Subgroups — C₂×C₃₁×#2564 #21491590

Order 7936.

3374 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2564	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#400	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#399	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#395	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#393	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#387	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#386	C₂	1	2	yes	maximal
10	C₂×C₃₁×#385	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#380	C₂	1	2	yes	maximal, complemented
12	C₂²×C₃₁×#95	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₂²⋊Q₈	—	2	8	no	—
19	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	8	yes	complemented
20	C₂×C₄×D₄×C₃₁	—	2	8	no	—
21	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	—
22	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
31	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#400	C₂²	1	8	yes	complemented
34	C₃₁×#399	C₂²	1	8	yes	complemented
35	C₃₁×#395	C₂²	1	8	yes	complemented
36	C₃₁×#393	C₂²	1	8	yes	complemented
37	C₃₁×#387	C₂²	1	16	yes	complemented
38	C₃₁×#386	C₂²	1	16	yes	—
39	C₃₁×#385	C₂²	1	16	yes	complemented
40	C₃₁×#380	C₂²	1	16	yes	complemented
41	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#95	C₂²	1	2	yes	—
43	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
44	C₂×C₃₁×#95	C₂²	1	2	yes	—
45	C₂×C₃₁×#95	C₂²	1	2	yes	—
46	C₂×C₃₁×#95	C₂²	1	2	yes	—
47	C₂×C₃₁×#95	C₂²	1	2	yes	—
48	C₂×C₃₁×#95	C₂²	1	8	yes	complemented
49	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
50	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×Q₈×C₃₁	—	2	4	no	—
52	C₂²×Q₈×C₃₁	—	4	4	no	—
53	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
54	C₂²×D₄×C₃₁	—	2	4	no	—
55	C₂²×D₄×C₃₁	—	2	4	no	—
56	C₂²×D₄×C₃₁	Q₈	1	8	yes	complemented
57	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
58	C₂³×C₄×C₃₁	—	2	8	no	—
59	C₂³×C₄×C₃₁	—	2	4	no	—
60	C₂³×C₄×C₃₁	—	2	8	no	—
61	C₂³×C₄×C₃₁	—	4	4	no	—
62	C₂³×C₄×C₃₁	—	4	4	no	—
63	C₂³×C₄×C₃₁	—	4	4	no	—
64	C₂³×C₄×C₃₁	D₄	1	8	yes	—
65	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
67	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
68	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
70	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
71	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
76	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
77	C₃₁×C₂²⋊Q₈	—	2	32	no	—
78	C₄×D₄×C₃₁	—	2	32	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
80	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
81	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
82	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
83	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
89	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
90	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
91	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
92	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
93	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
94	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
96	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
98	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
99	C₂×C₄²×C₃₁	—	2	4	no	—
