Subgroups — C₃₁×#17857 #21474616

Order 7936.

714 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2988	C₂	1	2	yes	maximal, complemented
2	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
3	C₃₁×SD₁₆⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₈×Q₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
5	C₈×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1477	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#135	C₄	1	1	yes	characteristic
10	C₄×#41×C₃₁	C₂²	1	2	yes	—
11	C₈×Q₈×C₃₁	C₂²	1	4	yes	—
12	C₄×Q₁₆×C₃₁	C₄	1	1	yes	characteristic
13	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₄×SD₁₆×C₃₁	C₄	1	2	yes	—
15	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
16	C₄×D₈×C₃₁	C₄	1	1	yes	characteristic
17	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₈×D₄×C₃₁	—	2	4	no	—
19	C₈×D₄×C₃₁	C₂²	1	4	yes	—
20	C₃₁×#431	—	2	2	no	—
21	C₃₁×#431	C₂²	1	1	yes	characteristic
22	C₃₁×#431	C₂²	1	1	yes	characteristic
23	C₃₁×#427	C₄	1	1	yes	characteristic
24	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
25	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
26	C₃₁×#416	C₄	1	2	yes	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#335	C₂²	1	2	yes	—
29	C₃₁×#335	C₂²	1	2	yes	—
30	C₃₁×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₃₁×D₄⋊C₈	C₂²	1	4	yes	—
32	C₃₁×#322	C₂²	1	2	yes	—
33	C₈²×C₃₁	C₂²	1	2	yes	—
34	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
35	C₃₁×#135	C₈	1	8	yes	complemented
36	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
38	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	—	2	2	no	—
40	C₂²×C₈×C₃₁	—	2	4	no	—
41	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
44	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
45	C₄×D₄×C₃₁	—	2	2	no	—
46	C₄×D₄×C₃₁	—	2	2	no	—
47	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
48	C₄×D₄×C₃₁	C₂³	1	2	yes	—
49	C₃₁×#117	—	2	2	no	—
50	C₃₁×#117	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₃₁	—	2	2	no	—
52	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
54	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
55	C₃₁×#106	C₂×C₄	1	2	yes	—
56	C₃₁×C₄⋊C₈	—	2	4	no	—
57	C₃₁×C₄⋊C₈	—	2	2	no	—
58	C₃₁×C₄⋊C₈	—	2	1	no	—
59	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
60	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
61	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
62	C₃₁×#103	C₂×C₄	1	2	yes	—
63	C₃₁×#103	C₂×C₄	1	2	yes	—
64	C₃₁×#102	C₂×C₄	1	2	yes	—
65	C₃₁×#102	C₂×C₄	1	2	yes	—
66	C₃₁×C₂²⋊C₈	—	2	1	no	—
67	C₃₁×C₂²⋊C₈	—	2	4	no	—
68	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
69	C₄×C₈×C₃₁	—	2	4	no	—
70	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₃₁	C₂³	1	2	yes	—
73	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₃₁	C₂³	1	2	yes	—
76	#41×C₃₁	—	2	4	no	—
77	#41×C₃₁	C₂×C₈	1	8	yes	complemented
78	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
79	C₂×D₄×C₃₁	—	2	2	no	—
80	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₃₁	—	2	2	no	—
82	C₂²×C₄×C₃₁	—	2	2	no	—
83	C₂²×C₄×C₃₁	—	2	2	no	—
84	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
87	Q₁₆×C₃₁	C₂×C₈	1	4	yes	complemented
88	SD₁₆×C₃₁	C₂×C₈	1	8	yes	complemented
89	D₈×C₃₁	C₂×C₈	1	4	yes	complemented
90	C₂×C₈×C₃₁	—	2	2	no	—
91	C₂×C₈×C₃₁	—	2	4	no	—
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	—	8	4	no	—
94	C₂×C₈×C₃₁	—	8	2	no	—
95	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	complemented
96	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
102	C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
105	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
106	C₂²⋊C₄×C₃₁	—	2	2	no	—
107	C₂²⋊C₄×C₃₁	—	2	2	no	—
108	C₂²⋊C₄×C₃₁	—	2	2	no	—
109	C₄²×C₃₁	—	2	2	no	—
110	C₄²×C₃₁	—	2	2	no	—
111	C₄²×C₃₁	—	2	2	no	—
112	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₃₁	#41	1	1	yes	characteristic
115	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
116	#17857	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₃₁	—	2	2	no	—
