Subgroups — C₂×C₃×C₁₆×C₂₅₆ #72201825

Order 24576.

1628 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₁₆×C₁₂₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₈×C₂₅₆	C₂	1	2	yes	maximal
3	C₃×C₁₆×C₂₅₆	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₁₆×C₂₅₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂×C₃×C₁₆×C₆₄	C₄	1	1	yes	characteristic
6	C₂×C₃×C₈×C₁₂₈	C₄	1	1	yes	characteristic
7	C₂×C₃×C₈×C₁₂₈	C₂²	1	1	yes	characteristic
8	C₃×C₁₆×C₁₂₈	C₄	1	2	yes	—
9	C₃×C₁₆×C₁₂₈	C₂²	1	2	yes	—
10	C₂×C₃×C₄×C₂₅₆	C₄	1	4	yes	—
11	C₃×C₈×C₂₅₆	C₄	1	4	yes	—
12	C₃×C₈×C₂₅₆	C₂²	1	4	yes	—
13	C₂×C₁₆×C₁₂₈	C₂×C₃	1	1	yes	characteristic
14	C₂×C₈×C₂₅₆	C₂×C₃	1	2	yes	—
15	C₁₆×C₂₅₆	C₂×C₃	1	4	yes	direct product factor
16	C₂×C₃×C₁₆×C₃₂	C₈	1	1	yes	characteristic
17	C₂×C₃×C₈×C₆₄	C₈	1	1	yes	characteristic
18	C₂×C₃×C₈×C₆₄	C₂×C₄	1	1	yes	characteristic
19	C₃×C₁₆×C₆₄	C₈	1	2	yes	—
20	C₃×C₁₆×C₆₄	C₂×C₄	1	2	yes	—
21	C₂×C₃×C₄×C₁₂₈	C₈	1	2	yes	—
22	C₂×C₃×C₄×C₁₂₈	C₂×C₄	1	2	yes	—
23	C₃×C₈×C₁₂₈	C₈	1	2	yes	—
24	C₃×C₈×C₁₂₈	C₂×C₄	1	1	yes	characteristic
25	C₃×C₈×C₁₂₈	C₂×C₄	1	2	yes	—
26	C₃×C₈×C₁₂₈	C₂×C₄	1	2	yes	—
27	C₃×C₈×C₁₂₈	C₂³	1	1	yes	characteristic
28	C₂²×C₃×C₂₅₆	C₈	1	8	yes	—
29	C₃×C₄×C₂₅₆	C₈	1	8	yes	—
30	C₃×C₄×C₂₅₆	C₂×C₄	1	8	yes	—
31	C₂×C₁₆×C₆₄	C₃×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₂₈	C₃×C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₂₈	C₂²×C₃	1	1	yes	characteristic
34	C₁₆×C₁₂₈	C₃×C₄	1	2	yes	—
35	C₁₆×C₁₂₈	C₂²×C₃	1	2	yes	—
36	C₂×C₄×C₂₅₆	C₃×C₄	1	4	yes	—
37	C₈×C₂₅₆	C₃×C₄	1	4	yes	—
38	C₈×C₂₅₆	C₂²×C₃	1	4	yes	—
39	C₂×C₃×C₁₆²	C₁₆	1	1	yes	characteristic
40	C₂×C₃×C₈×C₃₂	C₁₆	1	1	yes	characteristic
41	C₂×C₃×C₈×C₃₂	C₂×C₈	1	1	yes	characteristic
42	C₃×C₁₆×C₃₂	C₁₆	1	2	yes	—
43	C₃×C₁₆×C₃₂	C₂×C₈	1	2	yes	—
44	C₂×C₃×C₄×C₆₄	C₁₆	1	2	yes	—
45	C₂×C₃×C₄×C₆₄	C₄²	1	1	yes	characteristic
46	C₂×C₃×C₄×C₆₄	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₃×C₁₂₈	C₁₆	1	4	yes	—
48	C₂²×C₃×C₁₂₈	C₂×C₈	1	4	yes	—
49	C₃×C₈×C₆₄	C₁₆	1	2	yes	—
50	C₃×C₈×C₆₄	C₄²	1	1	yes	characteristic
51	C₃×C₈×C₆₄	C₂×C₈	1	1	yes	characteristic
52	C₃×C₈×C₆₄	C₂×C₈	1	1	yes	characteristic
53	C₃×C₈×C₆₄	C₂×C₈	1	2	yes	—
54	C₃×C₈×C₆₄	C₂²×C₄	1	1	yes	characteristic
55	C₃×C₄×C₁₂₈	C₁₆	1	4	yes	—
56	C₃×C₄×C₁₂₈	C₄²	1	2	yes	—
