Subgroups — C₃₁×#24330 #21481089

Order 7936.

1262 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#475	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2582	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2058	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2024	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1877	C₂	1	2	yes	maximal
10	C₃₁×#1864	C₂	1	2	yes	maximal
11	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic
12	C₂³×Q₈×C₃₁	—	2	1	no	—
13	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	D₄×Q₈×C₃₁	—	2	4	no	—
15	D₄×Q₈×C₃₁	—	2	4	no	—
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
19	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
20	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#475	—	2	4	no	—
25	C₄×SD₁₆×C₃₁	—	2	4	no	—
26	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#106	C₂²	1	2	yes	—
28	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#103	C₂²	1	2	yes	—
30	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#102	C₂²	1	2	yes	—
32	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#398	—	2	2	no	—
35	C₃₁×#391	—	2	1	no	—
36	C₃₁×#391	C₂²	1	2	yes	—
37	C₃₁×#389	—	2	1	no	—
38	C₃₁×#389	C₂²	1	2	yes	—
39	C₃₁×#384	C₂²	1	1	yes	characteristic
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₃₁×#336	C₂²	1	2	yes	—
45	C₂²×Q₈×C₃₁	—	2	8	no	—
46	C₂²×Q₈×C₃₁	—	2	2	no	—
47	C₂²×Q₈×C₃₁	—	2	1	no	—
48	C₂²×Q₈×C₃₁	—	4	4	no	—
49	C₂²×Q₈×C₃₁	—	4	4	no	—
50	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
51	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₃₁	—	2	2	no	—
54	C₂³×C₄×C₃₁	—	2	1	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×SD₁₆×C₃₁	—	2	2	no	—
57	C₂×SD₁₆×C₃₁	Q₈	1	4	yes	complemented
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×C₄⋊Q₈	—	2	4	no	—
62	C₃₁×C₄⋊Q₈	—	2	4	no	—
63	C₃₁×C₄⋊Q₈	—	2	2	no	—
64	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
65	C₃₁×C₂²⋊Q₈	—	2	8	no	—
66	C₃₁×C₂²⋊Q₈	—	2	16	no	—
67	C₄×Q₈×C₃₁	—	2	2	no	—
68	C₄×Q₈×C₃₁	—	2	2	no	—
69	C₄×Q₈×C₃₁	—	2	2	no	—
70	C₄×Q₈×C₃₁	—	2	2	no	—
71	C₄×D₄×C₃₁	—	2	4	no	—
72	C₄×D₄×C₃₁	—	2	4	no	—
73	C₄×D₄×C₃₁	—	2	2	no	—
74	C₄×D₄×C₃₁	D₄	1	4	yes	—
75	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
82	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
89	C₂×C₄²×C₃₁	—	2	1	no	—
90	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
91	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
92	C₃₁×#106	—	2	2	no	—
93	C₃₁×#106	—	2	4	no	—
94	C₃₁×C₄⋊C₈	—	2	2	no	—
95	C₃₁×#103	—	2	2	no	—
96	C₃₁×#103	—	2	4	no	—
97	C₃₁×#102	—	2	2	no	—
98	C₃₁×#102	—	2	4	no	—
99	C₄×C₈×C₃₁	—	2	2	no	—
100	C₃₁×#95	—	2	2	no	—
101	C₃₁×#95	—	2	2	no	—
102	C₃₁×#95	—	2	2	no	—
103	C₂⁴×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	—	2	8	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	4	no	—
107	C₂×Q₈×C₃₁	—	2	2	no	—
108	C₂×Q₈×C₃₁	—	2	1	no	—
109	C₂×Q₈×C₃₁	—	2	1	no	—
110	C₂×Q₈×C₃₁	—	2	1	no	—
111	C₂×Q₈×C₃₁	—	2	1	no	—
112	C₂×Q₈×C₃₁	—	2	1	no	—
113	C₂×Q₈×C₃₁	—	4	4	no	—
114	C₂×Q₈×C₃₁	—	4	32	no	—
115	C₂×Q₈×C₃₁	—	4	4	no	—
116	C₂×Q₈×C₃₁	—	4	2	no	—
117	C₂×Q₈×C₃₁	—	4	2	no	—
118	C₂×Q₈×C₃₁	SD₁₆	1	4	yes	complemented
119	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	complemented
120	C₂×D₄×C₃₁	—	2	1	no	—
