Subgroups — C₂×C₃₁×#2946 #21516918

Order 7936.

2946 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#518	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2946	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₃₁×#442	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#440	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#428	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#416	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₃₁×#102	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×AGL(1,ℤ/8)	C₄	1	16	yes	complemented
16	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	8	yes	—
17	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
18	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
19	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#520	—	2	4	no	—
22	C₃₁×#520	C₂²	1	8	yes	complemented
23	C₃₁×#518	—	2	4	no	—
24	C₃₁×#518	C₂²	1	8	yes	complemented
25	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
26	C₂×C₄×D₄×C₃₁	—	2	1	no	—
27	C₂×C₄×D₄×C₃₁	—	2	1	no	—
28	C₂×C₄×D₄×C₃₁	—	2	2	no	—
29	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
32	C₂×C₃₁×#117	—	2	1	no	—
33	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
35	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
36	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₂²×C₂²⋊C₄×C₃₁	—	2	2	no	—
38	C₃₁×#442	C₂²	1	16	yes	complemented
39	C₃₁×#440	C₂²	1	16	yes	complemented
40	C₄×SD₁₆×C₃₁	C₂²	1	16	yes	complemented
41	C₄×D₈×C₃₁	C₂²	1	16	yes	complemented
42	C₃₁×#428	C₂²	1	8	yes	complemented
43	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
44	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
45	C₃₁×#416	C₂²	1	8	yes	complemented
46	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic, complemented
47	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic, complemented
48	C₂×C₃₁×#102	C₂²	1	2	yes	—
49	C₂×C₃₁×#102	C₂²	1	2	yes	—
50	C₂×C₃₁×#102	C₂²	1	8	yes	complemented
51	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic, complemented
52	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic, complemented
53	C₃₁×#405	C₂²	1	8	yes	complemented
54	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
55	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
56	C₂⁵×C₃₁	—	2	1	no	—
57	C₂×#41×C₃₁	—	2	2	no	—
58	C₂×#41×C₃₁	—	2	4	no	—
59	C₂×#41×C₃₁	C₂×C₄	1	8	yes	complemented
60	C₂×#41×C₃₁	C₂³	1	4	yes	—
61	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₃₁	—	2	1	no	—
63	C₂²×D₄×C₃₁	—	2	1	no	—
64	C₂²×D₄×C₃₁	—	2	2	no	—
65	C₂²×D₄×C₃₁	—	2	4	no	—
66	C₂²×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
67	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
70	C₂³×C₄×C₃₁	—	2	1	no	—
71	C₂³×C₄×C₃₁	—	2	1	no	—
72	C₂³×C₄×C₃₁	—	2	2	no	—
73	C₂³×C₄×C₃₁	—	2	2	no	—
74	C₂³×C₄×C₃₁	—	2	2	no	—
75	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×AGL(1,ℤ/8)	C₂×C₄	1	64	yes	complemented
77	C₂×SD₁₆×C₃₁	C₂×C₄	1	16	yes	complemented
78	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
79	C₂×D₈×C₃₁	C₂×C₄	1	16	yes	complemented
80	C₂×D₈×C₃₁	C₂³	1	8	yes	—
81	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
82	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
