Subgroups — C₃₁×#61365 #21518124

Order 7936.

1766 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₃₁	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3390	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3389	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3388	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3387	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3386	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3385	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3384	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3162	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3076	C₂	1	4	yes	maximal, complemented
12	C₃₁×#3076	C₂	1	2	yes	maximal, complemented
13	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2943	C₂	1	2	yes	maximal
15	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
16	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
17	Q₈²×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#557	C₂²	1	2	yes	—
19	C₃₁×#557	C₂²	1	4	yes	—
20	C₃₁×#554	—	2	4	no	—
21	C₃₁×#554	—	2	2	no	—
22	C₃₁×#554	C₂²	1	2	yes	—
23	C₃₁×#554	C₂²	1	4	yes	—
24	D₄×Q₈×C₃₁	—	2	2	no	—
25	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#533	—	2	2	no	—
27	C₃₁×#533	—	2	4	no	—
28	C₃₁×#533	C₂²	1	2	yes	—
29	C₃₁×#533	C₂²	1	4	yes	—
30	C₄×#41×C₃₁	C₂²	1	2	yes	—
31	C₄×#41×C₃₁	C₂²	1	4	yes	—
32	C₂×C₄×Q₈×C₃₁	—	2	2	no	—
33	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#501	C₂²	1	4	yes	—
35	C₃₁×#501	C₂²	1	4	yes	—
36	C₃₁×#501	C₂²	1	4	yes	—
37	C₃₁×#501	C₂²	1	4	yes	—
38	C₃₁×#500	C₂²	1	1	yes	characteristic
39	C₃₁×#500	C₂²	1	1	yes	characteristic
40	C₃₁×#499	C₂²	1	2	yes	—
41	C₃₁×#499	C₂²	1	2	yes	—
42	C₃₁×#498	C₂²	1	2	yes	—
43	C₃₁×#479	C₂²	1	4	yes	—
44	C₃₁×#479	C₂²	1	4	yes	—
45	C₃₁×#479	C₂²	1	4	yes	—
46	C₃₁×#478	C₂²	1	4	yes	—
47	C₃₁×#478	C₂²	1	4	yes	—
48	C₃₁×#478	C₂²	1	4	yes	—
49	C₃₁×#477	C₂²	1	2	yes	—
50	C₃₁×#477	C₂²	1	4	yes	—
51	C₃₁×#476	C₂²	1	2	yes	—
52	C₃₁×#476	C₂²	1	4	yes	—
53	C₃₁×#475	C₂²	1	2	yes	—
54	C₃₁×#475	C₂²	1	4	yes	—
55	C₃₁×#474	C₂²	1	2	yes	—
56	C₃₁×#474	C₂²	1	4	yes	—
57	C₃₁×#446	C₂²	1	2	yes	—
58	C₃₁×#446	C₂²	1	2	yes	—
59	C₃₁×#446	C₂²	1	2	yes	—
60	C₃₁×#446	C₂²	1	2	yes	—
61	C₃₁×#442	C₂²	1	4	yes	—
62	C₃₁×#441	C₂²	1	4	yes	—
63	C₃₁×#440	C₂²	1	4	yes	—
64	C₃₁×#440	C₂²	1	4	yes	—
65	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
66	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
67	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
68	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
69	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
70	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
71	C₃₁×#427	C₂²	1	2	yes	—
72	C₃₁×#427	C₂²	1	4	yes	—
73	C₃₁×#427	C₂²	1	2	yes	—
74	C₃₁×#427	C₂²	1	1	yes	characteristic
75	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
76	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
77	C₃₁×#416	C₂²	1	2	yes	—
78	C₃₁×#416	C₂²	1	4	yes	—
79	C₂×C₃₁×#97	C₂²	1	2	yes	—
80	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
81	C₂×#41×C₃₁	C₂³	1	2	yes	—