121	C₂×D₄×C₃₁	—	2	1	no	—
122	C₂×D₄×C₃₁	—	2	1	no	—
123	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
124	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
125	C₂×D₄×C₃₁	#41	1	2	yes	—
126	C₂²×C₄×C₃₁	—	2	8	no	—
127	C₂²×C₄×C₃₁	—	2	1	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	1	no	—
130	C₂²×C₄×C₃₁	—	2	4	no	—
131	C₂²×C₄×C₃₁	—	2	2	no	—
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	—	4	4	no	—
134	C₂²×C₄×C₃₁	—	4	4	no	—
135	C₂²×C₄×C₃₁	—	4	2	no	—
136	C₂²×C₄×C₃₁	—	4	2	no	—
137	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
142	SD₁₆×C₃₁	—	2	8	no	—
143	C₂×C₈×C₃₁	—	2	1	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	1	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	—	2	2	no	—
148	C₂×C₈×C₃₁	—	2	1	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	complemented
152	C₄⋊C₄×C₃₁	—	2	8	no	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₄⋊C₄×C₃₁	—	2	4	no	—
155	C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	—	2	1	no	—
158	C₄⋊C₄×C₃₁	—	2	1	no	—
159	C₄⋊C₄×C₃₁	—	2	1	no	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₄⋊C₄×C₃₁	—	2	2	no	—
162	C₄⋊C₄×C₃₁	—	4	2	no	—
163	C₄⋊C₄×C₃₁	—	4	2	no	—
164	C₄⋊C₄×C₃₁	—	8	1	no	—
165	C₄⋊C₄×C₃₁	—	8	2	no	—
166	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
167	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
168	C₂²⋊C₄×C₃₁	—	2	8	no	—
169	C₂²⋊C₄×C₃₁	—	2	4	no	—
170	C₄²×C₃₁	—	2	2	no	—
171	C₄²×C₃₁	—	2	2	no	—
172	C₄²×C₃₁	—	4	2	no	—
173	C₄²×C₃₁	C₂×D₄	1	2	yes	—
174	C₄²×C₃₁	C₂×D₄	1	2	yes	—
175	#24330	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂³×C₃₁	—	2	2	no	—
177	C₂³×C₃₁	—	2	2	no	—
178	C₂³×C₃₁	—	2	2	no	—
179	C₂³×C₃₁	—	4	1	no	—
180	C₂³×C₃₁	—	4	1	no	—
181	C₂³×C₃₁	—	4	1	no	—
182	C₂³×C₃₁	—	4	1	no	—
183	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
184	Q₈×C₃₁	—	2	8	no	—
185	Q₈×C₃₁	—	2	2	no	—
186	Q₈×C₃₁	—	2	2	no	—
187	Q₈×C₃₁	—	4	16	no	—
188	Q₈×C₃₁	—	4	1	no	—
189	Q₈×C₃₁	—	4	1	no	—
190	Q₈×C₃₁	—	4	8	no	—
191	D₄×C₃₁	—	2	2	no	—
192	D₄×C₃₁	—	2	2	no	—
193	D₄×C₃₁	—	2	2	no	—
194	D₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	1	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	4	16	no	—
206	C₂×C₄×C₃₁	—	4	1	no	—
207	C₂×C₄×C₃₁	—	4	1	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	1	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	8	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	2	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	—	4	1	no	—
218	C₂×C₄×C₃₁	—	4	1	no	—
219	C₂×C₄×C₃₁	#120	1	2	yes	—
220	C₂×C₄×C₃₁	#120	1	2	yes	—
221	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
222	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
223	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
224	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
227	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
228	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
229	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
230	C₈×C₃₁	—	2	2	no	—
231	C₈×C₃₁	—	4	1	no	—
232	C₈×C₃₁	—	4	2	no	—
233	C₂×D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
234	C₂×#475	C₂×C₃₁	1	1	yes	characteristic, complemented
235	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
