Subgroups — C₃₁×#19809 #21476568

Order 7936.

632 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3362	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3278	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1714	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1695	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1678	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1672	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1666	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1645	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1637	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1631	C₂	1	1	yes	maximal, characteristic
13	C₃₁×#1623	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1583	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1579	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#137	—	2	1	no	—
17	Q₈²×C₃₁	C₂²	1	1	yes	characteristic
18	Q₈²×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#550	C₂²	1	1	yes	characteristic
20	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#500	—	2	1	no	—
22	C₃₁×#500	C₂²	1	1	yes	characteristic
23	C₃₁×#498	—	2	1	no	—
24	C₃₁×#498	C₂²	1	1	yes	characteristic
25	C₃₁×#492	C₂²	1	1	yes	characteristic
26	C₃₁×#492	C₂²	1	1	yes	characteristic
27	C₃₁×#487	C₂²	1	1	yes	characteristic
28	C₃₁×#486	—	2	1	no	—
29	C₃₁×#486	C₂²	1	1	yes	characteristic
30	C₃₁×#477	—	2	1	no	—
31	C₃₁×#477	—	2	1	no	—
32	C₃₁×#477	C₂²	1	1	yes	characteristic
33	C₃₁×#477	C₂²	1	1	yes	characteristic
34	C₃₁×#476	—	2	1	no	—
35	C₃₁×#476	—	2	1	no	—
36	C₃₁×#476	C₂²	1	1	yes	characteristic
37	C₃₁×#476	C₂²	1	1	yes	characteristic
38	C₃₁×#475	—	2	1	no	—
39	C₃₁×#475	C₂²	1	1	yes	characteristic
40	C₃₁×#468	—	2	1	no	—
41	C₃₁×#462	C₂²	1	1	yes	characteristic
42	C₃₁×#460	—	2	1	no	—
43	C₃₁×#460	C₂²	1	1	yes	characteristic
44	C₃₁×#460	C₂²	1	1	yes	characteristic
45	C₃₁×#452	—	2	1	no	—
46	C₃₁×#448	—	2	1	no	—
47	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
48	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
49	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
50	C₄×Q₁₆×C₃₁	—	2	1	no	—
51	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
52	C₄×SD₁₆×C₃₁	—	2	1	no	—
53	C₄×SD₁₆×C₃₁	—	2	1	no	—
54	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
55	C₃₁×#436	C₂²	1	1	yes	characteristic
56	C₃₁×#334	C₂²	1	1	yes	characteristic
57	C₃₁×#334	C₂²	1	1	yes	characteristic
58	C₃₁×#333	C₂²	1	1	yes	characteristic
59	C₃₁×#333	C₂²	1	1	yes	characteristic
60	C₃₁×#332	C₂²	1	1	yes	characteristic
61	C₃₁×#331	C₂²	1	1	yes	characteristic
62	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₃₁×#322	C₂²	1	1	yes	characteristic
