Subgroups — C₂×C₃₁×#3062 #21517210

Order 7936.

2934 conjugacy classes of subgroups.

The subgroup lattice has 666 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, complemented
2	C₂²×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#537	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#535	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3062	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₃₁×#469	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#468	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#466	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#465	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×#428	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#416	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₃₁×#102	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
17	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
18	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
19	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
20	C₂×C₃₁×#131	—	2	8	no	—
21	C₂×C₃₁×#131	C₂²	1	8	yes	complemented
22	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
23	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	complemented
24	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₂³×C₈×C₃₁	C₂²	1	2	yes	complemented
26	C₃₁×#537	—	2	4	no	—
27	C₃₁×#537	C₂²	1	8	yes	complemented
28	C₃₁×#535	—	2	4	no	—
29	C₃₁×#535	C₂²	1	8	yes	complemented
30	C₂×C₃₁×#127	—	2	1	no	—
31	C₂×C₃₁×#124	—	2	1	no	—
32	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
33	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
34	C₂×C₃₁×#121	—	2	1	no	—
35	C₂×C₃₁×#121	—	2	2	no	—
36	C₂×C₃₁×#121	C₂²	1	2	yes	—
37	C₂×C₃₁×#121	C₂²	1	2	yes	—
38	C₂×C₃₁×#121	C₂²	1	2	yes	complemented
39	C₂×C₃₁×#120	—	2	2	no	—
40	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
41	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#469	C₂²	1	16	yes	complemented
43	C₃₁×#468	C₂²	1	16	yes	complemented
44	C₃₁×#466	C₂²	1	16	yes	complemented
45	C₃₁×#465	C₂²	1	16	yes	complemented
46	C₃₁×#428	C₂²	1	8	yes	complemented
47	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
48	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
49	C₃₁×#416	C₂²	1	8	yes	complemented
50	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic, complemented
51	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic, complemented
52	C₂×C₃₁×#102	C₂²	1	2	yes	—
53	C₂×C₃₁×#102	C₂²	1	2	yes	—
54	C₂×C₃₁×#102	C₂²	1	8	yes	complemented
55	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic, complemented
56	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic, complemented
57	C₂⁵×C₃₁	—	2	1	no	—
58	C₂×#41×C₃₁	—	2	4	no	—
59	C₂×#41×C₃₁	—	2	2	no	—
60	C₂×#41×C₃₁	—	2	4	no	—
61	C₂×#41×C₃₁	—	2	4	no	—
62	C₂×#41×C₃₁	—	2	4	no	—
63	C₂×#41×C₃₁	D₄	1	4	yes	—
64	C₂×#41×C₃₁	C₂³	1	4	yes	—
65	C₂×#41×C₃₁	C₂³	1	4	yes	—
66	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
67	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₃₁	—	2	1	no	—
69	C₂²×D₄×C₃₁	—	2	2	no	—
70	C₂²×D₄×C₃₁	—	2	2	no	—
71	C₂²×D₄×C₃₁	—	2	2	no	—
72	C₂²×D₄×C₃₁	—	2	4	no	—
73	C₂²×D₄×C₃₁	—	2	1	no	—
74	C₂²×D₄×C₃₁	—	2	2	no	—
75	C₂²×D₄×C₃₁	—	2	4	no	—
76	C₂²×D₄×C₃₁	D₄	1	2	yes	—
77	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