57	C₃×C₄×C₁₂₈	C₂×C₈	1	4	yes	—
58	C₃×C₄×C₁₂₈	C₂×C₈	1	4	yes	—
59	C₃×C₄×C₁₂₈	C₂²×C₄	1	2	yes	—
60	C₂×C₃×C₂₅₆	C₁₆	1	32	yes	direct product factor
61	C₂×C₃×C₂₅₆	C₂×C₈	1	16	yes	—
62	C₂×C₁₆×C₃₂	C₃×C₈	1	1	yes	characteristic
63	C₂×C₈×C₆₄	C₃×C₈	1	1	yes	characteristic
64	C₂×C₈×C₆₄	C₂×C₃×C₄	1	1	yes	characteristic
65	C₁₆×C₆₄	C₃×C₈	1	2	yes	—
66	C₁₆×C₆₄	C₂×C₃×C₄	1	2	yes	—
67	C₂×C₄×C₁₂₈	C₃×C₈	1	2	yes	—
68	C₂×C₄×C₁₂₈	C₂×C₃×C₄	1	2	yes	—
69	C₈×C₁₂₈	C₃×C₈	1	2	yes	—
70	C₈×C₁₂₈	C₂×C₃×C₄	1	1	yes	characteristic
71	C₈×C₁₂₈	C₂×C₃×C₄	1	2	yes	—
72	C₈×C₁₂₈	C₂×C₃×C₄	1	2	yes	—
73	C₈×C₁₂₈	C₂³×C₃	1	1	yes	characteristic
74	C₂²×C₂₅₆	C₃×C₈	1	8	yes	—
75	C₄×C₂₅₆	C₃×C₈	1	8	yes	—
76	C₄×C₂₅₆	C₂×C₃×C₄	1	8	yes	—
77	C₂²×C₃×C₆₄	C₃₂	1	4	yes	—
78	C₂²×C₃×C₆₄	C₄×C₈	1	2	yes	—
79	C₂²×C₃×C₆₄	C₂×C₁₆	1	2	yes	—
80	C₂×C₃×C₄×C₃₂	C₃₂	1	2	yes	—
81	C₂×C₃×C₄×C₃₂	C₄×C₈	1	1	yes	characteristic
82	C₂×C₃×C₄×C₃₂	C₂×C₁₆	1	1	yes	characteristic
83	C₂×C₃×C₈×C₁₆	C₃₂	1	2	yes	—
84	C₂×C₃×C₈×C₁₆	C₂×C₁₆	1	1	yes	characteristic
85	C₂×C₃×C₁₂₈	C₃₂	1	16	yes	—
86	C₂×C₃×C₁₂₈	C₄×C₈	1	4	yes	—
87	C₂×C₃×C₁₂₈	C₂×C₁₆	1	8	yes	—
88	C₂×C₃×C₁₂₈	C₂×C₁₆	1	16	yes	—
89	C₂×C₃×C₁₂₈	C₂²×C₈	1	4	yes	—
90	C₃×C₄×C₆₄	C₃₂	1	4	yes	—
91	C₃×C₄×C₆₄	C₄×C₈	1	1	yes	characteristic
92	C₃×C₄×C₆₄	C₄×C₈	1	1	yes	characteristic
93	C₃×C₄×C₆₄	C₄×C₈	1	2	yes	—
94	C₃×C₄×C₆₄	C₂×C₁₆	1	2	yes	—
95	C₃×C₄×C₆₄	C₂×C₁₆	1	4	yes	—
96	C₃×C₄×C₆₄	C₂×C₄²	1	1	yes	characteristic
97	C₃×C₄×C₆₄	C₂²×C₈	1	1	yes	characteristic
98	C₃×C₈×C₃₂	C₃₂	1	2	yes	—
99	C₃×C₈×C₃₂	C₄×C₈	1	1	yes	characteristic
100	C₃×C₈×C₃₂	C₂×C₁₆	1	1	yes	characteristic
101	C₃×C₈×C₃₂	C₂×C₁₆	1	1	yes	characteristic
102	C₃×C₈×C₃₂	C₂×C₁₆	1	2	yes	—
103	C₃×C₈×C₃₂	C₂²×C₈	1	1	yes	characteristic
104	C₃×C₁₆²	C₃₂	1	2	yes	—
105	C₃×C₁₆²	C₂×C₁₆	1	2	yes	—
106	C₃×C₂₅₆	C₂×C₁₆	1	32	yes	direct product factor
107	C₂²×C₁₂₈	C₃×C₁₆	1	4	yes	—
108	C₂²×C₁₂₈	C₂×C₃×C₈	1	4	yes	—
109	C₂×C₄×C₆₄	C₃×C₁₆	1	2	yes	—
110	C₂×C₄×C₆₄	C₃×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₆₄	C₂×C₃×C₈	1	1	yes	characteristic
112	C₂×C₈×C₃₂	C₃×C₁₆	1	1	yes	characteristic
113	C₂×C₈×C₃₂	C₂×C₃×C₈	1	1	yes	characteristic
114	C₂×C₁₆²	C₃×C₁₆	1	1	yes	characteristic
115	C₂×C₂₅₆	C₃×C₁₆	1	32	yes	direct product factor
116	C₂×C₂₅₆	C₂×C₃×C₈	1	16	yes	—
117	C₄×C₁₂₈	C₃×C₁₆	1	4	yes	—