83	C₂²×C₈×C₃₁	—	2	4	no	—
84	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
86	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
87	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
88	C₄×Q₈×C₃₁	—	2	4	no	—
89	C₄×Q₈×C₃₁	C₂³	1	8	yes	complemented
90	C₄×D₄×C₃₁	—	2	8	no	—
91	C₄×D₄×C₃₁	—	2	8	no	—
92	C₄×D₄×C₃₁	—	2	16	no	—
93	C₄×D₄×C₃₁	—	2	4	no	—
94	C₄×D₄×C₃₁	—	2	4	no	—
95	C₄×D₄×C₃₁	—	2	4	no	—
96	C₄×D₄×C₃₁	C₂³	1	8	yes	—
97	C₄×D₄×C₃₁	C₂³	1	8	yes	—
98	C₄×D₄×C₃₁	C₂³	1	8	yes	complemented
99	C₃₁×#117	—	2	2	no	—
100	C₃₁×#117	—	2	8	no	—
101	C₃₁×#117	D₄	1	4	yes	—
102	C₃₁×#117	C₂³	1	4	yes	—
103	C₃₁×#117	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
108	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
114	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
117	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
119	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
120	C₂×C₄²×C₃₁	—	2	2	no	—
121	C₂×C₄²×C₃₁	—	2	1	no	—
122	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
123	C₃₁×#107	C₂³	1	8	yes	complemented
124	C₃₁×#106	C₂³	1	8	yes	complemented
125	C₃₁×#103	C₂³	1	4	yes	complemented
126	C₃₁×#103	C₂³	1	4	yes	complemented
127	C₃₁×#102	C₂³	1	8	yes	—
128	C₃₁×#102	C₂³	1	8	yes	—
129	C₃₁×#102	C₂³	1	4	yes	complemented
130	C₃₁×#102	C₂³	1	4	yes	complemented
131	C₃₁×#97	C₂³	1	8	yes	complemented
132	C₄×C₈×C₃₁	C₂³	1	8	yes	complemented
133	C₂⁴×C₃₁	—	2	1	no	—
134	C₂⁴×C₃₁	—	2	2	no	—
135	C₂⁴×C₃₁	—	2	8	no	—
136	C₂⁴×C₃₁	—	2	4	no	—
137	C₂⁴×C₃₁	—	8	2	no	—
138	C₂⁴×C₃₁	—	8	2	no	—
139	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
140	#41×C₃₁	—	2	16	no	—
141	#41×C₃₁	—	2	8	no	—
142	#41×C₃₁	C₂²×C₄	1	16	yes	complemented
143	C₂×Q₈×C₃₁	—	2	1	no	—
144	C₂×Q₈×C₃₁	—	2	2	no	—
145	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
146	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
147	C₂×D₄×C₃₁	—	2	8	no	—
148	C₂×D₄×C₃₁	—	2	4	no	—
149	C₂×D₄×C₃₁	—	2	8	no	—
150	C₂×D₄×C₃₁	—	2	16	no	—
151	C₂×D₄×C₃₁	—	2	4	no	—
152	C₂×D₄×C₃₁	—	2	2	no	—
153	C₂×D₄×C₃₁	—	2	1	no	—
154	C₂×D₄×C₃₁	—	2	4	no	—
155	C₂×D₄×C₃₁	—	2	8	no	—
156	C₂×D₄×C₃₁	—	2	2	no	—
157	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
158	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
159	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	complemented
160	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
161	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₃₁	—	2	1	no	—
163	C₂²×C₄×C₃₁	—	2	8	no	—
164	C₂²×C₄×C₃₁	—	2	4	no	—
165	C₂²×C₄×C₃₁	—	2	1	no	—
166	C₂²×C₄×C₃₁	—	2	1	no	—
167	C₂²×C₄×C₃₁	—	2	2	no	—
168	C₂²×C₄×C₃₁	—	2	4	no	—
169	C₂²×C₄×C₃₁	—	2	8	no	—
170	C₂²×C₄×C₃₁	—	2	8	no	—
171	C₂²×C₄×C₃₁	—	2	8	no	—
172	C₂²×C₄×C₃₁	—	8	4	no	—
173	C₂²×C₄×C₃₁	—	8	4	no	—
174	C₂²×C₄×C₃₁	—	8	4	no	—
175	C₂²×C₄×C₃₁	—	8	2	no	—
176	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
177	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
178	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
180	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₃₁	#41	1	4	yes	—
182	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
184	SD₁₆×C₃₁	C₂²×C₄	1	32	yes	complemented
185	D₈×C₃₁	C₂²×C₄	1	32	yes	complemented
