Subgroups — C₃₁×#61321 #21518080

Order 7936.

2126 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3368	C₂	1	6	yes	maximal, complemented
4	C₃₁×#3366	C₂	1	6	yes	maximal, complemented
5	C₃₁×#3362	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3358	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3138	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3077	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3038	C₂	1	3	yes	maximal, complemented
10	Q₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×AGL(1,ℤ/8)	C₂	1	3	yes	maximal, complemented
12	C₂²×#41×C₃₁	—	2	1	no	—
13	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	3	yes	—
14	Q₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
15	C₃₁×#557	C₂²	1	3	yes	complemented
16	C₃₁×#554	—	2	3	no	—
17	C₃₁×#554	C₂²	1	3	yes	complemented
18	C₃₁×#550	—	2	1	no	—
19	C₃₁×#550	—	2	3	no	—
20	C₃₁×#550	C₂²	1	2	yes	—
21	C₃₁×#550	C₂²	1	6	yes	—
22	D₄×Q₈×C₃₁	—	2	1	no	—
23	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×#533	—	2	3	no	—
25	C₃₁×#533	C₂²	1	3	yes	—
26	C₂×C₃₁×#127	C₂²	1	3	yes	—
27	C₂×C₃₁×#121	—	2	3	no	—
28	C₄×#41×C₃₁	C₂²	1	3	yes	—
29	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
30	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
32	C₃₁×#490	C₂²	1	12	yes	complemented
33	C₃₁×#486	C₂²	1	6	yes	complemented
34	C₃₁×#486	C₂²	1	6	yes	complemented
35	C₃₁×#478	C₂²	1	12	yes	complemented
36	C₃₁×#476	C₂²	1	6	yes	complemented
37	C₃₁×#474	C₂²	1	6	yes	complemented
38	C₃₁×#460	C₂²	1	6	yes	complemented
39	C₃₁×#460	C₂²	1	6	yes	complemented
40	C₃₁×#459	C₂²	1	6	yes	complemented
41	C₃₁×#459	C₂²	1	6	yes	complemented
42	C₃₁×#446	C₂²	1	6	yes	complemented
43	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
44	C₃₁×#442	C₂²	1	6	yes	complemented
45	C₃₁×#440	C₂²	1	6	yes	complemented
46	C₄×SD₁₆×C₃₁	C₂²	1	6	yes	complemented
47	C₄×D₈×C₃₁	C₂²	1	6	yes	complemented
48	C₃₁×#428	C₂²	1	3	yes	—
49	C₃₁×#423	C₂²	1	3	yes	—
50	C₃₁×#418	C₂²	1	6	yes	—
51	C₃₁×#418	C₂²	1	3	yes	—
52	C₃₁×#417	C₂²	1	1	yes	characteristic
53	C₄×M₁₆×C₃₁	C₂²	1	3	yes	—
54	C₂×#41×C₃₁	—	2	6	no	—
55	C₂×#41×C₃₁	—	2	2	no	—
56	C₂×#41×C₃₁	—	4	8	no	—
57	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×Q₈×C₃₁	—	2	1	no	—
59	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
60	C₂²×D₄×C₃₁	—	2	3	no	—
61	C₂²×D₄×C₃₁	—	2	3	no	—
62	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
63	C₂³×C₄×C₃₁	—	2	3	no	—
64	C₃₁×AGL(1,ℤ/8)	—	2	12	no	—
65	C₂×SD₁₆×C₃₁	C₂³	1	6	yes	—
66	C₂×D₈×C₃₁	C₂³	1	6	yes	—
67	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×M₁₆×C₃₁	C₂³	1	3	yes	—
69	C₃₁×C₄⋊Q₈	—	2	3	no	—
70	C₃₁×C₄⋊Q₈	—	2	3	no	—
71	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
72	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
73	C₃₁×#127	—	2	3	no	—
74	C₃₁×#127	—	2	3	no	—
75	C₃₁×#127	C₂³	1	6	yes	—
