Subgroups — C₃₁×#62148 #21518907

Order 7936.

1942 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3482	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3359	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3353	C₂	1	6	yes	maximal, complemented
6	C₃₁×#3330	C₂	1	6	yes	maximal, complemented
7	C₃₁×#3254	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3230	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3106	C₂	1	3	yes	maximal, complemented
10	C₃₁×#3079	C₂	1	3	yes	maximal, complemented
11	C₃₁×#2991	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×2₋¹⁺⁴	—	2	1	no	—
13	C₃₁×#557	C₂²	1	1	yes	characteristic
14	C₃₁×#556	C₂²	1	3	yes	—
15	C₃₁×#552	—	2	3	no	—
16	D₄×Q₈×C₃₁	—	2	1	no	—
17	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
18	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#548	—	2	1	no	—
20	C₃₁×#547	—	2	3	no	—
21	C₃₁×#547	C₂²	1	3	yes	—
22	C₃₁×#547	C₂²	1	6	yes	—
23	C₃₁×#536	—	2	3	no	—
24	C₃₁×#533	C₂²	1	3	yes	—
25	C₃₁×#532	C₂²	1	3	yes	—
26	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
27	C₃₁×#529	—	2	3	no	—
28	C₄×#41×C₃₁	C₂²	1	3	yes	—
29	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
31	C₃₁×#500	C₂²	1	6	yes	—
32	C₃₁×#487	C₂²	1	6	yes	—
33	C₃₁×#484	C₂²	1	12	yes	—
34	C₃₁×#482	C₂²	1	6	yes	—
35	C₃₁×#477	C₂²	1	6	yes	—
36	C₃₁×#477	C₂²	1	3	yes	—
37	C₃₁×#476	C₂²	1	3	yes	—
38	C₃₁×#475	C₂²	1	6	yes	—
39	C₃₁×#474	C₂²	1	6	yes	—
40	C₃₁×#474	C₂²	1	6	yes	—
41	C₃₁×#468	C₂²	1	12	yes	—
42	C₃₁×#461	C₂²	1	6	yes	—
43	C₃₁×#460	C₂²	1	3	yes	—
44	C₃₁×#459	C₂²	1	1	yes	characteristic
45	C₃₁×#449	C₂²	1	6	yes	—
46	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
47	C₄×SD₁₆×C₃₁	C₂²	1	6	yes	—
48	C₄×D₈×C₃₁	C₂²	1	6	yes	—
49	C₃₁×#436	C₂²	1	6	yes	—
50	C₃₁×#431	C₂²	1	3	yes	—
51	C₃₁×#428	C₂²	1	6	yes	—
52	C₃₁×#423	C₂²	1	3	yes	—
53	C₃₁×#417	C₂²	1	6	yes	—
54	C₃₁×2₋¹⁺⁴	—	4	8	no	—
55	C₂×#41×C₃₁	—	2	6	no	—
56	C₂×#41×C₃₁	—	2	3	no	—
57	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
58	C₂×#41×C₃₁	C₂³	1	3	yes	—
59	C₂²×Q₈×C₃₁	—	2	1	no	—
60	C₂²×Q₈×C₃₁	—	2	2	no	—
61	C₂²×Q₈×C₃₁	—	2	3	no	—
62	C₂×SD₁₆×C₃₁	C₂³	1	6	yes	—
63	C₂×D₈×C₃₁	C₂³	1	2	yes	—
64	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
65	C₃₁×C₄⋊Q₈	—	2	3	no	—
66	C₃₁×C₄⋊Q₈	—	2	3	no	—
67	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
68	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
69	C₃₁×C₄⋊Q₈	C₂³	1	6	yes	—
70	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
71	C₃₁×#127	C₂³	1	1	yes	characteristic
72	C₃₁×#126	—	2	6	no	—
73	C₃₁×#125	—	2	3	no	—
74	C₃₁×#125	—	2	3	no	—
75	C₃₁×#124	—	2	3	no	—
76	C₃₁×#124	—	2	3	no	—
77	C₃₁×#123	—	2	6	no	—
78	C₃₁×#123	—	2	12	no	—
79	C₃₁×C₂²⋊Q₈	—	2	3	no	—
80	C₃₁×C₂²⋊Q₈	—	2	3	no	—
81	C₃₁×C₂²⋊Q₈	—	2	6	no	—
82	C₃₁×C₂²⋊Q₈	—	2	6	no	—
83	C₃₁×C₂²⋊Q₈	C₂³	1	6	yes	—
84	C₃₁×#121	—	2	3	no	—
85	C₃₁×#121	—	2	3	no	—
86	C₃₁×#121	C₂³	1	6	yes	—