100	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
101	C₃₁×#95	C₂³	1	4	yes	—
102	C₃₁×#95	C₂³	1	8	yes	—
103	C₃₁×#95	C₂³	1	4	yes	—
104	C₃₁×#95	C₂³	1	8	yes	—
105	C₃₁×#95	C₂³	1	8	yes	—
106	C₃₁×#95	C₂³	1	8	yes	—
107	C₃₁×#95	C₂³	1	8	yes	—
108	C₂⁴×C₃₁	—	2	4	no	—
109	C₂⁴×C₃₁	—	2	4	no	—
110	C₂⁴×C₃₁	—	2	2	no	—
111	C₂⁴×C₃₁	—	2	4	no	—
112	C₂⁴×C₃₁	—	2	2	no	—
113	C₂⁴×C₃₁	—	2	4	no	—
114	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
115	C₂⁴×C₃₁	C₂×Q₈	1	4	yes	—
116	C₂⁴×C₃₁	C₂×Q₈	1	4	yes	—
117	C₂⁴×C₃₁	#41	1	4	yes	—
118	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
119	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
120	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
121	C₂×Q₈×C₃₁	—	2	8	no	—
122	C₂×Q₈×C₃₁	—	4	8	no	—
123	C₂×Q₈×C₃₁	—	4	16	no	—
124	C₂×D₄×C₃₁	—	2	32	no	—
125	C₂×D₄×C₃₁	—	2	8	no	—
126	C₂×D₄×C₃₁	—	2	8	no	—
127	C₂×D₄×C₃₁	C₂×Q₈	1	16	yes	complemented
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	4	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	2	4	no	—
132	C₂²×C₄×C₃₁	—	2	4	no	—
133	C₂²×C₄×C₃₁	—	2	4	no	—
134	C₂²×C₄×C₃₁	—	2	2	no	—
135	C₂²×C₄×C₃₁	—	2	4	no	—
136	C₂²×C₄×C₃₁	—	2	2	no	—
137	C₂²×C₄×C₃₁	—	2	4	no	—
138	C₂²×C₄×C₃₁	—	2	4	no	—
139	C₂²×C₄×C₃₁	—	2	4	no	—
140	C₂²×C₄×C₃₁	—	2	4	no	—
141	C₂²×C₄×C₃₁	—	2	16	no	—
142	C₂²×C₄×C₃₁	—	2	4	no	—
143	C₂²×C₄×C₃₁	—	2	4	no	—
144	C₂²×C₄×C₃₁	—	2	4	no	—
145	C₂²×C₄×C₃₁	—	2	4	no	—
146	C₂²×C₄×C₃₁	—	2	4	no	—
147	C₂²×C₄×C₃₁	—	2	4	no	—
148	C₂²×C₄×C₃₁	—	2	8	no	—
149	C₂²×C₄×C₃₁	—	2	16	no	—
150	C₂²×C₄×C₃₁	—	4	16	no	—
151	C₂²×C₄×C₃₁	—	4	8	no	—
152	C₂²×C₄×C₃₁	—	4	8	no	—
153	C₂²×C₄×C₃₁	—	4	8	no	—
154	C₂²×C₄×C₃₁	—	4	32	no	—
155	C₂²×C₄×C₃₁	—	4	16	no	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
157	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
158	C₂²×C₄×C₃₁	C₂×D₄	1	16	yes	—
159	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
160	C₂²×C₄×C₃₁	#41	1	8	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
170	C₄⋊C₄×C₃₁	—	2	8	no	—
171	C₄⋊C₄×C₃₁	—	2	8	no	—
172	C₄⋊C₄×C₃₁	—	2	16	no	—
173	C₄⋊C₄×C₃₁	—	2	16	no	—
174	C₄⋊C₄×C₃₁	—	4	8	no	—
175	C₂²⋊C₄×C₃₁	—	2	8	no	—
176	C₂²⋊C₄×C₃₁	—	2	16	no	—
177	C₂²⋊C₄×C₃₁	—	2	8	no	—
178	C₂²⋊C₄×C₃₁	—	2	16	no	—
179	C₂²⋊C₄×C₃₁	—	4	8	no	—
180	C₄²×C₃₁	—	2	8	no	—
181	C₂×#2564	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
182	C₂³×C₃₁	—	2	8	no	—
183	C₂³×C₃₁	—	2	8	no	—
184	C₂³×C₃₁	—	2	16	no	—
185	C₂³×C₃₁	—	2	16	no	—
186	C₂³×C₃₁	—	2	4	no	—
187	C₂³×C₃₁	—	2	4	no	—
188	C₂³×C₃₁	—	2	8	no	—
189	C₂³×C₃₁	—	2	16	no	—
190	C₂³×C₃₁	—	2	2	no	—
191	C₂³×C₃₁	—	2	4	no	—
192	C₂³×C₃₁	—	2	2	no	—
193	C₂³×C₃₁	—	2	4	no	—
194	C₂³×C₃₁	—	4	4	no	—
195	C₂³×C₃₁	—	4	4	no	—
196	C₂³×C₃₁	—	4	8	no	—
197	C₂³×C₃₁	C₂²⋊Q₈	1	16	yes	—
198	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
199	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
200	C₂³×C₃₁	C₂²×Q₈	1	4	yes	—
201	C₂³×C₃₁	C₂²×Q₈	1	4	yes	—
202	C₂³×C₃₁	C₂²×Q₈	1	2	yes	—
203	C₂³×C₃₁	C₂×#41	1	4	yes	—
204	C₂³×C₃₁	C₂×#41	1	2	yes	—
205	C₂³×C₃₁	C₂×#41	1	4	yes	—
206	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
207	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
208	Q₈×C₃₁	—	4	16	no	—
209	D₄×C₃₁	—	2	32	no	—
210	C₂×C₄×C₃₁	—	2	8	no	—
211	C₂×C₄×C₃₁	—	2	8	no	—
212	C₂×C₄×C₃₁	—	2	8	no	—