76	C₃₁×#127	C₂³	1	6	yes	—
77	C₃₁×#125	—	2	6	no	—
78	C₃₁×#125	C₂³	1	6	yes	—
79	C₃₁×C₂²⋊Q₈	—	2	3	no	—
80	C₃₁×C₂²⋊Q₈	—	2	3	no	—
81	C₃₁×C₂²⋊Q₈	—	2	3	no	—
82	C₃₁×C₂²⋊Q₈	—	2	3	no	—
83	C₃₁×C₂²⋊Q₈	—	2	6	no	—
84	C₃₁×#121	—	2	6	no	—
85	C₃₁×#121	—	2	12	no	—
86	C₃₁×#121	—	2	6	no	—
87	C₄×Q₈×C₃₁	—	2	1	no	—
88	C₄×Q₈×C₃₁	—	2	3	no	—
89	C₄×Q₈×C₃₁	—	2	3	no	—
90	C₄×Q₈×C₃₁	—	2	3	no	—
91	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
92	C₄×Q₈×C₃₁	D₄	1	3	yes	—
93	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
94	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
95	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
96	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
97	C₄×D₄×C₃₁	—	2	3	no	—
98	C₄×D₄×C₃₁	—	2	3	no	—
99	C₄×D₄×C₃₁	—	2	3	no	—
100	C₄×D₄×C₃₁	—	2	3	no	—
101	C₄×D₄×C₃₁	C₂³	1	3	yes	—
102	C₄×D₄×C₃₁	C₂³	1	6	yes	—
103	C₄×D₄×C₃₁	C₂³	1	6	yes	—
104	C₄×D₄×C₃₁	C₂³	1	3	yes	—
105	C₃₁×#117	—	2	3	no	—
106	C₃₁×#117	C₂³	1	3	yes	—
107	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
108	C₂×C₄⋊C₄×C₃₁	D₄	1	3	yes	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
110	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
111	C₂×C₄²×C₃₁	—	2	3	no	—
112	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
113	C₃₁×#107	C₂³	1	6	yes	—
114	C₃₁×#106	C₂³	1	6	yes	—
115	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
116	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
117	C₃₁×#103	C₂³	1	2	yes	—
118	C₃₁×#102	C₂³	1	2	yes	—
119	C₃₁×#102	C₂³	1	6	yes	—
120	C₃₁×#102	C₂³	1	6	yes	—
121	C₃₁×#102	C₂³	1	12	yes	—
122	C₃₁×#102	C₂³	1	12	yes	—
123	C₃₁×#97	C₂³	1	6	yes	—
124	C₄×C₈×C₃₁	C₂³	1	6	yes	—
125	C₂⁴×C₃₁	—	2	3	no	—
126	#41×C₃₁	—	2	2	no	—
127	#41×C₃₁	—	4	24	no	—
128	#41×C₃₁	—	4	8	no	—
129	#41×C₃₁	#41	1	4	yes	complemented
130	C₂×Q₈×C₃₁	—	2	4	no	—
131	C₂×Q₈×C₃₁	—	2	3	no	—
132	C₂×Q₈×C₃₁	—	2	3	no	—
133	C₂×Q₈×C₃₁	—	4	4	no	—
134	C₂×Q₈×C₃₁	C₂×D₄	1	3	yes	—
135	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
137	C₂×D₄×C₃₁	—	2	1	no	—
138	C₂×D₄×C₃₁	—	2	3	no	—
139	C₂×D₄×C₃₁	—	2	6	no	—
140	C₂×D₄×C₃₁	—	2	6	no	—
141	C₂×D₄×C₃₁	—	2	6	no	—
142	C₂×D₄×C₃₁	—	2	6	no	—
143	C₂×D₄×C₃₁	—	2	6	no	—
144	C₂×D₄×C₃₁	—	2	6	no	—
145	C₂×D₄×C₃₁	—	4	12	no	—
146	C₂×D₄×C₃₁	—	8	6	no	—
147	C₂×D₄×C₃₁	C₂⁴	1	6	yes	—
148	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	—	2	1	no	—
150	C₂²×C₄×C₃₁	—	2	1	no	—
151	C₂²×C₄×C₃₁	—	2	3	no	—
152	C₂²×C₄×C₃₁	—	2	6	no	—
153	C₂²×C₄×C₃₁	—	2	6	no	—
154	C₂²×C₄×C₃₁	—	2	3	no	—
155	C₂²×C₄×C₃₁	—	4	12	no	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
157	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
158	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
159	SD₁₆×C₃₁	—	2	12	no	—
160	D₈×C₃₁	—	2	12	no	—
161	M₁₆×C₃₁	—	2	6	no	—
162	M₁₆×C₃₁	—	2	2	no	—
163	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