82	C₂²×Q₈×C₃₁	—	2	2	no	—
83	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
84	C₃₁×#135	Q₈	1	8	yes	complemented
85	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
87	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
90	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
91	C₃₁×C₄⋊Q₈	—	2	2	no	—
92	C₃₁×C₄⋊Q₈	—	2	4	no	—
93	C₃₁×C₄⋊Q₈	—	2	2	no	—
94	C₃₁×C₄⋊Q₈	—	2	4	no	—
95	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
96	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
97	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
98	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
99	C₃₁×#125	—	2	4	no	—
100	C₃₁×#125	—	2	4	no	—
101	C₃₁×#125	—	2	4	no	—
102	C₃₁×#125	C₂³	1	4	yes	—
103	C₃₁×#125	C₂³	1	4	yes	—
104	C₃₁×#125	C₂³	1	4	yes	—
105	C₃₁×C₂²⋊Q₈	—	2	2	no	—
106	C₃₁×C₂²⋊Q₈	—	2	4	no	—
107	C₃₁×C₂²⋊Q₈	—	2	2	no	—
108	C₃₁×C₂²⋊Q₈	—	2	4	no	—
109	C₃₁×C₂²⋊Q₈	—	2	2	no	—
110	C₃₁×C₂²⋊Q₈	—	2	4	no	—
111	C₃₁×C₂²⋊Q₈	—	2	2	no	—
112	C₃₁×C₂²⋊Q₈	—	2	4	no	—
113	C₃₁×C₂²⋊Q₈	—	2	4	no	—
114	C₃₁×C₂²⋊Q₈	—	2	4	no	—
115	C₃₁×C₂²⋊Q₈	—	2	4	no	—
116	C₄×Q₈×C₃₁	—	2	2	no	—
117	C₄×Q₈×C₃₁	—	2	4	no	—
118	C₄×Q₈×C₃₁	—	2	2	no	—
119	C₄×Q₈×C₃₁	—	2	2	no	—
120	C₄×Q₈×C₃₁	—	2	4	no	—
121	C₄×Q₈×C₃₁	—	2	4	no	—
122	C₄×Q₈×C₃₁	—	2	2	no	—
123	C₄×Q₈×C₃₁	D₄	1	2	yes	—
124	C₄×Q₈×C₃₁	D₄	1	2	yes	—
125	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
126	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
127	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
128	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
129	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
130	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
131	C₄×D₄×C₃₁	—	2	2	no	—
132	C₄×D₄×C₃₁	—	2	4	no	—
133	C₄×D₄×C₃₁	—	2	2	no	—
134	C₄×D₄×C₃₁	—	2	4	no	—
135	C₄×D₄×C₃₁	C₂³	1	2	yes	—
136	C₄×D₄×C₃₁	C₂³	1	2	yes	—
137	C₄×D₄×C₃₁	C₂³	1	4	yes	—
138	C₄×D₄×C₃₁	C₂³	1	4	yes	—
139	C₃₁×#117	—	2	4	no	—
140	C₃₁×#117	—	2	2	no	—
141	C₃₁×#117	C₂³	1	2	yes	—
142	C₃₁×#117	C₂³	1	4	yes	—
143	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
144	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
146	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
147	C₂×C₄²×C₃₁	—	2	2	no	—
148	C₂×C₄²×C₃₁	—	2	4	no	—
149	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
150	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
151	C₃₁×#107	C₂³	1	1	yes	characteristic
152	C₃₁×#107	C₂³	1	1	yes	characteristic
153	C₃₁×#107	C₂³	1	2	yes	—
154	C₃₁×#107	C₂³	1	2	yes	—
155	C₃₁×#107	C₂³	1	2	yes	—
156	C₃₁×#107	C₂³	1	4	yes	—
157	C₃₁×#107	C₂³	1	4	yes	—
158	C₃₁×#107	C₂³	1	2	yes	—
159	C₃₁×#106	C₂³	1	2	yes	—
160	C₃₁×#106	C₂³	1	4	yes	—
161	C₃₁×#106	C₂³	1	4	yes	—
162	C₃₁×#106	C₂³	1	4	yes	—
163	C₃₁×#106	C₂³	1	4	yes	—
164	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
165	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
166	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
167	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
168	C₃₁×#103	C₂³	1	2	yes	—
169	C₃₁×#103	C₂³	1	2	yes	—
170	C₃₁×#103	C₂³	1	4	yes	—
171	C₃₁×#103	C₂³	1	4	yes	—
