Subgroups — C₂×C₃×#63389 #13143508

Order 1728.

842 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₉⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₃×#63389	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×#830	—	3	1	no	maximal
5	C₂×#63389	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂⁴×C₃×C₉	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₉	—	2	4	no	—
8	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
10	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₉⋊C₈	—	2	4	no	—
13	C₂×C₃×C₉⋊C₈	C₂²	1	4	yes	complemented
14	C₂³×C₃²×C₄	D₃	1	1	yes	characteristic
15	C₂²×C₃×C₃⋊C₈	—	3	2	no	—
16	C₂³×C₄×C₉	—	2	1	no	—
17	C₂³×C₄×C₉	C₂×C₃	1	1	yes	characteristic
18	C₃×#830	—	3	4	no	—
19	C₂²×C₉⋊C₈	C₂×C₃	1	2	yes	complemented
20	#63389	C₂×C₃	1	4	yes	direct product factor
21	C₂³×C₃×C₉	—	2	2	no	—
22	C₂³×C₃×C₉	—	2	2	no	—
23	C₂³×C₃×C₉	C₈	1	4	yes	complemented
24	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
25	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	—	2	2	no	—
27	C₂×C₃×C₄×C₉	—	2	4	no	—
28	C₂×C₃×C₄×C₉	—	2	4	no	—
29	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	D₄	1	4	yes	—
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₃×C₉⋊C₈	—	2	4	no	—
34	C₂×C₃×C₂²⋊C₈	—	9	1	no	—
35	C₂×#830	—	3	1	no	—
36	C₂⁴×C₃²	Dic₃	1	1	yes	characteristic
37	C₂²×C₃²×C₄	—	2	4	no	—
38	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
39	C₂²×C₃²×C₄	Dic₃	1	2	yes	—
40	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
41	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
42	C₂⁴×C₉	—	2	1	no	—
43	C₂⁴×C₉	C₃×C₄	1	1	yes	characteristic
44	C₂×C₃×C₃⋊C₈	—	3	4	no	—
45	C₂×C₃×C₃⋊C₈	—	6	4	no	—
46	C₂²×C₄×C₉	—	2	8	no	—
47	C₂²×C₄×C₉	—	2	1	no	—
48	C₂²×C₄×C₉	—	2	2	no	—
49	C₂²×C₄×C₉	—	2	2	no	—
50	C₂²×C₄×C₉	—	2	1	no	—
51	C₂²×C₄×C₉	—	2	4	no	—
52	C₂²×C₄×C₉	C₃×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₉	C₃×C₄	1	2	yes	—
54	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
55	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₉⋊C₈	—	2	4	no	—
57	C₂×C₉⋊C₈	C₂²×C₃	1	4	yes	complemented
58	C₂²×C₃×C₉	—	2	8	no	—
59	C₂²×C₃×C₉	—	2	2	no	—
60	C₂²×C₃×C₉	—	2	2	no	—
61	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₃×C₉	C₂×C₈	1	2	yes	—
63	C₂²×C₃×C₉	C₂×C₈	1	4	yes	complemented
64	C₂²×C₃×C₉	M₁₆	1	2	yes	—
65	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
66	C₃×C₄×C₉	—	2	2	no	—
67	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
68	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
69	C₂³×C₃×C₄	—	2	1	no	—
70	C₂³×C₃×C₄	D₉	1	1	yes	characteristic
71	C₂³×C₃×C₄	C₃×D₃	1	1	yes	characteristic
72	C₂²×C₃×C₈	—	9	2	no	—
73	C₂²×C₃⋊C₈	—	3	2	no	—
74	C₃×C₂²⋊C₈	—	9	4	no	—
75	#830	—	3	4	no	—
76	C₂³×C₃²	—	2	2	no	—
77	C₂³×C₃²	—	2	2	no	—
78	C₂³×C₃²	C₃⋊C₈	1	4	yes	—
79	C₂³×C₃²	C₂×Dic₃	1	2	yes	—
80	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
81	C₂×C₃²×C₄	—	2	2	no	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	2	4	no	—