186	M₁₆×C₃₁	C₂²×C₄	1	16	yes	complemented
187	C₂×C₈×C₃₁	—	2	2	no	—
188	C₂×C₈×C₃₁	—	2	2	no	—
189	C₂×C₈×C₃₁	—	2	8	no	—
190	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
191	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
192	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
193	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
194	C₄⋊C₄×C₃₁	—	2	4	no	—
195	C₄⋊C₄×C₃₁	—	2	8	no	—
196	C₄⋊C₄×C₃₁	—	2	2	no	—
197	C₄⋊C₄×C₃₁	—	2	1	no	—
198	C₄⋊C₄×C₃₁	—	2	1	no	—
199	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
200	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
201	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
202	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
203	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
204	C₂²⋊C₄×C₃₁	—	2	4	no	—
205	C₂²⋊C₄×C₃₁	—	2	8	no	—
206	C₂²⋊C₄×C₃₁	—	2	8	no	—
207	C₂²⋊C₄×C₃₁	—	2	8	no	—
208	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
209	C₄²×C₃₁	—	2	8	no	—
210	C₄²×C₃₁	—	2	4	no	—
211	C₄²×C₃₁	C₂×D₄	1	4	yes	—
212	C₄²×C₃₁	C₂⁴	1	4	yes	—
213	C₂×#2946	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
214	C₂³×C₃₁	—	2	2	no	—
215	C₂³×C₃₁	—	2	2	no	—
216	C₂³×C₃₁	—	2	4	no	—
217	C₂³×C₃₁	—	2	8	no	—
218	C₂³×C₃₁	—	2	2	no	—
219	C₂³×C₃₁	—	2	4	no	—
220	C₂³×C₃₁	—	2	16	no	—
221	C₂³×C₃₁	—	4	2	no	—
222	C₂³×C₃₁	—	4	4	no	—
223	C₂³×C₃₁	—	4	4	no	—
224	C₂³×C₃₁	—	4	2	no	—
225	C₂³×C₃₁	—	8	16	no	—
226	C₂³×C₃₁	—	8	4	no	—
227	C₂³×C₃₁	—	8	4	no	—
228	C₂³×C₃₁	C₄×D₄	1	4	yes	—
229	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
230	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
231	Q₈×C₃₁	—	2	4	no	—
232	Q₈×C₃₁	—	2	2	no	—
233	Q₈×C₃₁	C₂³×C₄	1	4	yes	—
234	D₄×C₃₁	—	2	16	no	—
235	D₄×C₃₁	—	2	8	no	—
236	D₄×C₃₁	—	2	16	no	—
237	D₄×C₃₁	—	2	4	no	—
238	D₄×C₃₁	—	2	4	no	—
239	D₄×C₃₁	—	2	2	no	—
240	D₄×C₃₁	C₂³×C₄	1	4	yes	—
241	D₄×C₃₁	C₂³×C₄	1	8	yes	—
242	C₂×C₄×C₃₁	—	2	4	no	—
243	C₂×C₄×C₃₁	—	2	4	no	—
244	C₂×C₄×C₃₁	—	2	4	no	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	16	no	—
247	C₂×C₄×C₃₁	—	2	2	no	—
248	C₂×C₄×C₃₁	—	2	2	no	—
249	C₂×C₄×C₃₁	—	2	4	no	—
250	C₂×C₄×C₃₁	—	4	2	no	—
251	C₂×C₄×C₃₁	—	4	4	no	—
252	C₂×C₄×C₃₁	—	4	2	no	—
253	C₂×C₄×C₃₁	—	8	8	no	—
254	C₂×C₄×C₃₁	—	8	8	no	—
255	C₂×C₄×C₃₁	—	8	8	no	—
256	C₂×C₄×C₃₁	—	8	4	no	—
257	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
258	C₂×C₄×C₃₁	C₄×D₄	1	8	yes	—
259	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
260	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
261	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
262	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
263	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
264	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
266	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
267	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
268	C₈×C₃₁	—	2	2	no	—
269	C₈×C₃₁	—	2	2	no	—
270	C₈×C₃₁	C₂³×C₄	1	8	yes	—
271	C₂²×AGL(1,ℤ/8)	C₂×C₃₁	1	1	yes	characteristic
272	C₂×#520	C₂×C₃₁	1	1	yes	characteristic, complemented
273	C₂×#518	C₂×C₃₁	1	1	yes	characteristic, complemented
274	#2946	C₂×C₃₁	1	16	yes	direct product factor
275	C₂×#442	C₂×C₃₁	1	2	yes	complemented
