Subgroups — C₂×C₃₁×#3386 #21518119

Order 7936.

2006 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3386	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#501	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#477	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#475	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#478	C₂	1	2	yes	maximal
9	C₂×C₃₁×#474	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#440	C₂	1	2	yes	maximal
12	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
13	C₂³×Q₈×C₃₁	—	2	1	no	—
14	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#557	—	2	4	no	—
16	C₃₁×#557	C₂²	1	8	yes	complemented
17	D₄×Q₈×C₃₁	—	2	4	no	—
18	D₄×Q₈×C₃₁	C₂²	1	8	yes	—
19	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
20	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
23	C₂×C₃₁×#125	—	2	2	no	—
24	C₂×C₃₁×#125	C₂²	1	2	yes	—
25	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
26	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
27	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
28	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
33	C₃₁×#501	C₂²	1	16	yes	complemented
34	C₃₁×#500	C₂²	1	8	yes	complemented
35	C₃₁×#478	C₂²	1	16	yes	—
36	C₃₁×#477	C₂²	1	16	yes	complemented
37	C₃₁×#475	C₂²	1	8	yes	complemented
38	C₃₁×#474	C₂²	1	8	yes	—
39	C₃₁×#446	C₂²	1	8	yes	complemented
40	C₃₁×#440	C₂²	1	16	yes	—
41	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	—
42	C₂×C₃₁×#107	C₂²	1	2	yes	—
43	C₂×C₃₁×#107	C₂²	1	2	yes	—
44	C₂×C₃₁×#107	C₂²	1	2	yes	—
45	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
46	C₂×C₃₁×#106	C₂²	1	2	yes	—
47	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
48	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
49	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
50	C₂×C₃₁×#103	C₂²	1	2	yes	—
51	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
52	C₂×C₃₁×#102	C₂²	1	2	yes	—
53	C₂×C₃₁×#97	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
55	C₂²×Q₈×C₃₁	—	2	8	no	—
56	C₂²×Q₈×C₃₁	—	2	1	no	—
57	C₂²×Q₈×C₃₁	—	2	2	no	—
58	C₂²×Q₈×C₃₁	—	2	1	no	—
59	C₂²×Q₈×C₃₁	—	4	8	no	—
60	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
61	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₃₁	—	2	2	no	—
64	C₂³×C₄×C₃₁	—	2	1	no	—
65	C₂×SD₁₆×C₃₁	Q₈	1	8	yes	complemented
66	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
67	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
69	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×C₄⋊Q₈	—	2	8	no	—
71	C₃₁×C₄⋊Q₈	—	2	4	no	—
72	C₃₁×C₄⋊Q₈	—	2	2	no	—
73	C₃₁×C₄⋊Q₈	—	2	2	no	—
74	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
75	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
76	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	—
77	C₃₁×#125	—	2	16	no	—
78	C₃₁×#125	—	2	4	no	—
79	C₃₁×#125	C₂³	1	8	yes	—
80	C₃₁×C₂²⋊Q₈	—	2	8	no	—
81	C₃₁×C₂²⋊Q₈	—	2	16	no	—
82	C₄×Q₈×C₃₁	—	2	8	no	—
83	C₄×Q₈×C₃₁	—	2	4	no	—
84	C₄×Q₈×C₃₁	—	2	2	no	—
85	C₄×Q₈×C₃₁	D₄	1	4	yes	—
86	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
87	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
88	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
89	C₄×D₄×C₃₁	—	2	4	no	—
90	C₄×D₄×C₃₁	—	2	2	no	—
91	C₄×D₄×C₃₁	C₂³	1	4	yes	—
92	C₄×D₄×C₃₁	C₂³	1	8	yes	—
93	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
95	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
98	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
99	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
106	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
107	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
108	C₂×C₄²×C₃₁	—	2	2	no	—
109	C₂×C₄²×C₃₁	—	2	1	no	—
110	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
112	C₃₁×#107	C₂³	1	8	yes	—
113	C₃₁×#107	C₂³	1	8	yes	—
114	C₃₁×#107	C₂³	1	8	yes	—
115	C₃₁×#106	C₂³	1	4	yes	—