87	C₄×Q₈×C₃₁	—	2	1	no	—
88	C₄×Q₈×C₃₁	—	2	3	no	—
89	C₄×Q₈×C₃₁	—	2	3	no	—
90	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
91	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
92	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
93	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₄×D₄×C₃₁	—	2	3	no	—
95	C₄×D₄×C₃₁	—	2	3	no	—
96	C₄×D₄×C₃₁	—	2	3	no	—
97	C₄×D₄×C₃₁	D₄	1	3	yes	—
98	C₄×D₄×C₃₁	C₂³	1	3	yes	—
99	C₄×D₄×C₃₁	C₂³	1	6	yes	—
100	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×D₄×C₃₁	C₂³	1	3	yes	—
102	C₃₁×#117	—	2	3	no	—
103	C₃₁×#117	—	2	3	no	—
104	C₃₁×#117	D₄	1	3	yes	—
105	C₃₁×#117	C₂³	1	3	yes	—
106	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
107	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
109	C₂×C₄²×C₃₁	—	2	3	no	—
110	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
111	C₃₁×#107	C₂³	1	12	yes	—
112	C₃₁×#107	C₂³	1	6	yes	—
113	C₃₁×#106	C₂³	1	6	yes	—
114	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
115	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
116	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₃₁×#103	C₂³	1	12	yes	—
118	C₃₁×#103	C₂³	1	6	yes	—
119	C₃₁×#102	C₂³	1	6	yes	—
120	C₃₁×#102	C₂³	1	6	yes	—
121	C₃₁×#102	C₂³	1	2	yes	—
122	C₃₁×C₂²⋊C₈	C₂³	1	6	yes	—
123	C₄×C₈×C₃₁	C₂³	1	6	yes	—
124	#41×C₃₁	—	2	4	no	—
125	#41×C₃₁	—	2	6	no	—
126	#41×C₃₁	—	4	3	no	—
127	#41×C₃₁	—	4	12	no	—
128	#41×C₃₁	—	4	24	no	—
129	C₂×Q₈×C₃₁	—	2	3	no	—
130	C₂×Q₈×C₃₁	—	2	6	no	—
131	C₂×Q₈×C₃₁	—	2	3	no	—
132	C₂×Q₈×C₃₁	—	4	12	no	—
133	C₂×Q₈×C₃₁	—	4	4	no	—
134	C₂×Q₈×C₃₁	—	4	8	no	—
135	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂×Q₈×C₃₁	C₂⁴	1	3	yes	—
137	C₂×D₄×C₃₁	—	2	3	no	—
138	C₂×D₄×C₃₁	—	2	6	no	—
139	C₂×D₄×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	—	2	1	no	—
141	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
142	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
143	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	—	2	6	no	—
145	C₂²×C₄×C₃₁	—	2	1	no	—
146	C₂²×C₄×C₃₁	—	2	3	no	—
147	C₂²×C₄×C₃₁	—	2	3	no	—
148	C₂²×C₄×C₃₁	—	2	6	no	—
149	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
150	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
151	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
152	SD₁₆×C₃₁	—	4	6	no	—
153	D₈×C₃₁	—	2	4	no	—
154	M₁₆×C₃₁	—	4	6	no	—
155	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
156	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
157	C₄⋊C₄×C₃₁	—	2	3	no	—
158	C₄⋊C₄×C₃₁	—	2	3	no	—
159	C₄⋊C₄×C₃₁	—	2	6	no	—
160	C₄⋊C₄×C₃₁	—	2	6	no	—
161	C₄⋊C₄×C₃₁	—	2	6	no	—
162	C₄⋊C₄×C₃₁	—	2	3	no	—
163	C₄⋊C₄×C₃₁	—	2	3	no	—
164	C₄⋊C₄×C₃₁	—	2	3	no	—
165	C₄⋊C₄×C₃₁	—	2	3	no	—
166	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
167	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
168	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