78	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
79	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
81	C₂³×C₄×C₃₁	—	2	1	no	—
82	C₂³×C₄×C₃₁	—	2	1	no	—
83	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
84	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₃₁×AGL(1,ℤ/8)	—	2	32	no	—
86	C₂×SD₁₆×C₃₁	—	2	8	no	—
87	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
88	C₂×D₈×C₃₁	—	2	8	no	—
89	C₂×D₈×C₃₁	C₂³	1	8	yes	—
90	C₃₁×#131	—	2	32	no	—
91	C₂×M₁₆×C₃₁	—	2	8	no	—
92	C₂×M₁₆×C₃₁	D₄	1	8	yes	complemented
93	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
94	C₂×M₁₆×C₃₁	C₂³	1	8	yes	complemented
95	C₂²×C₈×C₃₁	—	2	4	no	—
96	C₂²×C₈×C₃₁	—	2	8	no	—
97	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
98	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
99	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
100	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
101	C₂²×C₈×C₃₁	C₂³	1	8	yes	complemented
102	C₃₁×#127	—	2	8	no	—
103	C₃₁×#124	—	2	8	no	—
104	C₃₁×C₂²⋊Q₈	—	2	8	no	—
105	C₃₁×C₂²⋊Q₈	—	2	4	no	—
106	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	complemented
107	C₃₁×#121	—	2	8	no	—
108	C₃₁×#121	—	2	16	no	—
109	C₃₁×#121	—	2	4	no	—
110	C₃₁×#121	—	2	4	no	—
111	C₃₁×#121	—	2	4	no	—
112	C₃₁×#121	C₂³	1	8	yes	—
113	C₃₁×#121	C₂³	1	8	yes	—
114	C₃₁×#121	C₂³	1	8	yes	complemented
115	C₃₁×#120	—	2	16	no	—
116	C₃₁×#117	—	2	2	no	—
117	C₃₁×#117	C₂³	1	4	yes	—
118	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
119	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
122	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
123	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
124	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
125	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
126	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
127	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
128	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
129	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
130	C₂×C₄²×C₃₁	—	2	1	no	—
131	C₃₁×#107	C₂³	1	8	yes	complemented
132	C₃₁×#106	C₂³	1	8	yes	complemented
133	C₃₁×#103	C₂³	1	4	yes	complemented
134	C₃₁×#103	C₂³	1	4	yes	complemented
135	C₃₁×#102	C₂³	1	8	yes	—
136	C₃₁×#102	C₂³	1	8	yes	—
137	C₃₁×#102	C₂³	1	4	yes	complemented
138	C₃₁×#102	C₂³	1	4	yes	complemented
139	C₂⁴×C₃₁	—	2	1	no	—
140	C₂⁴×C₃₁	—	2	2	no	—
141	C₂⁴×C₃₁	—	2	4	no	—
142	C₂⁴×C₃₁	—	4	4	no	—
143	C₂⁴×C₃₁	—	8	2	no	—
144	C₂⁴×C₃₁	—	8	2	no	—
145	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
146	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
147	#41×C₃₁	—	2	16	no	—
148	#41×C₃₁	—	2	8	no	—
149	#41×C₃₁	—	2	8	no	—
150	#41×C₃₁	—	2	8	no	—
151	#41×C₃₁	—	2	8	no	—
152	#41×C₃₁	—	4	8	no	—
153	C₂×Q₈×C₃₁	—	2	4	no	—
154	C₂×Q₈×C₃₁	—	2	1	no	—
155	C₂×Q₈×C₃₁	—	2	2	no	—
156	C₂×Q₈×C₃₁	—	2	2	no	—
157	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
158	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
159	C₂×D₄×C₃₁	—	2	8	no	—
160	C₂×D₄×C₃₁	—	2	4	no	—
161	C₂×D₄×C₃₁	—	2	8	no	—
162	C₂×D₄×C₃₁	—	2	8	no	—
163	C₂×D₄×C₃₁	—	2	8	no	—
164	C₂×D₄×C₃₁	—	2	4	no	—
165	C₂×D₄×C₃₁	—	2	2	no	—
166	C₂×D₄×C₃₁	—	2	4	no	—
167	C₂×D₄×C₃₁	—	2	8	no	—
168	C₂×D₄×C₃₁	—	2	2	no	—
169	C₂×D₄×C₃₁	—	2	1	no	—
170	C₂×D₄×C₃₁	—	2	4	no	—
171	C₂×D₄×C₃₁	—	2	4	no	—