276	C₂×#440	C₂×C₃₁	1	2	yes	complemented
277	C₂×C₄×SD₁₆	C₂×C₃₁	1	2	yes	complemented
278	C₂×C₄×D₈	C₂×C₃₁	1	2	yes	complemented
279	C₂×#428	C₂×C₃₁	1	1	yes	characteristic, complemented
280	C₂×#416	C₂×C₃₁	1	1	yes	characteristic, complemented
281	C₂²×#102	C₂×C₃₁	1	1	yes	characteristic, complemented
282	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
283	C₂²×C₃₁	—	2	8	no	—
284	C₂²×C₃₁	—	2	2	no	—
285	C₂²×C₃₁	—	2	2	no	—
286	C₂²×C₃₁	—	2	4	no	—
287	C₂²×C₃₁	—	2	8	no	—
288	C₂²×C₃₁	—	4	8	no	—
289	C₂²×C₃₁	—	4	16	no	—
290	C₂²×C₃₁	—	4	2	no	—
291	C₂²×C₃₁	—	4	4	no	—
292	C₂²×C₃₁	—	4	4	no	—
293	C₂²×C₃₁	—	4	2	no	—
294	C₂²×C₃₁	—	8	16	no	—
295	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
296	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
297	C₂²×C₃₁	#576	1	2	yes	—
298	C₂²×C₃₁	#577	1	2	yes	—
299	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
300	C₄×C₃₁	—	2	4	no	—
301	C₄×C₃₁	—	2	4	no	—
302	C₄×C₃₁	—	2	4	no	—
303	C₄×C₃₁	—	4	2	no	—
304	C₄×C₃₁	—	4	4	no	—
305	C₄×C₃₁	—	4	2	no	—
306	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
307	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
308	C₄×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
309	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
310	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
311	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
312	C₂×AGL(1,ℤ/8)	C₄×C₃₁	1	16	yes	complemented
313	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	8	yes	—
314	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
315	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
316	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
317	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
318	#520	—	2	4	no	—
319	#520	C₂²×C₃₁	1	8	yes	complemented
320	#518	—	2	4	no	—
321	#518	C₂²×C₃₁	1	8	yes	complemented
322	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
323	C₂×C₄×D₄	—	2	1	no	—
324	C₂×C₄×D₄	—	2	1	no	—
325	C₂×C₄×D₄	—	2	2	no	—
326	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
327	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
328	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
329	C₂×#117	—	2	1	no	—
330	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
332	C₂²×C₄⋊C₄	—	2	1	no	—
333	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
334	C₂²×C₂²⋊C₄	—	2	2	no	—
335	#442	C₂²×C₃₁	1	16	yes	complemented
336	#440	C₂²×C₃₁	1	16	yes	complemented
337	C₄×SD₁₆	C₂²×C₃₁	1	16	yes	complemented
338	C₄×D₈	C₂²×C₃₁	1	16	yes	complemented
339	#428	C₂²×C₃₁	1	8	yes	complemented
340	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
341	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
342	#416	C₂²×C₃₁	1	8	yes	complemented
343	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic, complemented
344	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic, complemented
345	C₂×#102	C₂²×C₃₁	1	2	yes	—
346	C₂×#102	C₂²×C₃₁	1	2	yes	—
347	C₂×#102	C₂²×C₃₁	1	8	yes	complemented
348	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic, complemented
349	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic, complemented
350	#405	C₂²×C₃₁	1	8	yes	complemented