118	C₄×C₁₂₈	C₃×C₄²	1	2	yes	—
119	C₄×C₁₂₈	C₂×C₃×C₈	1	4	yes	—
120	C₄×C₁₂₈	C₂×C₃×C₈	1	4	yes	—
121	C₄×C₁₂₈	C₂²×C₃×C₄	1	2	yes	—
122	C₈×C₆₄	C₃×C₁₆	1	2	yes	—
123	C₈×C₆₄	C₃×C₄²	1	1	yes	characteristic
124	C₈×C₆₄	C₂×C₃×C₈	1	1	yes	characteristic
125	C₈×C₆₄	C₂×C₃×C₈	1	1	yes	characteristic
126	C₈×C₆₄	C₂×C₃×C₈	1	2	yes	—
127	C₈×C₆₄	C₂²×C₃×C₄	1	1	yes	characteristic
128	C₁₆×C₃₂	C₃×C₁₆	1	2	yes	—
129	C₁₆×C₃₂	C₂×C₃×C₈	1	2	yes	—
130	C₂²×C₃×C₃₂	C₆₄	1	4	yes	—
131	C₂²×C₃×C₃₂	C₈²	1	1	yes	characteristic
132	C₂²×C₃×C₃₂	C₄×C₁₆	1	1	yes	characteristic
133	C₂²×C₃×C₃₂	C₂×C₃₂	1	2	yes	—
134	C₂×C₃×C₄×C₁₆	C₆₄	1	4	yes	—
135	C₂×C₃×C₄×C₁₆	C₄×C₁₆	1	1	yes	characteristic
136	C₂×C₃×C₄×C₁₆	C₂×C₃₂	1	1	yes	characteristic
137	C₂×C₃×C₈²	C₂×C₃₂	1	1	yes	characteristic
138	C₂×C₃×C₆₄	C₆₄	1	16	yes	—
139	C₂×C₃×C₆₄	C₈²	1	2	yes	—
140	C₂×C₃×C₆₄	C₄×C₁₆	1	2	yes	—
141	C₂×C₃×C₆₄	C₄×C₁₆	1	8	yes	—
142	C₂×C₃×C₆₄	C₂×C₃₂	1	8	yes	—
143	C₂×C₃×C₆₄	C₂×C₃₂	1	8	yes	—
144	C₂×C₃×C₆₄	C₂×C₄×C₈	1	2	yes	—
145	C₂×C₃×C₆₄	C₂²×C₁₆	1	2	yes	—
146	C₃×C₄×C₃₂	C₆₄	1	4	yes	—
147	C₃×C₄×C₃₂	C₈²	1	1	yes	characteristic
148	C₃×C₄×C₃₂	C₄×C₁₆	1	1	yes	characteristic
149	C₃×C₄×C₃₂	C₄×C₁₆	1	1	yes	characteristic
150	C₃×C₄×C₃₂	C₄×C₁₆	1	1	yes	characteristic
151	C₃×C₄×C₃₂	C₂×C₃₂	1	2	yes	—
152	C₃×C₄×C₃₂	C₂×C₃₂	1	4	yes	—
153	C₃×C₄×C₃₂	C₂×C₄×C₈	1	1	yes	characteristic
154	C₃×C₄×C₃₂	C₂²×C₁₆	1	1	yes	characteristic
155	C₃×C₈×C₁₆	C₆₄	1	4	yes	—
156	C₃×C₈×C₁₆	C₄×C₁₆	1	1	yes	characteristic
157	C₃×C₈×C₁₆	C₂×C₃₂	1	1	yes	characteristic
158	C₃×C₈×C₁₆	C₂×C₃₂	1	1	yes	characteristic
159	C₃×C₈×C₁₆	C₂×C₃₂	1	4	yes	—
160	C₃×C₈×C₁₆	C₂²×C₁₆	1	1	yes	characteristic
161	C₃×C₁₂₈	C₄×C₁₆	1	8	yes	—
162	C₃×C₁₂₈	C₂×C₃₂	1	16	yes	—
163	C₃×C₁₂₈	C₂²×C₁₆	1	8	yes	—
164	C₂²×C₆₄	C₃×C₃₂	1	4	yes	—
165	C₂²×C₆₄	C₃×C₄×C₈	1	2	yes	—
166	C₂²×C₆₄	C₂×C₃×C₁₆	1	2	yes	—
167	C₂×C₄×C₃₂	C₃×C₃₂	1	2	yes	—
168	C₂×C₄×C₃₂	C₃×C₄×C₈	1	1	yes	characteristic
169	C₂×C₄×C₃₂	C₂×C₃×C₁₆	1	1	yes	characteristic
170	C₂×C₈×C₁₆	C₃×C₃₂	1	2	yes	—
171	C₂×C₈×C₁₆	C₂×C₃×C₁₆	1	1	yes	characteristic
172	C₂×C₁₂₈	C₃×C₃₂	1	16	yes	—
173	C₂×C₁₂₈	C₃×C₄×C₈	1	4	yes	—
174	C₂×C₁₂₈	C₂×C₃×C₁₆	1	8	yes	—
175	C₂×C₁₂₈	C₂×C₃×C₁₆	1	16	yes	—
176	C₂×C₁₂₈	C₂²×C₃×C₈	1	4	yes	—
177	C₄×C₆₄	C₃×C₃₂	1	4	yes	—
178	C₄×C₆₄	C₃×C₄×C₈	1	1	yes	characteristic
