Subgroups — C₃₁×#22510 #21479269

Order 7936.

1292 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#446	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1985	C₂	1	4	yes	maximal, complemented
4	C₃₁×#1984	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1797	C₂	1	2	yes	maximal
6	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1749	C₂	1	2	yes	maximal
8	C₄×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₃₁	C₄	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₃₁	—	2	8	no	—
11	C₂×C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₃₁	C₄	1	2	yes	—
13	C₂×C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
18	C₂²×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
19	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#446	—	2	8	no	—
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
24	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
26	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#97	C₂²	1	4	yes	—
28	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#336	C₂²	1	4	yes	—
30	C₃₁×#336	C₂²	1	2	yes	—
31	C₃₁×#336	C₂²	1	4	yes	—
32	C₃₁×#336	C₂²	1	2	yes	—
33	C₂²×Q₈×C₃₁	—	2	4	no	—
34	C₂²×Q₈×C₃₁	—	2	4	no	—
35	C₂²×Q₈×C₃₁	—	2	4	no	—
36	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
37	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
42	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₄×Q₈×C₃₁	—	2	4	no	—
48	C₄×Q₈×C₃₁	—	2	8	no	—
49	C₄×Q₈×C₃₁	—	2	4	no	—
50	C₄×Q₈×C₃₁	—	2	4	no	—
51	C₄×Q₈×C₃₁	—	2	8	no	—
52	C₄×Q₈×C₃₁	D₄	1	8	yes	—
53	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
55	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₃₁	—	2	4	no	—
63	C₂×C₄²×C₃₁	—	2	8	no	—
64	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
65	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
68	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
72	C₃₁×C₄⋊C₈	—	2	4	no	—
73	C₃₁×C₄⋊C₈	—	2	8	no	—
74	C₃₁×C₂²⋊C₈	—	2	2	no	—
75	C₃₁×C₂²⋊C₈	—	2	4	no	—
76	C₃₁×C₂²⋊C₈	Q₈	1	4	yes	complemented
77	C₃₁×#97	—	2	8	no	—
78	C₄×C₈×C₃₁	—	2	4	no	—
79	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×Q₈×C₃₁	—	2	32	no	—
81	C₂×Q₈×C₃₁	—	2	4	no	—
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	—	2	4	no	—
84	C₂×Q₈×C₃₁	—	2	4	no	—
85	C₂×Q₈×C₃₁	—	2	2	no	—
86	C₂×Q₈×C₃₁	—	2	2	no	—
87	C₂×Q₈×C₃₁	—	2	2	no	—
88	C₂×Q₈×C₃₁	C₂²⋊C₄	1	4	yes	—
89	C₂×Q₈×C₃₁	M₁₆	1	4	yes	—
90	C₂²×C₄×C₃₁	—	2	4	no	—
91	C₂²×C₄×C₃₁	—	2	4	no	—
92	C₂²×C₄×C₃₁	—	2	2	no	—
93	C₂²×C₄×C₃₁	—	2	4	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	4	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	2	4	no	—
101	C₂²×C₄×C₃₁	—	4	1	no	—
102	C₂²×C₄×C₃₁	—	4	2	no	—
103	C₂²×C₄×C₃₁	M₁₆	1	4	yes	—
104	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
114	C₂²×C₄×C₃₁	#41	1	2	yes	—
115	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	4	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	4	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	4	no	—
124	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	complemented
125	C₄⋊C₄×C₃₁	—	2	4	no	—
126	C₄⋊C₄×C₃₁	—	2	4	no	—
127	C₄⋊C₄×C₃₁	—	2	4	no	—
128	C₄⋊C₄×C₃₁	—	2	4	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
131	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
132	C₄²×C₃₁	—	2	2	no	—
133	C₄²×C₃₁	—	2	4	no	—
134	C₄²×C₃₁	—	2	4	no	—
135	C₄²×C₃₁	—	2	2	no	—
136	C₄²×C₃₁	—	2	2	no	—
137	C₄²×C₃₁	—	2	4	no	—
138	C₄²×C₃₁	C₂×D₄	1	4	yes	—
139	C₄²×C₃₁	C₂×D₄	1	8	yes	—
140	#22510	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₃₁	—	2	1	no	—
142	C₂³×C₃₁	—	2	1	no	—
143	C₂³×C₃₁	—	2	1	no	—
144	C₂³×C₃₁	—	2	1	no	—
145	C₂³×C₃₁	C₄×Q₈	1	2	yes	—
146	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
147	C₂³×C₃₁	#131	1	2	yes	—
148	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
149	Q₈×C₃₁	—	2	16	no	—
150	Q₈×C₃₁	—	2	8	no	—
151	Q₈×C₃₁	—	2	8	no	—
152	C₂×C₄×C₃₁	—	2	16	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	1	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	4	no	—
166	C₂×C₄×C₃₁	—	2	4	no	—
167	C₂×C₄×C₃₁	—	2	8	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	4	4	no	—
175	C₂×C₄×C₃₁	—	4	8	no	—
176	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
177	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
178	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
179	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
180	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
181	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
182	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
183	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
184	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
185	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
186	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
189	C₈×C₃₁	—	2	4	no	—
190	C₈×C₃₁	—	4	2	no	—
