Subgroups — C₃₁×#24342 #21481101

Order 7936.

1248 conjugacy classes of subgroups.

The subgroup lattice has 498 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×#477	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#440	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2581	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2057	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2024	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1877	C₂	1	2	yes	maximal
10	C₃₁×#1862	C₂	1	2	yes	maximal
11	C₃₁×#1804	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	1	no	—
19	C₂×C₃₁×C₂²⋊Q₈	—	2	2	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×#477	—	2	4	no	—
25	C₃₁×#440	—	2	4	no	—
26	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#107	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×C₃₁×#97	C₂²	1	1	yes	characteristic
34	C₃₁×#400	—	2	1	no	—
35	C₃₁×#395	—	2	1	no	—
36	C₃₁×#391	C₂²	1	2	yes	—
37	C₃₁×#389	C₂²	1	2	yes	—
38	C₃₁×#387	—	2	2	no	—
39	C₃₁×#382	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	1	no	—
47	C₂²×Q₈×C₃₁	—	2	2	no	—
48	C₂²×Q₈×C₃₁	—	4	8	no	—
49	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
50	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₃₁	—	2	2	no	—
53	C₂³×C₄×C₃₁	—	2	1	no	—
54	C₂×SD₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×SD₁₆×C₃₁	Q₈	1	4	yes	complemented
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
59	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×C₄⋊Q₈	—	2	4	no	—
61	C₃₁×C₄⋊Q₈	—	2	4	no	—
62	C₃₁×C₄⋊Q₈	—	2	2	no	—
63	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
64	C₃₁×C₂²⋊Q₈	—	2	8	no	—
65	C₃₁×C₂²⋊Q₈	—	2	16	no	—
66	C₄×Q₈×C₃₁	—	2	2	no	—
67	C₄×Q₈×C₃₁	—	2	2	no	—
68	C₄×Q₈×C₃₁	—	2	2	no	—
69	C₄×Q₈×C₃₁	—	2	2	no	—
70	C₄×D₄×C₃₁	—	2	4	no	—
71	C₄×D₄×C₃₁	—	2	4	no	—
72	C₄×D₄×C₃₁	—	2	2	no	—
73	C₄×D₄×C₃₁	D₄	1	4	yes	—
74	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₄²×C₃₁	—	2	1	no	—
87	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
88	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
89	C₃₁×#107	—	2	2	no	—
90	C₃₁×#107	—	2	4	no	—
91	C₃₁×C₄⋊C₈	—	2	2	no	—
92	C₃₁×#103	—	2	2	no	—
93	C₃₁×#103	—	2	4	no	—
94	C₃₁×#102	—	2	2	no	—
95	C₃₁×#102	—	2	4	no	—
96	C₃₁×#97	—	2	2	no	—
97	C₃₁×#95	—	2	1	no	—
98	C₃₁×#95	—	2	2	no	—
99	C₃₁×#95	—	2	2	no	—
100	C₃₁×#95	—	2	2	no	—
101	C₃₁×#95	—	2	1	no	—
102	C₂⁴×C₃₁	—	2	1	no	—
103	C₂×Q₈×C₃₁	—	2	8	no	—
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	—	2	4	no	—
106	C₂×Q₈×C₃₁	—	2	4	no	—
107	C₂×Q₈×C₃₁	—	2	1	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	—	4	4	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	C₂×Q₈	1	2	yes	complemented
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	1	no	—
119	C₂×D₄×C₃₁	—	2	1	no	—
120	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
121	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
122	C₂×D₄×C₃₁	#41	1	2	yes	—
123	C₂²×C₄×C₃₁	—	2	8	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	2	1	no	—
127	C₂²×C₄×C₃₁	—	2	4	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	1	no	—
130	C₂²×C₄×C₃₁	—	4	4	no	—
131	C₂²×C₄×C₃₁	—	4	4	no	—
132	C₂²×C₄×C₃₁	—	4	4	no	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
138	SD₁₆×C₃₁	—	2	8	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	—	2	2	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	C₂×Q₈	1	2	yes	complemented
148	C₄⋊C₄×C₃₁	—	2	8	no	—