84	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
85	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
86	C₂×C₃²×C₄	#67	1	4	yes	—
87	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
88	C₂³×C₉	—	2	4	no	—
89	C₂³×C₉	—	2	4	no	—
90	C₂³×C₉	—	2	2	no	—
91	C₂³×C₉	—	2	2	no	—
92	C₂³×C₉	—	2	4	no	—
93	C₂³×C₉	—	2	2	no	—
94	C₂³×C₉	—	2	1	no	—
95	C₂³×C₉	C₃×C₈	1	4	yes	complemented
96	C₂³×C₉	C₂×C₃×C₄	1	2	yes	—
97	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
98	C₃×C₃⋊C₈	—	6	4	no	—
99	C₂×C₄×C₉	—	2	4	no	—
100	C₂×C₄×C₉	—	2	8	no	—
101	C₂×C₄×C₉	—	2	8	no	—
102	C₂×C₄×C₉	—	2	2	no	—
103	C₂×C₄×C₉	—	2	4	no	—
104	C₂×C₄×C₉	—	2	4	no	—
105	C₂×C₄×C₉	—	2	2	no	—
106	C₂×C₄×C₉	—	2	1	no	—
107	C₂×C₄×C₉	—	2	4	no	—
108	C₂×C₄×C₉	—	2	1	no	—
109	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
110	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
111	C₂×C₄×C₉	C₃×D₄	1	4	yes	—
112	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
113	C₉⋊C₈	—	2	4	no	—
114	C₂×C₂²⋊C₈	—	9	1	no	Sylow
115	C₂×C₃×C₉	—	2	4	no	—
116	C₂×C₃×C₉	C₂²⋊C₈	1	4	yes	complemented
117	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
120	C₂⁴×C₃	—	2	1	no	—
121	C₂⁴×C₃	Dic₉	1	1	yes	characteristic
122	C₂⁴×C₃	C₃×Dic₃	1	1	yes	characteristic
123	C₂²×C₃×C₄	—	2	8	no	—
124	C₂²×C₃×C₄	—	2	4	no	—
125	C₂²×C₃×C₄	—	2	4	no	—
126	C₂²×C₃×C₄	—	2	1	no	—
127	C₂²×C₃×C₄	—	2	2	no	—
128	C₂²×C₃×C₄	—	2	2	no	—
129	C₂²×C₃×C₄	—	2	1	no	—
130	C₂²×C₃×C₄	Dic₉	1	1	yes	characteristic
131	C₂²×C₃×C₄	Dic₉	1	2	yes	—
132	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
133	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
134	C₂²×C₃×C₄	C₃×Dic₃	1	1	yes	characteristic
135	C₂²×C₃×C₄	C₃×Dic₃	1	2	yes	—
136	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
137	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
138	C₂×C₃×C₈	—	9	4	no	—
139	C₂×C₃×C₈	—	18	4	no	—
140	C₂×C₃⋊C₈	—	3	4	no	—
141	C₂×C₃⋊C₈	—	6	4	no	—
142	C₂²×C₃²	—	2	8	no	—
143	C₂²×C₃²	—	2	2	no	—
144	C₂²×C₃²	—	2	2	no	—
145	C₂²×C₃²	C₂×C₃⋊C₈	1	2	yes	—
146	C₂²×C₃²	C₂×C₃⋊C₈	1	4	yes	—
147	C₂²×C₃²	#208	1	2	yes	—
148	C₂²×C₃²	#217	1	2	yes	—
149	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
150	C₃²×C₄	—	2	2	no	—
151	C₃²×C₄	#217	1	2	yes	—
152	C₃²×C₄	C₂×#67	1	2	yes	—
153	C₂²×C₉	—	2	4	no	—
154	C₂²×C₉	—	2	4	no	—
155	C₂²×C₉	—	2	16	no	—
156	C₂²×C₉	—	2	8	no	—
157	C₂²×C₉	—	2	2	no	—
158	C₂²×C₉	—	2	2	no	—
159	C₂²×C₉	—	2	2	no	—
160	C₂²×C₉	—	2	4	no	—
161	C₂²×C₉	—	2	2	no	—
162	C₂²×C₉	—	2	2	no	—
163	C₂²×C₉	—	2	1	no	—
164	C₂²×C₉	C₃×C₂²⋊C₄	1	2	yes	—
165	C₂²×C₉	C₂×C₃×C₈	1	2	yes	—
166	C₂²×C₉	C₂×C₃×C₈	1	4	yes	complemented
167	C₂²×C₉	C₃×M₁₆	1	2	yes	—
168	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
169	C₄×C₉	—	2	4	no	—
170	C₄×C₉	—	2	2	no	—
171	C₄×C₉	—	2	2	no	—
172	C₄×C₉	—	2	2	no	—
173	C₄×C₉	C₃×C₂²⋊C₄	1	2	yes	—
174	C₄×C₉	C₂×C₃×D₄	1	2	yes	—
175	C₂³×C₄	C₃×D₉	1	1	yes	characteristic
176	C₂²×C₈	—	9	2	no	—
177	C₂²⋊C₈	—	9	4	no	—