116	C₃₁×#106	C₂³	1	8	yes	—
117	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
118	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
119	C₃₁×#103	C₂³	1	4	yes	—
120	C₃₁×#103	C₂³	1	8	yes	—
121	C₃₁×#102	C₂³	1	4	yes	—
122	C₃₁×#102	C₂³	1	8	yes	—
123	C₃₁×#97	C₂³	1	8	yes	—
124	C₄×C₈×C₃₁	C₂³	1	4	yes	—
125	C₂⁴×C₃₁	—	2	1	no	—
126	C₂×Q₈×C₃₁	—	2	4	no	—
127	C₂×Q₈×C₃₁	—	2	1	no	—
128	C₂×Q₈×C₃₁	—	2	4	no	—
129	C₂×Q₈×C₃₁	—	2	8	no	—
130	C₂×Q₈×C₃₁	—	2	4	no	—
131	C₂×Q₈×C₃₁	—	2	2	no	—
132	C₂×Q₈×C₃₁	—	4	8	no	—
133	C₂×Q₈×C₃₁	—	4	32	no	—
134	C₂×Q₈×C₃₁	—	4	4	no	—
135	C₂×Q₈×C₃₁	—	4	4	no	—
136	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
137	C₂×Q₈×C₃₁	C₂×Q₈	1	4	yes	complemented
138	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
139	C₂×D₄×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	—	2	2	no	—
141	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	—
142	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
143	C₂²×C₄×C₃₁	—	2	1	no	—
144	C₂²×C₄×C₃₁	—	2	1	no	—
145	C₂²×C₄×C₃₁	—	2	2	no	—
146	C₂²×C₄×C₃₁	—	2	2	no	—
147	C₂²×C₄×C₃₁	—	2	1	no	—
148	C₂²×C₄×C₃₁	—	2	8	no	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	4	8	no	—
151	C₂²×C₄×C₃₁	—	4	4	no	—
152	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
155	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
157	SD₁₆×C₃₁	C₂×Q₈	1	16	yes	complemented
158	C₂×C₈×C₃₁	—	2	2	no	—
159	C₂×C₈×C₃₁	—	2	4	no	—
160	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	complemented
161	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
162	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
163	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
164	C₄⋊C₄×C₃₁	—	2	8	no	—
165	C₄⋊C₄×C₃₁	—	2	4	no	—
166	C₄⋊C₄×C₃₁	—	2	8	no	—
167	C₄⋊C₄×C₃₁	—	2	8	no	—
168	C₄⋊C₄×C₃₁	—	2	4	no	—
169	C₄⋊C₄×C₃₁	—	2	4	no	—
170	C₄⋊C₄×C₃₁	—	2	8	no	—
171	C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₄⋊C₄×C₃₁	—	2	2	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	2	no	—
175	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
176	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
177	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
178	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
179	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
180	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
181	C₂²⋊C₄×C₃₁	—	2	8	no	—
182	C₂²⋊C₄×C₃₁	—	2	4	no	—
183	C₄²×C₃₁	—	2	8	no	—
184	C₄²×C₃₁	—	2	4	no	—
185	C₄²×C₃₁	C₂×D₄	1	4	yes	—
186	C₄²×C₃₁	C₂×D₄	1	2	yes	—
187	C₄²×C₃₁	C₂⁴	1	2	yes	—
188	C₄²×C₃₁	C₂⁴	1	4	yes	—
189	C₂×#3386	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂³×C₃₁	—	2	4	no	—
191	C₂³×C₃₁	—	2	2	no	—
192	C₂³×C₃₁	—	4	2	no	—
193	C₂³×C₃₁	—	4	2	no	—
194	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
195	Q₈×C₃₁	—	2	4	no	—
196	Q₈×C₃₁	—	2	8	no	—
197	Q₈×C₃₁	—	2	2	no	—
198	Q₈×C₃₁	—	4	16	no	—
199	Q₈×C₃₁	—	4	8	no	—
200	Q₈×C₃₁	—	4	8	no	—
201	Q₈×C₃₁	C₂²×Q₈	1	4	yes	complemented
202	D₄×C₃₁	—	2	4	no	—
203	D₄×C₃₁	—	2	2	no	—
204	D₄×C₃₁	C₂²×Q₈	1	4	yes	—
205	C₂×C₄×C₃₁	—	2	4	no	—
206	C₂×C₄×C₃₁	—	2	4	no	—
207	C₂×C₄×C₃₁	—	2	4	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	4	no	—
212	C₂×C₄×C₃₁	—	2	4	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	4	16	no	—
215	C₂×C₄×C₃₁	—	4	4	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	—	4	4	no	—
218	C₂×C₄×C₃₁	—	4	2	no	—
219	C₂×C₄×C₃₁	—	4	8	no	—
220	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
221	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
222	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
223	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
224	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
226	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
227	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
228	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
