Subgroups — C₃₁×#18040 #21474799

Order 7936.

708 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	D₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1592	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1588	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1585	C₂	1	2	yes	maximal, complemented
7	C₃₁×SD₁₆⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1487	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1486	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₄	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	—	2	1	no	—
14	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#446	—	2	2	no	—
17	C₃₁×#446	C₂²	1	2	yes	complemented
18	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
19	C₄×SD₁₆×C₃₁	C₄	1	1	yes	characteristic
20	C₄×SD₁₆×C₃₁	C₄	1	1	yes	characteristic
21	C₄×SD₁₆×C₃₁	C₄	1	1	yes	characteristic
22	C₄×SD₁₆×C₃₁	C₄	1	1	yes	characteristic
23	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
24	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
26	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#436	C₂²	1	2	yes	—
28	C₃₁×#435	—	2	2	no	—
29	C₈×D₄×C₃₁	C₂²	1	2	yes	—
30	C₃₁×#431	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#106	C₄	1	1	yes	characteristic
32	C₃₁×#423	—	2	1	no	—
33	C₃₁×#423	C₂²	1	1	yes	characteristic
34	C₃₁×#423	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#103	C₄	1	1	yes	characteristic
36	C₂×C₃₁×#102	C₄	1	1	yes	characteristic
37	C₃₁×#407	—	2	1	no	—
38	C₄×M₁₆×C₃₁	—	2	1	no	—
39	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#335	C₂²	1	2	yes	complemented
42	C₃₁×#335	C₂²	1	2	yes	complemented
43	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
44	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
45	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
46	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
47	C₃₁×#322	C₂²	1	2	yes	complemented
48	C₃₁×#322	C₂²	1	2	yes	complemented
49	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	—	2	1	no	—
52	C₂×SD₁₆×C₃₁	—	2	4	no	—
53	C₂×SD₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×SD₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×SD₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×SD₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂×M₁₆×C₃₁	—	2	1	no	—
58	C₂×M₁₆×C₃₁	—	2	1	no	—
59	C₂×M₁₆×C₃₁	—	2	2	no	—
60	C₂×M₁₆×C₃₁	—	4	4	no	—
61	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
62	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	—	2	2	no	—
64	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₄×Q₈×C₃₁	—	2	1	no	—
67	C₄×Q₈×C₃₁	—	2	1	no	—
68	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₄×D₄×C₃₁	—	2	1	no	—
73	C₄×D₄×C₃₁	—	2	1	no	—
74	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
75	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
76	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
82	C₂×C₄²×C₃₁	—	2	1	no	—
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×#106	C₂×C₄	1	1	yes	characteristic
85	C₃₁×#106	C₂×C₄	1	1	yes	characteristic
86	C₃₁×#106	C₂×C₄	1	1	yes	characteristic
87	C₃₁×#106	C₂×C₄	1	1	yes	characteristic
88	C₃₁×C₄⋊C₈	—	2	2	no	—
89	C₃₁×C₄⋊C₈	—	2	2	no	—
90	C₃₁×C₄⋊C₈	—	2	1	no	—
91	C₃₁×C₄⋊C₈	—	2	1	no	—
92	C₃₁×C₄⋊C₈	—	2	1	no	—
93	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
94	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
95	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
96	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
97	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
98	C₃₁×#103	C₂×C₄	1	1	yes	characteristic
99	C₃₁×#102	C₂×C₄	1	1	yes	characteristic
100	C₃₁×#102	C₂×C₄	1	1	yes	characteristic
101	C₃₁×#102	C₂×C₄	1	1	yes	characteristic
102	C₃₁×#102	C₂×C₄	1	1	yes	characteristic
103	C₃₁×C₂²⋊C₈	—	2	1	no	—