178	C₃×C₉	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₃	—	2	4	no	—
180	C₂³×C₃	—	2	4	no	—
181	C₂³×C₃	—	2	2	no	—
182	C₂³×C₃	—	2	2	no	—
183	C₂³×C₃	—	2	2	no	—
184	C₂³×C₃	—	2	2	no	—
185	C₂³×C₃	—	2	4	no	—
186	C₂³×C₃	—	2	2	no	—
187	C₂³×C₃	—	2	1	no	—
188	C₂³×C₃	C₉⋊C₈	1	4	yes	complemented
189	C₂³×C₃	C₂×Dic₉	1	2	yes	—
190	C₂³×C₃	C₂×Dic₉	1	1	yes	characteristic
191	C₂³×C₃	C₃×C₃⋊C₈	1	4	yes	—
192	C₂³×C₃	C₂×C₃×Dic₃	1	2	yes	—
193	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	2	8	no	—
196	C₂×C₃×C₄	—	2	8	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	2	4	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	2	4	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	4	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
208	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
209	C₂×C₃×C₄	#611	1	4	yes	—
210	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
212	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
213	C₂×C₃×C₄	C₃×#67	1	4	yes	—
214	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
215	C₃×C₈	—	18	4	no	—
216	C₃⋊C₈	—	6	4	no	—
217	C₂×C₃²	—	2	4	no	—
218	C₂×C₃²	#830	1	4	yes	—
219	C₂×C₃²	C₂²×C₃⋊C₈	1	1	yes	characteristic
220	C₂×C₃²	C₂×#208	1	1	yes	characteristic
221	C₂×C₃²	C₂×#217	1	1	yes	characteristic
222	C₂×C₉	—	2	8	no	—
223	C₂×C₉	—	2	4	no	—
224	C₂×C₉	—	2	1	no	—
225	C₂×C₉	—	2	1	no	—
226	C₂×C₉	—	2	1	no	—
227	C₂×C₉	—	2	4	no	—
228	C₂×C₉	C₃×C₂²⋊C₈	1	4	yes	complemented
229	C₂×C₉	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
230	C₂×C₉	C₂²×C₃×C₈	1	1	yes	characteristic
231	C₂×C₉	C₂×C₃×M₁₆	1	1	yes	characteristic
232	C₂⁴	C₃×Dic₉	1	1	yes	characteristic
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	C₃×Dic₉	1	1	yes	characteristic
235	C₂²×C₄	C₃×Dic₉	1	2	yes	—
236	C₂²×C₄	C₂×C₃×D₉	1	2	yes	—
237	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
238	C₂×C₈	—	9	4	no	—
239	C₂×C₈	—	18	4	no	—
240	C₂²×C₃	—	2	4	no	—
241	C₂²×C₃	—	2	16	no	—
242	C₂²×C₃	—	2	4	no	—
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	2	2	no	—
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	—	2	8	no	—
247	C₂²×C₃	—	2	8	no	—
248	C₂²×C₃	—	2	2	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	2	no	—
251	C₂²×C₃	—	2	4	no	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	—	2	2	no	—
254	C₂²×C₃	C₂×C₉⋊C₈	1	2	yes	—
255	C₂²×C₃	C₂×C₉⋊C₈	1	4	yes	complemented
256	C₂²×C₃	#3667	1	2	yes	—
257	C₂²×C₃	#3676	1	2	yes	—
258	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
259	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	2	yes	—
260	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	4	yes	—
261	C₂²×C₃	C₃×#208	1	2	yes	—
262	C₂²×C₃	C₃×#217	1	2	yes	—
263	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
264	C₃×C₄	—	2	4	no	—
265	C₃×C₄	—	2	2	no	—
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	2	2	no	—
269	C₃×C₄	#3676	1	2	yes	—
270	C₃×C₄	C₂×#611	1	2	yes	—
271	C₃×C₄	C₃×#217	1	2	yes	—
272	C₃×C₄	C₂×C₃×#67	1	2	yes	—
273	C₃²	C₂×#830	1	1	yes	characteristic
