Subgroups — C₃₁×#22523 #21479282

Order 7936.

1250 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2355	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1993	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1988	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1987	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1804	C₂	1	2	yes	maximal
9	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1762	C₂	1	2	yes	maximal
11	C₄×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
12	C₂³×Q₈×C₃₁	—	2	1	no	—
13	D₄×Q₈×C₃₁	—	2	4	no	—
14	D₄×Q₈×C₃₁	C₄	1	8	yes	complemented
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
18	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
19	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#446	—	2	4	no	—
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#97	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#391	—	2	1	no	—
31	C₃₁×#387	—	2	2	no	—
32	C₃₁×#384	C₂²	1	1	yes	characteristic
33	C₃₁×#382	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	1	yes	characteristic
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#340	C₂²	1	1	yes	characteristic
44	C₂²×Q₈×C₃₁	—	2	8	no	—
45	C₂²×Q₈×C₃₁	—	2	2	no	—
46	C₂²×Q₈×C₃₁	—	2	1	no	—
47	C₂²×Q₈×C₃₁	—	4	4	no	—
48	C₂²×Q₈×C₃₁	—	4	4	no	—
49	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
50	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	—	2	2	no	—
52	C₂³×C₄×C₃₁	—	2	1	no	—
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×C₄⋊Q₈	—	2	2	no	—
57	C₃₁×C₄⋊Q₈	—	2	4	no	—
58	C₃₁×C₄⋊Q₈	C₂×C₄	1	8	yes	—
59	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
60	C₃₁×C₂²⋊Q₈	—	2	16	no	—
61	C₃₁×C₂²⋊Q₈	—	2	8	no	—
62	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
63	C₄×Q₈×C₃₁	D₄	1	4	yes	—
64	C₄×D₄×C₃₁	—	2	2	no	—
65	C₄×D₄×C₃₁	—	2	4	no	—
66	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
67	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
68	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
81	C₂×C₄²×C₃₁	—	2	1	no	—
82	C₂×C₄²×C₃₁	—	2	2	no	—
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
85	C₃₁×C₄⋊C₈	—	2	2	no	—
86	C₃₁×C₄⋊C₈	—	2	4	no	—
87	C₃₁×#102	—	2	4	no	—
88	C₃₁×#102	—	2	2	no	—
89	C₃₁×#102	Q₈	1	4	yes	complemented
90	C₃₁×#97	—	2	4	no	—
91	C₄×C₈×C₃₁	—	2	2	no	—
92	C₃₁×#95	—	2	2	no	—
93	C₃₁×#95	—	2	1	no	—
94	C₃₁×#95	—	2	2	no	—
95	C₃₁×#95	—	2	2	no	—
96	C₂⁴×C₃₁	—	2	1	no	—
97	C₂×Q₈×C₃₁	—	2	8	no	—
98	C₂×Q₈×C₃₁	—	2	4	no	—
99	C₂×Q₈×C₃₁	—	2	2	no	—
100	C₂×Q₈×C₃₁	—	2	2	no	—
101	C₂×Q₈×C₃₁	—	4	4	no	—
102	C₂×Q₈×C₃₁	—	4	4	no	—
103	C₂×Q₈×C₃₁	—	4	32	no	—
104	C₂×Q₈×C₃₁	—	4	2	no	—
105	C₂×Q₈×C₃₁	—	4	2	no	—
106	C₂×Q₈×C₃₁	C₂²⋊C₄	1	4	yes	—
107	C₂×D₄×C₃₁	—	2	1	no	—
108	C₂×D₄×C₃₁	—	2	1	no	—
109	C₂×D₄×C₃₁	—	2	1	no	—
110	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
112	C₂×D₄×C₃₁	#41	1	2	yes	—
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	8	no	—
117	C₂²×C₄×C₃₁	—	2	4	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	4	4	no	—
121	C₂²×C₄×C₃₁	—	4	4	no	—
122	C₂²×C₄×C₃₁	—	4	2	no	—
123	C₂²×C₄×C₃₁	—	4	2	no	—
124	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
129	C₂×C₈×C₃₁	—	2	1	no	—
130	C₂×C₈×C₃₁	—	2	1	no	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	complemented
138	C₄⋊C₄×C₃₁	—	2	4	no	—
139	C₄⋊C₄×C₃₁	—	2	8	no	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	2	2	no	—
142	C₄⋊C₄×C₃₁	—	2	1	no	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	2	1	no	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	4	2	no	—
148	C₄⋊C₄×C₃₁	—	8	1	no	—
149	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
150	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
151	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
152	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
153	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	complemented