172	C₃₁×#102	C₂³	1	2	yes	—
173	C₃₁×#102	C₂³	1	2	yes	—
174	C₃₁×#102	C₂³	1	4	yes	—
175	C₃₁×#102	C₂³	1	4	yes	—
176	C₃₁×#97	C₂³	1	4	yes	—
177	C₃₁×#97	C₂³	1	4	yes	—
178	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
179	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
180	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
181	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
182	#41×C₃₁	—	2	4	no	—
183	#41×C₃₁	C₂×Q₈	1	8	yes	complemented
184	C₂×Q₈×C₃₁	—	2	4	no	—
185	C₂×Q₈×C₃₁	—	2	2	no	—
186	C₂×Q₈×C₃₁	—	2	4	no	—
187	C₂×Q₈×C₃₁	—	2	2	no	—
188	C₂×Q₈×C₃₁	—	4	2	no	—
189	C₂×Q₈×C₃₁	—	4	8	no	—
190	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
191	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
192	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
193	C₂×D₄×C₃₁	—	2	2	no	—
194	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
195	C₂²×C₄×C₃₁	—	2	2	no	—
196	C₂²×C₄×C₃₁	—	2	4	no	—
197	C₂²×C₄×C₃₁	—	2	2	no	—
198	C₂²×C₄×C₃₁	—	2	2	no	—
199	C₂²×C₄×C₃₁	—	2	4	no	—
200	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
201	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
202	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
203	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
204	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
205	Q₁₆×C₃₁	C₂×Q₈	1	4	yes	complemented
206	SD₁₆×C₃₁	C₂×Q₈	1	8	yes	complemented
207	D₈×C₃₁	C₂×Q₈	1	4	yes	complemented
208	C₂×C₈×C₃₁	—	2	2	no	—
209	C₂×C₈×C₃₁	—	2	4	no	—
210	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	complemented
211	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
212	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
213	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
214	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
215	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
216	C₄⋊C₄×C₃₁	—	2	4	no	—
217	C₄⋊C₄×C₃₁	—	2	4	no	—
218	C₄⋊C₄×C₃₁	—	2	4	no	—
219	C₄⋊C₄×C₃₁	—	2	4	no	—
220	C₄⋊C₄×C₃₁	—	2	2	no	—
221	C₄⋊C₄×C₃₁	—	2	4	no	—
222	C₄⋊C₄×C₃₁	—	2	4	no	—
223	C₄⋊C₄×C₃₁	—	2	2	no	—
224	C₄⋊C₄×C₃₁	—	2	2	no	—
225	C₄⋊C₄×C₃₁	—	2	4	no	—
226	C₄⋊C₄×C₃₁	—	2	2	no	—
227	C₄⋊C₄×C₃₁	—	2	4	no	—
228	C₄⋊C₄×C₃₁	—	2	4	no	—
229	C₄⋊C₄×C₃₁	—	2	4	no	—
230	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
231	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
232	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
233	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
234	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
235	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
236	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
237	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
238	C₂²⋊C₄×C₃₁	—	2	2	no	—
239	C₂²⋊C₄×C₃₁	—	2	4	no	—
240	C₂²⋊C₄×C₃₁	—	2	4	no	—
241	C₂²⋊C₄×C₃₁	—	2	4	no	—
242	C₂²⋊C₄×C₃₁	—	2	2	no	—
243	C₂²⋊C₄×C₃₁	—	2	2	no	—
244	C₄²×C₃₁	—	2	2	no	—
245	C₄²×C₃₁	—	2	4	no	—
246	C₄²×C₃₁	—	2	4	no	—
247	C₄²×C₃₁	—	2	2	no	—
248	C₄²×C₃₁	—	2	2	no	—
249	C₄²×C₃₁	—	2	4	no	—
250	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
