Subgroups — C₂×C₃₁×#2451 #21491449

Order 7936.

2702 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2451	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#398	C₂	1	2	yes	maximal
5	C₂×C₃₁×#395	C₂	1	2	yes	maximal
6	C₂×C₃₁×#391	C₂	1	2	yes	maximal
7	C₂×C₃₁×#391	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃₁×#391	C₂	1	1	yes	maximal, characteristic
9	C₂×C₃₁×#389	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×#389	C₂	1	1	yes	maximal, characteristic
11	C₂×C₃₁×#387	C₂	1	2	yes	maximal
12	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
13	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
14	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	—	2	8	no	—
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	8	yes	—
17	C₂×C₄×Q₈×C₃₁	—	2	8	no	—
18	C₂×C₄×Q₈×C₃₁	C₂²	1	8	yes	—
19	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
30	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
31	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#398	C₂²	1	16	yes	—
33	C₃₁×#395	C₂²	1	16	yes	—
34	C₃₁×#391	C₂²	1	8	yes	—
35	C₃₁×#391	C₂²	1	8	yes	—
36	C₃₁×#391	C₂²	1	16	yes	—
37	C₃₁×#389	C₂²	1	8	yes	—
38	C₃₁×#389	C₂²	1	8	yes	—
39	C₃₁×#387	C₂²	1	16	yes	—
40	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	—
41	C₂×C₃₁×#95	C₂²	1	2	yes	—
42	C₂×C₃₁×#95	C₂²	1	2	yes	—
43	C₂×C₃₁×#95	C₂²	1	2	yes	—
44	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
45	C₂×C₃₁×#95	C₂²	1	2	yes	—
46	C₂×C₃₁×#95	C₂²	1	2	yes	—
47	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
48	C₂×C₃₁×#95	C₂²	1	2	yes	—
49	C₂²×Q₈×C₃₁	—	2	4	no	—
50	C₂²×Q₈×C₃₁	—	2	8	no	—
51	C₂²×Q₈×C₃₁	—	2	4	no	—
52	C₂²×Q₈×C₃₁	—	2	8	no	—
53	C₂²×Q₈×C₃₁	—	4	4	no	—
54	C₂²×Q₈×C₃₁	Q₈	1	8	yes	complemented
55	C₂²×Q₈×C₃₁	C₂³	1	8	yes	—
56	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
57	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
60	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
61	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
62	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
66	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×C₄⋊Q₈	—	2	32	no	—
69	C₄×Q₈×C₃₁	—	2	32	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
75	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
76	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
77	C₂×C₄⋊C₄×C₃₁	D₄	1	8	yes	—
78	C₂×C₄⋊C₄×C₃₁	Q₈	1	8	yes	complemented
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
88	C₂×C₄²×C₃₁	—	2	8	no	—
89	C₂×C₄²×C₃₁	—	2	4	no	—
90	C₂×C₄²×C₃₁	—	2	4	no	—
91	C₂×C₄²×C₃₁	—	2	4	no	—
92	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
93	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
94	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
95	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
96	C₃₁×#95	C₂³	1	8	yes	—
97	C₃₁×#95	C₂³	1	8	yes	—
98	C₃₁×#95	C₂³	1	8	yes	—
99	C₃₁×#95	C₂³	1	4	yes	—
100	C₃₁×#95	C₂³	1	8	yes	—
101	C₃₁×#95	C₂³	1	8	yes	—
102	C₃₁×#95	C₂³	1	4	yes	—
103	C₃₁×#95	C₂³	1	8	yes	—
104	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
105	C₂×Q₈×C₃₁	—	2	32	no	—
106	C₂×Q₈×C₃₁	—	2	8	no	—
107	C₂×Q₈×C₃₁	—	2	16	no	—
108	C₂×Q₈×C₃₁	—	2	8	no	—
109	C₂×Q₈×C₃₁	—	2	16	no	—
110	C₂×Q₈×C₃₁	—	4	8	no	—
111	C₂×Q₈×C₃₁	—	4	32	no	—
112	C₂×Q₈×C₃₁	C₂×Q₈	1	16	yes	complemented
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	—	2	4	no	—
116	C₂²×C₄×C₃₁	—	2	2	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	4	no	—
120	C₂²×C₄×C₃₁	—	2	4	no	—
121	C₂²×C₄×C₃₁	—	2	4	no	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	2	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	2	4	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	—	2	4	no	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	2	4	no	—
131	C₂²×C₄×C₃₁	—	2	2	no	—
132	C₂²×C₄×C₃₁	—	2	4	no	—
133	C₂²×C₄×C₃₁	—	2	2	no	—
134	C₂²×C₄×C₃₁	—	2	4	no	—
135	C₂²×C₄×C₃₁	—	2	2	no	—
136	C₂²×C₄×C₃₁	—	2	4	no	—
137	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
138	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
139	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
140	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
141	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
142	C₂²×C₄×C₃₁	C₂×Q₈	1	8	yes	—
143	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
144	C₂²×C₄×C₃₁	#41	1	4	yes	—
145	C₂²×C₄×C₃₁	#41	1	4	yes	—
146	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
151	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
152	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
154	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
157	C₄⋊C₄×C₃₁	—	2	16	no	—
158	C₄⋊C₄×C₃₁	—	2	16	no	—
