Subgroups — C₃₁×#19396 #21476155

Order 7936.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3208	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3078	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#460	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2982	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1678	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1666	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1570	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1542	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₃₁	—	2	1	no	—
11	C₃₁×#533	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#492	C₂²	1	2	yes	—
15	C₃₁×#486	C₂²	1	2	yes	—
16	C₃₁×#479	—	2	2	no	—
17	C₃₁×#477	C₂²	1	2	yes	complemented
18	C₃₁×#475	C₂²	1	2	yes	complemented
19	C₃₁×#460	—	2	2	no	—
20	C₃₁×#460	—	2	2	no	—
21	C₃₁×#460	C₂²	1	4	yes	—
22	C₃₁×#450	—	2	2	no	—
23	C₈×Q₈×C₃₁	—	2	2	no	—
24	C₈×Q₈×C₃₁	C₂²	1	4	yes	complemented
25	C₃₁×#431	—	2	2	no	—
26	C₃₁×#431	C₂²	1	1	yes	characteristic
27	C₃₁×#431	C₂²	1	1	yes	characteristic
28	C₃₁×#428	—	2	1	no	—
29	C₃₁×#423	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₃₁×#419	—	2	1	no	—
32	C₂×C₃₁×#102	—	2	1	no	—
33	C₃₁×#334	C₂²	1	4	yes	complemented
34	C₃₁×#333	C₂²	1	2	yes	complemented
35	C₃₁×#332	C₂²	1	2	yes	complemented
36	C₃₁×#331	C₂²	1	4	yes	—
37	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×D₄×C₃₁	—	2	1	no	—
39	C₂²×D₄×C₃₁	—	2	1	no	—
40	C₂²×D₄×C₃₁	—	2	1	no	—
41	C₂×SD₁₆×C₃₁	—	2	4	no	—
42	C₂×SD₁₆×C₃₁	—	2	4	no	—
43	C₂×SD₁₆×C₃₁	—	4	4	no	—
44	C₂×M₁₆×C₃₁	—	2	1	no	—
45	C₂²×C₈×C₃₁	—	2	1	no	—
46	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
47	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
48	C₃₁×#127	—	2	1	no	—
49	C₃₁×#127	—	2	1	no	—
50	C₃₁×#127	—	2	1	no	—
51	C₃₁×#127	C₂³	1	2	yes	—
52	C₃₁×#125	—	2	1	no	—
53	C₃₁×C₂²⋊Q₈	—	2	1	no	—
54	C₃₁×C₂²⋊Q₈	—	2	1	no	—
55	C₄×Q₈×C₃₁	—	2	1	no	—
56	C₄×Q₈×C₃₁	—	2	1	no	—
57	C₄×Q₈×C₃₁	—	2	1	no	—
58	C₄×Q₈×C₃₁	—	2	1	no	—
59	C₄×Q₈×C₃₁	D₄	1	2	yes	—
60	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
61	C₃₁×#117	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₃₁	—	2	1	no	—
65	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
66	C₃₁×#107	—	2	2	no	—
67	C₃₁×#106	—	2	2	no	—
68	C₃₁×C₄⋊C₈	—	2	4	no	—
69	C₃₁×C₄⋊C₈	—	2	1	no	—
70	C₃₁×C₄⋊C₈	—	2	1	no	—
71	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
72	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
73	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
74	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
75	C₃₁×#103	—	2	2	no	—
76	C₃₁×#103	—	2	2	no	—
77	C₃₁×#102	—	2	4	no	—
78	C₃₁×#102	—	2	4	no	—
79	C₃₁×C₂²⋊C₈	—	2	1	no	—
80	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
81	C₄×C₈×C₃₁	—	2	4	no	—
82	C₄×C₈×C₃₁	—	2	1	no	—
83	C₄×C₈×C₃₁	C₂³	1	2	yes	—
84	C₂⁴×C₃₁	—	4	1	no	—
85	C₂×Q₈×C₃₁	—	2	1	no	—
86	C₂×Q₈×C₃₁	—	2	2	no	—
87	C₂×Q₈×C₃₁	—	2	1	no	—
88	C₂×Q₈×C₃₁	—	2	1	no	—
89	C₂×Q₈×C₃₁	SD₁₆	1	4	yes	complemented
90	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	4	1	no	—
95	C₂×D₄×C₃₁	—	4	1	no	—