67	C₂×#41×C₃₁	—	2	1	no	—
68	C₂²×Q₈×C₃₁	—	2	1	no	—
69	C₃₁×#137	—	4	4	no	—
70	C₂×Q₁₆×C₃₁	—	2	1	no	—
71	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
72	C₂×SD₁₆×C₃₁	—	2	1	no	—
73	C₂×SD₁₆×C₃₁	—	2	1	no	—
74	C₂×SD₁₆×C₃₁	—	2	1	no	—
75	C₂×SD₁₆×C₃₁	—	2	1	no	—
76	C₂×SD₁₆×C₃₁	—	4	1	no	—
77	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
78	C₂×M₁₆×C₃₁	—	2	1	no	—
79	C₃₁×C₄⋊Q₈	—	2	1	no	—
80	C₃₁×C₄⋊Q₈	—	2	1	no	—
81	C₃₁×C₄⋊Q₈	—	2	1	no	—
82	C₃₁×C₄⋊Q₈	—	2	1	no	—
83	C₃₁×C₄⋊Q₈	—	2	1	no	—
84	C₃₁×C₄⋊Q₈	—	4	1	no	—
85	C₃₁×C₄⋊Q₈	—	4	1	no	—
86	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₃₁×#127	—	2	1	no	—
90	C₃₁×#127	C₂³	1	1	yes	characteristic
91	C₃₁×C₂²⋊Q₈	—	2	1	no	—
92	C₃₁×C₂²⋊Q₈	—	4	1	no	—
93	C₃₁×#121	—	2	1	no	—
94	C₃₁×#121	—	4	1	no	—
95	C₄×Q₈×C₃₁	—	2	1	no	—
96	C₄×Q₈×C₃₁	—	2	1	no	—
97	C₄×Q₈×C₃₁	—	2	1	no	—
98	C₄×Q₈×C₃₁	—	2	1	no	—
99	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
102	C₄×D₄×C₃₁	—	2	1	no	—
103	C₄×D₄×C₃₁	—	2	1	no	—
104	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
105	C₃₁×#107	—	2	1	no	—
106	C₃₁×#107	—	2	1	no	—
107	C₃₁×#107	—	2	1	no	—
108	C₃₁×#107	—	2	1	no	—
109	C₃₁×#107	D₄	1	1	yes	characteristic
110	C₃₁×#106	—	2	1	no	—
111	C₃₁×#106	—	2	1	no	—
112	C₃₁×#106	—	2	1	no	—
113	C₃₁×#106	—	2	1	no	—
114	C₃₁×#106	—	2	1	no	—
115	C₃₁×#106	D₄	1	1	yes	characteristic
116	C₃₁×C₄⋊C₈	—	2	1	no	—
117	C₃₁×C₄⋊C₈	—	2	1	no	—
118	C₃₁×C₄⋊C₈	—	2	1	no	—
119	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
120	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
121	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
122	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
123	C₃₁×#103	—	2	1	no	—
124	C₃₁×#103	—	2	1	no	—
125	C₃₁×#103	—	2	1	no	—
126	C₃₁×#103	—	2	1	no	—
127	C₃₁×#103	—	4	1	no	—
128	C₃₁×#103	D₄	1	1	yes	characteristic
129	C₃₁×#103	D₄	1	1	yes	characteristic
130	C₃₁×#103	D₄	1	1	yes	characteristic
131	C₃₁×#102	—	2	1	no	—
132	C₃₁×#102	—	2	1	no	—
133	C₃₁×#102	—	2	1	no	—
134	C₃₁×#102	—	2	1	no	—
135	C₃₁×#102	—	2	1	no	—
136	C₃₁×#102	—	2	1	no	—
137	C₃₁×#102	D₄	1	1	yes	characteristic
138	C₃₁×C₂²⋊C₈	—	2	1	no	—
139	C₄×C₈×C₃₁	—	2	1	no	—
140	C₄×C₈×C₃₁	—	2	1	no	—
141	C₄×C₈×C₃₁	—	2	1	no	—
142	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
145	#41×C₃₁	—	4	2	no	—
146	#41×C₃₁	—	8	1	no	—
147	C₂×Q₈×C₃₁	—	2	1	no	—
148	C₂×Q₈×C₃₁	—	2	1	no	—
149	C₂×Q₈×C₃₁	—	2	1	no	—
150	C₂×Q₈×C₃₁	—	4	1	no	—
151	C₂×Q₈×C₃₁	—	4	1	no	—
152	C₂×Q₈×C₃₁	—	4	1	no	—
153	C₂×Q₈×C₃₁	—	4	2	no	—
154	C₂×Q₈×C₃₁	—	8	1	no	—
155	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂×D₄×C₃₁	—	2	1	no	—
159	C₂×D₄×C₃₁	—	2	1	no	—
160	C₂×D₄×C₃₁	—	2	1	no	—
161	C₂×D₄×C₃₁	—	4	1	no	—