236	#2582	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#2058	C₂×C₃₁	1	2	yes	complemented
238	#2024	C₂×C₃₁	1	2	yes	complemented
239	#1990	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#1877	C₂×C₃₁	1	2	yes	—
242	#1864	C₂×C₃₁	1	2	yes	—
243	#1802	C₂×C₃₁	1	1	yes	characteristic
244	C₂²×C₃₁	—	2	4	no	—
245	C₂²×C₃₁	—	4	2	no	—
246	C₂²×C₃₁	—	4	2	no	—
247	C₂²×C₃₁	—	4	2	no	—
248	C₂²×C₃₁	—	4	2	no	—
249	C₂²×C₃₁	—	4	2	no	—
250	C₂²×C₃₁	—	4	2	no	—
251	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
252	C₂²×C₃₁	#548	1	1	yes	characteristic
253	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
254	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
255	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
256	C₂²×C₃₁	#577	1	1	yes	characteristic
257	C₂²×C₃₁	#578	1	1	yes	characteristic
258	C₄×C₃₁	—	2	4	no	—
259	C₄×C₃₁	—	2	2	no	—
260	C₄×C₃₁	—	4	2	no	—
261	C₄×C₃₁	—	4	4	no	—
262	C₄×C₃₁	—	4	2	no	—
263	C₄×C₃₁	—	4	2	no	—
264	C₄×C₃₁	—	8	2	no	—
265	C₄×C₃₁	—	8	1	no	—
266	C₄×C₃₁	#450	1	4	yes	—
267	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
268	C₄×C₃₁	#536	1	1	yes	characteristic
269	C₄×C₃₁	#548	1	1	yes	characteristic
270	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
271	C₂³×Q₈	—	2	1	no	—
272	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
273	D₄×Q₈	—	2	4	no	—
274	D₄×Q₈	—	2	4	no	—
275	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
276	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×C₂²⋊Q₈	—	2	1	no	—
278	C₂×C₂²⋊Q₈	—	2	2	no	—
279	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
282	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
283	#475	—	2	4	no	—
284	C₄×SD₁₆	—	2	4	no	—
285	C₂×#106	C₂²×C₃₁	1	2	yes	—
286	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×#103	C₂²×C₃₁	1	2	yes	—
289	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×#102	C₂²×C₃₁	1	2	yes	—
291	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
293	#398	—	2	2	no	—
294	#391	—	2	1	no	—
295	#391	C₂²×C₃₁	1	2	yes	—
296	#389	—	2	1	no	—
297	#389	C₂²×C₃₁	1	2	yes	—
298	#384	C₂²×C₃₁	1	1	yes	characteristic
299	#340	C₂²×C₃₁	1	2	yes	—
300	#340	C₂²×C₃₁	1	2	yes	—
301	#340	C₂²×C₃₁	1	2	yes	—
302	#336	C₂²×C₃₁	1	2	yes	—
303	#336	C₂²×C₃₁	1	2	yes	—
304	C₂×C₃₁	—	4	4	no	—
305	C₂×C₃₁	#2409	1	1	yes	characteristic
306	C₂×C₃₁	C₂×#450	1	1	yes	characteristic
307	C₂×C₃₁	#3013	1	1	yes	characteristic
308	C₂×C₃₁	#3359	1	1	yes	characteristic
309	C₂×C₃₁	#3373	1	1	yes	characteristic
310	C₂×C₃₁	Q₈×SD₁₆	1	1	yes	characteristic
311	C₂×C₃₁	#3386	1	1	yes	characteristic
312	C₂²×Q₈	—	2	8	no	—
313	C₂²×Q₈	—	2	2	no	—
314	C₂²×Q₈	—	2	1	no	—
315	C₂²×Q₈	—	4	4	no	—
316	C₂²×Q₈	—	4	4	no	—
317	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
318	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
319	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
320	C₂³×C₄	—	2	2	no	—
321	C₂³×C₄	—	2	1	no	—
322	C₂×SD₁₆	—	2	2	no	—
323	C₂×SD₁₆	—	2	2	no	—
324	C₂×SD₁₆	Q₈×C₃₁	1	4	yes	complemented
325	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
326	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
327	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
328	C₄⋊Q₈	—	2	4	no	—
329	C₄⋊Q₈	—	2	4	no	—
330	C₄⋊Q₈	—	2	2	no	—
331	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