164	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
165	C₄⋊C₄×C₃₁	—	2	6	no	—
166	C₄⋊C₄×C₃₁	—	2	3	no	—
167	C₄⋊C₄×C₃₁	—	2	1	no	—
168	C₄⋊C₄×C₃₁	—	2	3	no	—
169	C₄⋊C₄×C₃₁	—	2	3	no	—
170	C₄⋊C₄×C₃₁	—	2	3	no	—
171	C₄⋊C₄×C₃₁	—	2	6	no	—
172	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
173	C₄⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
174	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
175	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
176	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
177	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
178	C₂²⋊C₄×C₃₁	—	2	3	no	—
179	C₂²⋊C₄×C₃₁	—	2	6	no	—
180	C₂²⋊C₄×C₃₁	—	2	6	no	—
181	C₂²⋊C₄×C₃₁	—	2	3	no	—
182	C₂²⋊C₄×C₃₁	—	2	3	no	—
183	C₄²×C₃₁	—	2	3	no	—
184	C₄²×C₃₁	—	2	3	no	—
185	C₄²×C₃₁	—	2	3	no	—
186	C₄²×C₃₁	C₂×D₄	1	3	yes	—
187	C₄²×C₃₁	C₂×D₄	1	3	yes	—
188	C₄²×C₃₁	C₂⁴	1	3	yes	—
189	C₄²×C₃₁	C₂⁴	1	3	yes	—
190	#61321	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂³×C₃₁	—	2	1	no	—
192	C₂³×C₃₁	—	2	6	no	—
193	C₂³×C₃₁	—	4	6	no	—
194	C₂³×C₃₁	—	4	6	no	—
195	C₂³×C₃₁	—	4	6	no	—
196	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
197	Q₈×C₃₁	—	2	6	no	—
198	Q₈×C₃₁	—	2	1	no	—
199	Q₈×C₃₁	—	4	6	no	—
200	Q₈×C₃₁	—	8	3	no	—
201	Q₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	complemented
202	Q₈×C₃₁	C₂×#41	1	2	yes	complemented
203	D₄×C₃₁	—	2	4	no	—
204	D₄×C₃₁	—	2	6	no	—
205	D₄×C₃₁	—	2	1	no	—
206	D₄×C₃₁	—	2	6	no	—
207	D₄×C₃₁	—	4	12	no	—
208	D₄×C₃₁	—	4	6	no	—
209	D₄×C₃₁	—	4	12	no	—
210	D₄×C₃₁	—	8	6	no	—
211	D₄×C₃₁	—	8	6	no	—
212	D₄×C₃₁	C₂×#41	1	2	yes	complemented
213	C₂×C₄×C₃₁	—	2	1	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	2	6	no	—
216	C₂×C₄×C₃₁	—	2	3	no	—
217	C₂×C₄×C₃₁	—	2	3	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	4	3	no	—
220	C₂×C₄×C₃₁	—	4	12	no	—
221	C₂×C₄×C₃₁	—	4	2	no	—
222	C₂×C₄×C₃₁	—	4	12	no	—
223	C₂×C₄×C₃₁	—	4	6	no	—
224	C₂×C₄×C₃₁	—	8	3	no	—
225	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
226	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
227	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
228	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
230	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
231	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
232	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
233	C₈×C₃₁	—	2	6	no	—
234	C₈×C₃₁	—	2	2	no	—
235	Q₈×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
236	C₂×#550	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#3368	C₂×C₃₁	1	6	yes	complemented
238	#3366	C₂×C₃₁	1	6	yes	complemented
239	#3362	C₂×C₃₁	1	2	yes	complemented
240	#3358	C₂×C₃₁	1	2	yes	complemented
241	#3138	C₂×C₃₁	1	3	yes	complemented
242	#3077	C₂×C₃₁	1	3	yes	complemented
243	#3038	C₂×C₃₁	1	3	yes	complemented
244	Q₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