172	C₂×D₄×C₃₁	—	2	8	no	—
173	C₂×D₄×C₃₁	—	2	2	no	—
174	C₂×D₄×C₃₁	—	4	4	no	—
175	C₂×D₄×C₃₁	—	4	4	no	—
176	C₂×D₄×C₃₁	—	4	8	no	—
177	C₂×D₄×C₃₁	—	8	4	no	—
178	C₂×D₄×C₃₁	—	8	4	no	—
179	C₂×D₄×C₃₁	—	8	4	no	—
180	C₂×D₄×C₃₁	C₂×D₄	1	8	yes	—
181	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
182	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
183	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₃₁	—	2	4	no	—
185	C₂²×C₄×C₃₁	—	2	4	no	—
186	C₂²×C₄×C₃₁	—	2	1	no	—
187	C₂²×C₄×C₃₁	—	2	1	no	—
188	C₂²×C₄×C₃₁	—	2	2	no	—
189	C₂²×C₄×C₃₁	—	2	1	no	—
190	C₂²×C₄×C₃₁	—	2	4	no	—
191	C₂²×C₄×C₃₁	—	2	4	no	—
192	C₂²×C₄×C₃₁	—	4	4	no	—
193	C₂²×C₄×C₃₁	—	8	2	no	—
194	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
195	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
196	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
197	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
198	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
199	C₂²×C₄×C₃₁	#41	1	4	yes	—
200	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
201	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
202	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
203	SD₁₆×C₃₁	—	2	16	no	—
204	D₈×C₃₁	—	2	16	no	—
205	M₁₆×C₃₁	—	2	16	no	—
206	M₁₆×C₃₁	C₂×D₄	1	16	yes	complemented
207	C₂×C₈×C₃₁	—	2	2	no	—
208	C₂×C₈×C₃₁	—	2	2	no	—
209	C₂×C₈×C₃₁	—	2	8	no	—
210	C₂×C₈×C₃₁	—	2	16	no	—
211	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
212	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
213	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
214	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
215	C₄⋊C₄×C₃₁	—	2	4	no	—
216	C₄⋊C₄×C₃₁	—	2	2	no	—
217	C₄⋊C₄×C₃₁	—	2	8	no	—
218	C₄⋊C₄×C₃₁	—	2	1	no	—
219	C₄⋊C₄×C₃₁	—	2	1	no	—
220	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
221	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
222	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
223	C₂²⋊C₄×C₃₁	—	2	4	no	—
224	C₂²⋊C₄×C₃₁	—	2	8	no	—
225	C₂²⋊C₄×C₃₁	—	2	8	no	—
226	C₂²⋊C₄×C₃₁	—	2	8	no	—
227	C₂²⋊C₄×C₃₁	—	2	8	no	—
228	C₄²×C₃₁	—	2	4	no	—
229	C₂×#3062	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
230	C₂³×C₃₁	—	2	2	no	—
231	C₂³×C₃₁	—	2	2	no	—
232	C₂³×C₃₁	—	2	4	no	—
233	C₂³×C₃₁	—	2	4	no	—
234	C₂³×C₃₁	—	2	4	no	—
235	C₂³×C₃₁	—	2	2	no	—
236	C₂³×C₃₁	—	2	4	no	—
237	C₂³×C₃₁	—	4	2	no	—
238	C₂³×C₃₁	—	4	8	no	—
239	C₂³×C₃₁	—	4	4	no	—
240	C₂³×C₃₁	—	4	2	no	—
241	C₂³×C₃₁	—	4	4	no	—
242	C₂³×C₃₁	—	4	4	no	—
243	C₂³×C₃₁	—	4	2	no	—
244	C₂³×C₃₁	—	8	4	no	—
245	C₂³×C₃₁	—	8	4	no	—
246	C₂³×C₃₁	—	8	16	no	—
247	C₂³×C₃₁	#121	1	4	yes	—
248	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
249	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
250	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
251	Q₈×C₃₁	—	2	8	no	—
252	Q₈×C₃₁	—	2	2	no	—
253	Q₈×C₃₁	—	2	2	no	—
254	Q₈×C₃₁	—	4	2	no	—
255	D₄×C₃₁	—	2	16	no	—
256	D₄×C₃₁	—	2	8	no	—
257	D₄×C₃₁	—	2	2	no	—
258	D₄×C₃₁	—	2	4	no	—
259	D₄×C₃₁	—	2	4	no	—
260	D₄×C₃₁	—	2	2	no	—
261	D₄×C₃₁	—	4	8	no	—
262	D₄×C₃₁	—	4	4	no	—
263	D₄×C₃₁	—	4	8	no	—
264	D₄×C₃₁	—	4	2	no	—
265	D₄×C₃₁	—	8	8	no	—
266	D₄×C₃₁	—	8	8	no	—
267	D₄×C₃₁	—	8	8	no	—
268	C₂×C₄×C₃₁	—	2	4	no	—
269	C₂×C₄×C₃₁	—	2	8	no	—
270	C₂×C₄×C₃₁	—	2	4	no	—
271	C₂×C₄×C₃₁	—	2	4	no	—
272	C₂×C₄×C₃₁	—	2	2	no	—
273	C₂×C₄×C₃₁	—	2	2	no	—