213	C₂×C₄×C₃₁	—	2	16	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	4	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	4	no	—
218	C₂×C₄×C₃₁	—	2	4	no	—
219	C₂×C₄×C₃₁	—	2	4	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	4	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	4	no	—
224	C₂×C₄×C₃₁	—	2	4	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	4	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	4	no	—
230	C₂×C₄×C₃₁	—	2	4	no	—
231	C₂×C₄×C₃₁	—	2	4	no	—
232	C₂×C₄×C₃₁	—	2	4	no	—
233	C₂×C₄×C₃₁	—	4	16	no	—
234	C₂×C₄×C₃₁	—	4	32	no	—
235	C₂×C₄×C₃₁	—	4	16	no	—
236	C₂×C₄×C₃₁	—	4	4	no	—
237	C₂×C₄×C₃₁	—	4	8	no	—
238	C₂×C₄×C₃₁	—	4	4	no	—
239	C₂×C₄×C₃₁	—	4	8	no	—
240	C₂×C₄×C₃₁	—	4	8	no	—
241	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
242	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
243	C₂×C₄×C₃₁	C₂²×Q₈	1	4	yes	—
244	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
245	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
246	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
247	#2564	C₂×C₃₁	1	16	yes	direct product factor
248	C₂×#400	C₂×C₃₁	1	1	yes	characteristic, complemented
249	C₂×#399	C₂×C₃₁	1	1	yes	characteristic, complemented
250	C₂×#395	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₂×#393	C₂×C₃₁	1	1	yes	characteristic, complemented
252	C₂×#387	C₂×C₃₁	1	2	yes	complemented
253	C₂×#386	C₂×C₃₁	1	2	yes	—
254	C₂×#385	C₂×C₃₁	1	2	yes	complemented
255	C₂×#380	C₂×C₃₁	1	2	yes	complemented
256	C₂²×#95	C₂×C₃₁	1	1	yes	characteristic, complemented
257	C₂²×C₃₁	—	2	32	no	—
258	C₂²×C₃₁	—	2	4	no	—
259	C₂²×C₃₁	—	2	4	no	—
260	C₂²×C₃₁	—	2	8	no	—
261	C₂²×C₃₁	—	2	8	no	—
262	C₂²×C₃₁	—	2	4	no	—
263	C₂²×C₃₁	—	2	8	no	—
264	C₂²×C₃₁	—	4	16	no	—
265	C₂²×C₃₁	—	4	2	no	—
266	C₂²×C₃₁	—	4	2	no	—
267	C₂²×C₃₁	—	4	4	no	—
268	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	8	yes	—
269	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	4	yes	—
270	C₂²×C₃₁	#532	1	4	yes	—
271	C₂²×C₃₁	#534	1	4	yes	—
272	C₂²×C₃₁	#551	1	8	yes	—
273	C₂²×C₃₁	#554	1	8	yes	—
274	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
275	C₂²×C₃₁	C₂³×Q₈	1	1	yes	characteristic
276	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
277	C₂²×C₃₁	C₂²×#41	1	2	yes	—
278	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
279	C₄×C₃₁	—	2	4	no	—
280	C₄×C₃₁	—	2	4	no	—
281	C₄×C₃₁	—	2	4	no	—
282	C₄×C₃₁	—	2	8	no	—
283	C₄×C₃₁	—	4	2	no	—
284	C₄×C₃₁	—	4	4	no	—
285	C₄×C₃₁	—	4	2	no	—
286	C₄×C₃₁	—	4	4	no	—
287	C₄×C₃₁	—	4	4	no	—
288	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
289	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
290	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
291	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
292	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
293	C₂×C₂²⋊Q₈	—	2	8	no	—
294	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	8	yes	complemented
295	C₂×C₄×D₄	—	2	8	no	—
296	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	—
297	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
298	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
299	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
300	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
301	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