191	C₈×C₃₁	—	4	4	no	—
192	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
193	C₂×#446	C₂×C₃₁	1	2	yes	complemented
194	#1985	C₂×C₃₁	1	4	yes	complemented
195	#1984	C₂×C₃₁	1	2	yes	complemented
196	#1797	C₂×C₃₁	1	2	yes	—
197	#1797	C₂×C₃₁	1	1	yes	characteristic
198	#1749	C₂×C₃₁	1	2	yes	—
199	C₄×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic
200	C₂²×C₃₁	—	2	2	no	—
201	C₂²×C₃₁	—	2	2	no	—
202	C₂²×C₃₁	—	2	2	no	—
203	C₂²×C₃₁	—	2	2	no	—
204	C₂²×C₃₁	—	2	2	no	—
205	C₂²×C₃₁	—	2	2	no	—
206	C₂²×C₃₁	#446	1	4	yes	complemented
207	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
208	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
209	C₂²×C₃₁	#519	1	1	yes	characteristic
210	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
211	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
212	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
213	C₂²×C₃₁	#568	1	1	yes	characteristic
214	C₄×C₃₁	—	2	2	no	—
215	C₄×C₃₁	—	2	4	no	—
216	C₄×C₃₁	—	2	4	no	—
217	C₄×C₃₁	—	2	2	no	—
218	C₄×C₃₁	—	2	2	no	—
219	C₄×C₃₁	—	2	4	no	—
220	C₄×C₃₁	—	4	2	no	—
221	C₄×C₃₁	—	4	1	no	—
222	C₄×C₃₁	#407	1	4	yes	—
223	C₄×C₃₁	#548	1	2	yes	—
224	C₄×C₃₁	D₄×Q₈	1	2	yes	—
225	C₂³×Q₈	C₄×C₃₁	1	1	yes	characteristic
226	C₂×C₄×Q₈	—	2	8	no	—
227	C₂×C₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
228	C₂×C₄×Q₈	C₄×C₃₁	1	2	yes	—
229	C₂×C₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
230	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
231	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
232	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
233	C₂²×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
234	C₂²×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
235	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
236	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
237	#446	—	2	8	no	—
238	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
239	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
240	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
242	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×#97	C₂²×C₃₁	1	4	yes	—
244	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
245	#336	C₂²×C₃₁	1	4	yes	—
246	#336	C₂²×C₃₁	1	2	yes	—
247	#336	C₂²×C₃₁	1	4	yes	—
248	#336	C₂²×C₃₁	1	2	yes	—
249	C₂×C₃₁	—	2	4	no	—
250	C₂×C₃₁	Q₈×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₃₁	C₂×#407	1	1	yes	characteristic
252	C₂×C₃₁	#2880	1	1	yes	characteristic
253	C₂×C₃₁	C₂×#446	1	1	yes	characteristic
254	C₂×C₃₁	Q₈×M₁₆	1	1	yes	characteristic
255	C₂×C₃₁	#2983	1	1	yes	characteristic
256	C₂×C₃₁	#2984	1	1	yes	characteristic
257	C₂²×Q₈	—	2	4	no	—
258	C₂²×Q₈	—	2	4	no	—
259	C₂²×Q₈	—	2	4	no	—
260	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
261	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
262	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
263	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
264	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
265	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
266	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
267	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
269	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
270	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
271	C₄×Q₈	—	2	4	no	—
272	C₄×Q₈	—	2	8	no	—
273	C₄×Q₈	—	2	4	no	—
274	C₄×Q₈	—	2	4	no	—
275	C₄×Q₈	—	2	8	no	—
276	C₄×Q₈	D₄×C₃₁	1	8	yes	—
277	C₂×C₄⋊C₄	—	2	8	no	—
278	C₂×C₄⋊C₄	—	2	4	no	—
279	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
280	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
281	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
282	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
283	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
284	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
285	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
286	C₂×C₄²	—	2	4	no	—
287	C₂×C₄²	—	2	8	no	—
288	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
289	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
290	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
291	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
292	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
293	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
294	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
295	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
296	C₄⋊C₈	—	2	4	no	—
297	C₄⋊C₈	—	2	8	no	—
298	C₂²⋊C₈	—	2	2	no	—
299	C₂²⋊C₈	—	2	4	no	—
300	C₂²⋊C₈	Q₈×C₃₁	1	4	yes	complemented
301	#97	—	2	8	no	—
302	C₄×C₈	—	2	4	no	—
303	C₃₁	#22510	1	1	yes	characteristic, direct product factor, Sylow
304	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
305	C₂×Q₈	—	2	32	no	—
306	C₂×Q₈	—	2	4	no	—
307	C₂×Q₈	—	2	2	no	—
308	C₂×Q₈	—	2	4	no	—