274	C₂×C₄×C₃₁	—	2	4	no	—
275	C₂×C₄×C₃₁	—	2	2	no	—
276	C₂×C₄×C₃₁	—	4	2	no	—
277	C₂×C₄×C₃₁	—	4	4	no	—
278	C₂×C₄×C₃₁	—	4	2	no	—
279	C₂×C₄×C₃₁	—	4	2	no	—
280	C₂×C₄×C₃₁	—	4	8	no	—
281	C₂×C₄×C₃₁	—	8	4	no	—
282	C₂×C₄×C₃₁	#121	1	4	yes	—
283	C₂×C₄×C₃₁	#121	1	8	yes	—
284	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
285	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
286	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
288	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
289	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
290	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
291	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
292	C₈×C₃₁	—	2	2	no	—
293	C₈×C₃₁	—	2	2	no	—
294	C₈×C₃₁	C₂²×D₄	1	8	yes	—
295	C₂²×AGL(1,ℤ/8)	C₂×C₃₁	1	1	yes	characteristic, complemented
296	C₂²×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
297	C₂×#537	C₂×C₃₁	1	1	yes	characteristic, complemented
298	C₂×#535	C₂×C₃₁	1	1	yes	characteristic, complemented
299	#3062	C₂×C₃₁	1	16	yes	direct product factor
300	C₂×#469	C₂×C₃₁	1	2	yes	complemented
301	C₂×#468	C₂×C₃₁	1	2	yes	complemented
302	C₂×#466	C₂×C₃₁	1	2	yes	complemented
303	C₂×#465	C₂×C₃₁	1	2	yes	complemented
304	C₂×#428	C₂×C₃₁	1	1	yes	characteristic, complemented
305	C₂×#416	C₂×C₃₁	1	1	yes	characteristic, complemented
306	C₂²×#102	C₂×C₃₁	1	1	yes	characteristic, complemented
307	C₂²×C₃₁	—	2	8	no	—
308	C₂²×C₃₁	—	2	2	no	—
309	C₂²×C₃₁	—	2	2	no	—
310	C₂²×C₃₁	—	2	4	no	—
311	C₂²×C₃₁	—	2	4	no	—
312	C₂²×C₃₁	—	2	4	no	—
313	C₂²×C₃₁	—	4	8	no	—
314	C₂²×C₃₁	—	4	2	no	—
315	C₂²×C₃₁	—	4	4	no	—
316	C₂²×C₃₁	—	4	2	no	—
317	C₂²×C₃₁	—	4	4	no	—
318	C₂²×C₃₁	—	4	4	no	—
319	C₂²×C₃₁	—	4	2	no	—
320	C₂²×C₃₁	—	8	16	no	—
321	C₂²×C₃₁	—	8	4	no	—
322	C₂²×C₃₁	—	8	8	no	—
323	C₂²×C₃₁	C₂×#121	1	4	yes	—
324	C₂²×C₃₁	C₂×#121	1	2	yes	—
325	C₂²×C₃₁	#576	1	2	yes	—
326	C₂²×C₃₁	#577	1	2	yes	—
327	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
328	C₄×C₃₁	—	2	4	no	—
329	C₄×C₃₁	—	4	2	no	—
330	C₄×C₃₁	—	4	4	no	—
331	C₄×C₃₁	—	4	2	no	—
332	C₄×C₃₁	—	4	2	no	—
333	C₄×C₃₁	C₂×#121	1	2	yes	—
334	C₄×C₃₁	C₂×#121	1	4	yes	—
335	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
336	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
337	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
338	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
339	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
340	C₂×AGL(1,ℤ/8)	—	2	8	no	—
341	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	8	yes	complemented
342	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
343	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
344	C₂×#131	—	2	8	no	—
345	C₂×#131	C₂²×C₃₁	1	8	yes	complemented
346	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
347	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	complemented
348	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
349	C₂³×C₈	C₂²×C₃₁	1	2	yes	complemented
350	#537	—	2	4	no	—
351	#537	C₂²×C₃₁	1	8	yes	complemented
352	#535	—	2	4	no	—
353	#535	C₂²×C₃₁	1	8	yes	complemented
354	C₂×#127	—	2	1	no	—
355	C₂×#124	—	2	1	no	—
356	C₂×C₂²⋊Q₈	—	2	1	no	—
357	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
358	C₂×#121	—	2	1	no	—
359	C₂×#121	—	2	2	no	—
360	C₂×#121	C₂²×C₃₁	1	2	yes	—
361	C₂×#121	C₂²×C₃₁	1	2	yes	—