162	C₂×D₄×C₃₁	—	4	1	no	—
163	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	—	2	1	no	—
165	C₂²×C₄×C₃₁	—	4	1	no	—
166	C₂²×C₄×C₃₁	—	4	1	no	—
167	Q₁₆×C₃₁	—	4	2	no	—
168	Q₁₆×C₃₁	SD₁₆	1	4	yes	complemented
169	SD₁₆×C₃₁	—	2	2	no	—
170	SD₁₆×C₃₁	—	4	2	no	—
171	SD₁₆×C₃₁	—	4	2	no	—
172	SD₁₆×C₃₁	—	4	2	no	—
173	SD₁₆×C₃₁	—	8	1	no	—
174	SD₁₆×C₃₁	—	8	2	no	—
175	M₁₆×C₃₁	—	4	2	no	—
176	C₂×C₈×C₃₁	—	2	1	no	—
177	C₂×C₈×C₃₁	—	2	1	no	—
178	C₂×C₈×C₃₁	—	2	1	no	—
179	C₂×C₈×C₃₁	—	2	1	no	—
180	C₂×C₈×C₃₁	—	2	1	no	—
181	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
185	C₄⋊C₄×C₃₁	—	2	1	no	—
186	C₄⋊C₄×C₃₁	—	2	1	no	—
187	C₄⋊C₄×C₃₁	—	2	1	no	—
188	C₄⋊C₄×C₃₁	—	2	1	no	—
189	C₄⋊C₄×C₃₁	—	2	1	no	—
190	C₄⋊C₄×C₃₁	—	2	1	no	—
191	C₄⋊C₄×C₃₁	—	2	1	no	—
192	C₄⋊C₄×C₃₁	—	2	1	no	—
193	C₄⋊C₄×C₃₁	—	2	1	no	—
194	C₄⋊C₄×C₃₁	—	4	1	no	—
195	C₄⋊C₄×C₃₁	—	4	1	no	—
196	C₄⋊C₄×C₃₁	—	4	1	no	—
197	C₄⋊C₄×C₃₁	—	4	1	no	—
198	C₄⋊C₄×C₃₁	—	4	1	no	—
199	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
200	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
201	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
202	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
203	C₂²⋊C₄×C₃₁	—	2	1	no	—
204	C₂²⋊C₄×C₃₁	—	4	1	no	—
205	C₂²⋊C₄×C₃₁	—	4	1	no	—
206	C₄²×C₃₁	—	2	1	no	—
207	C₄²×C₃₁	—	2	1	no	—
208	C₄²×C₃₁	—	2	1	no	—
209	C₄²×C₃₁	—	4	1	no	—
210	C₄²×C₃₁	—	4	1	no	—
211	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
212	#19809	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂³×C₃₁	—	2	1	no	—
214	C₂³×C₃₁	—	4	1	no	—
215	Q₈×C₃₁	—	2	1	no	—
216	Q₈×C₃₁	—	2	1	no	—
217	Q₈×C₃₁	—	2	1	no	—
218	Q₈×C₃₁	—	4	1	no	—
219	Q₈×C₃₁	—	4	1	no	—
220	Q₈×C₃₁	—	8	2	no	—
221	Q₈×C₃₁	—	8	1	no	—
222	Q₈×C₃₁	—	8	1	no	—
223	Q₈×C₃₁	—	16	1	no	—
224	Q₈×C₃₁	—	16	1	no	—
225	Q₈×C₃₁	C₂×SD₁₆	1	2	yes	—
226	Q₈×C₃₁	C₂×SD₁₆	1	2	yes	—
227	Q₈×C₃₁	#137	1	2	yes	complemented
228	D₄×C₃₁	—	2	1	no	—
229	D₄×C₃₁	—	2	1	no	—
230	D₄×C₃₁	—	4	1	no	—
231	D₄×C₃₁	—	4	1	no	—
232	D₄×C₃₁	—	4	1	no	—
233	D₄×C₃₁	—	4	1	no	—
234	D₄×C₃₁	—	8	2	no	—
235	D₄×C₃₁	—	8	1	no	—
236	D₄×C₃₁	—	16	1	no	—
237	C₂×C₄×C₃₁	—	2	1	no	—
238	C₂×C₄×C₃₁	—	2	1	no	—
239	C₂×C₄×C₃₁	—	2	1	no	—
240	C₂×C₄×C₃₁	—	2	1	no	—
241	C₂×C₄×C₃₁	—	2	1	no	—
242	C₂×C₄×C₃₁	—	2	1	no	—
243	C₂×C₄×C₃₁	—	2	1	no	—
244	C₂×C₄×C₃₁	—	4	1	no	—
245	C₂×C₄×C₃₁	—	4	1	no	—
246	C₂×C₄×C₃₁	—	4	1	no	—
247	C₂×C₄×C₃₁	—	4	1	no	—
248	C₂×C₄×C₃₁	—	4	1	no	—
249	C₂×C₄×C₃₁	—	8	2	no	—
250	C₂×C₄×C₃₁	—	8	2	no	—
251	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
254	C₈×C₃₁	—	2	1	no	—
255	C₈×C₃₁	—	2	1	no	—
256	C₈×C₃₁	—	2	1	no	—
257	C₈×C₃₁	—	4	1	no	—
258	C₈×C₃₁	—	4	1	no	—
259	C₈×C₃₁	—	4	1	no	—
260	C₈×C₃₁	—	4	1	no	—