104	C₃₁×C₂²⋊C₈	—	2	2	no	—
105	C₃₁×C₂²⋊C₈	—	2	2	no	—
106	C₃₁×#97	—	2	2	no	—
107	C₃₁×#97	D₄	1	2	yes	—
108	C₄×C₈×C₃₁	—	2	2	no	—
109	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
110	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
111	C₄×C₈×C₃₁	C₂³	1	2	yes	—
112	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
114	C₄×C₈×C₃₁	C₂³	1	2	yes	—
115	C₂⁴×C₃₁	—	2	1	no	—
116	C₂×Q₈×C₃₁	—	2	1	no	—
117	C₂×Q₈×C₃₁	—	2	2	no	—
118	C₂×Q₈×C₃₁	—	4	1	no	—
119	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
120	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₂×D₄×C₃₁	—	2	2	no	—
122	C₂×D₄×C₃₁	—	2	1	no	—
123	C₂×D₄×C₃₁	—	4	1	no	—
124	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
125	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	—	2	1	no	—
127	C₂²×C₄×C₃₁	—	2	1	no	—
128	C₂²×C₄×C₃₁	—	2	1	no	—
129	C₂²×C₄×C₃₁	—	2	1	no	—
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	—	4	2	no	—
134	C₂²×C₄×C₃₁	—	4	2	no	—
135	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
138	SD₁₆×C₃₁	—	2	2	no	—
139	SD₁₆×C₃₁	—	2	2	no	—
140	SD₁₆×C₃₁	—	2	2	no	—
141	SD₁₆×C₃₁	M₁₆	1	4	yes	complemented
142	M₁₆×C₃₁	—	2	2	no	—
143	M₁₆×C₃₁	—	2	2	no	—
144	M₁₆×C₃₁	—	4	4	no	—
145	M₁₆×C₃₁	—	4	2	no	—
146	M₁₆×C₃₁	—	4	2	no	—
147	C₂×C₈×C₃₁	—	2	2	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	—	2	2	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	4	1	no	—
152	C₂×C₈×C₃₁	—	4	4	no	—
153	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
154	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
155	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
156	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
157	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
158	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
159	C₄⋊C₄×C₃₁	—	2	1	no	—
160	C₄⋊C₄×C₃₁	—	2	1	no	—
161	C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₄⋊C₄×C₃₁	—	2	1	no	—
163	C₄⋊C₄×C₃₁	—	2	1	no	—
164	C₄⋊C₄×C₃₁	—	2	1	no	—
165	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
166	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
167	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
168	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
169	C₂²⋊C₄×C₃₁	—	2	1	no	—
170	C₂²⋊C₄×C₃₁	—	2	1	no	—
171	C₂²⋊C₄×C₃₁	—	2	1	no	—
172	C₂²⋊C₄×C₃₁	—	2	1	no	—
173	C₄²×C₃₁	—	2	1	no	—
174	C₄²×C₃₁	—	2	1	no	—
175	C₄²×C₃₁	—	2	1	no	—
176	C₄²×C₃₁	—	2	1	no	—
177	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
178	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
179	C₄²×C₃₁	#41	1	1	yes	characteristic
180	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
181	#18040	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
182	C₂³×C₃₁	—	2	1	no	—
183	C₂³×C₃₁	—	2	1	no	—
184	C₂³×C₃₁	—	4	2	no	—
185	C₂³×C₃₁	—	4	2	no	—
186	C₂³×C₃₁	—	8	1	no	—
187	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
188	Q₈×C₃₁	—	2	1	no	—
189	Q₈×C₃₁	—	2	1	no	—
190	Q₈×C₃₁	—	2	1	no	—
191	Q₈×C₃₁	—	4	2	no	—
192	Q₈×C₃₁	C₂×M₁₆	1	2	yes	complemented
193	D₄×C₃₁	—	2	1	no	—
194	D₄×C₃₁	—	2	1	no	—
195	D₄×C₃₁	—	2	1	no	—
196	D₄×C₃₁	—	4	2	no	—
197	D₄×C₃₁	C₂×M₁₆	1	2	yes	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	2	1	no	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	2	1	no	—
203	C₂×C₄×C₃₁	—	2	1	no	—
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	4	1	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	1	no	—
210	C₂×C₄×C₃₁	—	4	1	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	8	1	no	—
216	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
217	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
218	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
220	C₂×C₄×C₃₁	#137	1	2	yes	—
221	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
224	C₈×C₃₁	—	2	1	no	—
225	C₈×C₃₁	—	2	1	no	—
226	C₈×C₃₁	—	2	1	no	—