96	C₂×D₄×C₃₁	—	4	1	no	—
97	C₂×D₄×C₃₁	—	4	2	no	—
98	C₂×D₄×C₃₁	—	4	2	no	—
99	C₂×D₄×C₃₁	—	8	2	no	—
100	C₂×D₄×C₃₁	—	8	2	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
106	SD₁₆×C₃₁	—	4	8	no	—
107	SD₁₆×C₃₁	—	8	4	no	—
108	M₁₆×C₃₁	—	4	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	4	2	no	—
113	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	1	no	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	2	1	no	—
118	C₄⋊C₄×C₃₁	—	2	1	no	—
119	C₄⋊C₄×C₃₁	—	2	1	no	—
120	C₄⋊C₄×C₃₁	—	2	1	no	—
121	C₄⋊C₄×C₃₁	—	4	1	no	—
122	C₄⋊C₄×C₃₁	—	4	1	no	—
123	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
124	C₂²⋊C₄×C₃₁	—	4	1	no	—
125	C₄²×C₃₁	—	2	2	no	—
126	C₄²×C₃₁	—	2	2	no	—
127	C₄²×C₃₁	—	4	1	no	—
128	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₃₁	#41	1	1	yes	characteristic
131	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
132	#19396	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂³×C₃₁	—	4	2	no	—
134	C₂³×C₃₁	—	8	2	no	—
135	C₂³×C₃₁	—	8	2	no	—
136	C₂³×C₃₁	—	8	2	no	—
137	C₂³×C₃₁	#121	1	1	yes	characteristic
138	Q₈×C₃₁	—	2	2	no	—
139	Q₈×C₃₁	—	2	4	no	—
140	Q₈×C₃₁	—	4	2	no	—
141	Q₈×C₃₁	—	8	1	no	—
142	Q₈×C₃₁	C₂×SD₁₆	1	4	yes	—
143	D₄×C₃₁	—	4	2	no	—
144	D₄×C₃₁	—	4	2	no	—
145	D₄×C₃₁	—	4	4	no	—
146	D₄×C₃₁	—	8	2	no	—
147	D₄×C₃₁	—	8	2	no	—
148	D₄×C₃₁	—	8	2	no	—
149	D₄×C₃₁	—	8	2	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	2	4	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	4	1	no	—
155	C₂×C₄×C₃₁	—	4	1	no	—
156	C₂×C₄×C₃₁	—	4	1	no	—
157	C₂×C₄×C₃₁	—	8	1	no	—
158	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
159	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
160	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
161	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
162	C₂×C₄×C₃₁	#135	1	2	yes	—
163	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
166	C₈×C₃₁	—	2	4	no	—
167	C₈×C₃₁	—	4	2	no	—
168	C₈×C₃₁	—	4	2	no	—
169	C₈×C₃₁	—	4	2	no	—
170	#3208	C₂×C₃₁	1	1	yes	characteristic
171	#3078	C₂×C₃₁	1	1	yes	characteristic, complemented
172	C₂×#460	C₂×C₃₁	1	1	yes	characteristic
173	#2982	C₂×C₃₁	1	1	yes	characteristic, complemented
174	#1678	C₂×C₃₁	1	4	yes	complemented
175	#1666	C₂×C₃₁	1	4	yes	complemented
176	#1570	C₂×C₃₁	1	1	yes	characteristic, complemented
177	#1545	C₂×C₃₁	1	1	yes	characteristic
178	#1542	C₂×C₃₁	1	1	yes	characteristic, complemented
179	C₂²×C₃₁	—	2	1	no	—
180	C₂²×C₃₁	—	2	1	no	—
181	C₂²×C₃₁	—	8	2	no	—
182	C₂²×C₃₁	—	8	2	no	—
183	C₂²×C₃₁	—	8	2	no	—
184	C₂²×C₃₁	—	16	4	no	—
185	C₂²×C₃₁	#465	1	2	yes	—
186	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
187	C₄×C₃₁	—	2	1	no	—
188	C₄×C₃₁	—	2	4	no	—
189	C₄×C₃₁	—	2	1	no	—
190	C₄×C₃₁	—	2	1	no	—
191	C₄×C₃₁	—	4	2	no	—
192	C₄×C₃₁	—	8	1	no	—
193	C₄×C₃₁	—	8	1	no	—
194	C₄×C₃₁	#465	1	2	yes	—
195	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
196	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
197	C₄×C₃₁	#577	1	2	yes	—
198	C₂²×SD₁₆	—	2	1	no	—
199	#533	C₂²×C₃₁	1	1	yes	characteristic
200	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