149	C₄⋊C₄×C₃₁	—	2	4	no	—
150	C₄⋊C₄×C₃₁	—	2	4	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	2	2	no	—
153	C₄⋊C₄×C₃₁	—	2	1	no	—
154	C₄⋊C₄×C₃₁	—	2	1	no	—
155	C₄⋊C₄×C₃₁	—	2	1	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	—	2	2	no	—
158	C₄⋊C₄×C₃₁	—	4	2	no	—
159	C₄⋊C₄×C₃₁	—	4	4	no	—
160	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
161	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
162	C₂²⋊C₄×C₃₁	—	2	8	no	—
163	C₂²⋊C₄×C₃₁	—	2	4	no	—
164	C₄²×C₃₁	—	2	2	no	—
165	C₄²×C₃₁	—	2	2	no	—
166	C₄²×C₃₁	—	4	2	no	—
167	C₄²×C₃₁	C₂×D₄	1	2	yes	—
168	C₄²×C₃₁	C₂×D₄	1	2	yes	—
169	#24342	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂³×C₃₁	—	2	2	no	—
171	C₂³×C₃₁	—	2	2	no	—
172	C₂³×C₃₁	—	2	2	no	—
173	C₂³×C₃₁	—	4	2	no	—
174	C₂³×C₃₁	—	4	2	no	—
175	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
176	Q₈×C₃₁	—	2	8	no	—
177	Q₈×C₃₁	—	2	2	no	—
178	Q₈×C₃₁	—	2	2	no	—
179	Q₈×C₃₁	—	4	8	no	—
180	Q₈×C₃₁	—	4	16	no	—
181	Q₈×C₃₁	—	4	1	no	—
182	Q₈×C₃₁	—	4	1	no	—
183	D₄×C₃₁	—	2	2	no	—
184	D₄×C₃₁	—	2	2	no	—
185	D₄×C₃₁	—	2	2	no	—
186	D₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	4	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	4	no	—
196	C₂×C₄×C₃₁	—	4	8	no	—
197	C₂×C₄×C₃₁	—	4	16	no	—
198	C₂×C₄×C₃₁	—	4	1	no	—
199	C₂×C₄×C₃₁	—	4	1	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	#120	1	2	yes	—
209	C₂×C₄×C₃₁	#120	1	2	yes	—
210	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
211	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
216	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
217	C₈×C₃₁	—	2	2	no	—
218	C₈×C₃₁	—	4	1	no	—
219	C₈×C₃₁	—	4	2	no	—
220	C₂×D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
221	C₂×#477	C₂×C₃₁	1	1	yes	characteristic, complemented
222	C₂×#440	C₂×C₃₁	1	1	yes	characteristic
223	#2581	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#2057	C₂×C₃₁	1	2	yes	complemented
225	#2024	C₂×C₃₁	1	2	yes	complemented
226	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#1985	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#1877	C₂×C₃₁	1	2	yes	—
229	#1862	C₂×C₃₁	1	2	yes	—
230	#1804	C₂×C₃₁	1	1	yes	characteristic
231	C₂²×C₃₁	—	2	4	no	—
232	C₂²×C₃₁	—	4	4	no	—
233	C₂²×C₃₁	—	4	4	no	—
234	C₂²×C₃₁	—	4	2	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
237	C₂²×C₃₁	#548	1	1	yes	characteristic
238	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
239	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
240	C₂²×C₃₁	#578	1	2	yes	—
241	C₄×C₃₁	—	2	2	no	—
242	C₄×C₃₁	—	2	4	no	—
243	C₄×C₃₁	—	2	2	no	—
244	C₄×C₃₁	—	4	2	no	—
245	C₄×C₃₁	—	4	4	no	—
246	C₄×C₃₁	—	4	2	no	—
247	C₄×C₃₁	—	4	2	no	—
248	C₄×C₃₁	—	8	2	no	—
249	C₄×C₃₁	—	8	1	no	—
250	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
251	C₄×C₃₁	#536	1	1	yes	characteristic
252	C₄×C₃₁	#548	1	1	yes	characteristic
253	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
254	C₂³×Q₈	—	2	1	no	—
255	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
256	D₄×Q₈	—	2	4	no	—
257	D₄×Q₈	—	2	4	no	—
258	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
259	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₂²⋊Q₈	—	2	2	no	—
261	C₂×C₂²⋊Q₈	—	2	1	no	—
262	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
265	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
266	#477	—	2	4	no	—
267	#440	—	2	4	no	—
268	C₂×#107	C₂²×C₃₁	1	2	yes	—
269	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×#103	C₂²×C₃₁	1	2	yes	—