169	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
170	C₄⋊C₄×C₃₁	C₂⁴	1	6	yes	—
171	C₂²⋊C₄×C₃₁	—	2	6	no	—
172	C₂²⋊C₄×C₃₁	—	2	3	no	—
173	C₂²⋊C₄×C₃₁	—	2	1	no	—
174	C₂²⋊C₄×C₃₁	—	2	3	no	—
175	C₂²⋊C₄×C₃₁	—	2	6	no	—
176	C₂²⋊C₄×C₃₁	—	2	6	no	—
177	C₂²⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
178	C₄²×C₃₁	—	2	3	no	—
179	C₄²×C₃₁	—	2	3	no	—
180	C₄²×C₃₁	—	2	3	no	—
181	C₄²×C₃₁	C₂×D₄	1	3	yes	—
182	C₄²×C₃₁	C₂×D₄	1	3	yes	—
183	C₄²×C₃₁	C₂⁴	1	3	yes	—
184	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
185	#62148	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂³×C₃₁	—	2	1	no	—
187	C₂³×C₃₁	—	2	2	no	—
188	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
189	Q₈×C₃₁	—	2	3	no	—
190	Q₈×C₃₁	—	2	3	no	—
191	Q₈×C₃₁	—	4	6	no	—
192	Q₈×C₃₁	—	4	6	no	—
193	Q₈×C₃₁	—	4	12	no	—
194	Q₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
195	D₄×C₃₁	—	2	6	no	—
196	D₄×C₃₁	—	2	2	no	—
197	D₄×C₃₁	—	2	1	no	—
198	D₄×C₃₁	—	2	2	no	—
199	D₄×C₃₁	—	4	2	no	—
200	D₄×C₃₁	—	4	6	no	—
201	D₄×C₃₁	—	4	12	no	—
202	D₄×C₃₁	—	4	2	no	—
203	D₄×C₃₁	2₋¹⁺⁴	1	2	yes	complemented
204	C₂×C₄×C₃₁	—	2	6	no	—
205	C₂×C₄×C₃₁	—	2	6	no	—
206	C₂×C₄×C₃₁	—	2	3	no	—
207	C₂×C₄×C₃₁	—	2	3	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	2	3	no	—
210	C₂×C₄×C₃₁	—	2	3	no	—
211	C₂×C₄×C₃₁	—	2	6	no	—
212	C₂×C₄×C₃₁	—	4	6	no	—
213	C₂×C₄×C₃₁	—	4	12	no	—
214	C₂×C₄×C₃₁	—	4	6	no	—
215	C₂×C₄×C₃₁	—	4	12	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
218	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
220	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
221	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
223	C₈×C₃₁	—	2	2	no	—
224	C₈×C₃₁	—	2	6	no	—
225	#3499	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#3482	C₂×C₃₁	1	1	yes	characteristic, complemented
227	Q₈×D₈	C₂×C₃₁	1	2	yes	complemented
228	#3359	C₂×C₃₁	1	2	yes	complemented
229	#3353	C₂×C₃₁	1	6	yes	complemented
230	#3330	C₂×C₃₁	1	6	yes	complemented
231	#3254	C₂×C₃₁	1	3	yes	complemented
232	#3230	C₂×C₃₁	1	3	yes	complemented
233	#3106	C₂×C₃₁	1	3	yes	complemented
234	#3079	C₂×C₃₁	1	3	yes	complemented
235	#2991	C₂×C₃₁	1	1	yes	characteristic, complemented
236	C₂²×C₃₁	—	2	3	no	—
237	C₂²×C₃₁	—	2	2	no	—
238	C₂²×C₃₁	—	2	3	no	—
239	C₂²×C₃₁	—	2	3	no	—
240	C₂²×C₃₁	—	4	1	no	—
241	C₂²×C₃₁	—	4	1	no	—
242	C₂²×C₃₁	—	4	2	no	—
243	C₂²×C₃₁	—	4	2	no	—
244	C₂²×C₃₁	—	4	2	no	—
245	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
246	C₄×C₃₁	—	2	3	no	—
247	C₄×C₃₁	—	2	1	no	—
248	C₄×C₃₁	—	2	3	no	—
249	C₄×C₃₁	—	4	3	no	—
250	C₄×C₃₁	—	4	6	no	—
251	C₄×C₃₁	—	4	3	no	—
252	C₄×C₃₁	—	4	1	no	—
253	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	6	yes	—
254	C₄×C₃₁	#578	1	2	yes	—
255	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
256	C₄×C₃₁	#585	1	1	yes	characteristic
257	C₂×2₋¹⁺⁴	—	2	1	no	—
258	#557	C₂²×C₃₁	1	1	yes	characteristic