201	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
202	#492	C₂²×C₃₁	1	2	yes	—
203	#486	C₂²×C₃₁	1	2	yes	—
204	#479	—	2	2	no	—
205	#477	C₂²×C₃₁	1	2	yes	complemented
206	#475	C₂²×C₃₁	1	2	yes	complemented
207	#460	—	2	2	no	—
208	#460	—	2	2	no	—
209	#460	C₂²×C₃₁	1	4	yes	—
210	#450	—	2	2	no	—
211	C₈×Q₈	—	2	2	no	—
212	C₈×Q₈	C₂²×C₃₁	1	4	yes	complemented
213	#431	—	2	2	no	—
214	#431	C₂²×C₃₁	1	1	yes	characteristic
215	#431	C₂²×C₃₁	1	1	yes	characteristic
216	#428	—	2	1	no	—
217	#423	C₂²×C₃₁	1	1	yes	characteristic
218	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
219	#419	—	2	1	no	—
220	C₂×#102	—	2	1	no	—
221	#334	C₂²×C₃₁	1	4	yes	complemented
222	#333	C₂²×C₃₁	1	2	yes	complemented
223	#332	C₂²×C₃₁	1	2	yes	complemented
224	#331	C₂²×C₃₁	1	4	yes	—
225	C₂×C₃₁	—	2	2	no	—
226	C₂×C₃₁	—	16	2	no	—
227	C₂×C₃₁	#3043	1	1	yes	characteristic
228	C₂×C₃₁	C₂×#465	1	1	yes	characteristic
229	C₂×C₃₁	#3068	1	1	yes	characteristic
230	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
231	C₂²×D₄	—	2	1	no	—
232	C₂²×D₄	—	2	1	no	—
233	C₂²×D₄	—	2	1	no	—
234	C₂×SD₁₆	—	2	4	no	—
235	C₂×SD₁₆	—	2	4	no	—
236	C₂×SD₁₆	—	4	4	no	—
237	C₂×M₁₆	—	2	1	no	—
238	C₂²×C₈	—	2	1	no	—
239	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
240	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
241	#127	—	2	1	no	—
242	#127	—	2	1	no	—
243	#127	—	2	1	no	—
244	#127	C₂³×C₃₁	1	2	yes	—
245	#125	—	2	1	no	—
246	C₂²⋊Q₈	—	2	1	no	—
247	C₂²⋊Q₈	—	2	1	no	—
248	C₄×Q₈	—	2	1	no	—
249	C₄×Q₈	—	2	1	no	—
250	C₄×Q₈	—	2	1	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	D₄×C₃₁	1	2	yes	—
253	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
254	#117	—	2	1	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄²	—	2	1	no	—
258	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
259	#107	—	2	2	no	—
260	#106	—	2	2	no	—
261	C₄⋊C₈	—	2	4	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	2	1	no	—
264	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
265	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
266	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
267	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
268	#103	—	2	2	no	—
269	#103	—	2	2	no	—
270	#102	—	2	4	no	—
271	#102	—	2	4	no	—
272	C₂²⋊C₈	—	2	1	no	—
273	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
274	C₄×C₈	—	2	4	no	—
275	C₄×C₈	—	2	1	no	—
276	C₄×C₈	C₂³×C₃₁	1	2	yes	—
277	C₃₁	#19396	1	1	yes	characteristic, direct product factor, Sylow
278	C₂⁴	—	4	1	no	—
279	C₂×Q₈	—	2	1	no	—
280	C₂×Q₈	—	2	2	no	—
281	C₂×Q₈	—	2	1	no	—
282	C₂×Q₈	—	2	1	no	—
283	C₂×Q₈	SD₁₆×C₃₁	1	4	yes	complemented
284	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	2	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂×D₄	—	4	1	no	—
290	C₂×D₄	—	4	1	no	—
291	C₂×D₄	—	4	2	no	—
292	C₂×D₄	—	4	2	no	—
293	C₂×D₄	—	8	2	no	—
294	C₂×D₄	—	8	2	no	—
295	C₂²×C₄	—	2	1	no	—
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
298	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
299	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
300	SD₁₆	—	4	8	no	—
301	SD₁₆	—	8	4	no	—