272	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×#102	C₂²×C₃₁	1	2	yes	—
274	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
276	#400	—	2	1	no	—
277	#395	—	2	1	no	—
278	#391	C₂²×C₃₁	1	2	yes	—
279	#389	C₂²×C₃₁	1	2	yes	—
280	#387	—	2	2	no	—
281	#382	C₂²×C₃₁	1	1	yes	characteristic
282	#340	C₂²×C₃₁	1	2	yes	—
283	#340	C₂²×C₃₁	1	2	yes	—
284	#340	C₂²×C₃₁	1	2	yes	—
285	#336	C₂²×C₃₁	1	2	yes	—
286	#336	C₂²×C₃₁	1	2	yes	—
287	C₂×C₃₁	—	4	4	no	—
288	C₂×C₃₁	#2409	1	1	yes	characteristic
289	C₂×C₃₁	#3003	1	1	yes	characteristic
290	C₂×C₃₁	#3011	1	1	yes	characteristic
291	C₂×C₃₁	#3367	1	2	yes	—
292	C₂×C₃₁	#3386	1	2	yes	—
293	C₂²×Q₈	—	2	8	no	—
294	C₂²×Q₈	—	2	1	no	—
295	C₂²×Q₈	—	2	2	no	—
296	C₂²×Q₈	—	4	8	no	—
297	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
298	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
299	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
300	C₂³×C₄	—	2	2	no	—
301	C₂³×C₄	—	2	1	no	—
302	C₂×SD₁₆	—	2	2	no	—
303	C₂×SD₁₆	—	2	2	no	—
304	C₂×SD₁₆	Q₈×C₃₁	1	4	yes	complemented
305	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
306	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
307	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
308	C₄⋊Q₈	—	2	4	no	—
309	C₄⋊Q₈	—	2	4	no	—
310	C₄⋊Q₈	—	2	2	no	—
311	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
312	C₂²⋊Q₈	—	2	8	no	—
313	C₂²⋊Q₈	—	2	16	no	—
314	C₄×Q₈	—	2	2	no	—
315	C₄×Q₈	—	2	2	no	—
316	C₄×Q₈	—	2	2	no	—
317	C₄×Q₈	—	2	2	no	—
318	C₄×D₄	—	2	4	no	—
319	C₄×D₄	—	2	4	no	—
320	C₄×D₄	—	2	2	no	—
321	C₄×D₄	D₄×C₃₁	1	4	yes	—
322	C₂×C₄⋊C₄	—	2	2	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	—	2	2	no	—
326	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
327	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
328	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
329	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
330	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
331	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
332	C₂×C₂²⋊C₄	—	2	2	no	—
333	C₂×C₂²⋊C₄	—	2	1	no	—
334	C₂×C₄²	—	2	1	no	—
335	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
336	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
337	#107	—	2	2	no	—
338	#107	—	2	4	no	—
339	C₄⋊C₈	—	2	2	no	—
340	#103	—	2	2	no	—
341	#103	—	2	4	no	—
342	#102	—	2	2	no	—
343	#102	—	2	4	no	—
344	#97	—	2	2	no	—
345	#95	—	2	1	no	—
346	#95	—	2	1	no	—
347	#95	—	2	2	no	—
348	#95	—	2	2	no	—
349	#95	—	2	2	no	—
350	C₃₁	#24342	1	1	yes	characteristic, direct product factor, Sylow
351	C₂⁴	—	2	1	no	—
352	C₂×Q₈	—	2	8	no	—
353	C₂×Q₈	—	2	2	no	—
354	C₂×Q₈	—	2	4	no	—
355	C₂×Q₈	—	2	4	no	—
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	2	1	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	—	4	4	no	—
362	C₂×Q₈	—	4	4	no	—
363	C₂×Q₈	—	4	32	no	—
364	C₂×Q₈	—	4	4	no	—
365	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	complemented
366	C₂×D₄	—	2	1	no	—
367	C₂×D₄	—	2	1	no	—
368	C₂×D₄	—	2	1	no	—
369	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
370	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
371	C₂×D₄	#41×C₃₁	1	2	yes	—
372	C₂²×C₄	—	2	8	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	2	4	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	4	4	no	—
380	C₂²×C₄	—	4	4	no	—
381	C₂²×C₄	—	4	4	no	—
382	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