302	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
303	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
304	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
305	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
306	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
307	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
308	#400	C₂²×C₃₁	1	8	yes	complemented
309	#399	C₂²×C₃₁	1	8	yes	complemented
310	#395	C₂²×C₃₁	1	8	yes	complemented
311	#393	C₂²×C₃₁	1	8	yes	complemented
312	#387	C₂²×C₃₁	1	16	yes	complemented
313	#386	C₂²×C₃₁	1	16	yes	—
314	#385	C₂²×C₃₁	1	16	yes	complemented
315	#380	C₂²×C₃₁	1	16	yes	complemented
316	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
317	C₂×#95	C₂²×C₃₁	1	2	yes	—
318	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
319	C₂×#95	C₂²×C₃₁	1	2	yes	—
320	C₂×#95	C₂²×C₃₁	1	2	yes	—
321	C₂×#95	C₂²×C₃₁	1	2	yes	—
322	C₂×#95	C₂²×C₃₁	1	2	yes	—
323	C₂×#95	C₂²×C₃₁	1	8	yes	complemented
324	C₂×C₃₁	—	2	8	no	—
325	C₂×C₃₁	—	4	4	no	—
326	C₂×C₃₁	#2564	1	8	yes	direct product factor
327	C₂×C₃₁	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
328	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
329	C₂×C₃₁	C₂×#534	1	1	yes	characteristic
330	C₂×C₃₁	C₂×#551	1	2	yes	—
331	C₂×C₃₁	C₂×#554	1	2	yes	—
332	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
333	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
334	C₂²×Q₈	—	2	4	no	—
335	C₂²×Q₈	—	4	4	no	—
336	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
337	C₂²×D₄	—	2	4	no	—
338	C₂²×D₄	—	2	4	no	—
339	C₂²×D₄	Q₈×C₃₁	1	8	yes	complemented
340	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
341	C₂³×C₄	—	2	8	no	—
342	C₂³×C₄	—	2	4	no	—
343	C₂³×C₄	—	2	8	no	—
344	C₂³×C₄	—	4	4	no	—
345	C₂³×C₄	—	4	4	no	—
346	C₂³×C₄	—	4	4	no	—
347	C₂³×C₄	D₄×C₃₁	1	8	yes	—
348	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
349	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
350	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
351	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
352	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
353	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
354	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
355	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
357	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
358	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
359	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
360	C₂²⋊Q₈	—	2	32	no	—
361	C₄×D₄	—	2	32	no	—
362	C₂×C₄⋊C₄	—	2	4	no	—
363	C₂×C₄⋊C₄	—	2	4	no	—
364	C₂×C₄⋊C₄	—	2	8	no	—
365	C₂×C₄⋊C₄	—	2	8	no	—
366	C₂×C₄⋊C₄	—	4	4	no	—
367	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
368	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
369	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
370	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
371	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
372	C₂×C₂²⋊C₄	—	2	4	no	—
373	C₂×C₂²⋊C₄	—	2	8	no	—
374	C₂×C₂²⋊C₄	—	2	4	no	—
375	C₂×C₂²⋊C₄	—	2	8	no	—
376	C₂×C₂²⋊C₄	—	4	4	no	—
377	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
382	C₂×C₄²	—	2	4	no	—
383	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
384	#95	C₂³×C₃₁	1	8	yes	—
385	#95	C₂³×C₃₁	1	4	yes	—