261	C₈×C₃₁	—	4	1	no	—
262	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
263	Q₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
264	Q₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#3362	C₂×C₃₁	1	1	yes	characteristic
266	#3278	C₂×C₃₁	1	1	yes	characteristic
267	#1714	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#1695	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#1678	C₂×C₃₁	1	1	yes	characteristic, complemented
270	#1672	C₂×C₃₁	1	1	yes	characteristic, complemented
271	#1666	C₂×C₃₁	1	1	yes	characteristic, complemented
272	#1645	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#1637	C₂×C₃₁	1	1	yes	characteristic, complemented
274	#1631	C₂×C₃₁	1	1	yes	characteristic
275	#1623	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#1583	C₂×C₃₁	1	1	yes	characteristic, complemented
277	#1579	C₂×C₃₁	1	1	yes	characteristic, complemented
278	C₂²×C₃₁	—	4	1	no	—
279	C₂²×C₃₁	—	4	1	no	—
280	C₂²×C₃₁	—	4	1	no	—
281	C₂²×C₃₁	—	8	1	no	—
282	C₂²×C₃₁	—	8	1	no	—
283	C₂²×C₃₁	—	8	1	no	—
284	C₂²×C₃₁	D₄²	1	1	yes	characteristic
285	C₄×C₃₁	—	2	2	no	—
286	C₄×C₃₁	—	2	2	no	—
287	C₄×C₃₁	—	2	2	no	—
288	C₄×C₃₁	—	2	1	no	—
289	C₄×C₃₁	—	4	1	no	—
290	C₄×C₃₁	—	4	1	no	—
291	C₄×C₃₁	—	4	1	no	—
292	C₄×C₃₁	—	4	1	no	—
293	C₄×C₃₁	—	8	1	no	—
294	C₄×C₃₁	—	8	1	no	—
295	C₄×C₃₁	—	8	1	no	—
296	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
297	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
298	C₄×C₃₁	#577	1	1	yes	characteristic
299	C₄×C₃₁	#578	1	1	yes	characteristic
300	C₂×#137	—	2	1	no	—
301	Q₈²	C₂²×C₃₁	1	1	yes	characteristic
302	Q₈²	C₂²×C₃₁	1	1	yes	characteristic
303	#550	C₂²×C₃₁	1	1	yes	characteristic
304	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
305	#500	—	2	1	no	—
306	#500	C₂²×C₃₁	1	1	yes	characteristic
307	#498	—	2	1	no	—
308	#498	C₂²×C₃₁	1	1	yes	characteristic
309	#492	C₂²×C₃₁	1	1	yes	characteristic
310	#492	C₂²×C₃₁	1	1	yes	characteristic
311	#487	C₂²×C₃₁	1	1	yes	characteristic
312	#486	—	2	1	no	—
313	#486	C₂²×C₃₁	1	1	yes	characteristic
314	#477	—	2	1	no	—
315	#477	—	2	1	no	—
316	#477	C₂²×C₃₁	1	1	yes	characteristic
317	#477	C₂²×C₃₁	1	1	yes	characteristic
318	#476	—	2	1	no	—
319	#476	—	2	1	no	—
320	#476	C₂²×C₃₁	1	1	yes	characteristic
321	#476	C₂²×C₃₁	1	1	yes	characteristic
322	#475	—	2	1	no	—
323	#475	C₂²×C₃₁	1	1	yes	characteristic
324	#468	—	2	1	no	—
325	#462	C₂²×C₃₁	1	1	yes	characteristic
326	#460	—	2	1	no	—
327	#460	C₂²×C₃₁	1	1	yes	characteristic
328	#460	C₂²×C₃₁	1	1	yes	characteristic
329	#452	—	2	1	no	—
330	#448	—	2	1	no	—
331	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
332	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