362	C₂×#121	C₂²×C₃₁	1	2	yes	complemented
363	C₂×#120	—	2	2	no	—
364	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
365	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
366	#469	C₂²×C₃₁	1	16	yes	complemented
367	#468	C₂²×C₃₁	1	16	yes	complemented
368	#466	C₂²×C₃₁	1	16	yes	complemented
369	#465	C₂²×C₃₁	1	16	yes	complemented
370	#428	C₂²×C₃₁	1	8	yes	complemented
371	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
372	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
373	#416	C₂²×C₃₁	1	8	yes	complemented
374	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic, complemented
375	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic, complemented
376	C₂×#102	C₂²×C₃₁	1	2	yes	—
377	C₂×#102	C₂²×C₃₁	1	2	yes	—
378	C₂×#102	C₂²×C₃₁	1	8	yes	complemented
379	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic, complemented
380	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic, complemented
381	C₂×C₃₁	—	2	4	no	—
382	C₂×C₃₁	—	4	4	no	—
383	C₂×C₃₁	—	8	2	no	—
384	C₂×C₃₁	—	8	4	no	—
385	C₂×C₃₁	#3062	1	4	yes	direct product factor
386	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
387	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
388	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
389	C₂⁵	—	2	1	no	—
390	C₂×#41	—	2	4	no	—
391	C₂×#41	—	2	2	no	—
392	C₂×#41	—	2	4	no	—
393	C₂×#41	—	2	4	no	—
394	C₂×#41	—	2	4	no	—
395	C₂×#41	D₄×C₃₁	1	4	yes	—
396	C₂×#41	C₂³×C₃₁	1	4	yes	—
397	C₂×#41	C₂³×C₃₁	1	4	yes	—
398	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
399	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
400	C₂²×D₄	—	2	1	no	—
401	C₂²×D₄	—	2	2	no	—
402	C₂²×D₄	—	2	2	no	—
403	C₂²×D₄	—	2	2	no	—
404	C₂²×D₄	—	2	4	no	—
405	C₂²×D₄	—	2	1	no	—
406	C₂²×D₄	—	2	2	no	—
407	C₂²×D₄	—	2	4	no	—
408	C₂²×D₄	D₄×C₃₁	1	2	yes	—
409	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
410	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
411	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
412	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
413	C₂³×C₄	—	2	1	no	—
414	C₂³×C₄	—	2	1	no	—
415	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
416	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
417	AGL(1,ℤ/8)	—	2	32	no	—
418	C₂×SD₁₆	—	2	8	no	—
419	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
420	C₂×D₈	—	2	8	no	—
421	C₂×D₈	C₂³×C₃₁	1	8	yes	—
422	#131	—	2	32	no	—
423	C₂×M₁₆	—	2	8	no	—
424	C₂×M₁₆	D₄×C₃₁	1	8	yes	complemented
425	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
426	C₂×M₁₆	C₂³×C₃₁	1	8	yes	complemented
427	C₂²×C₈	—	2	4	no	—
428	C₂²×C₈	—	2	8	no	—
429	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
430	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
431	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
432	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
433	C₂²×C₈	C₂³×C₃₁	1	8	yes	complemented
434	#127	—	2	8	no	—
435	#124	—	2	8	no	—
436	C₂²⋊Q₈	—	2	8	no	—
437	C₂²⋊Q₈	—	2	4	no	—
438	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	complemented
439	#121	—	2	8	no	—
440	#121	—	2	16	no	—
441	#121	—	2	4	no	—
442	#121	—	2	4	no	—
443	#121	—	2	4	no	—
444	#121	C₂³×C₃₁	1	8	yes	—
445	#121	C₂³×C₃₁	1	8	yes	—
446	#121	C₂³×C₃₁	1	8	yes	complemented
447	#120	—	2	16	no	—
448	#117	—	2	2	no	—