332	C₂²⋊Q₈	—	2	8	no	—
333	C₂²⋊Q₈	—	2	16	no	—
334	C₄×Q₈	—	2	2	no	—
335	C₄×Q₈	—	2	2	no	—
336	C₄×Q₈	—	2	2	no	—
337	C₄×Q₈	—	2	2	no	—
338	C₄×D₄	—	2	4	no	—
339	C₄×D₄	—	2	4	no	—
340	C₄×D₄	—	2	2	no	—
341	C₄×D₄	D₄×C₃₁	1	4	yes	—
342	C₂×C₄⋊C₄	—	2	2	no	—
343	C₂×C₄⋊C₄	—	2	1	no	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	—	2	1	no	—
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	—	2	2	no	—
348	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
349	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
350	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
351	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
352	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
353	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
354	C₂×C₂²⋊C₄	—	2	2	no	—
355	C₂×C₂²⋊C₄	—	2	1	no	—
356	C₂×C₄²	—	2	1	no	—
357	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
358	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
359	#106	—	2	2	no	—
360	#106	—	2	4	no	—
361	C₄⋊C₈	—	2	2	no	—
362	#103	—	2	2	no	—
363	#103	—	2	4	no	—
364	#102	—	2	2	no	—
365	#102	—	2	4	no	—
366	C₄×C₈	—	2	2	no	—
367	#95	—	2	2	no	—
368	#95	—	2	2	no	—
369	#95	—	2	2	no	—
370	C₃₁	#24330	1	1	yes	characteristic, direct product factor, Sylow
371	C₂⁴	—	2	1	no	—
372	C₂×Q₈	—	2	8	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	2	4	no	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	—	2	1	no	—
377	C₂×Q₈	—	2	1	no	—
378	C₂×Q₈	—	2	1	no	—
379	C₂×Q₈	—	2	1	no	—
380	C₂×Q₈	—	2	1	no	—
381	C₂×Q₈	—	4	4	no	—
382	C₂×Q₈	—	4	32	no	—
383	C₂×Q₈	—	4	4	no	—
384	C₂×Q₈	—	4	2	no	—
385	C₂×Q₈	—	4	2	no	—
386	C₂×Q₈	SD₁₆×C₃₁	1	4	yes	complemented
387	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	complemented
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	1	no	—
391	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
392	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
393	C₂×D₄	#41×C₃₁	1	2	yes	—
394	C₂²×C₄	—	2	8	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	4	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	4	4	no	—
402	C₂²×C₄	—	4	4	no	—
403	C₂²×C₄	—	4	2	no	—
404	C₂²×C₄	—	4	2	no	—
405	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
406	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
407	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
408	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
409	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
410	SD₁₆	—	2	8	no	—
411	C₂×C₈	—	2	1	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	complemented
420	C₄⋊C₄	—	2	8	no	—
421	C₄⋊C₄	—	2	4	no	—
422	C₄⋊C₄	—	2	4	no	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	2	no	—
430	C₄⋊C₄	—	4	2	no	—
431	C₄⋊C₄	—	4	2	no	—
432	C₄⋊C₄	—	8	1	no	—
433	C₄⋊C₄	—	8	2	no	—
434	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
435	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
436	C₂²⋊C₄	—	2	8	no	—
437	C₂²⋊C₄	—	2	4	no	—
438	C₄²	—	2	2	no	—
439	C₄²	—	2	2	no	—
440	C₄²	—	4	2	no	—
441	C₄²	C₂×D₄×C₃₁	1	2	yes	—
442	C₄²	C₂×D₄×C₃₁	1	2	yes	—
443	C₂³	—	2	2	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	2	2	no	—
446	C₂³	—	4	1	no	—
447	C₂³	—	4	1	no	—
448	C₂³	—	4	1	no	—
449	C₂³	—	4	1	no	—