333	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
334	C₄×Q₁₆	—	2	1	no	—
335	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
336	C₄×SD₁₆	—	2	1	no	—
337	C₄×SD₁₆	—	2	1	no	—
338	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
339	#436	C₂²×C₃₁	1	1	yes	characteristic
340	#334	C₂²×C₃₁	1	1	yes	characteristic
341	#334	C₂²×C₃₁	1	1	yes	characteristic
342	#333	C₂²×C₃₁	1	1	yes	characteristic
343	#333	C₂²×C₃₁	1	1	yes	characteristic
344	#332	C₂²×C₃₁	1	1	yes	characteristic
345	#331	C₂²×C₃₁	1	1	yes	characteristic
346	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
347	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
348	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
349	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
350	#322	C₂²×C₃₁	1	1	yes	characteristic
351	C₂×C₃₁	—	8	1	no	—
352	C₂×C₃₁	—	16	1	no	—
353	C₂×C₃₁	#3270	1	1	yes	characteristic
354	C₂×C₃₁	D₄×SD₁₆	1	1	yes	characteristic
355	C₂×C₃₁	#3288	1	1	yes	characteristic
356	C₂×#41	—	2	1	no	—
357	C₂²×Q₈	—	2	1	no	—
358	#137	—	4	4	no	—
359	C₂×Q₁₆	—	2	1	no	—
360	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
361	C₂×SD₁₆	—	2	1	no	—
362	C₂×SD₁₆	—	2	1	no	—
363	C₂×SD₁₆	—	2	1	no	—
364	C₂×SD₁₆	—	2	1	no	—
365	C₂×SD₁₆	—	4	1	no	—
366	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
367	C₂×M₁₆	—	2	1	no	—
368	C₄⋊Q₈	—	2	1	no	—
369	C₄⋊Q₈	—	2	1	no	—
370	C₄⋊Q₈	—	2	1	no	—
371	C₄⋊Q₈	—	2	1	no	—
372	C₄⋊Q₈	—	2	1	no	—
373	C₄⋊Q₈	—	4	1	no	—
374	C₄⋊Q₈	—	4	1	no	—
375	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
376	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
377	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
378	#127	—	2	1	no	—
379	#127	C₂³×C₃₁	1	1	yes	characteristic
380	C₂²⋊Q₈	—	2	1	no	—
381	C₂²⋊Q₈	—	4	1	no	—
382	#121	—	2	1	no	—
383	#121	—	4	1	no	—
384	C₄×Q₈	—	2	1	no	—
385	C₄×Q₈	—	2	1	no	—
386	C₄×Q₈	—	2	1	no	—
387	C₄×Q₈	—	2	1	no	—
388	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
389	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
390	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
391	C₄×D₄	—	2	1	no	—
392	C₄×D₄	—	2	1	no	—
393	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
394	#107	—	2	1	no	—
395	#107	—	2	1	no	—
396	#107	—	2	1	no	—
397	#107	—	2	1	no	—
398	#107	D₄×C₃₁	1	1	yes	characteristic
399	#106	—	2	1	no	—
400	#106	—	2	1	no	—
401	#106	—	2	1	no	—
402	#106	—	2	1	no	—
403	#106	—	2	1	no	—
404	#106	D₄×C₃₁	1	1	yes	characteristic
405	C₄⋊C₈	—	2	1	no	—
406	C₄⋊C₈	—	2	1	no	—
407	C₄⋊C₈	—	2	1	no	—
408	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
409	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