245	C₄×AGL(1,ℤ/8)	C₂×C₃₁	1	3	yes	complemented
246	C₂²×C₃₁	—	2	1	no	—
247	C₂²×C₃₁	—	2	1	no	—
248	C₂²×C₃₁	—	2	2	no	—
249	C₂²×C₃₁	—	4	1	no	—
250	C₂²×C₃₁	—	4	1	no	—
251	C₂²×C₃₁	—	4	6	no	—
252	C₂²×C₃₁	—	4	6	no	—
253	C₂²×C₃₁	—	4	6	no	—
254	C₂²×C₃₁	—	8	12	no	—
255	C₂²×C₃₁	#548	1	2	yes	—
256	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
257	C₄×C₃₁	—	2	3	no	—
258	C₄×C₃₁	—	2	3	no	—
259	C₄×C₃₁	—	2	2	no	—
260	C₄×C₃₁	—	2	3	no	—
261	C₄×C₃₁	—	4	3	no	—
262	C₄×C₃₁	—	4	2	no	—
263	C₄×C₃₁	—	4	3	no	—
264	C₄×C₃₁	#548	1	2	yes	—
265	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	6	yes	—
266	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
267	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
268	C₂²×#41	—	2	1	no	—
269	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	3	yes	—
270	Q₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
271	#557	C₂²×C₃₁	1	3	yes	complemented
272	#554	—	2	3	no	—
273	#554	C₂²×C₃₁	1	3	yes	complemented
274	#550	—	2	1	no	—
275	#550	—	2	3	no	—
276	#550	C₂²×C₃₁	1	2	yes	—
277	#550	C₂²×C₃₁	1	6	yes	—
278	D₄×Q₈	—	2	1	no	—
279	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
280	#533	—	2	3	no	—
281	#533	C₂²×C₃₁	1	3	yes	—
282	C₂×#127	C₂²×C₃₁	1	3	yes	—
283	C₂×#121	—	2	3	no	—
284	C₄×#41	C₂²×C₃₁	1	3	yes	—
285	C₂×C₄×Q₈	—	2	1	no	—
286	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
288	#490	C₂²×C₃₁	1	12	yes	complemented
289	#486	C₂²×C₃₁	1	6	yes	complemented
290	#486	C₂²×C₃₁	1	6	yes	complemented
291	#478	C₂²×C₃₁	1	12	yes	complemented
292	#476	C₂²×C₃₁	1	6	yes	complemented
293	#474	C₂²×C₃₁	1	6	yes	complemented
294	#460	C₂²×C₃₁	1	6	yes	complemented
295	#460	C₂²×C₃₁	1	6	yes	complemented
296	#459	C₂²×C₃₁	1	6	yes	complemented
297	#459	C₂²×C₃₁	1	6	yes	complemented
298	#446	C₂²×C₃₁	1	6	yes	complemented
299	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
300	#442	C₂²×C₃₁	1	6	yes	complemented
301	#440	C₂²×C₃₁	1	6	yes	complemented
302	C₄×SD₁₆	C₂²×C₃₁	1	6	yes	complemented
303	C₄×D₈	C₂²×C₃₁	1	6	yes	complemented
304	#428	C₂²×C₃₁	1	3	yes	—
305	#423	C₂²×C₃₁	1	3	yes	—
306	#418	C₂²×C₃₁	1	6	yes	—
307	#418	C₂²×C₃₁	1	3	yes	—
308	#417	C₂²×C₃₁	1	1	yes	characteristic
309	C₄×M₁₆	C₂²×C₃₁	1	3	yes	—
310	C₂×C₃₁	—	2	2	no	—
311	C₂×C₃₁	—	4	2	no	—
312	C₂×C₃₁	—	8	6	no	—
313	C₂×C₃₁	C₂×#548	1	1	yes	characteristic
314	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
315	C₂×C₃₁	#3585	1	1	yes	characteristic
316	C₂×#41	—	2	6	no	—
317	C₂×#41	—	2	2	no	—
318	C₂×#41	—	4	8	no	—
319	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
320	C₂²×Q₈	—	2	1	no	—
321	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
322	C₂²×D₄	—	2	3	no	—
323	C₂²×D₄	—	2	3	no	—
324	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
325	C₂³×C₄	—	2	3	no	—
326	AGL(1,ℤ/8)	—	2	12	no	—
327	C₂×SD₁₆	C₂³×C₃₁	1	6	yes	—
328	C₂×D₈	C₂³×C₃₁	1	6	yes	—