386	#95	C₂³×C₃₁	1	4	yes	—
387	#95	C₂³×C₃₁	1	8	yes	—
388	#95	C₂³×C₃₁	1	8	yes	—
389	#95	C₂³×C₃₁	1	8	yes	—
390	#95	C₂³×C₃₁	1	8	yes	—
391	C₃₁	C₂×#2564	1	1	yes	characteristic, direct product factor, Sylow
392	C₂⁴	—	2	4	no	—
393	C₂⁴	—	2	4	no	—
394	C₂⁴	—	2	2	no	—
395	C₂⁴	—	2	4	no	—
396	C₂⁴	—	2	2	no	—
397	C₂⁴	—	2	4	no	—
398	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
399	C₂⁴	C₂×Q₈×C₃₁	1	4	yes	—
400	C₂⁴	C₂×Q₈×C₃₁	1	4	yes	—
401	C₂⁴	#41×C₃₁	1	4	yes	—
402	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
403	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
404	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
405	C₂×Q₈	—	2	8	no	—
406	C₂×Q₈	—	4	8	no	—
407	C₂×Q₈	—	4	16	no	—
408	C₂×D₄	—	2	32	no	—
409	C₂×D₄	—	2	8	no	—
410	C₂×D₄	—	2	8	no	—
411	C₂×D₄	C₂×Q₈×C₃₁	1	16	yes	complemented
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	16	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	2	8	no	—
433	C₂²×C₄	—	2	16	no	—
434	C₂²×C₄	—	4	16	no	—
435	C₂²×C₄	—	4	8	no	—
436	C₂²×C₄	—	4	8	no	—
437	C₂²×C₄	—	4	8	no	—
438	C₂²×C₄	—	4	32	no	—
439	C₂²×C₄	—	4	16	no	—
440	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
441	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
442	C₂²×C₄	C₂×D₄×C₃₁	1	16	yes	—
443	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
444	C₂²×C₄	#41×C₃₁	1	8	yes	—
445	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
446	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
447	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
448	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
449	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
450	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
451	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
452	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
453	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
454	C₄⋊C₄	—	2	8	no	—
455	C₄⋊C₄	—	2	8	no	—
456	C₄⋊C₄	—	2	16	no	—
457	C₄⋊C₄	—	2	16	no	—
458	C₄⋊C₄	—	4	8	no	—
459	C₂²⋊C₄	—	2	8	no	—
460	C₂²⋊C₄	—	2	16	no	—
461	C₂²⋊C₄	—	2	8	no	—
462	C₂²⋊C₄	—	2	16	no	—
463	C₂²⋊C₄	—	4	8	no	—
464	C₄²	—	2	8	no	—
465	C₂³	—	2	8	no	—
466	C₂³	—	2	8	no	—
467	C₂³	—	2	16	no	—
468	C₂³	—	2	16	no	—
469	C₂³	—	2	4	no	—
470	C₂³	—	2	4	no	—
471	C₂³	—	2	8	no	—
472	C₂³	—	2	16	no	—
473	C₂³	—	2	2	no	—
474	C₂³	—	2	4	no	—
475	C₂³	—	2	2	no	—
476	C₂³	—	2	4	no	—
477	C₂³	—	4	4	no	—
478	C₂³	—	4	4	no	—
479	C₂³	—	4	8	no	—
480	C₂³	C₃₁×C₂²⋊Q₈	1	16	yes	—
481	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
482	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
483	C₂³	C₂²×Q₈×C₃₁	1	4	yes	—
484	C₂³	C₂²×Q₈×C₃₁	1	4	yes	—
485	C₂³	C₂²×Q₈×C₃₁	1	2	yes	—
486	C₂³	C₂×#41×C₃₁	1	4	yes	—
487	C₂³	C₂×#41×C₃₁	1	2	yes	—
488	C₂³	C₂×#41×C₃₁	1	4	yes	—
489	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
490	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
491	Q₈	—	4	16	no	—
492	D₄	—	2	32	no	—
493	C₂×C₄	—	2	8	no	—
494	C₂×C₄	—	2	8	no	—
495	C₂×C₄	—	2	8	no	—
496	C₂×C₄	—	2	16	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	4	16	no	—
517	C₂×C₄	—	4	32	no	—
518	C₂×C₄	—	4	16	no	—
519	C₂×C₄	—	4	4	no	—
520	C₂×C₄	—	4	8	no	—
521	C₂×C₄	—	4	4	no	—
522	C₂×C₄	—	4	8	no	—
523	C₂×C₄	—	4	8	no	—
524	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