227	C₈×C₃₁	—	2	2	no	—
228	C₈×C₃₁	—	2	2	no	—
229	C₈×C₃₁	—	4	2	no	—
230	C₈×C₃₁	—	4	2	no	—
231	C₈×C₃₁	—	4	2	no	—
232	C₈×C₃₁	C₂×M₁₆	1	2	yes	complemented
233	Q₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
234	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
235	D₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
236	#1592	C₂×C₃₁	1	2	yes	complemented
237	#1588	C₂×C₃₁	1	2	yes	complemented
238	#1585	C₂×C₃₁	1	2	yes	complemented
239	SD₁₆⋊C₈	C₂×C₃₁	1	2	yes	complemented
240	#1567	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#1487	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#1486	C₂×C₃₁	1	1	yes	characteristic
243	#1451	C₂×C₃₁	1	1	yes	characteristic, complemented
244	C₂²×C₃₁	—	2	1	no	—
245	C₂²×C₃₁	—	2	1	no	—
246	C₂²×C₃₁	—	2	2	no	—
247	C₂²×C₃₁	—	4	1	no	—
248	C₂²×C₃₁	—	4	1	no	—
249	C₂²×C₃₁	—	4	1	no	—
250	C₂²×C₃₁	—	4	1	no	—
251	C₂²×C₃₁	—	4	1	no	—
252	C₂²×C₃₁	—	8	4	no	—
253	C₂²×C₃₁	#440	1	2	yes	—
254	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
255	C₄×C₃₁	—	2	1	no	—
256	C₄×C₃₁	—	2	2	no	—
257	C₄×C₃₁	—	2	1	no	—
258	C₄×C₃₁	—	2	1	no	—
259	C₄×C₃₁	—	4	1	no	—
260	C₄×C₃₁	—	4	1	no	—
261	C₄×C₃₁	—	4	1	no	—
262	C₄×C₃₁	—	4	1	no	—
263	C₄×C₃₁	—	4	1	no	—
264	C₄×C₃₁	#440	1	2	yes	—
265	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
266	C₄×C₃₁	#568	1	1	yes	characteristic
267	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
268	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
269	C₂²×SD₁₆	C₄×C₃₁	1	1	yes	characteristic
270	C₂²×M₁₆	—	2	1	no	—
271	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
273	#446	—	2	2	no	—
274	#446	C₂²×C₃₁	1	2	yes	complemented
275	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
276	C₄×SD₁₆	C₄×C₃₁	1	1	yes	characteristic
277	C₄×SD₁₆	C₄×C₃₁	1	1	yes	characteristic
278	C₄×SD₁₆	C₄×C₃₁	1	1	yes	characteristic
279	C₄×SD₁₆	C₄×C₃₁	1	1	yes	characteristic
280	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
281	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
282	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
283	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
284	#436	C₂²×C₃₁	1	2	yes	—
285	#435	—	2	2	no	—
286	C₈×D₄	C₂²×C₃₁	1	2	yes	—
287	#431	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×#106	C₄×C₃₁	1	1	yes	characteristic
289	#423	—	2	1	no	—
290	#423	C₂²×C₃₁	1	1	yes	characteristic
291	#423	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×#103	C₄×C₃₁	1	1	yes	characteristic
293	C₂×#102	C₄×C₃₁	1	1	yes	characteristic
294	#407	—	2	1	no	—
295	C₄×M₁₆	—	2	1	no	—
296	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
297	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
298	#335	C₂²×C₃₁	1	2	yes	complemented
299	#335	C₂²×C₃₁	1	2	yes	complemented
300	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	complemented
301	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	complemented
302	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
303	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
304	#322	C₂²×C₃₁	1	2	yes	complemented
305	#322	C₂²×C₃₁	1	2	yes	complemented
306	C₂×C₃₁	—	2	2	no	—
307	C₂×C₃₁	—	4	2	no	—
308	C₂×C₃₁	—	8	1	no	—
309	C₂×C₃₁	D₄×M₁₆	1	1	yes	characteristic
310	C₂×C₃₁	C₂×#440	1	1	yes	characteristic
311	C₂×C₃₁	#2957	1	1	yes	characteristic
312	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
313	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
314	C₂³×C₄	—	2	1	no	—
315	C₂×SD₁₆	—	2	4	no	—
316	C₂×SD₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
317	C₂×SD₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
318	C₂×SD₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
319	C₂×SD₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
320	C₂×M₁₆	—	2	1	no	—
321	C₂×M₁₆	—	2	1	no	—
322	C₂×M₁₆	—	2	2	no	—