154	C₂²⋊C₄×C₃₁	—	2	4	no	—
155	C₂²⋊C₄×C₃₁	—	2	8	no	—
156	C₄²×C₃₁	—	4	2	no	—
157	C₄²×C₃₁	—	8	2	no	—
158	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
159	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
160	C₄²×C₃₁	C₂×D₄	1	2	yes	—
161	C₄²×C₃₁	C₂×D₄	1	4	yes	—
162	#22523	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂³×C₃₁	—	2	2	no	—
164	C₂³×C₃₁	—	2	2	no	—
165	C₂³×C₃₁	—	2	2	no	—
166	C₂³×C₃₁	—	4	1	no	—
167	C₂³×C₃₁	—	4	1	no	—
168	C₂³×C₃₁	—	4	1	no	—
169	C₂³×C₃₁	—	4	1	no	—
170	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
171	Q₈×C₃₁	—	2	8	no	—
172	Q₈×C₃₁	—	4	16	no	—
173	Q₈×C₃₁	—	4	8	no	—
174	D₄×C₃₁	—	2	2	no	—
175	D₄×C₃₁	—	2	2	no	—
176	D₄×C₃₁	—	2	2	no	—
177	D₄×C₃₁	C₄×Q₈	1	4	yes	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	4	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	4	16	no	—
188	C₂×C₄×C₃₁	—	4	1	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	8	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	2	no	—
197	C₂×C₄×C₃₁	—	4	1	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	1	no	—
201	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
202	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
203	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
204	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
205	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
206	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
207	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
208	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
210	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
212	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
213	C₈×C₃₁	—	2	2	no	—
214	C₈×C₃₁	—	4	1	no	—
215	C₈×C₃₁	—	4	2	no	—
216	C₂×D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
217	C₂×#446	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#2355	C₂×C₃₁	1	1	yes	characteristic, complemented
219	#1993	C₂×C₃₁	1	2	yes	complemented
220	#1990	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#1988	C₂×C₃₁	1	1	yes	characteristic, complemented
222	#1987	C₂×C₃₁	1	2	yes	complemented
223	#1804	C₂×C₃₁	1	2	yes	—
224	#1802	C₂×C₃₁	1	1	yes	characteristic
225	#1762	C₂×C₃₁	1	2	yes	—
226	C₄×#102	C₂×C₃₁	1	1	yes	characteristic
227	C₂²×C₃₁	—	2	4	no	—
228	C₂²×C₃₁	—	4	2	no	—
229	C₂²×C₃₁	—	4	2	no	—
230	C₂²×C₃₁	—	4	2	no	—
231	C₂²×C₃₁	—	4	2	no	—
232	C₂²×C₃₁	—	4	2	no	—
233	C₂²×C₃₁	—	4	2	no	—
234	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
235	C₂²×C₃₁	#548	1	1	yes	characteristic
236	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
237	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
238	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
239	C₂²×C₃₁	#577	1	1	yes	characteristic
240	C₂²×C₃₁	#578	1	1	yes	characteristic
241	C₄×C₃₁	—	2	4	no	—
242	C₄×C₃₁	—	2	4	no	—
243	C₄×C₃₁	—	2	2	no	—
244	C₄×C₃₁	—	4	2	no	—
245	C₄×C₃₁	—	4	4	no	—
246	C₄×C₃₁	—	4	2	no	—
247	C₄×C₃₁	—	8	1	no	—
248	C₄×C₃₁	—	8	2	no	—
249	C₄×C₃₁	#418	1	4	yes	—
250	C₄×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
251	C₄×C₃₁	#519	1	1	yes	characteristic
252	C₄×C₃₁	#548	1	1	yes	characteristic
253	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
254	C₂³×Q₈	—	2	1	no	—
255	D₄×Q₈	—	2	4	no	—
256	D₄×Q₈	C₄×C₃₁	1	8	yes	complemented
257	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
258	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×C₂²⋊Q₈	—	2	1	no	—
260	C₂×C₂²⋊Q₈	—	2	2	no	—
261	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
264	#446	—	2	4	no	—
265	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
267	C₂×#102	C₂²×C₃₁	1	2	yes	—
268	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×#97	C₂²×C₃₁	1	2	yes	—
271	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
272	#391	—	2	1	no	—