329	C₂×M₁₆	C₂³×C₃₁	1	3	yes	—
330	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
331	C₄⋊Q₈	—	2	3	no	—
332	C₄⋊Q₈	—	2	3	no	—
333	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
334	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
335	#127	—	2	3	no	—
336	#127	—	2	3	no	—
337	#127	C₂³×C₃₁	1	6	yes	—
338	#127	C₂³×C₃₁	1	6	yes	—
339	#125	—	2	6	no	—
340	#125	C₂³×C₃₁	1	6	yes	—
341	C₂²⋊Q₈	—	2	3	no	—
342	C₂²⋊Q₈	—	2	3	no	—
343	C₂²⋊Q₈	—	2	3	no	—
344	C₂²⋊Q₈	—	2	3	no	—
345	C₂²⋊Q₈	—	2	6	no	—
346	#121	—	2	6	no	—
347	#121	—	2	12	no	—
348	#121	—	2	6	no	—
349	C₄×Q₈	—	2	1	no	—
350	C₄×Q₈	—	2	3	no	—
351	C₄×Q₈	—	2	3	no	—
352	C₄×Q₈	—	2	3	no	—
353	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
354	C₄×Q₈	D₄×C₃₁	1	3	yes	—
355	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
356	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
357	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
358	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
359	C₄×D₄	—	2	3	no	—
360	C₄×D₄	—	2	3	no	—
361	C₄×D₄	—	2	3	no	—
362	C₄×D₄	—	2	3	no	—
363	C₄×D₄	C₂³×C₃₁	1	6	yes	—
364	C₄×D₄	C₂³×C₃₁	1	6	yes	—
365	C₄×D₄	C₂³×C₃₁	1	3	yes	—
366	C₄×D₄	C₂³×C₃₁	1	3	yes	—
367	#117	—	2	3	no	—
368	#117	C₂³×C₃₁	1	3	yes	—
369	C₂×C₄⋊C₄	—	2	3	no	—
370	C₂×C₄⋊C₄	D₄×C₃₁	1	3	yes	—
371	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
372	C₂×C₂²⋊C₄	—	2	3	no	—
373	C₂×C₄²	—	2	3	no	—
374	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
375	#107	C₂³×C₃₁	1	6	yes	—
376	#106	C₂³×C₃₁	1	6	yes	—
377	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
378	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
379	#103	C₂³×C₃₁	1	2	yes	—
380	#102	C₂³×C₃₁	1	2	yes	—
381	#102	C₂³×C₃₁	1	6	yes	—
382	#102	C₂³×C₃₁	1	6	yes	—
383	#102	C₂³×C₃₁	1	12	yes	—
384	#102	C₂³×C₃₁	1	12	yes	—
385	#97	C₂³×C₃₁	1	6	yes	—
386	C₄×C₈	C₂³×C₃₁	1	6	yes	—
387	C₃₁	#61321	1	1	yes	characteristic, direct product factor, Sylow
388	C₂⁴	—	2	3	no	—
389	#41	—	2	2	no	—
390	#41	—	4	24	no	—
391	#41	—	4	8	no	—
392	#41	#41×C₃₁	1	4	yes	complemented
393	C₂×Q₈	—	2	4	no	—
394	C₂×Q₈	—	2	3	no	—
395	C₂×Q₈	—	2	3	no	—
396	C₂×Q₈	—	4	4	no	—
397	C₂×Q₈	C₂×D₄×C₃₁	1	3	yes	—
398	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
399	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
400	C₂×D₄	—	2	1	no	—
401	C₂×D₄	—	2	3	no	—
402	C₂×D₄	—	2	6	no	—
403	C₂×D₄	—	2	6	no	—
404	C₂×D₄	—	2	6	no	—
405	C₂×D₄	—	2	6	no	—
406	C₂×D₄	—	2	6	no	—
407	C₂×D₄	—	2	6	no	—
408	C₂×D₄	—	4	12	no	—
409	C₂×D₄	—	8	6	no	—
410	C₂×D₄	C₂⁴×C₃₁	1	6	yes	—
411	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	3	no	—
415	C₂²×C₄	—	2	6	no	—
416	C₂²×C₄	—	2	6	no	—
417	C₂²×C₄	—	2	3	no	—
418	C₂²×C₄	—	4	12	no	—
419	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
420	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