323	C₂×M₁₆	—	4	4	no	—
324	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
325	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
326	C₂²×C₈	—	2	2	no	—
327	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
328	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
329	C₄×Q₈	—	2	1	no	—
330	C₄×Q₈	—	2	1	no	—
331	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
332	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
333	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
334	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
335	C₄×D₄	—	2	1	no	—
336	C₄×D₄	—	2	1	no	—
337	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
338	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
339	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
340	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
341	C₂×C₄⋊C₄	—	2	1	no	—
342	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
343	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
344	C₂×C₂²⋊C₄	—	2	1	no	—
345	C₂×C₄²	—	2	1	no	—
346	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
347	#106	C₂×C₄×C₃₁	1	1	yes	characteristic
348	#106	C₂×C₄×C₃₁	1	1	yes	characteristic
349	#106	C₂×C₄×C₃₁	1	1	yes	characteristic
350	#106	C₂×C₄×C₃₁	1	1	yes	characteristic
351	C₄⋊C₈	—	2	2	no	—
352	C₄⋊C₈	—	2	2	no	—
353	C₄⋊C₈	—	2	1	no	—
354	C₄⋊C₈	—	2	1	no	—
355	C₄⋊C₈	—	2	1	no	—
356	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
357	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
358	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
359	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
360	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
361	#103	C₂×C₄×C₃₁	1	1	yes	characteristic
362	#102	C₂×C₄×C₃₁	1	1	yes	characteristic
363	#102	C₂×C₄×C₃₁	1	1	yes	characteristic
364	#102	C₂×C₄×C₃₁	1	1	yes	characteristic
365	#102	C₂×C₄×C₃₁	1	1	yes	characteristic
366	C₂²⋊C₈	—	2	1	no	—
367	C₂²⋊C₈	—	2	2	no	—
368	C₂²⋊C₈	—	2	2	no	—
369	#97	—	2	2	no	—
370	#97	D₄×C₃₁	1	2	yes	—
371	C₄×C₈	—	2	2	no	—
372	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
373	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
374	C₄×C₈	C₂³×C₃₁	1	2	yes	—
375	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
376	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
377	C₄×C₈	C₂³×C₃₁	1	2	yes	—
378	C₃₁	#18040	1	1	yes	characteristic, direct product factor, Sylow
379	C₂⁴	—	2	1	no	—
380	C₂×Q₈	—	2	1	no	—
381	C₂×Q₈	—	2	2	no	—
382	C₂×Q₈	—	4	1	no	—
383	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
384	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
385	C₂×D₄	—	2	2	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	4	1	no	—
388	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
389	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	4	2	no	—
398	C₂²×C₄	—	4	2	no	—
399	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
401	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
402	SD₁₆	—	2	2	no	—
403	SD₁₆	—	2	2	no	—
404	SD₁₆	—	2	2	no	—
405	SD₁₆	M₁₆×C₃₁	1	4	yes	complemented
406	M₁₆	—	2	2	no	—
407	M₁₆	—	2	2	no	—
408	M₁₆	—	4	4	no	—
409	M₁₆	—	4	2	no	—
410	M₁₆	—	4	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	4	1	no	—
416	C₂×C₈	—	4	4	no	—
417	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
418	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
419	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
420	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
421	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
422	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
430	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
431	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
432	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
433	C₂²⋊C₄	—	2	1	no	—
434	C₂²⋊C₄	—	2	1	no	—