273	#387	—	2	2	no	—
274	#384	C₂²×C₃₁	1	1	yes	characteristic
275	#382	C₂²×C₃₁	1	2	yes	—
276	C₄×C₄⋊C₄	—	2	1	no	—
277	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
278	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
279	#340	C₂²×C₃₁	1	2	yes	—
280	#340	C₂²×C₃₁	1	2	yes	—
281	#340	C₂²×C₃₁	1	2	yes	—
282	#340	C₂²×C₃₁	1	1	yes	characteristic
283	#340	C₂²×C₃₁	1	2	yes	—
284	#340	C₂²×C₃₁	1	2	yes	—
285	#340	C₂²×C₃₁	1	1	yes	characteristic
286	C₂×C₃₁	—	4	4	no	—
287	C₂×C₃₁	Q₈×C₂²⋊C₄	1	1	yes	characteristic
288	C₂×C₃₁	C₂×#418	1	1	yes	characteristic
289	C₂×C₃₁	#2898	1	1	yes	characteristic
290	C₂×C₃₁	#3367	1	1	yes	characteristic
291	C₂×C₃₁	#3373	1	1	yes	characteristic
292	C₂×C₃₁	#3384	1	1	yes	characteristic
293	C₂×C₃₁	#3386	1	1	yes	characteristic
294	C₂²×Q₈	—	2	8	no	—
295	C₂²×Q₈	—	2	2	no	—
296	C₂²×Q₈	—	2	1	no	—
297	C₂²×Q₈	—	4	4	no	—
298	C₂²×Q₈	—	4	4	no	—
299	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
300	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
301	C₂³×C₄	—	2	2	no	—
302	C₂³×C₄	—	2	1	no	—
303	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
304	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
305	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
306	C₄⋊Q₈	—	2	2	no	—
307	C₄⋊Q₈	—	2	4	no	—
308	C₄⋊Q₈	C₂×C₄×C₃₁	1	8	yes	—
309	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
310	C₂²⋊Q₈	—	2	16	no	—
311	C₂²⋊Q₈	—	2	8	no	—
312	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
313	C₄×Q₈	D₄×C₃₁	1	4	yes	—
314	C₄×D₄	—	2	2	no	—
315	C₄×D₄	—	2	4	no	—
316	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
317	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
318	C₂×C₄⋊C₄	—	2	2	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
323	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
324	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
325	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
326	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
327	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
328	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
329	C₂×C₂²⋊C₄	—	2	2	no	—
330	C₂×C₂²⋊C₄	—	2	1	no	—
331	C₂×C₄²	—	2	1	no	—
332	C₂×C₄²	—	2	2	no	—
333	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
334	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
335	C₄⋊C₈	—	2	2	no	—
336	C₄⋊C₈	—	2	4	no	—
337	#102	—	2	4	no	—
338	#102	—	2	2	no	—
339	#102	Q₈×C₃₁	1	4	yes	complemented
340	#97	—	2	4	no	—
341	C₄×C₈	—	2	2	no	—
342	#95	—	2	2	no	—
343	#95	—	2	1	no	—
344	#95	—	2	2	no	—
345	#95	—	2	2	no	—
346	C₃₁	#22523	1	1	yes	characteristic, direct product factor, Sylow
347	C₂⁴	—	2	1	no	—
348	C₂×Q₈	—	2	8	no	—
349	C₂×Q₈	—	2	4	no	—
350	C₂×Q₈	—	2	2	no	—
351	C₂×Q₈	—	2	2	no	—
352	C₂×Q₈	—	4	4	no	—
353	C₂×Q₈	—	4	4	no	—
354	C₂×Q₈	—	4	32	no	—
355	C₂×Q₈	—	4	2	no	—
356	C₂×Q₈	—	4	2	no	—
357	C₂×Q₈	C₂²⋊C₄×C₃₁	1	4	yes	—
358	C₂×D₄	—	2	1	no	—
359	C₂×D₄	—	2	1	no	—
360	C₂×D₄	—	2	1	no	—
361	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
362	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
363	C₂×D₄	#41×C₃₁	1	2	yes	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	8	no	—
368	C₂²×C₄	—	2	4	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	4	4	no	—
372	C₂²×C₄	—	4	4	no	—
373	C₂²×C₄	—	4	2	no	—
374	C₂²×C₄	—	4	2	no	—
375	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
376	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
377	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
379	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	complemented
389	C₄⋊C₄	—	2	4	no	—
390	C₄⋊C₄	—	2	8	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	4	2	no	—
399	C₄⋊C₄	—	8	1	no	—
400	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
401	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