450	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
451	Q₈	—	2	8	no	—
452	Q₈	—	2	2	no	—
453	Q₈	—	2	2	no	—
454	Q₈	—	4	16	no	—
455	Q₈	—	4	8	no	—
456	Q₈	—	4	1	no	—
457	Q₈	—	4	1	no	—
458	D₄	—	2	2	no	—
459	D₄	—	2	2	no	—
460	D₄	—	2	2	no	—
461	D₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	4	16	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	8	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	1	no	—
486	C₂×C₄	C₃₁×#120	1	2	yes	—
487	C₂×C₄	C₃₁×#120	1	2	yes	—
488	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
489	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
490	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
491	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
492	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
493	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
494	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
495	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
496	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
497	C₈	—	2	2	no	—
498	C₈	—	4	1	no	—
499	C₈	—	4	2	no	—
500	C₂²	—	2	4	no	—
501	C₂²	—	4	2	no	—
502	C₂²	—	4	2	no	—
503	C₂²	—	4	2	no	—
504	C₂²	—	4	2	no	—
505	C₂²	—	4	2	no	—
506	C₂²	—	4	2	no	—
507	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
508	C₂²	C₃₁×#548	1	1	yes	characteristic
509	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
510	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
511	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
512	C₂²	C₃₁×#577	1	1	yes	characteristic
513	C₂²	C₃₁×#578	1	1	yes	characteristic
514	C₄	—	2	4	no	—
515	C₄	—	2	2	no	—
516	C₄	—	4	2	no	—
517	C₄	—	4	4	no	—
518	C₄	—	4	2	no	—
519	C₄	—	4	2	no	—
520	C₄	—	8	2	no	—
521	C₄	—	8	1	no	—
522	C₄	C₃₁×#450	1	4	yes	—
523	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
524	C₄	C₃₁×#536	1	1	yes	characteristic
525	C₄	C₃₁×#548	1	1	yes	characteristic
526	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
527	C₂	—	4	4	no	—
528	C₂	C₃₁×#2409	1	1	yes	characteristic
529	C₂	C₂×C₃₁×#450	1	1	yes	characteristic
530	C₂	C₃₁×#3013	1	1	yes	characteristic
531	C₂	C₃₁×#3359	1	1	yes	characteristic
532	C₂	C₃₁×#3373	1	1	yes	characteristic
533	C₂	Q₈×SD₁₆×C₃₁	1	1	yes	characteristic
534	C₂	C₃₁×#3386	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#475	C₂	semidirect
4. C₃₁×#2582	C₂	semidirect
5. C₃₁×#2058	C₂	semidirect
6. C₃₁×#2024	C₂	semidirect
7. C₃₁×#1990	C₂	semidirect
8. C₃₁×#1984	C₂	semidirect
57. C₂×SD₁₆×C₃₁	Q₈	semidirect
118. C₂×Q₈×C₃₁	SD₁₆	semidirect
119. C₂×Q₈×C₃₁	C₂×Q₈	semidirect
151. C₂×C₈×C₃₁	C₂×Q₈	semidirect
175. #24330	C₃₁	direct (N × H)
234. C₂×#475	C₂×C₃₁	semidirect
236. #2582	C₂×C₃₁	semidirect
237. #2058	C₂×C₃₁	semidirect
238. #2024	C₂×C₃₁	semidirect
239. #1990	C₂×C₃₁	semidirect
240. #1984	C₂×C₃₁	semidirect
324. C₂×SD₁₆	Q₈×C₃₁	semidirect
370. C₃₁	#24330	direct (N × H)
386. C₂×Q₈	SD₁₆×C₃₁	semidirect
387. C₂×Q₈	C₂×Q₈×C₃₁	semidirect
419. 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
C₂×D₄×Q₈×C₃₁	2
C₂×C₃₁×#475	2
C₂×C₄×SD₁₆×C₃₁	2
C₃₁×#2582	2
C₃₁×#2058	2
C₃₁×#2024	2
C₃₁×#1990	2
C₃₁×#1984	2
C₃₁×#1877	2
C₃₁×#1864	2
C₃₁×#1802	2
#24330	31

Subgroups — C31×#24330 #21481089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24330 #21481089