179	C₄×C₆₄	C₃×C₄×C₈	1	1	yes	characteristic
180	C₄×C₆₄	C₃×C₄×C₈	1	2	yes	—
181	C₄×C₆₄	C₂×C₃×C₁₆	1	2	yes	—
182	C₄×C₆₄	C₂×C₃×C₁₆	1	4	yes	—
183	C₄×C₆₄	C₂×C₃×C₄²	1	1	yes	characteristic
184	C₄×C₆₄	C₂²×C₃×C₈	1	1	yes	characteristic
185	C₈×C₃₂	C₃×C₃₂	1	2	yes	—
186	C₈×C₃₂	C₃×C₄×C₈	1	1	yes	characteristic
187	C₈×C₃₂	C₂×C₃×C₁₆	1	1	yes	characteristic
188	C₈×C₃₂	C₂×C₃×C₁₆	1	1	yes	characteristic
189	C₈×C₃₂	C₂×C₃×C₁₆	1	2	yes	—
190	C₈×C₃₂	C₂²×C₃×C₈	1	1	yes	characteristic
191	C₁₆²	C₃×C₃₂	1	2	yes	—
192	C₁₆²	C₂×C₃×C₁₆	1	2	yes	—
193	C₂₅₆	C₂×C₃×C₁₆	1	32	yes	direct product factor
194	C₂²×C₃×C₁₆	C₁₂₈	1	8	yes	—
195	C₂²×C₃×C₁₆	C₈×C₁₆	1	1	yes	characteristic
196	C₂²×C₃×C₁₆	C₄×C₃₂	1	1	yes	characteristic
197	C₂²×C₃×C₁₆	C₂×C₆₄	1	2	yes	—
198	C₂×C₃×C₄×C₈	C₄×C₃₂	1	1	yes	characteristic
199	C₂×C₃×C₄×C₈	C₂×C₆₄	1	2	yes	—
200	C₂×C₃×C₃₂	C₁₂₈	1	16	yes	—
201	C₂×C₃×C₃₂	C₈×C₁₆	1	1	yes	characteristic
202	C₂×C₃×C₃₂	C₈×C₁₆	1	1	yes	characteristic
203	C₂×C₃×C₃₂	C₈×C₁₆	1	4	yes	—
204	C₂×C₃×C₃₂	C₄×C₃₂	1	2	yes	—
205	C₂×C₃×C₃₂	C₄×C₃₂	1	4	yes	—
206	C₂×C₃×C₃₂	C₂×C₆₄	1	8	yes	—
207	C₂×C₃×C₃₂	C₂×C₆₄	1	8	yes	—
208	C₂×C₃×C₃₂	C₂×C₈²	1	1	yes	characteristic
209	C₂×C₃×C₃₂	C₂×C₄×C₁₆	1	1	yes	characteristic
210	C₂×C₃×C₃₂	C₂²×C₃₂	1	2	yes	—
211	C₃×C₄×C₁₆	C₁₂₈	1	8	yes	—
212	C₃×C₄×C₁₆	C₈×C₁₆	1	1	yes	characteristic
213	C₃×C₄×C₁₆	C₄×C₃₂	1	1	yes	characteristic
214	C₃×C₄×C₁₆	C₄×C₃₂	1	1	yes	characteristic
215	C₃×C₄×C₁₆	C₄×C₃₂	1	1	yes	characteristic
216	C₃×C₄×C₁₆	C₂×C₆₄	1	2	yes	—
217	C₃×C₄×C₁₆	C₂×C₆₄	1	8	yes	—
218	C₃×C₄×C₁₆	C₂×C₄×C₁₆	1	1	yes	characteristic
219	C₃×C₄×C₁₆	C₂²×C₃₂	1	1	yes	characteristic
220	C₃×C₈²	C₄×C₃₂	1	1	yes	characteristic
221	C₃×C₈²	C₂×C₆₄	1	2	yes	—
222	C₃×C₈²	C₂²×C₃₂	1	1	yes	characteristic
223	C₃×C₆₄	C₈×C₁₆	1	4	yes	—
224	C₃×C₆₄	C₄×C₃₂	1	4	yes	—
225	C₃×C₆₄	C₂×C₆₄	1	16	yes	—
226	C₃×C₆₄	C₂×C₄×C₁₆	1	4	yes	—
227	C₃×C₆₄	C₂²×C₃₂	1	4	yes	—
228	C₂²×C₃₂	C₃×C₆₄	1	4	yes	—
229	C₂²×C₃₂	C₃×C₈²	1	1	yes	characteristic
230	C₂²×C₃₂	C₃×C₄×C₁₆	1	1	yes	characteristic
231	C₂²×C₃₂	C₂×C₃×C₃₂	1	2	yes	—
232	C₂×C₄×C₁₆	C₃×C₆₄	1	4	yes	—
233	C₂×C₄×C₁₆	C₃×C₄×C₁₆	1	1	yes	characteristic
234	C₂×C₄×C₁₆	C₂×C₃×C₃₂	1	1	yes	characteristic
235	C₂×C₈²	C₂×C₃×C₃₂	1	1	yes	characteristic
236	C₂×C₆₄	C₃×C₆₄	1	16	yes	—
237	C₂×C₆₄	C₃×C₈²	1	2	yes	—