251	C₄²×C₃₁	C₂×D₄	1	4	yes	—
252	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
253	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
254	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
255	C₄²×C₃₁	C₂⁴	1	4	yes	—
256	#61365	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
257	C₂³×C₃₁	—	2	2	no	—
258	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
259	Q₈×C₃₁	—	2	4	no	—
260	Q₈×C₃₁	—	2	2	no	—
261	Q₈×C₃₁	—	4	2	no	—
262	Q₈×C₃₁	—	4	8	no	—
263	Q₈×C₃₁	—	4	4	no	—
264	Q₈×C₃₁	—	8	4	no	—
265	Q₈×C₃₁	—	8	2	no	—
266	Q₈×C₃₁	C₂²×Q₈	1	4	yes	—
267	D₄×C₃₁	—	2	8	no	—
268	D₄×C₃₁	—	2	2	no	—
269	D₄×C₃₁	C₂²×Q₈	1	4	yes	—
270	C₂×C₄×C₃₁	—	2	2	no	—
271	C₂×C₄×C₃₁	—	2	8	no	—
272	C₂×C₄×C₃₁	—	2	2	no	—
273	C₂×C₄×C₃₁	—	2	4	no	—
274	C₂×C₄×C₃₁	—	2	2	no	—
275	C₂×C₄×C₃₁	—	2	4	no	—
276	C₂×C₄×C₃₁	—	2	2	no	—
277	C₂×C₄×C₃₁	—	4	1	no	—
278	C₂×C₄×C₃₁	—	4	2	no	—
279	C₂×C₄×C₃₁	—	4	2	no	—
280	C₂×C₄×C₃₁	—	4	8	no	—
281	C₂×C₄×C₃₁	—	4	4	no	—
282	C₂×C₄×C₃₁	—	8	2	no	—
283	C₂×C₄×C₃₁	—	8	4	no	—
284	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
285	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
286	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
287	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
288	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
289	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
290	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
291	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
292	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
293	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
294	C₈×C₃₁	—	2	1	no	—
295	C₈×C₃₁	—	2	1	no	—
296	C₈×C₃₁	—	2	4	no	—
297	C₈×C₃₁	C₂²×Q₈	1	2	yes	—
298	C₈×C₃₁	C₂²×Q₈	1	2	yes	—
299	Q₈×#41	C₂×C₃₁	1	2	yes	complemented
300	#3390	C₂×C₃₁	1	2	yes	complemented
301	#3389	C₂×C₃₁	1	2	yes	complemented
302	#3388	C₂×C₃₁	1	4	yes	complemented
303	#3387	C₂×C₃₁	1	2	yes	complemented
304	#3386	C₂×C₃₁	1	2	yes	complemented
305	#3385	C₂×C₃₁	1	2	yes	complemented
306	#3384	C₂×C₃₁	1	2	yes	complemented
307	#3162	C₂×C₃₁	1	2	yes	complemented
308	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
309	#3076	C₂×C₃₁	1	2	yes	complemented
310	#3076	C₂×C₃₁	1	4	yes	complemented
311	C₂×#446	C₂×C₃₁	1	1	yes	characteristic, complemented
312	#2943	C₂×C₃₁	1	2	yes	—
313	C₄×#135	C₂×C₃₁	1	1	yes	characteristic
314	C₂²×C₃₁	—	2	1	no	—
315	C₂²×C₃₁	—	2	1	no	—
316	C₂²×C₃₁	—	2	4	no	—
317	C₂²×C₃₁	—	4	2	no	—
318	C₂²×C₃₁	—	4	2	no	—
319	C₂²×C₃₁	D₄×Q₈	1	2	yes	—
320	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
321	C₄×C₃₁	—	2	1	no	—
322	C₄×C₃₁	—	2	4	no	—
323	C₄×C₃₁	—	2	1	no	—
324	C₄×C₃₁	—	2	4	no	—
325	C₄×C₃₁	—	2	1	no	—
326	C₄×C₃₁	—	2	4	no	—
327	C₄×C₃₁	—	4	2	no	—
328	C₄×C₃₁	—	4	4	no	—
329	C₄×C₃₁	—	4	4	no	—
330	C₄×C₃₁	—	4	2	no	—
331	C₄×C₃₁	D₄×Q₈	1	2	yes	—
332	C₄×C₃₁	#575	1	2	yes	—
333	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
334	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
335	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
336	Q₈²	C₂²×C₃₁	1	2	yes	—
337	#557	C₂²×C₃₁	1	2	yes	—