259	#556	C₂²×C₃₁	1	3	yes	—
260	#552	—	2	3	no	—
261	D₄×Q₈	—	2	1	no	—
262	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
263	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
264	#548	—	2	1	no	—
265	#547	—	2	3	no	—
266	#547	C₂²×C₃₁	1	3	yes	—
267	#547	C₂²×C₃₁	1	6	yes	—
268	#536	—	2	3	no	—
269	#533	C₂²×C₃₁	1	3	yes	—
270	#532	C₂²×C₃₁	1	3	yes	—
271	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	—
272	#529	—	2	3	no	—
273	C₄×#41	C₂²×C₃₁	1	3	yes	—
274	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×C₄×Q₈	—	2	1	no	—
276	#500	C₂²×C₃₁	1	6	yes	—
277	#487	C₂²×C₃₁	1	6	yes	—
278	#484	C₂²×C₃₁	1	12	yes	—
279	#482	C₂²×C₃₁	1	6	yes	—
280	#477	C₂²×C₃₁	1	6	yes	—
281	#477	C₂²×C₃₁	1	3	yes	—
282	#476	C₂²×C₃₁	1	3	yes	—
283	#475	C₂²×C₃₁	1	6	yes	—
284	#474	C₂²×C₃₁	1	6	yes	—
285	#474	C₂²×C₃₁	1	6	yes	—
286	#468	C₂²×C₃₁	1	12	yes	—
287	#461	C₂²×C₃₁	1	6	yes	—
288	#460	C₂²×C₃₁	1	3	yes	—
289	#459	C₂²×C₃₁	1	1	yes	characteristic
290	#449	C₂²×C₃₁	1	6	yes	—
291	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
292	C₄×SD₁₆	C₂²×C₃₁	1	6	yes	—
293	C₄×D₈	C₂²×C₃₁	1	6	yes	—
294	#436	C₂²×C₃₁	1	6	yes	—
295	#431	C₂²×C₃₁	1	3	yes	—
296	#428	C₂²×C₃₁	1	6	yes	—
297	#423	C₂²×C₃₁	1	3	yes	—
298	#417	C₂²×C₃₁	1	6	yes	—
299	C₂×C₃₁	—	4	3	no	—
300	C₂×C₃₁	—	4	2	no	—
301	C₂×C₃₁	—	8	2	no	—
302	C₂×C₃₁	#3489	1	1	yes	characteristic
303	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
304	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
305	2₋¹⁺⁴	—	4	8	no	—
306	C₂×#41	—	2	6	no	—
307	C₂×#41	—	2	3	no	—
308	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
309	C₂×#41	C₂³×C₃₁	1	3	yes	—
310	C₂²×Q₈	—	2	1	no	—
311	C₂²×Q₈	—	2	2	no	—
312	C₂²×Q₈	—	2	3	no	—
313	C₂×SD₁₆	C₂³×C₃₁	1	6	yes	—
314	C₂×D₈	C₂³×C₃₁	1	2	yes	—
315	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
316	C₄⋊Q₈	—	2	3	no	—
317	C₄⋊Q₈	—	2	3	no	—
318	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
319	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
320	C₄⋊Q₈	C₂³×C₃₁	1	6	yes	—
321	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
322	#127	C₂³×C₃₁	1	1	yes	characteristic
323	#126	—	2	6	no	—
324	#125	—	2	3	no	—
325	#125	—	2	3	no	—
326	#124	—	2	3	no	—
327	#124	—	2	3	no	—
328	#123	—	2	6	no	—
329	#123	—	2	12	no	—
330	C₂²⋊Q₈	—	2	3	no	—
331	C₂²⋊Q₈	—	2	3	no	—
332	C₂²⋊Q₈	—	2	6	no	—
333	C₂²⋊Q₈	—	2	6	no	—
334	C₂²⋊Q₈	C₂³×C₃₁	1	6	yes	—
335	#121	—	2	3	no	—
336	#121	—	2	3	no	—
337	#121	C₂³×C₃₁	1	6	yes	—
338	C₄×Q₈	—	2	1	no	—
339	C₄×Q₈	—	2	3	no	—
340	C₄×Q₈	—	2	3	no	—
341	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
342	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
343	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
344	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
345	C₄×D₄	—	2	3	no	—
346	C₄×D₄	—	2	3	no	—
347	C₄×D₄	—	2	3	no	—
348	C₄×D₄	D₄×C₃₁	1	3	yes	—
349	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