351	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
352	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
353	C₂×C₃₁	—	2	4	no	—
354	C₂×C₃₁	—	4	4	no	—
355	C₂×C₃₁	—	4	8	no	—
356	C₂×C₃₁	#2946	1	4	yes	direct product factor
357	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
358	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
359	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
360	C₂⁵	—	2	1	no	—
361	C₂×#41	—	2	2	no	—
362	C₂×#41	—	2	4	no	—
363	C₂×#41	C₂×C₄×C₃₁	1	8	yes	complemented
364	C₂×#41	C₂³×C₃₁	1	4	yes	—
365	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
366	C₂²×D₄	—	2	1	no	—
367	C₂²×D₄	—	2	1	no	—
368	C₂²×D₄	—	2	2	no	—
369	C₂²×D₄	—	2	4	no	—
370	C₂²×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
371	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
372	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
373	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
374	C₂³×C₄	—	2	1	no	—
375	C₂³×C₄	—	2	1	no	—
376	C₂³×C₄	—	2	2	no	—
377	C₂³×C₄	—	2	2	no	—
378	C₂³×C₄	—	2	2	no	—
379	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
380	AGL(1,ℤ/8)	C₂×C₄×C₃₁	1	64	yes	complemented
381	C₂×SD₁₆	C₂×C₄×C₃₁	1	16	yes	complemented
382	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
383	C₂×D₈	C₂×C₄×C₃₁	1	16	yes	complemented
384	C₂×D₈	C₂³×C₃₁	1	8	yes	—
385	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
386	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
387	C₂²×C₈	—	2	4	no	—
388	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
389	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
390	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
391	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
392	C₄×Q₈	—	2	4	no	—
393	C₄×Q₈	C₂³×C₃₁	1	8	yes	complemented
394	C₄×D₄	—	2	8	no	—
395	C₄×D₄	—	2	8	no	—
396	C₄×D₄	—	2	16	no	—
397	C₄×D₄	—	2	4	no	—
398	C₄×D₄	—	2	4	no	—
399	C₄×D₄	—	2	4	no	—
400	C₄×D₄	C₂³×C₃₁	1	8	yes	—
401	C₄×D₄	C₂³×C₃₁	1	8	yes	—
402	C₄×D₄	C₂³×C₃₁	1	8	yes	complemented
403	#117	—	2	2	no	—
404	#117	—	2	8	no	—
405	#117	D₄×C₃₁	1	4	yes	—
406	#117	C₂³×C₃₁	1	4	yes	—
407	#117	C₂³×C₃₁	1	4	yes	—
408	C₂×C₄⋊C₄	—	2	1	no	—
409	C₂×C₄⋊C₄	—	2	2	no	—
410	C₂×C₄⋊C₄	—	2	2	no	—
411	C₂×C₄⋊C₄	—	2	8	no	—
412	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
413	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
414	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
415	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
416	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
417	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
418	C₂×C₂²⋊C₄	—	2	1	no	—
419	C₂×C₂²⋊C₄	—	2	2	no	—
420	C₂×C₂²⋊C₄	—	2	2	no	—
421	C₂×C₂²⋊C₄	—	2	2	no	—
422	C₂×C₂²⋊C₄	—	2	16	no	—
423	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
424	C₂×C₄²	—	2	2	no	—
425	C₂×C₄²	—	2	1	no	—
426	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
427	#107	C₂³×C₃₁	1	8	yes	complemented
428	#106	C₂³×C₃₁	1	8	yes	complemented
429	#103	C₂³×C₃₁	1	4	yes	complemented
430	#103	C₂³×C₃₁	1	4	yes	complemented
431	#102	C₂³×C₃₁	1	8	yes	—
432	#102	C₂³×C₃₁	1	8	yes	—
433	#102	C₂³×C₃₁	1	4	yes	complemented
434	#102	C₂³×C₃₁	1	4	yes	complemented
435	#97	C₂³×C₃₁	1	8	yes	complemented