309	C₂×Q₈	—	2	4	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂×Q₈	—	2	2	no	—
312	C₂×Q₈	—	2	2	no	—
313	C₂×Q₈	C₂²⋊C₄×C₃₁	1	4	yes	—
314	C₂×Q₈	M₁₆×C₃₁	1	4	yes	—
315	C₂²×C₄	—	2	4	no	—
316	C₂²×C₄	—	2	4	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	4	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	—	2	4	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	4	no	—
326	C₂²×C₄	—	4	1	no	—
327	C₂²×C₄	—	4	2	no	—
328	C₂²×C₄	M₁₆×C₃₁	1	4	yes	—
329	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
330	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
331	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
332	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
333	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
334	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
335	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
336	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
337	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
338	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
339	C₂²×C₄	#41×C₃₁	1	2	yes	—
340	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	4	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	4	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	4	no	—
349	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	complemented
350	C₄⋊C₄	—	2	4	no	—
351	C₄⋊C₄	—	2	4	no	—
352	C₄⋊C₄	—	2	4	no	—
353	C₄⋊C₄	—	2	4	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
356	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
357	C₄²	—	2	2	no	—
358	C₄²	—	2	4	no	—
359	C₄²	—	2	4	no	—
360	C₄²	—	2	2	no	—
361	C₄²	—	2	2	no	—
362	C₄²	—	2	4	no	—
363	C₄²	C₂×D₄×C₃₁	1	4	yes	—
364	C₄²	C₂×D₄×C₃₁	1	8	yes	—
365	C₂³	—	2	1	no	—
366	C₂³	—	2	1	no	—
367	C₂³	—	2	1	no	—
368	C₂³	—	2	1	no	—
369	C₂³	C₄×Q₈×C₃₁	1	2	yes	—
370	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
371	C₂³	C₃₁×#131	1	2	yes	—
372	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
373	Q₈	—	2	16	no	—
374	Q₈	—	2	8	no	—
375	Q₈	—	2	8	no	—
376	C₂×C₄	—	2	16	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	8	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	4	4	no	—
399	C₂×C₄	—	4	8	no	—
400	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
401	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
402	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
403	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
404	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
405	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
406	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
407	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
408	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
409	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
410	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
411	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
412	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
413	C₈	—	2	4	no	—
414	C₈	—	4	2	no	—
415	C₈	—	4	4	no	—
416	C₂²	—	2	2	no	—
417	C₂²	—	2	2	no	—
418	C₂²	—	2	2	no	—
419	C₂²	—	2	2	no	—
420	C₂²	—	2	2	no	—
421	C₂²	—	2	2	no	—
422	C₂²	C₃₁×#446	1	4	yes	complemented
423	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
424	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
425	C₂²	C₃₁×#519	1	1	yes	characteristic
426	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
427	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
428	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
429	C₂²	C₃₁×#568	1	1	yes	characteristic
430	C₄	—	2	2	no	—
431	C₄	—	2	4	no	—
432	C₄	—	2	4	no	—
433	C₄	—	2	2	no	—
434	C₄	—	2	2	no	—
435	C₄	—	2	4	no	—
436	C₄	—	4	2	no	—
437	C₄	—	4	1	no	—
438	C₄	C₃₁×#407	1	4	yes	—
439	C₄	C₃₁×#548	1	2	yes	—
440	C₄	D₄×Q₈×C₃₁	1	2	yes	—
441	C₂	—	2	4	no	—
442	C₂	Q₈×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
443	C₂	C₂×C₃₁×#407	1	1	yes	characteristic
444	C₂	C₃₁×#2880	1	1	yes	characteristic
445	C₂	C₂×C₃₁×#446	1	1	yes	characteristic
446	C₂	Q₈×M₁₆×C₃₁	1	1	yes	characteristic
447	C₂	C₃₁×#2983	1	1	yes	characteristic
448	C₂	C₃₁×#2984	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#446	C₂	semidirect
3. C₃₁×#1985	C₂	semidirect
4. C₃₁×#1984	C₂	semidirect
76. C₃₁×C₂²⋊C₈	Q₈	semidirect
124. C₂×C₈×C₃₁	C₂×Q₈	semidirect
140. #22510	C₃₁	direct (N × H)
193. C₂×#446	C₂×C₃₁	semidirect
194. #1985	C₂×C₃₁	semidirect
195. #1984	C₂×C₃₁	semidirect
206. C₂²×C₃₁	#446	semidirect
300. C₂²⋊C₈	Q₈×C₃₁	semidirect
303. C₃₁	#22510	direct (N × H)
349. C₂×C₈	C₂×Q₈×C₃₁	semidirect
422. C₂²	C₃₁×#446	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₃₁	2
C₂×C₃₁×#446	2
C₃₁×#1985	2
C₃₁×#1984	2
C₃₁×#1797	2
C₃₁×#1749	2
C₄×C₃₁×C₂²⋊C₈	2
#22510	31

Subgroups — C31×#22510 #21479269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22510 #21479269