229	C₈×C₃₁	—	2	2	no	—
230	C₈×C₃₁	—	2	4	no	—
231	C₈×C₃₁	C₂²×Q₈	1	4	yes	complemented
232	C₂×#557	C₂×C₃₁	1	1	yes	characteristic, complemented
233	C₂×D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
234	#3386	C₂×C₃₁	1	16	yes	direct product factor
235	C₂×#501	C₂×C₃₁	1	2	yes	complemented
236	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
237	C₂×#477	C₂×C₃₁	1	2	yes	complemented
238	C₂×#475	C₂×C₃₁	1	1	yes	characteristic, complemented
239	C₂×#478	C₂×C₃₁	1	2	yes	—
240	C₂×#474	C₂×C₃₁	1	1	yes	characteristic
241	C₂×#446	C₂×C₃₁	1	1	yes	characteristic, complemented
242	C₂×#440	C₂×C₃₁	1	2	yes	—
243	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
244	C₂²×C₃₁	—	2	4	no	—
245	C₂²×C₃₁	—	4	8	no	—
246	C₂²×C₃₁	—	4	2	no	—
247	C₂²×C₃₁	—	4	2	no	—
248	C₂²×C₃₁	D₄×Q₈	1	2	yes	—
249	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
250	C₂²×C₃₁	#578	1	2	yes	—
251	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
252	C₄×C₃₁	—	2	4	no	—
253	C₄×C₃₁	—	2	4	no	—
254	C₄×C₃₁	—	2	4	no	—
255	C₄×C₃₁	—	2	2	no	—
256	C₄×C₃₁	—	4	4	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	4	4	no	—
259	C₄×C₃₁	—	4	2	no	—
260	C₄×C₃₁	D₄×Q₈	1	2	yes	—
261	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
262	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
263	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
264	C₂³×Q₈	—	2	1	no	—
265	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
266	#557	—	2	4	no	—
267	#557	C₂²×C₃₁	1	8	yes	complemented
268	D₄×Q₈	—	2	4	no	—
269	D₄×Q₈	C₂²×C₃₁	1	8	yes	—
270	C₂×C₄⋊Q₈	—	2	1	no	—
271	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
273	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×#125	—	2	2	no	—
275	C₂×#125	C₂²×C₃₁	1	2	yes	—
276	C₂×C₂²⋊Q₈	—	2	1	no	—
277	C₂×C₂²⋊Q₈	—	2	2	no	—
278	C₂×C₄×Q₈	—	2	1	no	—
279	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
281	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
282	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
283	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
284	#501	C₂²×C₃₁	1	16	yes	complemented
285	#500	C₂²×C₃₁	1	8	yes	complemented
286	#478	C₂²×C₃₁	1	16	yes	—
287	#477	C₂²×C₃₁	1	16	yes	complemented
288	#475	C₂²×C₃₁	1	8	yes	complemented
289	#474	C₂²×C₃₁	1	8	yes	—
290	#446	C₂²×C₃₁	1	8	yes	complemented
291	#440	C₂²×C₃₁	1	16	yes	—
292	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	—
293	C₂×#107	C₂²×C₃₁	1	2	yes	—
294	C₂×#107	C₂²×C₃₁	1	2	yes	—
295	C₂×#107	C₂²×C₃₁	1	2	yes	—
296	C₂×#106	C₂²×C₃₁	1	2	yes	—
297	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
298	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
299	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
300	C₂×#103	C₂²×C₃₁	1	2	yes	—
301	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
302	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×#102	C₂²×C₃₁	1	2	yes	—
304	C₂×#97	C₂²×C₃₁	1	2	yes	—
305	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
306	C₂×C₃₁	—	4	4	no	—
307	C₂×C₃₁	#3386	1	4	yes	direct product factor
308	C₂×C₃₁	C₂×D₄×Q₈	1	1	yes	characteristic
309	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
310	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
311	C₂²×Q₈	—	2	8	no	—
312	C₂²×Q₈	—	2	1	no	—
313	C₂²×Q₈	—	2	2	no	—
314	C₂²×Q₈	—	2	1	no	—
315	C₂²×Q₈	—	4	8	no	—
316	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
317	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
318	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
319	C₂³×C₄	—	2	2	no	—
320	C₂³×C₄	—	2	1	no	—
321	C₂×SD₁₆	Q₈×C₃₁	1	8	yes	complemented
322	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
323	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
324	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
325	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
326	C₄⋊Q₈	—	2	8	no	—
327	C₄⋊Q₈	—	2	4	no	—