338	#557	C₂²×C₃₁	1	4	yes	—
339	#554	—	2	4	no	—
340	#554	—	2	2	no	—
341	#554	C₂²×C₃₁	1	2	yes	—
342	#554	C₂²×C₃₁	1	4	yes	—
343	D₄×Q₈	—	2	2	no	—
344	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
345	#533	—	2	2	no	—
346	#533	—	2	4	no	—
347	#533	C₂²×C₃₁	1	2	yes	—
348	#533	C₂²×C₃₁	1	4	yes	—
349	C₄×#41	C₂²×C₃₁	1	2	yes	—
350	C₄×#41	C₂²×C₃₁	1	4	yes	—
351	C₂×C₄×Q₈	—	2	2	no	—
352	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
353	#501	C₂²×C₃₁	1	4	yes	—
354	#501	C₂²×C₃₁	1	4	yes	—
355	#501	C₂²×C₃₁	1	4	yes	—
356	#501	C₂²×C₃₁	1	4	yes	—
357	#500	C₂²×C₃₁	1	1	yes	characteristic
358	#500	C₂²×C₃₁	1	1	yes	characteristic
359	#499	C₂²×C₃₁	1	2	yes	—
360	#499	C₂²×C₃₁	1	2	yes	—
361	#498	C₂²×C₃₁	1	2	yes	—
362	#479	C₂²×C₃₁	1	4	yes	—
363	#479	C₂²×C₃₁	1	4	yes	—
364	#479	C₂²×C₃₁	1	4	yes	—
365	#478	C₂²×C₃₁	1	4	yes	—
366	#478	C₂²×C₃₁	1	4	yes	—
367	#478	C₂²×C₃₁	1	4	yes	—
368	#477	C₂²×C₃₁	1	2	yes	—
369	#477	C₂²×C₃₁	1	4	yes	—
370	#476	C₂²×C₃₁	1	2	yes	—
371	#476	C₂²×C₃₁	1	4	yes	—
372	#475	C₂²×C₃₁	1	2	yes	—
373	#475	C₂²×C₃₁	1	4	yes	—
374	#474	C₂²×C₃₁	1	2	yes	—
375	#474	C₂²×C₃₁	1	4	yes	—
376	#446	C₂²×C₃₁	1	2	yes	—
377	#446	C₂²×C₃₁	1	2	yes	—
378	#446	C₂²×C₃₁	1	2	yes	—
379	#446	C₂²×C₃₁	1	2	yes	—
380	#442	C₂²×C₃₁	1	4	yes	—
381	#441	C₂²×C₃₁	1	4	yes	—
382	#440	C₂²×C₃₁	1	4	yes	—
383	#440	C₂²×C₃₁	1	4	yes	—
384	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
385	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
386	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
387	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
388	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
389	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
390	#427	C₂²×C₃₁	1	2	yes	—
391	#427	C₂²×C₃₁	1	4	yes	—
392	#427	C₂²×C₃₁	1	2	yes	—
393	#427	C₂²×C₃₁	1	1	yes	characteristic
394	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
395	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
396	#416	C₂²×C₃₁	1	2	yes	—
397	#416	C₂²×C₃₁	1	4	yes	—
398	C₂×#97	C₂²×C₃₁	1	2	yes	—
399	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
400	C₂×C₃₁	—	2	2	no	—
401	C₂×C₃₁	—	4	4	no	—
402	C₂×C₃₁	C₂×D₄×Q₈	1	1	yes	characteristic
403	C₂×C₃₁	C₂×#575	1	1	yes	characteristic
404	C₂×C₃₁	#3584	1	1	yes	characteristic
405	C₂×#41	C₂³×C₃₁	1	2	yes	—
406	C₂²×Q₈	—	2	2	no	—
407	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
408	#135	Q₈×C₃₁	1	8	yes	complemented
409	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
410	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
411	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
412	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
413	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
414	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
415	C₄⋊Q₈	—	2	2	no	—
416	C₄⋊Q₈	—	2	4	no	—
417	C₄⋊Q₈	—	2	2	no	—
418	C₄⋊Q₈	—	2	4	no	—
419	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
420	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
421	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
422	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