436	C₄×C₈	C₂³×C₃₁	1	8	yes	complemented
437	C₃₁	C₂×#2946	1	1	yes	characteristic, direct product factor, Sylow
438	C₂⁴	—	2	1	no	—
439	C₂⁴	—	2	2	no	—
440	C₂⁴	—	2	8	no	—
441	C₂⁴	—	2	4	no	—
442	C₂⁴	—	8	2	no	—
443	C₂⁴	—	8	2	no	—
444	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
445	#41	—	2	16	no	—
446	#41	—	2	8	no	—
447	#41	C₂²×C₄×C₃₁	1	16	yes	complemented
448	C₂×Q₈	—	2	1	no	—
449	C₂×Q₈	—	2	2	no	—
450	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
451	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
452	C₂×D₄	—	2	8	no	—
453	C₂×D₄	—	2	4	no	—
454	C₂×D₄	—	2	8	no	—
455	C₂×D₄	—	2	16	no	—
456	C₂×D₄	—	2	4	no	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	1	no	—
459	C₂×D₄	—	2	4	no	—
460	C₂×D₄	—	2	8	no	—
461	C₂×D₄	—	2	2	no	—
462	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
463	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
464	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	complemented
465	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
466	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	8	no	—
469	C₂²×C₄	—	2	4	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	4	no	—
474	C₂²×C₄	—	2	8	no	—
475	C₂²×C₄	—	2	8	no	—
476	C₂²×C₄	—	2	8	no	—
477	C₂²×C₄	—	8	4	no	—
478	C₂²×C₄	—	8	4	no	—
479	C₂²×C₄	—	8	4	no	—
480	C₂²×C₄	—	8	2	no	—
481	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
482	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
483	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
484	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
485	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
486	C₂²×C₄	#41×C₃₁	1	4	yes	—
487	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
488	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
489	SD₁₆	C₂²×C₄×C₃₁	1	32	yes	complemented
490	D₈	C₂²×C₄×C₃₁	1	32	yes	complemented
491	M₁₆	C₂²×C₄×C₃₁	1	16	yes	complemented
492	C₂×C₈	—	2	2	no	—
493	C₂×C₈	—	2	2	no	—
494	C₂×C₈	—	2	8	no	—
495	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
496	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
497	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
498	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
499	C₄⋊C₄	—	2	4	no	—
500	C₄⋊C₄	—	2	8	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
505	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
506	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
507	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
508	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₂²⋊C₄	—	2	8	no	—
511	C₂²⋊C₄	—	2	8	no	—
512	C₂²⋊C₄	—	2	8	no	—
513	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
514	C₄²	—	2	8	no	—
515	C₄²	—	2	4	no	—
516	C₄²	C₂×D₄×C₃₁	1	4	yes	—
517	C₄²	C₂⁴×C₃₁	1	4	yes	—
518	C₂³	—	2	2	no	—
519	C₂³	—	2	2	no	—
520	C₂³	—	2	4	no	—
521	C₂³	—	2	8	no	—
522	C₂³	—	2	2	no	—
523	C₂³	—	2	4	no	—
524	C₂³	—	2	16	no	—
525	C₂³	—	4	2	no	—
526	C₂³	—	4	4	no	—
527	C₂³	—	4	4	no	—
528	C₂³	—	4	2	no	—
529	C₂³	—	8	16	no	—
530	C₂³	—	8	4	no	—
531	C₂³	—	8	4	no	—
532	C₂³	C₄×D₄×C₃₁	1	4	yes	—
533	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
534	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
535	Q₈	—	2	4	no	—