238	C₂×C₆₄	C₃×C₄×C₁₆	1	2	yes	—
239	C₂×C₆₄	C₃×C₄×C₁₆	1	8	yes	—
240	C₂×C₆₄	C₂×C₃×C₃₂	1	8	yes	—
241	C₂×C₆₄	C₂×C₃×C₃₂	1	8	yes	—
242	C₂×C₆₄	C₂×C₃×C₄×C₈	1	2	yes	—
243	C₂×C₆₄	C₂²×C₃×C₁₆	1	2	yes	—
244	C₄×C₃₂	C₃×C₆₄	1	4	yes	—
245	C₄×C₃₂	C₃×C₈²	1	1	yes	characteristic
246	C₄×C₃₂	C₃×C₄×C₁₆	1	1	yes	characteristic
247	C₄×C₃₂	C₃×C₄×C₁₆	1	1	yes	characteristic
248	C₄×C₃₂	C₃×C₄×C₁₆	1	1	yes	characteristic
249	C₄×C₃₂	C₂×C₃×C₃₂	1	2	yes	—
250	C₄×C₃₂	C₂×C₃×C₃₂	1	4	yes	—
251	C₄×C₃₂	C₂×C₃×C₄×C₈	1	1	yes	characteristic
252	C₄×C₃₂	C₂²×C₃×C₁₆	1	1	yes	characteristic
253	C₈×C₁₆	C₃×C₆₄	1	4	yes	—
254	C₈×C₁₆	C₃×C₄×C₁₆	1	1	yes	characteristic
255	C₈×C₁₆	C₂×C₃×C₃₂	1	1	yes	characteristic
256	C₈×C₁₆	C₂×C₃×C₃₂	1	1	yes	characteristic
257	C₈×C₁₆	C₂×C₃×C₃₂	1	4	yes	—
258	C₈×C₁₆	C₂²×C₃×C₁₆	1	1	yes	characteristic
259	C₁₂₈	C₃×C₄×C₁₆	1	8	yes	—
260	C₁₂₈	C₂×C₃×C₃₂	1	16	yes	—
261	C₁₂₈	C₂²×C₃×C₁₆	1	8	yes	—
262	C₂²×C₃×C₈	C₈×C₃₂	1	1	yes	characteristic
263	C₂²×C₃×C₈	C₄×C₆₄	1	1	yes	characteristic
264	C₂²×C₃×C₈	C₂×C₁₂₈	1	4	yes	—
265	C₂×C₃×C₄²	C₄×C₆₄	1	1	yes	characteristic
266	C₂×C₃×C₁₆	C₂₅₆	1	32	yes	direct product factor
267	C₂×C₃×C₁₆	C₁₆²	1	2	yes	—
268	C₂×C₃×C₁₆	C₈×C₃₂	1	1	yes	characteristic
269	C₂×C₃×C₁₆	C₈×C₃₂	1	1	yes	characteristic
270	C₂×C₃×C₁₆	C₈×C₃₂	1	2	yes	—
271	C₂×C₃×C₁₆	C₄×C₆₄	1	2	yes	—
272	C₂×C₃×C₁₆	C₄×C₆₄	1	4	yes	—
273	C₂×C₃×C₁₆	C₂×C₁₂₈	1	8	yes	—
274	C₂×C₃×C₁₆	C₂×C₁₂₈	1	16	yes	—
275	C₂×C₃×C₁₆	C₂×C₈×C₁₆	1	1	yes	characteristic
276	C₂×C₃×C₁₆	C₂×C₄×C₃₂	1	1	yes	characteristic
277	C₂×C₃×C₁₆	C₂²×C₆₄	1	2	yes	—
278	C₃×C₄×C₈	C₈×C₃₂	1	1	yes	characteristic
279	C₃×C₄×C₈	C₄×C₆₄	1	1	yes	characteristic
280	C₃×C₄×C₈	C₄×C₆₄	1	1	yes	characteristic
281	C₃×C₄×C₈	C₄×C₆₄	1	2	yes	—
282	C₃×C₄×C₈	C₂×C₁₂₈	1	4	yes	—
283	C₃×C₄×C₈	C₂×C₄×C₃₂	1	1	yes	characteristic
284	C₃×C₄×C₈	C₂²×C₆₄	1	2	yes	—
285	C₃×C₃₂	C₁₆²	1	2	yes	—
286	C₃×C₃₂	C₈×C₃₂	1	2	yes	—
287	C₃×C₃₂	C₄×C₆₄	1	4	yes	—
288	C₃×C₃₂	C₂×C₁₂₈	1	16	yes	—
289	C₃×C₃₂	C₂×C₈×C₁₆	1	2	yes	—
290	C₃×C₃₂	C₂×C₄×C₃₂	1	2	yes	—
291	C₃×C₃₂	C₂²×C₆₄	1	4	yes	—
292	C₂²×C₁₆	C₃×C₁₂₈	1	8	yes	—
293	C₂²×C₁₆	C₃×C₈×C₁₆	1	1	yes	characteristic
294	C₂²×C₁₆	C₃×C₄×C₃₂	1	1	yes	characteristic
295	C₂²×C₁₆	C₂×C₃×C₆₄	1	2	yes	—
296	C₂×C₄×C₈	C₃×C₄×C₃₂	1	1	yes	characteristic