423	#125	—	2	4	no	—
424	#125	—	2	4	no	—
425	#125	—	2	4	no	—
426	#125	C₂³×C₃₁	1	4	yes	—
427	#125	C₂³×C₃₁	1	4	yes	—
428	#125	C₂³×C₃₁	1	4	yes	—
429	C₂²⋊Q₈	—	2	2	no	—
430	C₂²⋊Q₈	—	2	4	no	—
431	C₂²⋊Q₈	—	2	2	no	—
432	C₂²⋊Q₈	—	2	4	no	—
433	C₂²⋊Q₈	—	2	2	no	—
434	C₂²⋊Q₈	—	2	4	no	—
435	C₂²⋊Q₈	—	2	2	no	—
436	C₂²⋊Q₈	—	2	4	no	—
437	C₂²⋊Q₈	—	2	4	no	—
438	C₂²⋊Q₈	—	2	4	no	—
439	C₂²⋊Q₈	—	2	4	no	—
440	C₄×Q₈	—	2	2	no	—
441	C₄×Q₈	—	2	4	no	—
442	C₄×Q₈	—	2	2	no	—
443	C₄×Q₈	—	2	2	no	—
444	C₄×Q₈	—	2	4	no	—
445	C₄×Q₈	—	2	4	no	—
446	C₄×Q₈	—	2	2	no	—
447	C₄×Q₈	D₄×C₃₁	1	2	yes	—
448	C₄×Q₈	D₄×C₃₁	1	2	yes	—
449	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
450	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
451	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
452	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
453	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
454	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
455	C₄×D₄	—	2	2	no	—
456	C₄×D₄	—	2	4	no	—
457	C₄×D₄	—	2	2	no	—
458	C₄×D₄	—	2	4	no	—
459	C₄×D₄	C₂³×C₃₁	1	2	yes	—
460	C₄×D₄	C₂³×C₃₁	1	4	yes	—
461	C₄×D₄	C₂³×C₃₁	1	2	yes	—
462	C₄×D₄	C₂³×C₃₁	1	4	yes	—
463	#117	—	2	4	no	—
464	#117	—	2	2	no	—
465	#117	C₂³×C₃₁	1	2	yes	—
466	#117	C₂³×C₃₁	1	4	yes	—
467	C₂×C₄⋊C₄	—	2	4	no	—
468	C₂×C₄⋊C₄	—	2	2	no	—
469	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
470	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
471	C₂×C₄²	—	2	2	no	—
472	C₂×C₄²	—	2	4	no	—
473	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
474	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
475	#107	C₂³×C₃₁	1	1	yes	characteristic
476	#107	C₂³×C₃₁	1	1	yes	characteristic
477	#107	C₂³×C₃₁	1	2	yes	—
478	#107	C₂³×C₃₁	1	2	yes	—
479	#107	C₂³×C₃₁	1	2	yes	—
480	#107	C₂³×C₃₁	1	4	yes	—
481	#107	C₂³×C₃₁	1	4	yes	—
482	#107	C₂³×C₃₁	1	2	yes	—
483	#106	C₂³×C₃₁	1	2	yes	—
484	#106	C₂³×C₃₁	1	4	yes	—
485	#106	C₂³×C₃₁	1	4	yes	—
486	#106	C₂³×C₃₁	1	4	yes	—
487	#106	C₂³×C₃₁	1	4	yes	—
488	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
489	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
490	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
491	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
492	#103	C₂³×C₃₁	1	2	yes	—
493	#103	C₂³×C₃₁	1	2	yes	—
494	#103	C₂³×C₃₁	1	4	yes	—
495	#103	C₂³×C₃₁	1	4	yes	—
496	#102	C₂³×C₃₁	1	2	yes	—
497	#102	C₂³×C₃₁	1	2	yes	—
498	#102	C₂³×C₃₁	1	4	yes	—
499	#102	C₂³×C₃₁	1	4	yes	—
500	#97	C₂³×C₃₁	1	4	yes	—
501	#97	C₂³×C₃₁	1	4	yes	—
502	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
503	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
504	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
505	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
506	C₃₁	#61365	1	1	yes	characteristic, direct product factor, Sylow
507	#41	—	2	4	no	—
508	#41	C₂×Q₈×C₃₁	1	8	yes	complemented
509	C₂×Q₈	—	2	4	no	—
510	C₂×Q₈	—	2	2	no	—
511	C₂×Q₈	—	2	4	no	—
512	C₂×Q₈	—	2	2	no	—
513	C₂×Q₈	—	4	2	no	—
514	C₂×Q₈	—	4	8	no	—
515	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