536	Q₈	—	2	2	no	—
537	Q₈	C₂³×C₄×C₃₁	1	4	yes	—
538	D₄	—	2	16	no	—
539	D₄	—	2	8	no	—
540	D₄	—	2	16	no	—
541	D₄	—	2	4	no	—
542	D₄	—	2	4	no	—
543	D₄	—	2	2	no	—
544	D₄	C₂³×C₄×C₃₁	1	4	yes	—
545	D₄	C₂³×C₄×C₃₁	1	8	yes	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	16	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	4	2	no	—
555	C₂×C₄	—	4	4	no	—
556	C₂×C₄	—	4	2	no	—
557	C₂×C₄	—	8	8	no	—
558	C₂×C₄	—	8	8	no	—
559	C₂×C₄	—	8	8	no	—
560	C₂×C₄	—	8	4	no	—
561	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
562	C₂×C₄	C₄×D₄×C₃₁	1	8	yes	—
563	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
564	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
565	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
566	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
567	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
568	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
569	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
570	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
571	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
572	C₈	—	2	2	no	—
573	C₈	—	2	2	no	—
574	C₈	C₂³×C₄×C₃₁	1	8	yes	—
575	C₂²	—	2	8	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	2	2	no	—
578	C₂²	—	2	4	no	—
579	C₂²	—	2	8	no	—
580	C₂²	—	4	16	no	—
581	C₂²	—	4	8	no	—
582	C₂²	—	4	2	no	—
583	C₂²	—	4	4	no	—
584	C₂²	—	4	4	no	—
585	C₂²	—	4	2	no	—
586	C₂²	—	8	16	no	—
587	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
588	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
589	C₂²	C₃₁×#576	1	2	yes	—
590	C₂²	C₃₁×#577	1	2	yes	—
591	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
592	C₄	—	2	4	no	—
593	C₄	—	2	4	no	—
594	C₄	—	2	4	no	—
595	C₄	—	4	2	no	—
596	C₄	—	4	4	no	—
597	C₄	—	4	2	no	—
598	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
599	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
600	C₄	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
601	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
602	C₂	—	2	4	no	—
603	C₂	—	4	8	no	—
604	C₂	—	4	4	no	—
605	C₂	C₃₁×#2946	1	4	yes	direct product factor
606	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
607	C₂	C₂×C₃₁×#576	1	1	yes	characteristic
608	C₂	C₂×C₃₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₄	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂²×C₃₁×AGL(1,ℤ/8)	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#520	C₂	semidirect
3. C₂×C₃₁×#518	C₂	semidirect
4. C₃₁×#2946	C₂	direct (N × H)
5. C₂×C₃₁×#442	C₂	semidirect
6. C₂×C₃₁×#440	C₂	semidirect
7. C₂×C₄×SD₁₆×C₃₁	C₂	semidirect
8. C₂×C₄×D₈×C₃₁	C₂	semidirect
9. C₂×C₃₁×#428	C₂	semidirect
10. C₂×C₃₁×#416	C₂	semidirect
11. C₂²×C₃₁×#102	C₂	semidirect
12. C₂×C₃₁×#405	C₂	semidirect
15. C₂×C₃₁×AGL(1,ℤ/8)	C₄	semidirect
22. C₃₁×#520	C₂²	semidirect
24. C₃₁×#518	C₂²	semidirect
25. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
31. C₂×C₄×D₄×C₃₁	C₂²	semidirect
38. C₃₁×#442	C₂²	semidirect
39. C₃₁×#440	C₂²	semidirect
40. C₄×SD₁₆×C₃₁	C₂²	semidirect
41. C₄×D₈×C₃₁	C₂²	semidirect
42. C₃₁×#428	C₂²	semidirect
43. C₂×C₃₁×#107	C₂²	semidirect
44. C₂×C₃₁×#106	C₂²	semidirect
45. C₃₁×#416	C₂²	semidirect
46. C₂×C₃₁×#103	C₂²	semidirect