525	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
526	C₂×C₄	C₂²×Q₈×C₃₁	1	4	yes	—
527	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
528	C₂²	—	2	32	no	—
529	C₂²	—	2	4	no	—
530	C₂²	—	2	4	no	—
531	C₂²	—	2	8	no	—
532	C₂²	—	2	8	no	—
533	C₂²	—	2	4	no	—
534	C₂²	—	2	8	no	—
535	C₂²	—	4	16	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	2	no	—
538	C₂²	—	4	4	no	—
539	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	8	yes	—
540	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	4	yes	—
541	C₂²	C₃₁×#532	1	4	yes	—
542	C₂²	C₃₁×#534	1	4	yes	—
543	C₂²	C₃₁×#551	1	8	yes	—
544	C₂²	C₃₁×#554	1	8	yes	—
545	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
546	C₂²	C₂³×Q₈×C₃₁	1	1	yes	characteristic
547	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
548	C₂²	C₂²×#41×C₃₁	1	2	yes	—
549	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
550	C₄	—	2	4	no	—
551	C₄	—	2	4	no	—
552	C₄	—	2	4	no	—
553	C₄	—	2	8	no	—
554	C₄	—	4	2	no	—
555	C₄	—	4	4	no	—
556	C₄	—	4	2	no	—
557	C₄	—	4	4	no	—
558	C₄	—	4	4	no	—
559	C₂	—	2	8	no	—
560	C₂	—	4	4	no	—
561	C₂	C₃₁×#2564	1	8	yes	direct product factor
562	C₂	C₂²×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
563	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
564	C₂	C₂×C₃₁×#534	1	1	yes	characteristic
565	C₂	C₂×C₃₁×#551	1	2	yes	—
566	C₂	C₂×C₃₁×#554	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂³×D₄×C₃₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
3. C₃₁×#2564	C₂	direct (N × H)
4. C₂×C₃₁×#400	C₂	semidirect
5. C₂×C₃₁×#399	C₂	semidirect
6. C₂×C₃₁×#395	C₂	semidirect
7. C₂×C₃₁×#393	C₂	semidirect
8. C₂×C₃₁×#387	C₂	semidirect
10. C₂×C₃₁×#385	C₂	semidirect
11. C₂×C₃₁×#380	C₂	semidirect
12. C₂²×C₃₁×#95	C₂	semidirect
19. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
33. C₃₁×#400	C₂²	semidirect
34. C₃₁×#399	C₂²	semidirect
35. C₃₁×#395	C₂²	semidirect
36. C₃₁×#393	C₂²	semidirect
37. C₃₁×#387	C₂²	semidirect
39. C₃₁×#385	C₂²	semidirect
40. C₃₁×#380	C₂²	semidirect
48. C₂×C₃₁×#95	C₂²	semidirect
56. C₂²×D₄×C₃₁	Q₈	semidirect
127. C₂×D₄×C₃₁	C₂×Q₈	semidirect
181. C₂×#2564	C₃₁	direct (N × H)
245. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
247. #2564	C₂×C₃₁	direct (N × H)
248. C₂×#400	C₂×C₃₁	semidirect
249. C₂×#399	C₂×C₃₁	semidirect
250. C₂×#395	C₂×C₃₁	semidirect
251. C₂×#393	C₂×C₃₁	semidirect
252. C₂×#387	C₂×C₃₁	semidirect
254. C₂×#385	C₂×C₃₁	semidirect
255. C₂×#380	C₂×C₃₁	semidirect
256. C₂²×#95	C₂×C₃₁	semidirect
294. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
308. #400	C₂²×C₃₁	semidirect
309. #399	C₂²×C₃₁	semidirect
310. #395	C₂²×C₃₁	semidirect
311. #393	C₂²×C₃₁	semidirect
312. #387	C₂²×C₃₁	semidirect
314. #385	C₂²×C₃₁	semidirect
315. #380	C₂²×C₃₁	semidirect
323. C₂×#95	C₂²×C₃₁	semidirect
326. C₂×C₃₁	#2564	direct (N × H)
339. C₂²×D₄	Q₈×C₃₁	semidirect
391. C₃₁	C₂×#2564	direct (N × H)
411. C₂×D₄	C₂×Q₈×C₃₁	semidirect
561. C₂	C₃₁×#2564	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×C₂²⋊Q₈	2
C₂²×C₄×D₄×C₃₁	2
C₃₁×#2564	2
C₂×C₃₁×#400	2
C₂×C₃₁×#399	2
C₂×C₃₁×#395	2
C₂×C₃₁×#393	2
C₂×C₃₁×#387	2
C₂×C₃₁×#386	2
C₂×C₃₁×#385	2
C₂×C₃₁×#380	2
C₂²×C₃₁×#95	2
C₂×#2564	31

Subgroups — C2×C31×#2564 #21491590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2564 #21491590