302	M₁₆	—	4	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	4	2	no	—
307	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	4	1	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
318	C₂²⋊C₄	—	4	1	no	—
319	C₄²	—	2	2	no	—
320	C₄²	—	2	2	no	—
321	C₄²	—	4	1	no	—
322	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
324	C₄²	#41×C₃₁	1	1	yes	characteristic
325	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
326	C₂³	—	4	2	no	—
327	C₂³	—	8	2	no	—
328	C₂³	—	8	2	no	—
329	C₂³	—	8	2	no	—
330	C₂³	C₃₁×#121	1	1	yes	characteristic
331	Q₈	—	2	4	no	—
332	Q₈	—	2	2	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	8	1	no	—
335	Q₈	C₂×SD₁₆×C₃₁	1	4	yes	—
336	D₄	—	4	2	no	—
337	D₄	—	4	2	no	—
338	D₄	—	4	4	no	—
339	D₄	—	8	2	no	—
340	D₄	—	8	2	no	—
341	D₄	—	8	2	no	—
342	D₄	—	8	2	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	8	1	no	—
351	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
352	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
353	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
354	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
355	C₂×C₄	C₃₁×#135	1	2	yes	—
356	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
357	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
358	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
359	C₈	—	2	4	no	—
360	C₈	—	4	2	no	—
361	C₈	—	4	2	no	—
362	C₈	—	4	2	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	2	no	—
368	C₂²	—	16	4	no	—
369	C₂²	C₃₁×#465	1	2	yes	—
370	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	4	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	8	1	no	—
377	C₄	—	8	1	no	—
378	C₄	C₃₁×#465	1	2	yes	—
379	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
380	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
381	C₄	C₃₁×#577	1	2	yes	—
382	C₂	—	2	2	no	—
383	C₂	—	16	2	no	—
384	C₂	C₃₁×#3043	1	1	yes	characteristic
385	C₂	C₂×C₃₁×#465	1	1	yes	characteristic
386	C₂	C₃₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3078	C₂	semidirect
4. C₃₁×#2982	C₂	semidirect
5. C₃₁×#1678	C₂	semidirect
6. C₃₁×#1666	C₂	semidirect
7. C₃₁×#1570	C₂	semidirect
9. C₃₁×#1542	C₂	semidirect
17. C₃₁×#477	C₂²	semidirect
18. C₃₁×#475	C₂²	semidirect
24. C₈×Q₈×C₃₁	C₂²	semidirect
33. C₃₁×#334	C₂²	semidirect
34. C₃₁×#333	C₂²	semidirect
35. C₃₁×#332	C₂²	semidirect
89. C₂×Q₈×C₃₁	SD₁₆	semidirect
132. #19396	C₃₁	direct (N × H)
171. #3078	C₂×C₃₁	semidirect
173. #2982	C₂×C₃₁	semidirect
174. #1678	C₂×C₃₁	semidirect
175. #1666	C₂×C₃₁	semidirect
176. #1570	C₂×C₃₁	semidirect
178. #1542	C₂×C₃₁	semidirect
205. #477	C₂²×C₃₁	semidirect
206. #475	C₂²×C₃₁	semidirect
212. C₈×Q₈	C₂²×C₃₁	semidirect
221. #334	C₂²×C₃₁	semidirect
222. #333	C₂²×C₃₁	semidirect
223. #332	C₂²×C₃₁	semidirect
277. C₃₁	#19396	direct (N × H)
283. C₂×Q₈	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3208	2
C₃₁×#3078	2
C₂×C₃₁×#460	2
C₃₁×#2982	2
C₃₁×#1678	2
C₃₁×#1666	2
C₃₁×#1570	2
C₃₁×#1545	2
C₃₁×#1542	2
#19396	31

Subgroups — C31×#19396 #21476155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19396 #21476155