350	C₄×D₄	C₂³×C₃₁	1	3	yes	—
351	C₄×D₄	C₂³×C₃₁	1	6	yes	—
352	C₄×D₄	C₂³×C₃₁	1	3	yes	—
353	#117	—	2	3	no	—
354	#117	—	2	3	no	—
355	#117	D₄×C₃₁	1	3	yes	—
356	#117	C₂³×C₃₁	1	3	yes	—
357	C₂×C₄⋊C₄	—	2	3	no	—
358	C₂×C₄⋊C₄	—	2	3	no	—
359	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
360	C₂×C₄²	—	2	3	no	—
361	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
362	#107	C₂³×C₃₁	1	12	yes	—
363	#107	C₂³×C₃₁	1	6	yes	—
364	#106	C₂³×C₃₁	1	6	yes	—
365	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
366	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
367	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
368	#103	C₂³×C₃₁	1	12	yes	—
369	#103	C₂³×C₃₁	1	6	yes	—
370	#102	C₂³×C₃₁	1	6	yes	—
371	#102	C₂³×C₃₁	1	6	yes	—
372	#102	C₂³×C₃₁	1	2	yes	—
373	C₂²⋊C₈	C₂³×C₃₁	1	6	yes	—
374	C₄×C₈	C₂³×C₃₁	1	6	yes	—
375	C₃₁	#62148	1	1	yes	characteristic, direct product factor, Sylow
376	#41	—	2	4	no	—
377	#41	—	2	6	no	—
378	#41	—	4	3	no	—
379	#41	—	4	12	no	—
380	#41	—	4	24	no	—
381	C₂×Q₈	—	2	3	no	—
382	C₂×Q₈	—	2	6	no	—
383	C₂×Q₈	—	2	3	no	—
384	C₂×Q₈	—	4	12	no	—
385	C₂×Q₈	—	4	4	no	—
386	C₂×Q₈	—	4	8	no	—
387	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₂×Q₈	C₂⁴×C₃₁	1	3	yes	—
389	C₂×D₄	—	2	3	no	—
390	C₂×D₄	—	2	6	no	—
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	—	2	1	no	—
393	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
394	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
395	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
396	C₂²×C₄	—	2	6	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	3	no	—
399	C₂²×C₄	—	2	3	no	—
400	C₂²×C₄	—	2	6	no	—
401	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
402	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
403	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
404	SD₁₆	—	4	6	no	—
405	D₈	—	2	4	no	—
406	M₁₆	—	4	6	no	—
407	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
408	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
409	C₄⋊C₄	—	2	3	no	—
410	C₄⋊C₄	—	2	3	no	—
411	C₄⋊C₄	—	2	6	no	—
412	C₄⋊C₄	—	2	6	no	—
413	C₄⋊C₄	—	2	6	no	—
414	C₄⋊C₄	—	2	3	no	—
415	C₄⋊C₄	—	2	3	no	—
416	C₄⋊C₄	—	2	3	no	—
417	C₄⋊C₄	—	2	3	no	—
418	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
419	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
420	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
421	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
422	C₄⋊C₄	C₂⁴×C₃₁	1	6	yes	—
423	C₂²⋊C₄	—	2	6	no	—
424	C₂²⋊C₄	—	2	3	no	—
425	C₂²⋊C₄	—	2	1	no	—
426	C₂²⋊C₄	—	2	3	no	—
427	C₂²⋊C₄	—	2	6	no	—
428	C₂²⋊C₄	—	2	6	no	—
429	C₂²⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
430	C₄²	—	2	3	no	—
431	C₄²	—	2	3	no	—
432	C₄²	—	2	3	no	—
433	C₄²	C₂×D₄×C₃₁	1	3	yes	—
434	C₄²	C₂×D₄×C₃₁	1	3	yes	—
435	C₄²	C₂⁴×C₃₁	1	3	yes	—
436	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