118	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
119	Q₈×C₃₁	—	2	2	no	—
120	Q₈×C₃₁	C₂²×C₈	1	4	yes	—
121	D₄×C₃₁	—	2	8	no	—
122	D₄×C₃₁	—	2	2	no	—
123	D₄×C₃₁	C₂²×C₈	1	4	yes	—
124	C₂×C₄×C₃₁	—	2	2	no	—
125	C₂×C₄×C₃₁	—	2	1	no	—
126	C₂×C₄×C₃₁	—	2	1	no	—
127	C₂×C₄×C₃₁	—	2	8	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	4	1	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	8	2	no	—
134	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₃₁	#131	1	2	yes	—
139	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
142	C₈×C₃₁	—	2	1	no	—
143	C₈×C₃₁	—	2	1	no	—
144	C₈×C₃₁	—	2	2	no	—
145	C₈×C₃₁	—	2	2	no	—
146	C₈×C₃₁	—	4	2	no	—
147	C₈×C₃₁	—	4	4	no	—
148	C₈×C₃₁	C₂²×C₈	1	2	yes	—
149	C₈×C₃₁	C₂²×C₈	1	2	yes	—
150	#2988	C₂×C₃₁	1	2	yes	complemented
151	C₄×#135	C₂×C₃₁	1	1	yes	characteristic
152	SD₁₆⋊C₈	C₂×C₃₁	1	4	yes	complemented
153	C₈×Q₁₆	C₂×C₃₁	1	2	yes	complemented
154	C₈×D₈	C₂×C₃₁	1	2	yes	complemented
155	C₂×#335	C₂×C₃₁	1	1	yes	characteristic, complemented
156	#1477	C₂×C₃₁	1	2	yes	complemented
157	#1450	C₂×C₃₁	1	1	yes	characteristic, complemented
158	C₂²×C₃₁	—	2	1	no	—
159	C₂²×C₃₁	—	2	1	no	—
160	C₂²×C₃₁	—	2	4	no	—
161	C₂²×C₃₁	—	4	2	no	—
162	C₂²×C₃₁	—	4	2	no	—
163	C₂²×C₃₁	C₈×D₄	1	2	yes	—
164	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
165	C₄×C₃₁	—	2	4	no	—
166	C₄×C₃₁	—	2	1	no	—
167	C₄×C₃₁	—	2	1	no	—
168	C₄×C₃₁	—	2	1	no	—
169	C₄×C₃₁	—	4	2	no	—
170	C₄×C₃₁	—	4	2	no	—
171	C₄×C₃₁	C₈×D₄	1	2	yes	—
172	C₄×C₃₁	C₂³×C₈	1	1	yes	characteristic
173	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
174	C₄×C₃₁	#576	1	1	yes	characteristic
175	C₄×C₃₁	#578	1	1	yes	characteristic
176	C₂×#135	C₄×C₃₁	1	1	yes	characteristic
177	C₄×#41	C₂²×C₃₁	1	2	yes	—
178	C₈×Q₈	C₂²×C₃₁	1	4	yes	—
179	C₄×Q₁₆	C₄×C₃₁	1	1	yes	characteristic
180	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
181	C₄×SD₁₆	C₄×C₃₁	1	2	yes	—
182	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
183	C₄×D₈	C₄×C₃₁	1	1	yes	characteristic
184	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
185	C₈×D₄	—	2	4	no	—
186	C₈×D₄	C₂²×C₃₁	1	4	yes	—
187	#431	—	2	2	no	—
188	#431	C₂²×C₃₁	1	1	yes	characteristic
189	#431	C₂²×C₃₁	1	1	yes	characteristic
190	#427	C₄×C₃₁	1	1	yes	characteristic
191	C₂×C₄⋊C₈	—	2	2	no	—
192	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
193	#416	C₄×C₃₁	1	2	yes	—
194	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
195	#335	C₂²×C₃₁	1	2	yes	—
196	#335	C₂²×C₃₁	1	2	yes	—
197	Q₈⋊C₈	C₂²×C₃₁	1	4	yes	—
198	D₄⋊C₈	C₂²×C₃₁	1	4	yes	—
199	#322	C₂²×C₃₁	1	2	yes	—
200	C₈²	C₂²×C₃₁	1	2	yes	—
201	C₂×C₃₁	—	2	2	no	—
202	C₂×C₃₁	—	4	4	no	—
203	C₂×C₃₁	C₂×C₈×D₄	1	1	yes	characteristic
204	C₂×C₃₁	#2951	1	1	yes	characteristic
205	C₂×C₃₁	#2957	1	1	yes	characteristic
206	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
207	#135	C₈×C₃₁	1	8	yes	complemented
208	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
209	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
210	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
211	C₂²×C₈	—	2	2	no	—
212	C₂²×C₈	—	2	4	no	—
213	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
214	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
215	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
216	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	—	2	2	no	—
219	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
220	C₄×D₄	C₂³×C₃₁	1	2	yes	—
221	#117	—	2	2	no	—
222	#117	C₂×C₄×C₃₁	1	2	yes	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
225	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
226	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
227	#106	C₂×C₄×C₃₁	1	2	yes	—
228	C₄⋊C₈	—	2	4	no	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
232	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
233	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
234	#103	C₂×C₄×C₃₁	1	2	yes	—
235	#103	C₂×C₄×C₃₁	1	2	yes	—
236	#102	C₂×C₄×C₃₁	1	2	yes	—
237	#102	C₂×C₄×C₃₁	1	2	yes	—