421	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
422	SD₁₆	—	2	12	no	—
423	D₈	—	2	12	no	—
424	M₁₆	—	2	6	no	—
425	M₁₆	—	2	2	no	—
426	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
427	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
428	C₄⋊C₄	—	2	6	no	—
429	C₄⋊C₄	—	2	3	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	3	no	—
432	C₄⋊C₄	—	2	3	no	—
433	C₄⋊C₄	—	2	3	no	—
434	C₄⋊C₄	—	2	6	no	—
435	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
436	C₄⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
437	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
438	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
439	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
440	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
441	C₂²⋊C₄	—	2	3	no	—
442	C₂²⋊C₄	—	2	6	no	—
443	C₂²⋊C₄	—	2	6	no	—
444	C₂²⋊C₄	—	2	3	no	—
445	C₂²⋊C₄	—	2	3	no	—
446	C₄²	—	2	3	no	—
447	C₄²	—	2	3	no	—
448	C₄²	—	2	3	no	—
449	C₄²	C₂×D₄×C₃₁	1	3	yes	—
450	C₄²	C₂×D₄×C₃₁	1	3	yes	—
451	C₄²	C₂⁴×C₃₁	1	3	yes	—
452	C₄²	C₂⁴×C₃₁	1	3	yes	—
453	C₂³	—	2	1	no	—
454	C₂³	—	2	6	no	—
455	C₂³	—	4	6	no	—
456	C₂³	—	4	6	no	—
457	C₂³	—	4	6	no	—
458	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
459	Q₈	—	2	6	no	—
460	Q₈	—	2	1	no	—
461	Q₈	—	4	6	no	—
462	Q₈	—	8	3	no	—
463	Q₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	complemented
464	Q₈	C₂×#41×C₃₁	1	2	yes	complemented
465	D₄	—	2	4	no	—
466	D₄	—	2	6	no	—
467	D₄	—	2	1	no	—
468	D₄	—	2	6	no	—
469	D₄	—	4	12	no	—
470	D₄	—	4	6	no	—
471	D₄	—	4	12	no	—
472	D₄	—	8	6	no	—
473	D₄	—	8	6	no	—
474	D₄	C₂×#41×C₃₁	1	2	yes	complemented
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	6	no	—
478	C₂×C₄	—	2	3	no	—
479	C₂×C₄	—	2	3	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	4	3	no	—
482	C₂×C₄	—	4	12	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	12	no	—
485	C₂×C₄	—	4	6	no	—
486	C₂×C₄	—	8	3	no	—
487	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
488	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
489	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
490	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
491	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
492	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
493	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
494	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
495	C₈	—	2	6	no	—
496	C₈	—	2	2	no	—
497	C₂²	—	2	1	no	—
498	C₂²	—	2	1	no	—
499	C₂²	—	2	2	no	—
500	C₂²	—	4	1	no	—
501	C₂²	—	4	1	no	—
502	C₂²	—	4	6	no	—
503	C₂²	—	4	6	no	—
504	C₂²	—	4	6	no	—
505	C₂²	—	8	12	no	—
506	C₂²	C₃₁×#548	1	2	yes	—
507	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
508	C₄	—	2	2	no	—
509	C₄	—	2	3	no	—
510	C₄	—	2	3	no	—
511	C₄	—	2	3	no	—
512	C₄	—	4	2	no	—
513	C₄	—	4	3	no	—
514	C₄	—	4	3	no	—
515	C₄	C₃₁×#548	1	2	yes	—