297	C₂×C₄×C₈	C₂×C₃×C₆₄	1	2	yes	—
298	C₂×C₃₂	C₃×C₁₂₈	1	16	yes	—
299	C₂×C₃₂	C₃×C₈×C₁₆	1	1	yes	characteristic
300	C₂×C₃₂	C₃×C₈×C₁₆	1	1	yes	characteristic
301	C₂×C₃₂	C₃×C₈×C₁₆	1	4	yes	—
302	C₂×C₃₂	C₃×C₄×C₃₂	1	2	yes	—
303	C₂×C₃₂	C₃×C₄×C₃₂	1	4	yes	—
304	C₂×C₃₂	C₂×C₃×C₆₄	1	8	yes	—
305	C₂×C₃₂	C₂×C₃×C₆₄	1	8	yes	—
306	C₂×C₃₂	C₂×C₃×C₈²	1	1	yes	characteristic
307	C₂×C₃₂	C₂×C₃×C₄×C₁₆	1	1	yes	characteristic
308	C₂×C₃₂	C₂²×C₃×C₃₂	1	2	yes	—
309	C₄×C₁₆	C₃×C₁₂₈	1	8	yes	—
310	C₄×C₁₆	C₃×C₈×C₁₆	1	1	yes	characteristic
311	C₄×C₁₆	C₃×C₄×C₃₂	1	1	yes	characteristic
312	C₄×C₁₆	C₃×C₄×C₃₂	1	1	yes	characteristic
313	C₄×C₁₆	C₃×C₄×C₃₂	1	1	yes	characteristic
314	C₄×C₁₆	C₂×C₃×C₆₄	1	2	yes	—
315	C₄×C₁₆	C₂×C₃×C₆₄	1	8	yes	—
316	C₄×C₁₆	C₂×C₃×C₄×C₁₆	1	1	yes	characteristic
317	C₄×C₁₆	C₂²×C₃×C₃₂	1	1	yes	characteristic
318	C₈²	C₃×C₄×C₃₂	1	1	yes	characteristic
319	C₈²	C₂×C₃×C₆₄	1	2	yes	—
320	C₈²	C₂²×C₃×C₃₂	1	1	yes	characteristic
321	C₆₄	C₃×C₈×C₁₆	1	4	yes	—
322	C₆₄	C₃×C₄×C₃₂	1	4	yes	—
323	C₆₄	C₂×C₃×C₆₄	1	16	yes	—
324	C₆₄	C₂×C₃×C₄×C₁₆	1	4	yes	—
325	C₆₄	C₂²×C₃×C₃₂	1	4	yes	—
326	C₂²×C₃×C₄	C₈×C₆₄	1	1	yes	characteristic
327	C₂²×C₃×C₄	C₄×C₁₂₈	1	2	yes	—
328	C₂×C₃×C₈	C₁₆×C₃₂	1	2	yes	—
329	C₂×C₃×C₈	C₈×C₆₄	1	1	yes	characteristic
330	C₂×C₃×C₈	C₈×C₆₄	1	1	yes	characteristic
331	C₂×C₃×C₈	C₈×C₆₄	1	2	yes	—
332	C₂×C₃×C₈	C₄×C₁₂₈	1	4	yes	—
333	C₂×C₃×C₈	C₄×C₁₂₈	1	4	yes	—
334	C₂×C₃×C₈	C₂×C₂₅₆	1	16	yes	—
335	C₂×C₃×C₈	C₂×C₈×C₃₂	1	1	yes	characteristic
336	C₂×C₃×C₈	C₂×C₄×C₆₄	1	1	yes	characteristic
337	C₂×C₃×C₈	C₂²×C₁₂₈	1	4	yes	—
338	C₃×C₄²	C₈×C₆₄	1	1	yes	characteristic
339	C₃×C₄²	C₄×C₁₂₈	1	2	yes	—
340	C₃×C₄²	C₂×C₄×C₆₄	1	1	yes	characteristic
341	C₃×C₁₆	C₁₆×C₃₂	1	2	yes	—
342	C₃×C₁₆	C₈×C₆₄	1	2	yes	—
343	C₃×C₁₆	C₄×C₁₂₈	1	4	yes	—
344	C₃×C₁₆	C₂×C₂₅₆	1	32	yes	direct product factor
345	C₃×C₁₆	C₂×C₁₆²	1	1	yes	characteristic
346	C₃×C₁₆	C₂×C₈×C₃₂	1	1	yes	characteristic
347	C₃×C₁₆	C₂×C₄×C₆₄	1	2	yes	—
348	C₃×C₁₆	C₂²×C₁₂₈	1	4	yes	—
349	C₂²×C₈	C₃×C₈×C₃₂	1	1	yes	characteristic
350	C₂²×C₈	C₃×C₄×C₆₄	1	1	yes	characteristic
351	C₂²×C₈	C₂×C₃×C₁₂₈	1	4	yes	—
352	C₂×C₄²	C₃×C₄×C₆₄	1	1	yes	characteristic
353	C₂×C₁₆	C₃×C₂₅₆	1	32	yes	direct product factor
354	C₂×C₁₆	C₃×C₁₆²	1	2	yes	—
355	C₂×C₁₆	C₃×C₈×C₃₂	1	1	yes	characteristic