402	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
403	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
404	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	complemented
405	C₂²⋊C₄	—	2	4	no	—
406	C₂²⋊C₄	—	2	8	no	—
407	C₄²	—	4	2	no	—
408	C₄²	—	8	2	no	—
409	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
410	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
411	C₄²	C₂×D₄×C₃₁	1	2	yes	—
412	C₄²	C₂×D₄×C₃₁	1	4	yes	—
413	C₂³	—	2	2	no	—
414	C₂³	—	2	2	no	—
415	C₂³	—	2	2	no	—
416	C₂³	—	4	1	no	—
417	C₂³	—	4	1	no	—
418	C₂³	—	4	1	no	—
419	C₂³	—	4	1	no	—
420	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
421	Q₈	—	2	8	no	—
422	Q₈	—	4	16	no	—
423	Q₈	—	4	8	no	—
424	D₄	—	2	2	no	—
425	D₄	—	2	2	no	—
426	D₄	—	2	2	no	—
427	D₄	C₄×Q₈×C₃₁	1	4	yes	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	4	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	4	16	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	8	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
452	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
453	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
454	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
455	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
456	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
457	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
458	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
459	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
460	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
461	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
462	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
463	C₈	—	2	2	no	—
464	C₈	—	4	1	no	—
465	C₈	—	4	2	no	—
466	C₂²	—	2	4	no	—
467	C₂²	—	4	2	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
474	C₂²	C₃₁×#548	1	1	yes	characteristic
475	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
476	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
477	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
478	C₂²	C₃₁×#577	1	1	yes	characteristic
479	C₂²	C₃₁×#578	1	1	yes	characteristic
480	C₄	—	2	4	no	—
481	C₄	—	2	4	no	—
482	C₄	—	2	2	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	2	no	—
486	C₄	—	8	1	no	—
487	C₄	—	8	2	no	—
488	C₄	C₃₁×#418	1	4	yes	—
489	C₄	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
490	C₄	C₃₁×#519	1	1	yes	characteristic
491	C₄	C₃₁×#548	1	1	yes	characteristic
492	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
493	C₂	—	4	4	no	—
494	C₂	Q₈×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
495	C₂	C₂×C₃₁×#418	1	1	yes	characteristic
496	C₂	C₃₁×#2898	1	1	yes	characteristic
497	C₂	C₃₁×#3367	1	1	yes	characteristic
498	C₂	C₃₁×#3373	1	1	yes	characteristic
499	C₂	C₃₁×#3384	1	1	yes	characteristic
500	C₂	C₃₁×#3386	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#446	C₂	semidirect
3. C₃₁×#2355	C₂	semidirect
4. C₃₁×#1993	C₂	semidirect
5. C₃₁×#1990	C₂	semidirect
6. C₃₁×#1988	C₂	semidirect
7. C₃₁×#1987	C₂	semidirect
14. D₄×Q₈×C₃₁	C₄	semidirect
89. C₃₁×#102	Q₈	semidirect
137. C₂×C₈×C₃₁	C₂×Q₈	semidirect
153. C₄⋊C₄×C₃₁	C₂×Q₈	semidirect
162. #22523	C₃₁	direct (N × H)
217. C₂×#446	C₂×C₃₁	semidirect
218. #2355	C₂×C₃₁	semidirect
219. #1993	C₂×C₃₁	semidirect
220. #1990	C₂×C₃₁	semidirect
221. #1988	C₂×C₃₁	semidirect
222. #1987	C₂×C₃₁	semidirect
256. D₄×Q₈	C₄×C₃₁	semidirect
339. #102	Q₈×C₃₁	semidirect
346. C₃₁	#22523	direct (N × H)
388. C₂×C₈	C₂×Q₈×C₃₁	semidirect
404. C₄⋊C₄	C₂×Q₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Q₈×C₃₁	2
C₂×C₃₁×#446	2
C₃₁×#2355	2
C₃₁×#1993	2
C₃₁×#1990	2
C₃₁×#1988	2
C₃₁×#1987	2
C₃₁×#1804	2
C₃₁×#1802	2
C₃₁×#1762	2
C₄×C₃₁×#102	2
#22523	31

Subgroups — C31×#22523 #21479282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22523 #21479282