516	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
517	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
518	C₂×D₄	—	2	2	no	—
519	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
520	C₂²×C₄	—	2	2	no	—
521	C₂²×C₄	—	2	4	no	—
522	C₂²×C₄	—	2	2	no	—
523	C₂²×C₄	—	2	2	no	—
524	C₂²×C₄	—	2	4	no	—
525	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
526	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
527	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
528	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
529	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
530	Q₁₆	C₂×Q₈×C₃₁	1	4	yes	complemented
531	SD₁₆	C₂×Q₈×C₃₁	1	8	yes	complemented
532	D₈	C₂×Q₈×C₃₁	1	4	yes	complemented
533	C₂×C₈	—	2	2	no	—
534	C₂×C₈	—	2	4	no	—
535	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	complemented
536	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
537	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
538	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
539	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
540	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
541	C₄⋊C₄	—	2	4	no	—
542	C₄⋊C₄	—	2	4	no	—
543	C₄⋊C₄	—	2	4	no	—
544	C₄⋊C₄	—	2	4	no	—
545	C₄⋊C₄	—	2	2	no	—
546	C₄⋊C₄	—	2	4	no	—
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	2	no	—
549	C₄⋊C₄	—	2	2	no	—
550	C₄⋊C₄	—	2	4	no	—
551	C₄⋊C₄	—	2	2	no	—
552	C₄⋊C₄	—	2	4	no	—
553	C₄⋊C₄	—	2	4	no	—
554	C₄⋊C₄	—	2	4	no	—
555	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
556	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
557	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
558	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
559	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
560	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
561	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
562	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
563	C₂²⋊C₄	—	2	2	no	—
564	C₂²⋊C₄	—	2	4	no	—
565	C₂²⋊C₄	—	2	4	no	—
566	C₂²⋊C₄	—	2	4	no	—
567	C₂²⋊C₄	—	2	2	no	—
568	C₂²⋊C₄	—	2	2	no	—
569	C₄²	—	2	2	no	—
570	C₄²	—	2	4	no	—
571	C₄²	—	2	2	no	—
572	C₄²	—	2	4	no	—
573	C₄²	—	2	4	no	—
574	C₄²	—	2	2	no	—
575	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
576	C₄²	C₂×D₄×C₃₁	1	4	yes	—
577	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
578	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
579	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
580	C₄²	C₂⁴×C₃₁	1	4	yes	—
581	C₂³	—	2	2	no	—
582	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
583	Q₈	—	2	4	no	—
584	Q₈	—	2	2	no	—
585	Q₈	—	4	2	no	—
586	Q₈	—	4	8	no	—
587	Q₈	—	4	4	no	—
588	Q₈	—	8	4	no	—
589	Q₈	—	8	2	no	—
590	Q₈	C₂²×Q₈×C₃₁	1	4	yes	—
591	D₄	—	2	8	no	—
592	D₄	—	2	2	no	—
593	D₄	C₂²×Q₈×C₃₁	1	4	yes	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	8	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	4	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	4	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	2	no	—
603	C₂×C₄	—	4	2	no	—
604	C₂×C₄	—	4	8	no	—
605	C₂×C₄	—	4	4	no	—