238	C₂²⋊C₈	—	2	1	no	—
239	C₂²⋊C₈	—	2	4	no	—
240	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
241	C₄×C₈	—	2	4	no	—
242	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
243	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
244	C₄×C₈	C₂³×C₃₁	1	2	yes	—
245	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₃₁	1	2	yes	—
248	C₃₁	#17857	1	1	yes	characteristic, direct product factor, Sylow
249	#41	—	2	4	no	—
250	#41	C₂×C₈×C₃₁	1	8	yes	complemented
251	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
259	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
260	Q₁₆	C₂×C₈×C₃₁	1	4	yes	complemented
261	SD₁₆	C₂×C₈×C₃₁	1	8	yes	complemented
262	D₈	C₂×C₈×C₃₁	1	4	yes	complemented
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	8	4	no	—
267	C₂×C₈	—	8	2	no	—
268	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	complemented
269	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
270	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
271	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
272	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
273	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
274	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
278	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
287	C₄²	#41×C₃₁	1	1	yes	characteristic
288	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
289	C₂³	—	2	2	no	—
290	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
291	Q₈	—	2	2	no	—
292	Q₈	C₂²×C₈×C₃₁	1	4	yes	—
293	D₄	—	2	8	no	—
294	D₄	—	2	2	no	—
295	D₄	C₂²×C₈×C₃₁	1	4	yes	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
307	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
308	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
310	C₂×C₄	C₃₁×#131	1	2	yes	—
311	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
313	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	2	2	no	—
317	C₈	—	2	2	no	—
318	C₈	—	4	2	no	—
319	C₈	—	4	4	no	—
320	C₈	C₂²×C₈×C₃₁	1	2	yes	—
321	C₈	C₂²×C₈×C₃₁	1	2	yes	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	4	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	C₈×D₄×C₃₁	1	2	yes	—
328	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
329	C₄	—	2	4	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	4	2	no	—
334	C₄	—	4	2	no	—
335	C₄	C₈×D₄×C₃₁	1	2	yes	—
336	C₄	C₂³×C₈×C₃₁	1	1	yes	characteristic
337	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
338	C₄	C₃₁×#576	1	1	yes	characteristic
339	C₄	C₃₁×#578	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	4	4	no	—
342	C₂	C₂×C₈×D₄×C₃₁	1	1	yes	characteristic
343	C₂	C₃₁×#2951	1	1	yes	characteristic
344	C₂	C₃₁×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₄	C₂³×C₈×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂×C₃₁×#135	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₃₁×#431 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2988	C₂	semidirect
3. C₃₁×SD₁₆⋊C₈	C₂	semidirect
4. C₈×Q₁₆×C₃₁	C₂	semidirect
5. C₈×D₈×C₃₁	C₂	semidirect
6. C₂×C₃₁×#335	C₂	semidirect
7. C₃₁×#1477	C₂	semidirect
8. C₃₁×#1450	C₂	semidirect
35. C₃₁×#135	C₈	semidirect
77. #41×C₃₁	C₂×C₈	semidirect
87. Q₁₆×C₃₁	C₂×C₈	semidirect
88. SD₁₆×C₃₁	C₂×C₈	semidirect
89. D₈×C₃₁	C₂×C₈	semidirect
95. C₂×C₈×C₃₁	C₂×C₈	semidirect
116. #17857	C₃₁	direct (N × H)
150. #2988	C₂×C₃₁	semidirect
152. SD₁₆⋊C₈	C₂×C₃₁	semidirect
153. C₈×Q₁₆	C₂×C₃₁	semidirect
154. C₈×D₈	C₂×C₃₁	semidirect
155. C₂×#335	C₂×C₃₁	semidirect
156. #1477	C₂×C₃₁	semidirect
157. #1450	C₂×C₃₁	semidirect
207. #135	C₈×C₃₁	semidirect
248. C₃₁	#17857	direct (N × H)
250. #41	C₂×C₈×C₃₁	semidirect
260. Q₁₆	C₂×C₈×C₃₁	semidirect
261. SD₁₆	C₂×C₈×C₃₁	semidirect
262. D₈	C₂×C₈×C₃₁	semidirect
268. C₂×C₈	C₂×C₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2988	2
C₄×C₃₁×#135	2
C₃₁×SD₁₆⋊C₈	2
C₈×Q₁₆×C₃₁	2
C₈×D₈×C₃₁	2
C₂×C₃₁×#335	2
C₃₁×#1477	2
C₃₁×#1450	2
#17857	31

Subgroups — C31×#17857 #21474616

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17857 #21474616