449	#117	C₂³×C₃₁	1	4	yes	—
450	C₂×C₄⋊C₄	—	2	1	no	—
451	C₂×C₄⋊C₄	—	2	2	no	—
452	C₂×C₄⋊C₄	—	2	2	no	—
453	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
454	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
455	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
456	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
457	C₂×C₂²⋊C₄	—	2	1	no	—
458	C₂×C₂²⋊C₄	—	2	2	no	—
459	C₂×C₂²⋊C₄	—	2	2	no	—
460	C₂×C₂²⋊C₄	—	2	2	no	—
461	C₂×C₂²⋊C₄	—	2	2	no	—
462	C₂×C₄²	—	2	1	no	—
463	#107	C₂³×C₃₁	1	8	yes	complemented
464	#106	C₂³×C₃₁	1	8	yes	complemented
465	#103	C₂³×C₃₁	1	4	yes	complemented
466	#103	C₂³×C₃₁	1	4	yes	complemented
467	#102	C₂³×C₃₁	1	8	yes	—
468	#102	C₂³×C₃₁	1	8	yes	—
469	#102	C₂³×C₃₁	1	4	yes	complemented
470	#102	C₂³×C₃₁	1	4	yes	complemented
471	C₃₁	C₂×#3062	1	1	yes	characteristic, direct product factor, Sylow
472	C₂⁴	—	2	1	no	—
473	C₂⁴	—	2	2	no	—
474	C₂⁴	—	2	4	no	—
475	C₂⁴	—	4	4	no	—
476	C₂⁴	—	8	2	no	—
477	C₂⁴	—	8	2	no	—
478	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
479	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
480	#41	—	2	16	no	—
481	#41	—	2	8	no	—
482	#41	—	2	8	no	—
483	#41	—	2	8	no	—
484	#41	—	2	8	no	—
485	#41	—	4	8	no	—
486	C₂×Q₈	—	2	4	no	—
487	C₂×Q₈	—	2	1	no	—
488	C₂×Q₈	—	2	2	no	—
489	C₂×Q₈	—	2	2	no	—
490	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
491	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
492	C₂×D₄	—	2	8	no	—
493	C₂×D₄	—	2	4	no	—
494	C₂×D₄	—	2	8	no	—
495	C₂×D₄	—	2	8	no	—
496	C₂×D₄	—	2	8	no	—
497	C₂×D₄	—	2	4	no	—
498	C₂×D₄	—	2	2	no	—
499	C₂×D₄	—	2	4	no	—
500	C₂×D₄	—	2	8	no	—
501	C₂×D₄	—	2	2	no	—
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	2	4	no	—
504	C₂×D₄	—	2	4	no	—
505	C₂×D₄	—	2	8	no	—
506	C₂×D₄	—	2	2	no	—
507	C₂×D₄	—	4	4	no	—
508	C₂×D₄	—	4	4	no	—
509	C₂×D₄	—	4	8	no	—
510	C₂×D₄	—	8	4	no	—
511	C₂×D₄	—	8	4	no	—
512	C₂×D₄	—	8	4	no	—
513	C₂×D₄	C₂×D₄×C₃₁	1	8	yes	—
514	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
515	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
516	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
517	C₂²×C₄	—	2	4	no	—
518	C₂²×C₄	—	2	4	no	—
519	C₂²×C₄	—	2	1	no	—
520	C₂²×C₄	—	2	1	no	—
521	C₂²×C₄	—	2	2	no	—
522	C₂²×C₄	—	2	1	no	—
523	C₂²×C₄	—	2	4	no	—
524	C₂²×C₄	—	2	4	no	—
525	C₂²×C₄	—	4	4	no	—
526	C₂²×C₄	—	8	2	no	—
527	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
528	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
529	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
530	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
531	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
532	C₂²×C₄	#41×C₃₁	1	4	yes	—
533	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
534	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
535	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
536	SD₁₆	—	2	16	no	—
537	D₈	—	2	16	no	—
538	M₁₆	—	2	16	no	—
539	M₁₆	C₂×D₄×C₃₁	1	16	yes	complemented
540	C₂×C₈	—	2	2	no	—
541	C₂×C₈	—	2	2	no	—
542	C₂×C₈	—	2	8	no	—
543	C₂×C₈	—	2	16	no	—
544	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
545	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
546	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
547	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
548	C₄⋊C₄	—	2	4	no	—
549	C₄⋊C₄	—	2	2	no	—