606	C₂×C₄	—	8	2	no	—
607	C₂×C₄	—	8	4	no	—
608	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
609	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
610	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
611	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
612	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
613	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
614	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
615	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
616	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
617	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
618	C₈	—	2	1	no	—
619	C₈	—	2	1	no	—
620	C₈	—	2	4	no	—
621	C₈	C₂²×Q₈×C₃₁	1	2	yes	—
622	C₈	C₂²×Q₈×C₃₁	1	2	yes	—
623	C₂²	—	2	1	no	—
624	C₂²	—	2	1	no	—
625	C₂²	—	2	4	no	—
626	C₂²	—	4	2	no	—
627	C₂²	—	4	2	no	—
628	C₂²	D₄×Q₈×C₃₁	1	2	yes	—
629	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
630	C₄	—	2	4	no	—
631	C₄	—	2	1	no	—
632	C₄	—	2	4	no	—
633	C₄	—	2	1	no	—
634	C₄	—	2	4	no	—
635	C₄	—	2	1	no	—
636	C₄	—	4	4	no	—
637	C₄	—	4	2	no	—
638	C₄	—	4	2	no	—
639	C₄	—	4	4	no	—
640	C₄	D₄×Q₈×C₃₁	1	2	yes	—
641	C₄	C₃₁×#575	1	2	yes	—
642	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
643	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
644	C₂	—	2	2	no	—
645	C₂	—	4	4	no	—
646	C₂	C₂×D₄×Q₈×C₃₁	1	1	yes	characteristic
647	C₂	C₂×C₃₁×#575	1	1	yes	characteristic
648	C₂	C₃₁×#3584	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂³×D₄
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×#41×C₃₁	C₂	semidirect
2. C₃₁×#3390	C₂	semidirect
3. C₃₁×#3389	C₂	semidirect
4. C₃₁×#3388	C₂	semidirect
5. C₃₁×#3387	C₂	semidirect
6. C₃₁×#3386	C₂	semidirect
7. C₃₁×#3385	C₂	semidirect
8. C₃₁×#3384	C₂	semidirect
9. C₃₁×#3162	C₂	semidirect
10. C₃₁×#3160	C₂	semidirect
11. C₃₁×#3076	C₂	semidirect
12. C₃₁×#3076	C₂	semidirect
13. C₂×C₃₁×#446	C₂	semidirect
84. C₃₁×#135	Q₈	semidirect
183. #41×C₃₁	C₂×Q₈	semidirect
205. Q₁₆×C₃₁	C₂×Q₈	semidirect
206. SD₁₆×C₃₁	C₂×Q₈	semidirect
207. D₈×C₃₁	C₂×Q₈	semidirect
210. C₂×C₈×C₃₁	C₂×Q₈	semidirect
256. #61365	C₃₁	direct (N × H)
299. Q₈×#41	C₂×C₃₁	semidirect
300. #3390	C₂×C₃₁	semidirect
301. #3389	C₂×C₃₁	semidirect
302. #3388	C₂×C₃₁	semidirect
303. #3387	C₂×C₃₁	semidirect
304. #3386	C₂×C₃₁	semidirect
305. #3385	C₂×C₃₁	semidirect
306. #3384	C₂×C₃₁	semidirect
307. #3162	C₂×C₃₁	semidirect
308. #3160	C₂×C₃₁	semidirect
309. #3076	C₂×C₃₁	semidirect
310. #3076	C₂×C₃₁	semidirect
311. C₂×#446	C₂×C₃₁	semidirect
408. #135	Q₈×C₃₁	semidirect
506. C₃₁	#61365	direct (N × H)
508. #41	C₂×Q₈×C₃₁	semidirect
530. Q₁₆	C₂×Q₈×C₃₁	semidirect
531. SD₁₆	C₂×Q₈×C₃₁	semidirect
532. D₈	C₂×Q₈×C₃₁	semidirect
535. C₂×C₈	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
Q₈×#41×C₃₁	2
C₃₁×#3390	2
C₃₁×#3389	2
C₃₁×#3388	2
C₃₁×#3387	2
C₃₁×#3386	2
C₃₁×#3385	2
C₃₁×#3384	2
C₃₁×#3162	2
C₃₁×#3160	2
C₃₁×#3076	2
C₂×C₃₁×#446	2
C₃₁×#2943	2
C₄×C₃₁×#135	2
#61365	31

Subgroups — C31×#61365 #21518124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61365 #21518124