328	C₄⋊Q₈	—	2	2	no	—
329	C₄⋊Q₈	—	2	2	no	—
330	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
331	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	—
332	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
333	#125	—	2	16	no	—
334	#125	—	2	4	no	—
335	#125	C₂³×C₃₁	1	8	yes	—
336	C₂²⋊Q₈	—	2	8	no	—
337	C₂²⋊Q₈	—	2	16	no	—
338	C₄×Q₈	—	2	8	no	—
339	C₄×Q₈	—	2	4	no	—
340	C₄×Q₈	—	2	2	no	—
341	C₄×Q₈	D₄×C₃₁	1	4	yes	—
342	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
343	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
344	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
345	C₄×D₄	—	2	4	no	—
346	C₄×D₄	—	2	2	no	—
347	C₄×D₄	C₂³×C₃₁	1	4	yes	—
348	C₄×D₄	C₂³×C₃₁	1	8	yes	—
349	C₂×C₄⋊C₄	—	2	2	no	—
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄⋊C₄	—	2	2	no	—
352	C₂×C₄⋊C₄	—	2	2	no	—
353	C₂×C₄⋊C₄	—	2	1	no	—
354	C₂×C₄⋊C₄	—	2	1	no	—
355	C₂×C₄⋊C₄	—	2	2	no	—
356	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
357	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
358	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
359	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
360	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
361	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
362	C₂×C₂²⋊C₄	—	2	2	no	—
363	C₂×C₂²⋊C₄	—	2	1	no	—
364	C₂×C₄²	—	2	2	no	—
365	C₂×C₄²	—	2	1	no	—
366	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
367	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
368	#107	C₂³×C₃₁	1	8	yes	—
369	#107	C₂³×C₃₁	1	8	yes	—
370	#107	C₂³×C₃₁	1	8	yes	—
371	#106	C₂³×C₃₁	1	4	yes	—
372	#106	C₂³×C₃₁	1	8	yes	—
373	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
374	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
375	#103	C₂³×C₃₁	1	4	yes	—
376	#103	C₂³×C₃₁	1	8	yes	—
377	#102	C₂³×C₃₁	1	4	yes	—
378	#102	C₂³×C₃₁	1	8	yes	—
379	#97	C₂³×C₃₁	1	8	yes	—
380	C₄×C₈	C₂³×C₃₁	1	4	yes	—
381	C₃₁	C₂×#3386	1	1	yes	characteristic, direct product factor, Sylow
382	C₂⁴	—	2	1	no	—
383	C₂×Q₈	—	2	4	no	—
384	C₂×Q₈	—	2	1	no	—
385	C₂×Q₈	—	2	4	no	—
386	C₂×Q₈	—	2	8	no	—
387	C₂×Q₈	—	2	4	no	—
388	C₂×Q₈	—	2	2	no	—
389	C₂×Q₈	—	4	8	no	—
390	C₂×Q₈	—	4	32	no	—
391	C₂×Q₈	—	4	4	no	—
392	C₂×Q₈	—	4	4	no	—
393	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
394	C₂×Q₈	C₂×Q₈×C₃₁	1	4	yes	complemented
395	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
396	C₂×D₄	—	2	1	no	—
397	C₂×D₄	—	2	2	no	—
398	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	—
399	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	8	no	—
406	C₂²×C₄	—	2	4	no	—
407	C₂²×C₄	—	4	8	no	—
408	C₂²×C₄	—	4	4	no	—
409	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
410	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
411	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
412	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
413	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
414	SD₁₆	C₂×Q₈×C₃₁	1	16	yes	complemented
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	4	no	—
417	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	complemented
418	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
419	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
420	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
421	C₄⋊C₄	—	2	8	no	—
422	C₄⋊C₄	—	2	4	no	—
423	C₄⋊C₄	—	2	8	no	—
424	C₄⋊C₄	—	2	8	no	—
425	C₄⋊C₄	—	2	4	no	—
426	C₄⋊C₄	—	2	4	no	—
427	C₄⋊C₄	—	2	8	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
433	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
434	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
435	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
436	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
437	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
438	C₂²⋊C₄	—	2	8	no	—
439	C₂²⋊C₄	—	2	4	no	—
440	C₄²	—	2	8	no	—
441	C₄²	—	2	4	no	—