274	C₉	—	2	1	no	—
275	C₉	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic, complemented
276	C₂³	—	2	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	C₃×C₉⋊C₈	1	4	yes	complemented
279	C₂³	C₂×C₃×Dic₉	1	2	yes	—
280	C₂³	C₂×C₃×Dic₉	1	1	yes	characteristic
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	C₂×C₃×Dic₉	1	2	yes	—
285	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
286	C₂×C₄	C₃×#611	1	4	yes	—
287	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
288	C₈	—	18	4	no	—
289	C₂×C₃	—	2	8	no	—
290	C₂×C₃	—	2	4	no	—
291	C₂×C₃	—	2	4	no	—
292	C₂×C₃	—	2	4	no	—
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	2	1	no	—
296	C₂×C₃	#63389	1	4	yes	direct product factor
297	C₂×C₃	C₂²×C₉⋊C₈	1	1	yes	characteristic
298	C₂×C₃	C₂×#3667	1	1	yes	characteristic
299	C₂×C₃	C₂×#3676	1	1	yes	characteristic
300	C₂×C₃	C₃×#830	1	4	yes	—
301	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
302	C₂×C₃	C₂×C₃×#208	1	1	yes	characteristic
303	C₂×C₃	C₂×C₃×#217	1	1	yes	characteristic
304	C₂²	—	2	8	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	C₂×C₃×C₉⋊C₈	1	2	yes	—
308	C₂²	C₂×C₃×C₉⋊C₈	1	4	yes	complemented
309	C₂²	C₃×#3667	1	2	yes	—
310	C₂²	C₃×#3676	1	2	yes	—
311	C₂²	C₂²×C₃×Dic₉	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	C₃×#3676	1	2	yes	—
314	C₄	C₂×C₃×#611	1	2	yes	—
315	C₃	—	2	1	no	—
316	C₃	C₂×#63389	1	1	yes	characteristic, direct product factor
317	C₃	C₂×C₃×#830	1	1	yes	characteristic
318	C₂	—	2	4	no	—
319	C₂	C₃×#63389	1	4	yes	direct product factor
320	C₂	C₂²×C₃×C₉⋊C₈	1	1	yes	characteristic
321	C₂	C₂×C₃×#3667	1	1	yes	characteristic
322	C₂	C₂×C₃×#3676	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂×D₉
commutator	C₂×C₉	C₂²×C₃×C₈
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂³×C₃×C₄×C₉	C₂
socle	C₂³×C₃²	C₂×Dic₃
prime radical	C₂⁴×C₃²	Dic₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₂³×C₃×C₄	D₉
hypocenter	C₉	C₂×C₃×C₂²⋊C₈
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂³×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₉⋊C₈	C₂	semidirect
3. C₃×#63389	C₂	direct (N × H)
5. C₂×#63389	C₃	direct (N × H)
13. C₂×C₃×C₉⋊C₈	C₂²	semidirect
19. C₂²×C₉⋊C₈	C₂×C₃	semidirect
20. #63389	C₂×C₃	direct (N × H)
23. C₂³×C₃×C₉	C₈	semidirect
57. C₂×C₉⋊C₈	C₂²×C₃	semidirect
63. C₂²×C₃×C₉	C₂×C₈	semidirect
95. C₂³×C₉	C₃×C₈	semidirect
116. C₂×C₃×C₉	C₂²⋊C₈	semidirect
166. C₂²×C₉	C₂×C₃×C₈	semidirect
178. C₃×C₉	C₂×C₂²⋊C₈	semidirect
188. C₂³×C₃	C₉⋊C₈	semidirect
228. C₂×C₉	C₃×C₂²⋊C₈	semidirect
255. C₂²×C₃	C₂×C₉⋊C₈	semidirect
275. C₉	C₂×C₃×C₂²⋊C₈	semidirect
278. C₂³	C₃×C₉⋊C₈	semidirect
296. C₂×C₃	#63389	direct (N × H)
308. C₂²	C₂×C₃×C₉⋊C₈	semidirect
316. C₃	C₂×#63389	direct (N × H)
319. C₂	C₃×#63389	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₉	2
C₂²×C₃×C₉⋊C₈	2
C₃×#63389	2
C₂×C₃×#830	3
C₂×#63389	3

Subgroups — C2×C3×#63389 #13143508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#63389 #13143508