semidirect
251. C₂×C₈	C₂×D₃×C₉	semidirect
267. C₂²×C₃	C₉×SD₁₆	semidirect
287. C₉	#4731	direct (N × H)
323. C₂²	C₉×#204	semidirect
330. C₃	C₉×#450	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×D₁₂	2
C₉×#924	2
C₂×C₉×#204	2
C₃×C₉×C₂²⋊C₈	2
C₉×#821	2
C₃×#4731	3
C₉×#450	3

Subgroups — C9×#4731 #13141591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4731 #13141591