516	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	6	yes	—
517	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
518	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
519	C₂	—	2	2	no	—
520	C₂	—	4	2	no	—
521	C₂	—	8	6	no	—
522	C₂	C₂×C₃₁×#548	1	1	yes	characteristic
523	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
524	C₂	C₃₁×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×#41×C₃₁	C₂	semidirect
2. C₂×C₃₁×#550	C₂	semidirect
3. C₃₁×#3368	C₂	semidirect
4. C₃₁×#3366	C₂	semidirect
5. C₃₁×#3362	C₂	semidirect
6. C₃₁×#3358	C₂	semidirect
7. C₃₁×#3138	C₂	semidirect
8. C₃₁×#3077	C₂	semidirect
9. C₃₁×#3038	C₂	semidirect
10. Q₈×M₁₆×C₃₁	C₂	semidirect
11. C₄×C₃₁×AGL(1,ℤ/8)	C₂	semidirect
14. Q₈²×C₃₁	C₂²	semidirect
15. C₃₁×#557	C₂²	semidirect
17. C₃₁×#554	C₂²	semidirect
23. D₄×Q₈×C₃₁	C₂²	semidirect
32. C₃₁×#490	C₂²	semidirect
33. C₃₁×#486	C₂²	semidirect
34. C₃₁×#486	C₂²	semidirect
35. C₃₁×#478	C₂²	semidirect
36. C₃₁×#476	C₂²	semidirect
37. C₃₁×#474	C₂²	semidirect
38. C₃₁×#460	C₂²	semidirect
39. C₃₁×#460	C₂²	semidirect
40. C₃₁×#459	C₂²	semidirect
41. C₃₁×#459	C₂²	semidirect
42. C₃₁×#446	C₂²	semidirect
43. C₈×Q₈×C₃₁	C₂²	semidirect
44. C₃₁×#442	C₂²	semidirect
45. C₃₁×#440	C₂²	semidirect
46. C₄×SD₁₆×C₃₁	C₂²	semidirect
47. C₄×D₈×C₃₁	C₂²	semidirect
129. #41×C₃₁	#41	semidirect
190. #61321	C₃₁	direct (N × H)
201. Q₈×C₃₁	AGL(1,ℤ/8)	semidirect
202. Q₈×C₃₁	C₂×#41	semidirect
212. D₄×C₃₁	C₂×#41	semidirect
235. Q₈×#41	C₂×C₃₁	semidirect
236. C₂×#550	C₂×C₃₁	semidirect
237. #3368	C₂×C₃₁	semidirect
238. #3366	C₂×C₃₁	semidirect
239. #3362	C₂×C₃₁	semidirect
240. #3358	C₂×C₃₁	semidirect
241. #3138	C₂×C₃₁	semidirect
242. #3077	C₂×C₃₁	semidirect
243. #3038	C₂×C₃₁	semidirect
244. Q₈×M₁₆	C₂×C₃₁	semidirect
245. C₄×AGL(1,ℤ/8)	C₂×C₃₁	semidirect
270. Q₈²	C₂²×C₃₁	semidirect
271. #557	C₂²×C₃₁	semidirect
273. #554	C₂²×C₃₁	semidirect
279. D₄×Q₈	C₂²×C₃₁	semidirect
288. #490	C₂²×C₃₁	semidirect
289. #486	C₂²×C₃₁	semidirect
290. #486	C₂²×C₃₁	semidirect
291. #478	C₂²×C₃₁	semidirect
292. #476	C₂²×C₃₁	semidirect
293. #474	C₂²×C₃₁	semidirect
294. #460	C₂²×C₃₁	semidirect
295. #460	C₂²×C₃₁	semidirect
296. #459	C₂²×C₃₁	semidirect
297. #459	C₂²×C₃₁	semidirect
298. #446	C₂²×C₃₁	semidirect
299. C₈×Q₈	C₂²×C₃₁	semidirect
300. #442	C₂²×C₃₁	semidirect
301. #440	C₂²×C₃₁	semidirect
302. C₄×SD₁₆	C₂²×C₃₁	semidirect
303. C₄×D₈	C₂²×C₃₁	semidirect
387. C₃₁	#61321	direct (N × H)
392. #41	#41×C₃₁	semidirect
463. Q₈	C₃₁×AGL(1,ℤ/8)	semidirect
464. Q₈	C₂×#41×C₃₁	semidirect
474. D₄	C₂×#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#41×C₃₁	2
C₂×C₃₁×#550	2
C₃₁×#3368	2
C₃₁×#3366	2
C₃₁×#3362	2
C₃₁×#3358	2
C₃₁×#3138	2
C₃₁×#3077	2
C₃₁×#3038	2
Q₈×M₁₆×C₃₁	2
C₄×C₃₁×AGL(1,ℤ/8)	2
#61321	31

Subgroups — C31×#61321 #21518080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61321 #21518080