410	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
411	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
412	#103	—	2	1	no	—
413	#103	—	2	1	no	—
414	#103	—	2	1	no	—
415	#103	—	2	1	no	—
416	#103	—	4	1	no	—
417	#103	D₄×C₃₁	1	1	yes	characteristic
418	#103	D₄×C₃₁	1	1	yes	characteristic
419	#103	D₄×C₃₁	1	1	yes	characteristic
420	#102	—	2	1	no	—
421	#102	—	2	1	no	—
422	#102	—	2	1	no	—
423	#102	—	2	1	no	—
424	#102	—	2	1	no	—
425	#102	—	2	1	no	—
426	#102	D₄×C₃₁	1	1	yes	characteristic
427	C₂²⋊C₈	—	2	1	no	—
428	C₄×C₈	—	2	1	no	—
429	C₄×C₈	—	2	1	no	—
430	C₄×C₈	—	2	1	no	—
431	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
432	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
433	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
434	C₃₁	#19809	1	1	yes	characteristic, direct product factor, Sylow
435	#41	—	4	2	no	—
436	#41	—	8	1	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	4	1	no	—
441	C₂×Q₈	—	4	1	no	—
442	C₂×Q₈	—	4	1	no	—
443	C₂×Q₈	—	4	2	no	—
444	C₂×Q₈	—	8	1	no	—
445	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
446	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
447	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	4	1	no	—
452	C₂×D₄	—	4	1	no	—
453	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	4	1	no	—
456	C₂²×C₄	—	4	1	no	—
457	Q₁₆	—	4	2	no	—
458	Q₁₆	SD₁₆×C₃₁	1	4	yes	complemented
459	SD₁₆	—	2	2	no	—
460	SD₁₆	—	4	2	no	—
461	SD₁₆	—	4	2	no	—
462	SD₁₆	—	4	2	no	—
463	SD₁₆	—	8	1	no	—
464	SD₁₆	—	8	2	no	—
465	M₁₆	—	4	2	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
472	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
473	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
474	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	4	1	no	—
485	C₄⋊C₄	—	4	1	no	—
486	C₄⋊C₄	—	4	1	no	—
487	C₄⋊C₄	—	4	1	no	—
488	C₄⋊C₄	—	4	1	no	—
489	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
490	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
491	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
492	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
493	C₂²⋊C₄	—	2	1	no	—
494	C₂²⋊C₄	—	4	1	no	—
495	C₂²⋊C₄	—	4	1	no	—
496	C₄²	—	2	1	no	—
497	C₄²	—	2	1	no	—
498	C₄²	—	2	1	no	—
499	C₄²	—	4	1	no	—
500	C₄²	—	4	1	no	—
501	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
502	C₂³	—	2	1	no	—
503	C₂³	—	4	1	no	—
504	Q₈	—	2	1	no	—
505	Q₈	—	2	1	no	—
506	Q₈	—	2	1	no	—
507	Q₈	—	4	1	no	—
508	Q₈	—	4	1	no	—
509	Q₈	—	8	2	no	—
510	Q₈	—	8	1	no	—
511	Q₈	—	8	1	no	—
512	Q₈	—	16	1	no	—
513	Q₈	—	16	1	no	—
514	Q₈	C₂×SD₁₆×C₃₁	1	2	yes	—
515	Q₈	C₂×SD₁₆×C₃₁	1	2	yes	—
516	Q₈	C₃₁×#137	1	2	yes	complemented