383	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
386	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
387	SD₁₆	—	2	8	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	complemented
397	C₄⋊C₄	—	2	8	no	—
398	C₄⋊C₄	—	2	4	no	—
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	4	2	no	—
408	C₄⋊C₄	—	4	4	no	—
409	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
410	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
411	C₂²⋊C₄	—	2	8	no	—
412	C₂²⋊C₄	—	2	4	no	—
413	C₄²	—	2	2	no	—
414	C₄²	—	2	2	no	—
415	C₄²	—	4	2	no	—
416	C₄²	C₂×D₄×C₃₁	1	2	yes	—
417	C₄²	C₂×D₄×C₃₁	1	2	yes	—
418	C₂³	—	2	2	no	—
419	C₂³	—	2	2	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	—	4	2	no	—
423	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
424	Q₈	—	2	8	no	—
425	Q₈	—	2	2	no	—
426	Q₈	—	2	2	no	—
427	Q₈	—	4	8	no	—
428	Q₈	—	4	16	no	—
429	Q₈	—	4	1	no	—
430	Q₈	—	4	1	no	—
431	D₄	—	2	2	no	—
432	D₄	—	2	2	no	—
433	D₄	—	2	2	no	—
434	D₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	4	8	no	—
445	C₂×C₄	—	4	16	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	C₃₁×#120	1	2	yes	—
457	C₂×C₄	C₃₁×#120	1	2	yes	—
458	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
459	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
460	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
461	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
462	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
463	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
464	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
465	C₈	—	2	2	no	—
466	C₈	—	4	1	no	—
467	C₈	—	4	2	no	—
468	C₂²	—	2	4	no	—
469	C₂²	—	4	4	no	—
470	C₂²	—	4	4	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
474	C₂²	C₃₁×#548	1	1	yes	characteristic
475	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
476	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
477	C₂²	C₃₁×#578	1	2	yes	—
478	C₄	—	2	2	no	—
479	C₄	—	2	4	no	—
480	C₄	—	2	2	no	—
481	C₄	—	4	2	no	—
482	C₄	—	4	4	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	2	no	—
485	C₄	—	8	2	no	—
486	C₄	—	8	1	no	—
487	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
488	C₄	C₃₁×#536	1	1	yes	characteristic
489	C₄	C₃₁×#548	1	1	yes	characteristic
490	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
491	C₂	—	4	4	no	—
492	C₂	C₃₁×#2409	1	1	yes	characteristic
493	C₂	C₃₁×#3003	1	1	yes	characteristic
494	C₂	C₃₁×#3011	1	1	yes	characteristic
495	C₂	C₃₁×#3367	1	2	yes	—
496	C₂	C₃₁×#3386	1	2	yes	—

	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×#477	C₂	semidirect
4. C₃₁×#2581	C₂	semidirect
5. C₃₁×#2057	C₂	semidirect
6. C₃₁×#2024	C₂	semidirect
7. C₃₁×#1993	C₂	semidirect
8. C₃₁×#1985	C₂	semidirect
56. C₂×SD₁₆×C₃₁	Q₈	semidirect
116. C₂×Q₈×C₃₁	C₂×Q₈	semidirect
147. C₂×C₈×C₃₁	C₂×Q₈	semidirect
169. #24342	C₃₁	direct (N × H)
221. C₂×#477	C₂×C₃₁	semidirect
223. #2581	C₂×C₃₁	semidirect
224. #2057	C₂×C₃₁	semidirect
225. #2024	C₂×C₃₁	semidirect
226. #1993	C₂×C₃₁	semidirect
227. #1985	C₂×C₃₁	semidirect
304. C₂×SD₁₆	Q₈×C₃₁	semidirect
350. C₃₁	#24342	direct (N × H)
365. C₂×Q₈	C₂×Q₈×C₃₁	semidirect
396. 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×#477	2
C₂×C₃₁×#440	2
C₃₁×#2581	2
C₃₁×#2057	2
C₃₁×#2024	2
C₃₁×#1993	2
C₃₁×#1985	2
C₃₁×#1877	2
C₃₁×#1862	2
C₃₁×#1804	2
#24342	31

Subgroups — C31×#24342 #21481101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24342 #21481101