437	C₂³	—	2	1	no	—
438	C₂³	—	2	2	no	—
439	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
440	Q₈	—	2	3	no	—
441	Q₈	—	2	3	no	—
442	Q₈	—	4	6	no	—
443	Q₈	—	4	6	no	—
444	Q₈	—	4	12	no	—
445	Q₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
446	D₄	—	2	6	no	—
447	D₄	—	2	2	no	—
448	D₄	—	2	1	no	—
449	D₄	—	2	2	no	—
450	D₄	—	4	2	no	—
451	D₄	—	4	6	no	—
452	D₄	—	4	12	no	—
453	D₄	—	4	2	no	—
454	D₄	C₃₁×2₋¹⁺⁴	1	2	yes	complemented
455	C₂×C₄	—	2	6	no	—
456	C₂×C₄	—	2	6	no	—
457	C₂×C₄	—	2	3	no	—
458	C₂×C₄	—	2	3	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	3	no	—
461	C₂×C₄	—	2	3	no	—
462	C₂×C₄	—	2	6	no	—
463	C₂×C₄	—	4	6	no	—
464	C₂×C₄	—	4	12	no	—
465	C₂×C₄	—	4	6	no	—
466	C₂×C₄	—	4	12	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
469	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
470	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
471	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
472	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
473	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
474	C₈	—	2	2	no	—
475	C₈	—	2	6	no	—
476	C₂²	—	2	3	no	—
477	C₂²	—	2	2	no	—
478	C₂²	—	2	3	no	—
479	C₂²	—	2	3	no	—
480	C₂²	—	4	1	no	—
481	C₂²	—	4	1	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
486	C₄	—	2	3	no	—
487	C₄	—	2	1	no	—
488	C₄	—	2	3	no	—
489	C₄	—	4	3	no	—
490	C₄	—	4	6	no	—
491	C₄	—	4	3	no	—
492	C₄	—	4	1	no	—
493	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	6	yes	—
494	C₄	C₃₁×#578	1	2	yes	—
495	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
496	C₄	C₃₁×#585	1	1	yes	characteristic
497	C₂	—	4	3	no	—
498	C₂	—	4	2	no	—
499	C₂	—	8	2	no	—
500	C₂	C₃₁×#3489	1	1	yes	characteristic
501	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
502	C₂	C₂×C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3499	C₂	semidirect
2. C₃₁×#3482	C₂	semidirect
3. Q₈×D₈×C₃₁	C₂	semidirect
4. C₃₁×#3359	C₂	semidirect
5. C₃₁×#3353	C₂	semidirect
6. C₃₁×#3330	C₂	semidirect
7. C₃₁×#3254	C₂	semidirect
8. C₃₁×#3230	C₂	semidirect
9. C₃₁×#3106	C₂	semidirect
10. C₃₁×#3079	C₂	semidirect
11. C₃₁×#2991	C₂	semidirect
185. #62148	C₃₁	direct (N × H)
203. D₄×C₃₁	2₋¹⁺⁴	semidirect
225. #3499	C₂×C₃₁	semidirect
226. #3482	C₂×C₃₁	semidirect
227. Q₈×D₈	C₂×C₃₁	semidirect
228. #3359	C₂×C₃₁	semidirect
229. #3353	C₂×C₃₁	semidirect
230. #3330	C₂×C₃₁	semidirect
231. #3254	C₂×C₃₁	semidirect
232. #3230	C₂×C₃₁	semidirect
233. #3106	C₂×C₃₁	semidirect
234. #3079	C₂×C₃₁	semidirect
235. #2991	C₂×C₃₁	semidirect
375. C₃₁	#62148	direct (N × H)
454. D₄	C₃₁×2₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3499	2
C₃₁×#3482	2
Q₈×D₈×C₃₁	2
C₃₁×#3359	2
C₃₁×#3353	2
C₃₁×#3330	2
C₃₁×#3254	2
C₃₁×#3230	2
C₃₁×#3106	2
C₃₁×#3079	2
C₃₁×#2991	2
#62148	31

Subgroups — C31×#62148 #21518907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62148 #21518907