550	C₄⋊C₄	—	2	8	no	—
551	C₄⋊C₄	—	2	1	no	—
552	C₄⋊C₄	—	2	1	no	—
553	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
554	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
555	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
556	C₂²⋊C₄	—	2	4	no	—
557	C₂²⋊C₄	—	2	8	no	—
558	C₂²⋊C₄	—	2	8	no	—
559	C₂²⋊C₄	—	2	8	no	—
560	C₂²⋊C₄	—	2	8	no	—
561	C₄²	—	2	4	no	—
562	C₂³	—	2	2	no	—
563	C₂³	—	2	2	no	—
564	C₂³	—	2	4	no	—
565	C₂³	—	2	4	no	—
566	C₂³	—	2	4	no	—
567	C₂³	—	2	2	no	—
568	C₂³	—	2	4	no	—
569	C₂³	—	4	2	no	—
570	C₂³	—	4	8	no	—
571	C₂³	—	4	4	no	—
572	C₂³	—	4	2	no	—
573	C₂³	—	4	4	no	—
574	C₂³	—	4	4	no	—
575	C₂³	—	4	2	no	—
576	C₂³	—	8	4	no	—
577	C₂³	—	8	4	no	—
578	C₂³	—	8	16	no	—
579	C₂³	C₃₁×#121	1	4	yes	—
580	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
581	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
582	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
583	Q₈	—	2	8	no	—
584	Q₈	—	2	2	no	—
585	Q₈	—	2	2	no	—
586	Q₈	—	4	2	no	—
587	D₄	—	2	16	no	—
588	D₄	—	2	8	no	—
589	D₄	—	2	2	no	—
590	D₄	—	2	4	no	—
591	D₄	—	2	4	no	—
592	D₄	—	2	2	no	—
593	D₄	—	4	8	no	—
594	D₄	—	4	4	no	—
595	D₄	—	4	8	no	—
596	D₄	—	4	2	no	—
597	D₄	—	8	8	no	—
598	D₄	—	8	8	no	—
599	D₄	—	8	8	no	—
600	C₂×C₄	—	2	4	no	—
601	C₂×C₄	—	2	8	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	2	4	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	4	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	4	2	no	—
609	C₂×C₄	—	4	4	no	—
610	C₂×C₄	—	4	2	no	—
611	C₂×C₄	—	4	2	no	—
612	C₂×C₄	—	4	8	no	—
613	C₂×C₄	—	8	4	no	—
614	C₂×C₄	C₃₁×#121	1	4	yes	—
615	C₂×C₄	C₃₁×#121	1	8	yes	—
616	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
617	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
618	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
619	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
620	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
621	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
622	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
623	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
624	C₈	—	2	2	no	—
625	C₈	—	2	2	no	—
626	C₈	C₂²×D₄×C₃₁	1	8	yes	—
627	C₂²	—	2	8	no	—
628	C₂²	—	2	2	no	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	4	no	—
631	C₂²	—	2	4	no	—
632	C₂²	—	2	4	no	—
633	C₂²	—	4	8	no	—
634	C₂²	—	4	2	no	—
635	C₂²	—	4	4	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	4	4	no	—
638	C₂²	—	4	4	no	—
639	C₂²	—	4	2	no	—
640	C₂²	—	8	16	no	—
641	C₂²	—	8	8	no	—
642	C₂²	—	8	4	no	—
643	C₂²	C₂×C₃₁×#121	1	4	yes	—
644	C₂²	C₂×C₃₁×#121	1	2	yes	—
645	C₂²	C₃₁×#576	1	2	yes	—
646	C₂²	C₃₁×#577	1	2	yes	—
647	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
648	C₄	—	2	4	no	—
649	C₄	—	4	2	no	—
650	C₄	—	4	4	no	—
651	C₄	—	4	2	no	—
652	C₄	—	4	2	no	—
653	C₄	C₂×C₃₁×#121	1	2	yes	—
654	C₄	C₂×C₃₁×#121	1	4	yes	—
655	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
656	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
657	C₂	—	2	4	no	—
658	C₂	—	4	4	no	—
659	C₂	—	8	4	no	—
660	C₂	—	8	2	no	—
661	C₂	C₃₁×#3062	1	4	yes	direct product factor
662	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic
663	C₂	C₂×C₃₁×#576	1	1	yes	characteristic
664	C₂	C₂×C₃₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×AGL(1,ℤ/8)	C₂	semidirect
2. C₂²×C₃₁×#131	C₂	semidirect