517	D₄	—	2	1	no	—
518	D₄	—	2	1	no	—
519	D₄	—	4	1	no	—
520	D₄	—	4	1	no	—
521	D₄	—	4	1	no	—
522	D₄	—	4	1	no	—
523	D₄	—	8	2	no	—
524	D₄	—	8	1	no	—
525	D₄	—	16	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	8	2	no	—
539	C₂×C₄	—	8	2	no	—
540	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
541	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
542	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
543	C₈	—	2	1	no	—
544	C₈	—	2	1	no	—
545	C₈	—	2	1	no	—
546	C₈	—	4	1	no	—
547	C₈	—	4	1	no	—
548	C₈	—	4	1	no	—
549	C₈	—	4	1	no	—
550	C₈	—	4	1	no	—
551	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
552	C₂²	—	4	1	no	—
553	C₂²	—	4	1	no	—
554	C₂²	—	4	1	no	—
555	C₂²	—	8	1	no	—
556	C₂²	—	8	1	no	—
557	C₂²	—	8	1	no	—
558	C₂²	D₄²×C₃₁	1	1	yes	characteristic
559	C₄	—	2	2	no	—
560	C₄	—	2	2	no	—
561	C₄	—	2	2	no	—
562	C₄	—	2	1	no	—
563	C₄	—	4	1	no	—
564	C₄	—	4	1	no	—
565	C₄	—	4	1	no	—
566	C₄	—	4	1	no	—
567	C₄	—	8	1	no	—
568	C₄	—	8	1	no	—
569	C₄	—	8	1	no	—
570	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
571	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
572	C₄	C₃₁×#577	1	1	yes	characteristic
573	C₄	C₃₁×#578	1	1	yes	characteristic
574	C₂	—	8	1	no	—
575	C₂	—	16	1	no	—
576	C₂	C₃₁×#3270	1	1	yes	characteristic
577	C₂	D₄×SD₁₆×C₃₁	1	1	yes	characteristic
578	C₂	C₃₁×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#550	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×Q₁₆×C₃₁	C₂	semidirect
2. Q₈×SD₁₆×C₃₁	C₂	semidirect
5. C₃₁×#1714	C₂	semidirect
6. C₃₁×#1695	C₂	semidirect
7. C₃₁×#1678	C₂	semidirect
8. C₃₁×#1672	C₂	semidirect
9. C₃₁×#1666	C₂	semidirect
10. C₃₁×#1645	C₂	semidirect
11. C₃₁×#1637	C₂	semidirect
13. C₃₁×#1623	C₂	semidirect
14. C₃₁×#1583	C₂	semidirect
15. C₃₁×#1579	C₂	semidirect
168. Q₁₆×C₃₁	SD₁₆	semidirect
212. #19809	C₃₁	direct (N × H)
227. Q₈×C₃₁	#137	semidirect
263. Q₈×Q₁₆	C₂×C₃₁	semidirect
264. Q₈×SD₁₆	C₂×C₃₁	semidirect
267. #1714	C₂×C₃₁	semidirect
268. #1695	C₂×C₃₁	semidirect
269. #1678	C₂×C₃₁	semidirect
270. #1672	C₂×C₃₁	semidirect
271. #1666	C₂×C₃₁	semidirect
272. #1645	C₂×C₃₁	semidirect
273. #1637	C₂×C₃₁	semidirect
275. #1623	C₂×C₃₁	semidirect
276. #1583	C₂×C₃₁	semidirect
277. #1579	C₂×C₃₁	semidirect
434. C₃₁	#19809	direct (N × H)
458. Q₁₆	SD₁₆×C₃₁	semidirect
516. Q₈	C₃₁×#137	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Q₁₆×C₃₁	2
Q₈×SD₁₆×C₃₁	2
C₃₁×#3362	2
C₃₁×#3278	2
C₃₁×#1714	2
C₃₁×#1695	2
C₃₁×#1678	2
C₃₁×#1672	2
C₃₁×#1666	2
C₃₁×#1645	2
C₃₁×#1637	2
C₃₁×#1631	2
C₃₁×#1623	2
C₃₁×#1583	2
C₃₁×#1579	2
#19809	31

Subgroups — C31×#19809 #21476568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19809 #21476568