356	C₂×C₁₆	C₃×C₈×C₃₂	1	1	yes	characteristic
357	C₂×C₁₆	C₃×C₈×C₃₂	1	2	yes	—
358	C₂×C₁₆	C₃×C₄×C₆₄	1	2	yes	—
359	C₂×C₁₆	C₃×C₄×C₆₄	1	4	yes	—
360	C₂×C₁₆	C₂×C₃×C₁₂₈	1	8	yes	—
361	C₂×C₁₆	C₂×C₃×C₁₂₈	1	16	yes	—
362	C₂×C₁₆	C₂×C₃×C₈×C₁₆	1	1	yes	characteristic
363	C₂×C₁₆	C₂×C₃×C₄×C₃₂	1	1	yes	characteristic
364	C₂×C₁₆	C₂²×C₃×C₆₄	1	2	yes	—
365	C₄×C₈	C₃×C₈×C₃₂	1	1	yes	characteristic
366	C₄×C₈	C₃×C₄×C₆₄	1	1	yes	characteristic
367	C₄×C₈	C₃×C₄×C₆₄	1	1	yes	characteristic
368	C₄×C₈	C₃×C₄×C₆₄	1	2	yes	—
369	C₄×C₈	C₂×C₃×C₁₂₈	1	4	yes	—
370	C₄×C₈	C₂×C₃×C₄×C₃₂	1	1	yes	characteristic
371	C₄×C₈	C₂²×C₃×C₆₄	1	2	yes	—
372	C₃₂	C₃×C₁₆²	1	2	yes	—
373	C₃₂	C₃×C₈×C₃₂	1	2	yes	—
374	C₃₂	C₃×C₄×C₆₄	1	4	yes	—
375	C₃₂	C₂×C₃×C₁₂₈	1	16	yes	—
376	C₃₂	C₂×C₃×C₈×C₁₆	1	2	yes	—
377	C₃₂	C₂×C₃×C₄×C₃₂	1	2	yes	—
378	C₃₂	C₂²×C₃×C₆₄	1	4	yes	—
379	C₂³×C₃	C₈×C₁₂₈	1	1	yes	characteristic
380	C₂×C₃×C₄	C₄×C₂₅₆	1	8	yes	—
381	C₂×C₃×C₄	C₈×C₁₂₈	1	1	yes	characteristic
382	C₂×C₃×C₄	C₈×C₁₂₈	1	2	yes	—
383	C₂×C₃×C₄	C₈×C₁₂₈	1	2	yes	—
384	C₂×C₃×C₄	C₂×C₄×C₁₂₈	1	2	yes	—
385	C₂×C₃×C₄	C₁₆×C₆₄	1	2	yes	—
386	C₂×C₃×C₄	C₂×C₈×C₆₄	1	1	yes	characteristic
387	C₃×C₈	C₄×C₂₅₆	1	8	yes	—
388	C₃×C₈	C₂²×C₂₅₆	1	8	yes	—
389	C₃×C₈	C₈×C₁₂₈	1	2	yes	—
390	C₃×C₈	C₂×C₄×C₁₂₈	1	2	yes	—
391	C₃×C₈	C₁₆×C₆₄	1	2	yes	—
392	C₃×C₈	C₂×C₈×C₆₄	1	1	yes	characteristic
393	C₃×C₈	C₂×C₁₆×C₃₂	1	1	yes	characteristic
394	C₂²×C₄	C₃×C₄×C₁₂₈	1	2	yes	—
395	C₂²×C₄	C₃×C₈×C₆₄	1	1	yes	characteristic
396	C₂×C₈	C₂×C₃×C₂₅₆	1	16	yes	—
397	C₂×C₈	C₃×C₄×C₁₂₈	1	4	yes	—
398	C₂×C₈	C₃×C₄×C₁₂₈	1	4	yes	—
399	C₂×C₈	C₃×C₈×C₆₄	1	1	yes	characteristic
400	C₂×C₈	C₃×C₈×C₆₄	1	1	yes	characteristic
401	C₂×C₈	C₃×C₈×C₆₄	1	2	yes	—
402	C₂×C₈	C₂²×C₃×C₁₂₈	1	4	yes	—
403	C₂×C₈	C₂×C₃×C₄×C₆₄	1	1	yes	characteristic
404	C₂×C₈	C₃×C₁₆×C₃₂	1	2	yes	—
405	C₂×C₈	C₂×C₃×C₈×C₃₂	1	1	yes	characteristic
406	C₄²	C₃×C₄×C₁₂₈	1	2	yes	—
407	C₄²	C₃×C₈×C₆₄	1	1	yes	characteristic
408	C₄²	C₂×C₃×C₄×C₆₄	1	1	yes	characteristic
409	C₁₆	C₂×C₃×C₂₅₆	1	32	yes	direct product factor
410	C₁₆	C₃×C₄×C₁₂₈	1	4	yes	—
411	C₁₆	C₃×C₈×C₆₄	1	2	yes	—
412	C₁₆	C₂²×C₃×C₁₂₈	1	4	yes	—
413	C₁₆	C₂×C₃×C₄×C₆₄	1	2	yes	—
414	C₁₆	C₃×C₁₆×C₃₂	1	2	yes	—
415	C₁₆	C₂×C₃×C₈×C₃₂	1	1	yes	characteristic
416	C₁₆	C₂×C₃×C₁₆²	1	1	yes	characteristic
417	C₂²×C₃	C₈×C₂₅₆	1	4	yes	—