3. C₂×C₃₁×#537	C₂	semidirect
4. C₂×C₃₁×#535	C₂	semidirect
5. C₃₁×#3062	C₂	direct (N × H)
6. C₂×C₃₁×#469	C₂	semidirect
7. C₂×C₃₁×#468	C₂	semidirect
8. C₂×C₃₁×#466	C₂	semidirect
9. C₂×C₃₁×#465	C₂	semidirect
10. C₂×C₃₁×#428	C₂	semidirect
11. C₂×C₃₁×#416	C₂	semidirect
12. C₂²×C₃₁×#102	C₂	semidirect
17. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
21. C₂×C₃₁×#131	C₂²	semidirect
23. C₂²×M₁₆×C₃₁	C₂²	semidirect
25. C₂³×C₈×C₃₁	C₂²	semidirect
27. C₃₁×#537	C₂²	semidirect
29. C₃₁×#535	C₂²	semidirect
33. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
38. C₂×C₃₁×#121	C₂²	semidirect
42. C₃₁×#469	C₂²	semidirect
43. C₃₁×#468	C₂²	semidirect
44. C₃₁×#466	C₂²	semidirect
45. C₃₁×#465	C₂²	semidirect
46. C₃₁×#428	C₂²	semidirect
47. C₂×C₃₁×#107	C₂²	semidirect
48. C₂×C₃₁×#106	C₂²	semidirect
49. C₃₁×#416	C₂²	semidirect
50. C₂×C₃₁×#103	C₂²	semidirect
51. C₂×C₃₁×#103	C₂²	semidirect
54. C₂×C₃₁×#102	C₂²	semidirect
55. C₂×C₃₁×#102	C₂²	semidirect
56. C₂×C₃₁×#102	C₂²	semidirect
92. C₂×M₁₆×C₃₁	D₄	semidirect
94. C₂×M₁₆×C₃₁	C₂³	semidirect
101. C₂²×C₈×C₃₁	C₂³	semidirect
106. C₃₁×C₂²⋊Q₈	C₂³	semidirect
114. C₃₁×#121	C₂³	semidirect
131. C₃₁×#107	C₂³	semidirect
132. C₃₁×#106	C₂³	semidirect
133. C₃₁×#103	C₂³	semidirect
134. C₃₁×#103	C₂³	semidirect
137. C₃₁×#102	C₂³	semidirect
138. C₃₁×#102	C₂³	semidirect
206. M₁₆×C₃₁	C₂×D₄	semidirect
229. C₂×#3062	C₃₁	direct (N × H)
295. C₂²×AGL(1,ℤ/8)	C₂×C₃₁	semidirect
296. C₂²×#131	C₂×C₃₁	semidirect
297. C₂×#537	C₂×C₃₁	semidirect
298. C₂×#535	C₂×C₃₁	semidirect
299. #3062	C₂×C₃₁	direct (N × H)
300. C₂×#469	C₂×C₃₁	semidirect
301. C₂×#468	C₂×C₃₁	semidirect
302. C₂×#466	C₂×C₃₁	semidirect
303. C₂×#465	C₂×C₃₁	semidirect
304. C₂×#428	C₂×C₃₁	semidirect
305. C₂×#416	C₂×C₃₁	semidirect
306. C₂²×#102	C₂×C₃₁	semidirect
341. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
345. C₂×#131	C₂²×C₃₁	semidirect
347. C₂²×M₁₆	C₂²×C₃₁	semidirect
349. C₂³×C₈	C₂²×C₃₁	semidirect
351. #537	C₂²×C₃₁	semidirect
353. #535	C₂²×C₃₁	semidirect
357. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
362. C₂×#121	C₂²×C₃₁	semidirect
366. #469	C₂²×C₃₁	semidirect
367. #468	C₂²×C₃₁	semidirect
368. #466	C₂²×C₃₁	semidirect
369. #465	C₂²×C₃₁	semidirect
370. #428	C₂²×C₃₁	semidirect
371. C₂×#107	C₂²×C₃₁	semidirect
372. C₂×#106	C₂²×C₃₁	semidirect
373. #416	C₂²×C₃₁	semidirect
374. C₂×#103	C₂²×C₃₁	semidirect
375. C₂×#103	C₂²×C₃₁	semidirect
378. C₂×#102	C₂²×C₃₁	semidirect
379. C₂×#102	C₂²×C₃₁	semidirect
380. C₂×#102	C₂²×C₃₁	semidirect
385. C₂×C₃₁	#3062	direct (N × H)
424. C₂×M₁₆	D₄×C₃₁	semidirect
426. C₂×M₁₆	C₂³×C₃₁	semidirect
433. C₂²×C₈	C₂³×C₃₁	semidirect
438. C₂²⋊Q₈	C₂³×C₃₁	semidirect
446. #121	C₂³×C₃₁	semidirect
463. #107	C₂³×C₃₁	semidirect
464. #106	C₂³×C₃₁	semidirect
465. #103	C₂³×C₃₁	semidirect
466. #103	C₂³×C₃₁	semidirect
469. #102	C₂³×C₃₁	semidirect
470. #102	C₂³×C₃₁	semidirect
471. C₃₁	C₂×#3062	direct (N × H)
539. M₁₆	C₂×D₄×C₃₁	semidirect
661. C₂	C₃₁×#3062	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
C₂²×C₃₁×#131	2
C₂×C₃₁×#537	2
C₂×C₃₁×#535	2
C₃₁×#3062	2
C₂×C₃₁×#469	2
C₂×C₃₁×#468	2
C₂×C₃₁×#466	2
C₂×C₃₁×#465	2
C₂×C₃₁×#428	2
C₂×C₃₁×#416	2
C₂²×C₃₁×#102	2
C₂×#3062	31

Subgroups — C2×C31×#3062 #21517210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3062 #21517210