47. C₂×C₃₁×#103	C₂²	semidirect
50. C₂×C₃₁×#102	C₂²	semidirect
51. C₂×C₃₁×#102	C₂²	semidirect
52. C₂×C₃₁×#102	C₂²	semidirect
53. C₃₁×#405	C₂²	semidirect
54. C₂×C₃₁×#97	C₂²	semidirect
55. C₂×C₄×C₈×C₃₁	C₂²	semidirect
59. C₂×#41×C₃₁	C₂×C₄	semidirect
66. C₂²×D₄×C₃₁	C₂×C₄	semidirect
76. C₃₁×AGL(1,ℤ/8)	C₂×C₄	semidirect
77. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
79. C₂×D₈×C₃₁	C₂×C₄	semidirect
81. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
89. C₄×Q₈×C₃₁	C₂³	semidirect
98. C₄×D₄×C₃₁	C₂³	semidirect
123. C₃₁×#107	C₂³	semidirect
124. C₃₁×#106	C₂³	semidirect
125. C₃₁×#103	C₂³	semidirect
126. C₃₁×#103	C₂³	semidirect
129. C₃₁×#102	C₂³	semidirect
130. C₃₁×#102	C₂³	semidirect
131. C₃₁×#97	C₂³	semidirect
132. C₄×C₈×C₃₁	C₂³	semidirect
142. #41×C₃₁	C₂²×C₄	semidirect
159. C₂×D₄×C₃₁	C₂²×C₄	semidirect
184. SD₁₆×C₃₁	C₂²×C₄	semidirect
185. D₈×C₃₁	C₂²×C₄	semidirect
186. M₁₆×C₃₁	C₂²×C₄	semidirect
213. C₂×#2946	C₃₁	direct (N × H)
272. C₂×#520	C₂×C₃₁	semidirect
273. C₂×#518	C₂×C₃₁	semidirect
274. #2946	C₂×C₃₁	direct (N × H)
275. C₂×#442	C₂×C₃₁	semidirect
276. C₂×#440	C₂×C₃₁	semidirect
277. C₂×C₄×SD₁₆	C₂×C₃₁	semidirect
278. C₂×C₄×D₈	C₂×C₃₁	semidirect
279. C₂×#428	C₂×C₃₁	semidirect
280. C₂×#416	C₂×C₃₁	semidirect
281. C₂²×#102	C₂×C₃₁	semidirect
282. C₂×#405	C₂×C₃₁	semidirect
312. C₂×AGL(1,ℤ/8)	C₄×C₃₁	semidirect
319. #520	C₂²×C₃₁	semidirect
321. #518	C₂²×C₃₁	semidirect
322. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
328. C₂×C₄×D₄	C₂²×C₃₁	semidirect
335. #442	C₂²×C₃₁	semidirect
336. #440	C₂²×C₃₁	semidirect
337. C₄×SD₁₆	C₂²×C₃₁	semidirect
338. C₄×D₈	C₂²×C₃₁	semidirect
339. #428	C₂²×C₃₁	semidirect
340. C₂×#107	C₂²×C₃₁	semidirect
341. C₂×#106	C₂²×C₃₁	semidirect
342. #416	C₂²×C₃₁	semidirect
343. C₂×#103	C₂²×C₃₁	semidirect
344. C₂×#103	C₂²×C₃₁	semidirect
347. C₂×#102	C₂²×C₃₁	semidirect
348. C₂×#102	C₂²×C₃₁	semidirect
349. C₂×#102	C₂²×C₃₁	semidirect
350. #405	C₂²×C₃₁	semidirect
351. C₂×#97	C₂²×C₃₁	semidirect
352. C₂×C₄×C₈	C₂²×C₃₁	semidirect
356. C₂×C₃₁	#2946	direct (N × H)
363. C₂×#41	C₂×C₄×C₃₁	semidirect
370. C₂²×D₄	C₂×C₄×C₃₁	semidirect
380. AGL(1,ℤ/8)	C₂×C₄×C₃₁	semidirect
381. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
383. C₂×D₈	C₂×C₄×C₃₁	semidirect
385. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
393. C₄×Q₈	C₂³×C₃₁	semidirect
402. C₄×D₄	C₂³×C₃₁	semidirect
427. #107	C₂³×C₃₁	semidirect
428. #106	C₂³×C₃₁	semidirect
429. #103	C₂³×C₃₁	semidirect
430. #103	C₂³×C₃₁	semidirect
433. #102	C₂³×C₃₁	semidirect
434. #102	C₂³×C₃₁	semidirect
435. #97	C₂³×C₃₁	semidirect
436. C₄×C₈	C₂³×C₃₁	semidirect
437. C₃₁	C₂×#2946	direct (N × H)
447. #41	C₂²×C₄×C₃₁	semidirect
464. C₂×D₄	C₂²×C₄×C₃₁	semidirect
489. SD₁₆	C₂²×C₄×C₃₁	semidirect
490. D₈	C₂²×C₄×C₃₁	semidirect
491. M₁₆	C₂²×C₄×C₃₁	semidirect
605. C₂	C₃₁×#2946	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×AGL(1,ℤ/8)	2
C₂×C₃₁×#520	2
C₂×C₃₁×#518	2
C₃₁×#2946	2
C₂×C₃₁×#442	2
C₂×C₃₁×#440	2
C₂×C₄×SD₁₆×C₃₁	2
C₂×C₄×D₈×C₃₁	2
C₂×C₃₁×#428	2
C₂×C₃₁×#416	2
C₂²×C₃₁×#102	2
C₂×C₃₁×#405	2
C₂×#2946	31

Subgroups — C2×C31×#2946 #21516918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2946 #21516918