442	C₄²	C₂×D₄×C₃₁	1	4	yes	—
443	C₄²	C₂×D₄×C₃₁	1	2	yes	—
444	C₄²	C₂⁴×C₃₁	1	2	yes	—
445	C₄²	C₂⁴×C₃₁	1	4	yes	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	4	2	no	—
449	C₂³	—	4	2	no	—
450	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
451	Q₈	—	2	4	no	—
452	Q₈	—	2	8	no	—
453	Q₈	—	2	2	no	—
454	Q₈	—	4	16	no	—
455	Q₈	—	4	8	no	—
456	Q₈	—	4	8	no	—
457	Q₈	C₂²×Q₈×C₃₁	1	4	yes	complemented
458	D₄	—	2	4	no	—
459	D₄	—	2	2	no	—
460	D₄	C₂²×Q₈×C₃₁	1	4	yes	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	4	16	no	—
471	C₂×C₄	—	4	4	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	4	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	8	no	—
476	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
477	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
478	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
479	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
480	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
481	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
482	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
483	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
484	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
485	C₈	—	2	2	no	—
486	C₈	—	2	4	no	—
487	C₈	C₂²×Q₈×C₃₁	1	4	yes	complemented
488	C₂²	—	2	4	no	—
489	C₂²	—	4	8	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	2	no	—
492	C₂²	D₄×Q₈×C₃₁	1	2	yes	—
493	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
494	C₂²	C₃₁×#578	1	2	yes	—
495	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
496	C₄	—	2	4	no	—
497	C₄	—	2	4	no	—
498	C₄	—	2	4	no	—
499	C₄	—	2	2	no	—
500	C₄	—	4	4	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	4	no	—
503	C₄	—	4	2	no	—
504	C₄	D₄×Q₈×C₃₁	1	2	yes	—
505	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
506	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
507	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
508	C₂	—	4	4	no	—
509	C₂	C₃₁×#3386	1	4	yes	direct product factor
510	C₂	C₂×D₄×Q₈×C₃₁	1	1	yes	characteristic
511	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
512	C₂	C₂×C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂×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
1. C₂×C₃₁×#557	C₂	semidirect
3. C₃₁×#3386	C₂	direct (N × H)
4. C₂×C₃₁×#501	C₂	semidirect
5. C₂×C₃₁×#500	C₂	semidirect
6. C₂×C₃₁×#477	C₂	semidirect
7. C₂×C₃₁×#475	C₂	semidirect
10. C₂×C₃₁×#446	C₂	semidirect
16. C₃₁×#557	C₂²	semidirect
33. C₃₁×#501	C₂²	semidirect
34. C₃₁×#500	C₂²	semidirect
36. C₃₁×#477	C₂²	semidirect
37. C₃₁×#475	C₂²	semidirect
39. C₃₁×#446	C₂²	semidirect
65. C₂×SD₁₆×C₃₁	Q₈	semidirect
137. C₂×Q₈×C₃₁	C₂×Q₈	semidirect
157. SD₁₆×C₃₁	C₂×Q₈	semidirect
160. C₂×C₈×C₃₁	C₂×Q₈	semidirect
189. C₂×#3386	C₃₁	direct (N × H)
201. Q₈×C₃₁	C₂²×Q₈	semidirect
231. C₈×C₃₁	C₂²×Q₈	semidirect
232. C₂×#557	C₂×C₃₁	semidirect
234. #3386	C₂×C₃₁	direct (N × H)
235. C₂×#501	C₂×C₃₁	semidirect
236. C₂×#500	C₂×C₃₁	semidirect
237. C₂×#477	C₂×C₃₁	semidirect
238. C₂×#475	C₂×C₃₁	semidirect
241. C₂×#446	C₂×C₃₁	semidirect
267. #557	C₂²×C₃₁	semidirect
284. #501	C₂²×C₃₁	semidirect
285. #500	C₂²×C₃₁	semidirect
287. #477	C₂²×C₃₁	semidirect
288. #475	C₂²×C₃₁	semidirect
290. #446	C₂²×C₃₁	semidirect
307. C₂×C₃₁	#3386	direct (N × H)
321. C₂×SD₁₆	Q₈×C₃₁	semidirect
381. C₃₁	C₂×#3386	direct (N × H)
394. C₂×Q₈	C₂×Q₈×C₃₁	semidirect
414. SD₁₆	C₂×Q₈×C₃₁	semidirect
417. C₂×C₈	C₂×Q₈×C₃₁	semidirect
457. Q₈	C₂²×Q₈×C₃₁	semidirect
487. C₈	C₂²×Q₈×C₃₁	semidirect
509. C₂	C₃₁×#3386	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#557	2
C₂×D₄×Q₈×C₃₁	2
C₃₁×#3386	2
C₂×C₃₁×#501	2
C₂×C₃₁×#500	2
C₂×C₃₁×#477	2
C₂×C₃₁×#475	2
C₂×C₃₁×#478	2
C₂×C₃₁×#474	2
C₂×C₃₁×#446	2
C₂×C₃₁×#440	2
C₂×C₄×SD₁₆×C₃₁	2
C₂×#3386	31

Subgroups — C2×C31×#3386 #21518119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3386 #21518119