435	C₂²⋊C₄	—	2	1	no	—
436	C₂²⋊C₄	—	2	1	no	—
437	C₄²	—	2	1	no	—
438	C₄²	—	2	1	no	—
439	C₄²	—	2	1	no	—
440	C₄²	—	2	1	no	—
441	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
442	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
443	C₄²	#41×C₃₁	1	1	yes	characteristic
444	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
445	C₂³	—	2	1	no	—
446	C₂³	—	2	1	no	—
447	C₂³	—	4	2	no	—
448	C₂³	—	4	2	no	—
449	C₂³	—	8	1	no	—
450	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
451	Q₈	—	2	1	no	—
452	Q₈	—	2	1	no	—
453	Q₈	—	2	1	no	—
454	Q₈	—	4	2	no	—
455	Q₈	C₂×M₁₆×C₃₁	1	2	yes	complemented
456	D₄	—	2	1	no	—
457	D₄	—	2	1	no	—
458	D₄	—	2	1	no	—
459	D₄	—	4	2	no	—
460	D₄	C₂×M₁₆×C₃₁	1	2	yes	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	8	1	no	—
479	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
480	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
481	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
482	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
483	C₂×C₄	C₃₁×#137	1	2	yes	—
484	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
486	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
487	C₈	—	2	1	no	—
488	C₈	—	2	1	no	—
489	C₈	—	2	1	no	—
490	C₈	—	2	2	no	—
491	C₈	—	2	2	no	—
492	C₈	—	4	2	no	—
493	C₈	—	4	2	no	—
494	C₈	—	4	2	no	—
495	C₈	C₂×M₁₆×C₃₁	1	2	yes	complemented
496	C₂²	—	2	1	no	—
497	C₂²	—	2	1	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	4	1	no	—
500	C₂²	—	4	1	no	—
501	C₂²	—	4	1	no	—
502	C₂²	—	4	1	no	—
503	C₂²	—	4	1	no	—
504	C₂²	—	8	4	no	—
505	C₂²	C₃₁×#440	1	2	yes	—
506	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
507	C₄	—	2	1	no	—
508	C₄	—	2	2	no	—
509	C₄	—	2	1	no	—
510	C₄	—	2	1	no	—
511	C₄	—	4	1	no	—
512	C₄	—	4	1	no	—
513	C₄	—	4	1	no	—
514	C₄	—	4	1	no	—
515	C₄	—	4	1	no	—
516	C₄	C₃₁×#440	1	2	yes	—
517	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
518	C₄	C₃₁×#568	1	1	yes	characteristic
519	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
520	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
521	C₂	—	2	2	no	—
522	C₂	—	4	2	no	—
523	C₂	—	8	1	no	—
524	C₂	D₄×M₁₆×C₃₁	1	1	yes	characteristic
525	C₂	C₂×C₃₁×#440	1	1	yes	characteristic
526	C₂	C₃₁×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂²×D₄×C₃₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×M₁₆×C₃₁	C₂	semidirect
4. C₃₁×#1592	C₂	semidirect
5. C₃₁×#1588	C₂	semidirect
6. C₃₁×#1585	C₂	semidirect
7. C₃₁×SD₁₆⋊C₈	C₂	semidirect
8. C₃₁×#1567	C₂	semidirect
9. C₃₁×#1487	C₂	semidirect
11. C₃₁×#1451	C₂	semidirect
17. C₃₁×#446	C₂²	semidirect
18. C₈×Q₈×C₃₁	C₂²	semidirect
41. C₃₁×#335	C₂²	semidirect
42. C₃₁×#335	C₂²	semidirect
43. C₃₁×Q₈⋊C₈	C₂²	semidirect
44. C₃₁×Q₈⋊C₈	C₂²	semidirect
47. C₃₁×#322	C₂²	semidirect
48. C₃₁×#322	C₂²	semidirect
141. SD₁₆×C₃₁	M₁₆	semidirect
181. #18040	C₃₁	direct (N × H)
192. Q₈×C₃₁	C₂×M₁₆	semidirect
232. C₈×C₃₁	C₂×M₁₆	semidirect
233. Q₈×M₁₆	C₂×C₃₁	semidirect
236. #1592	C₂×C₃₁	semidirect
237. #1588	C₂×C₃₁	semidirect
238. #1585	C₂×C₃₁	semidirect
239. SD₁₆⋊C₈	C₂×C₃₁	semidirect
240. #1567	C₂×C₃₁	semidirect
241. #1487	C₂×C₃₁	semidirect
243. #1451	C₂×C₃₁	semidirect
274. #446	C₂²×C₃₁	semidirect
275. C₈×Q₈	C₂²×C₃₁	semidirect
298. #335	C₂²×C₃₁	semidirect
299. #335	C₂²×C₃₁	semidirect
300. Q₈⋊C₈	C₂²×C₃₁	semidirect
301. Q₈⋊C₈	C₂²×C₃₁	semidirect
304. #322	C₂²×C₃₁	semidirect
305. #322	C₂²×C₃₁	semidirect
378. C₃₁	#18040	direct (N × H)
405. SD₁₆	M₁₆×C₃₁	semidirect
455. Q₈	C₂×M₁₆×C₃₁	semidirect
495. C₈	C₂×M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈×M₁₆×C₃₁	2
C₂×C₄×SD₁₆×C₃₁	2
D₄×M₁₆×C₃₁	2
C₃₁×#1592	2
C₃₁×#1588	2
C₃₁×#1585	2
C₃₁×SD₁₆⋊C₈	2
C₃₁×#1567	2
C₃₁×#1487	2
C₃₁×#1486	2
C₃₁×#1451	2
#18040	31

Subgroups — C31×#18040 #21474799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18040 #21474799