418	C₂²×C₃	C₁₆×C₁₂₈	1	2	yes	—
419	C₂²×C₃	C₂×C₈×C₁₂₈	1	1	yes	characteristic
420	C₃×C₄	C₈×C₂₅₆	1	4	yes	—
421	C₃×C₄	C₂×C₄×C₂₅₆	1	4	yes	—
422	C₃×C₄	C₁₆×C₁₂₈	1	2	yes	—
423	C₃×C₄	C₂×C₈×C₁₂₈	1	1	yes	characteristic
424	C₃×C₄	C₂×C₁₆×C₆₄	1	1	yes	characteristic
425	C₂³	C₃×C₈×C₁₂₈	1	1	yes	characteristic
426	C₂×C₄	C₃×C₄×C₂₅₆	1	8	yes	—
427	C₂×C₄	C₃×C₈×C₁₂₈	1	1	yes	characteristic
428	C₂×C₄	C₃×C₈×C₁₂₈	1	2	yes	—
429	C₂×C₄	C₃×C₈×C₁₂₈	1	2	yes	—
430	C₂×C₄	C₂×C₃×C₄×C₁₂₈	1	2	yes	—
431	C₂×C₄	C₃×C₁₆×C₆₄	1	2	yes	—
432	C₂×C₄	C₂×C₃×C₈×C₆₄	1	1	yes	characteristic
433	C₈	C₃×C₄×C₂₅₆	1	8	yes	—
434	C₈	C₂²×C₃×C₂₅₆	1	8	yes	—
435	C₈	C₃×C₈×C₁₂₈	1	2	yes	—
436	C₈	C₂×C₃×C₄×C₁₂₈	1	2	yes	—
437	C₈	C₃×C₁₆×C₆₄	1	2	yes	—
438	C₈	C₂×C₃×C₈×C₆₄	1	1	yes	characteristic
439	C₈	C₂×C₃×C₁₆×C₃₂	1	1	yes	characteristic
440	C₂×C₃	C₁₆×C₂₅₆	1	4	yes	direct product factor
441	C₂×C₃	C₂×C₈×C₂₅₆	1	2	yes	—
442	C₂×C₃	C₂×C₁₆×C₁₂₈	1	1	yes	characteristic
443	C₂²	C₃×C₈×C₂₅₆	1	4	yes	—
444	C₂²	C₃×C₁₆×C₁₂₈	1	2	yes	—
445	C₂²	C₂×C₃×C₈×C₁₂₈	1	1	yes	characteristic
446	C₄	C₃×C₈×C₂₅₆	1	4	yes	—
447	C₄	C₂×C₃×C₄×C₂₅₆	1	4	yes	—
448	C₄	C₃×C₁₆×C₁₂₈	1	2	yes	—
449	C₄	C₂×C₃×C₈×C₁₂₈	1	1	yes	characteristic
450	C₄	C₂×C₃×C₁₆×C₆₄	1	1	yes	characteristic
451	C₃	C₂×C₁₆×C₂₅₆	1	1	yes	characteristic, direct product factor, Sylow
452	C₂	C₃×C₁₆×C₂₅₆	1	4	yes	direct product factor
453	C₂	C₂×C₃×C₈×C₂₅₆	1	2	yes	—
454	C₂	C₂×C₃×C₁₆×C₁₂₈	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈×C₁₂₈	C₂³×C₃
socle, prime radical	C₂³×C₃	C₈×C₁₂₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₁₆×C₂₅₆	C₂	direct (N × H)
4. C₂×C₁₆×C₂₅₆	C₃	direct (N × H)
15. C₁₆×C₂₅₆	C₂×C₃	direct (N × H)
60. C₂×C₃×C₂₅₆	C₁₆	direct (N × H)
106. C₃×C₂₅₆	C₂×C₁₆	direct (N × H)
115. C₂×C₂₅₆	C₃×C₁₆	direct (N × H)
193. C₂₅₆	C₂×C₃×C₁₆	direct (N × H)
266. C₂×C₃×C₁₆	C₂₅₆	direct (N × H)
344. C₃×C₁₆	C₂×C₂₅₆	direct (N × H)
353. C₂×C₁₆	C₃×C₂₅₆	direct (N × H)
409. C₁₆	C₂×C₃×C₂₅₆	direct (N × H)
440. C₂×C₃	C₁₆×C₂₅₆	direct (N × H)
451. C₃	C₂×C₁₆×C₂₅₆	direct (N × H)
452. C₂	C₃×C₁₆×C₂₅₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₁₆×C₁₂₈	2
C₂×C₃×C₈×C₂₅₆	2
C₃×C₁₆×C₂₅₆	2
C₂×C₁₆×C₂₅₆	3

Subgroups — C2×C3×C16×C256 #72201825

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₁₆×C₂₅₆ #72201825