159	C₄⋊C₄×C₃₁	—	2	8	no	—
160	C₄⋊C₄×C₃₁	—	2	8	no	—
161	C₄⋊C₄×C₃₁	—	2	8	no	—
162	C₄⋊C₄×C₃₁	—	4	8	no	—
163	C₄⋊C₄×C₃₁	—	4	8	no	—
164	C₄⋊C₄×C₃₁	C₂×D₄	1	16	yes	—
165	C₄⋊C₄×C₃₁	C₂×Q₈	1	16	yes	complemented
166	C₄²×C₃₁	—	2	16	no	—
167	C₄²×C₃₁	—	2	8	no	—
168	C₄²×C₃₁	—	2	8	no	—
169	C₄²×C₃₁	—	2	8	no	—
170	C₂×#2451	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
172	C₂³×C₃₁	C₂²×Q₈	1	2	yes	—
173	C₂³×C₃₁	C₂²×Q₈	1	2	yes	—
174	C₂³×C₃₁	C₂×#41	1	2	yes	—
175	C₂³×C₃₁	C₂×#41	1	2	yes	—
176	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
177	C₂³×C₃₁	2₋¹⁺⁴	1	2	yes	—
178	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
179	Q₈×C₃₁	—	2	32	no	—
180	Q₈×C₃₁	—	4	32	no	—
181	C₂×C₄×C₃₁	—	2	16	no	—
182	C₂×C₄×C₃₁	—	2	8	no	—
183	C₂×C₄×C₃₁	—	2	16	no	—
184	C₂×C₄×C₃₁	—	2	8	no	—
185	C₂×C₄×C₃₁	—	2	8	no	—
186	C₂×C₄×C₃₁	—	2	8	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	4	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	4	no	—
194	C₂×C₄×C₃₁	—	2	4	no	—
195	C₂×C₄×C₃₁	—	2	4	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	4	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	4	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	4	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	4	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	4	4	no	—
212	C₂×C₄×C₃₁	—	4	8	no	—
213	C₂×C₄×C₃₁	—	4	4	no	—
214	C₂×C₄×C₃₁	—	4	8	no	—
215	C₂×C₄×C₃₁	C₂²⋊Q₈	1	16	yes	—
216	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
218	C₂×C₄×C₃₁	C₂²×Q₈	1	4	yes	—
219	C₂×C₄×C₃₁	C₂²×Q₈	1	4	yes	—
220	C₂×C₄×C₃₁	C₂²×Q₈	1	4	yes	—
221	C₂×C₄×C₃₁	C₂²×Q₈	1	8	yes	—
222	C₂×C₄×C₃₁	C₂²×Q₈	1	4	yes	—
223	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
224	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
225	C₂²×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
226	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
227	#2451	C₂×C₃₁	1	16	yes	direct product factor
228	C₂×#398	C₂×C₃₁	1	2	yes	—
229	C₂×#395	C₂×C₃₁	1	2	yes	—
230	C₂×#391	C₂×C₃₁	1	1	yes	characteristic
231	C₂×#391	C₂×C₃₁	1	1	yes	characteristic
232	C₂×#391	C₂×C₃₁	1	2	yes	—
233	C₂×#389	C₂×C₃₁	1	1	yes	characteristic
234	C₂×#389	C₂×C₃₁	1	1	yes	characteristic
235	C₂×#387	C₂×C₃₁	1	2	yes	—
236	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
237	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	4	yes	—
238	C₂²×C₃₁	#533	1	4	yes	—
239	C₂²×C₃₁	#542	1	4	yes	—
240	C₂²×C₃₁	#546	1	4	yes	—
241	C₂²×C₃₁	D₄×Q₈	1	4	yes	—
242	C₂²×C₃₁	#557	1	4	yes	—
243	C₂²×C₃₁	Q₈²	1	4	yes	—
244	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
245	C₂²×C₃₁	C₂³×Q₈	1	1	yes	characteristic
246	C₂²×C₃₁	C₂³×Q₈	1	1	yes	characteristic
247	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
248	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
249	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
250	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
251	C₄×C₃₁	—	2	8	no	—
252	C₄×C₃₁	—	2	4	no	—
253	C₄×C₃₁	—	2	8	no	—
254	C₄×C₃₁	—	2	4	no	—
255	C₄×C₃₁	—	2	4	no	—
256	C₄×C₃₁	—	2	4	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	4	4	no	—
259	C₄×C₃₁	—	4	2	no	—
260	C₄×C₃₁	—	4	4	no	—
261	C₄×C₃₁	C₂×C₂²⋊Q₈	1	8	yes	—
262	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
263	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×C₄⋊Q₈	—	2	8	no	—
265	C₂×C₄⋊Q₈	C₂²×C₃₁	1	8	yes	—
266	C₂×C₄×Q₈	—	2	8	no	—
267	C₂×C₄×Q₈	C₂²×C₃₁	1	8	yes	—
268	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
269	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
270	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
271	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
272	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
273	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
274	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
275	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
276	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
277	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
278	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
279	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
280	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
281	#398	C₂²×C₃₁	1	16	yes	—
282	#395	C₂²×C₃₁	1	16	yes	—
283	#391	C₂²×C₃₁	1	16	yes	—
284	#391	C₂²×C₃₁	1	8	yes	—
285	#391	C₂²×C₃₁	1	8	yes	—
286	#389	C₂²×C₃₁	1	8	yes	—
287	#389	C₂²×C₃₁	1	8	yes	—
288	#387	C₂²×C₃₁	1	16	yes	—
289	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	—
290	C₂×#95	C₂²×C₃₁	1	2	yes	—
291	C₂×#95	C₂²×C₃₁	1	2	yes	—
292	C₂×#95	C₂²×C₃₁	1	2	yes	—
293	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
294	C₂×#95	C₂²×C₃₁	1	2	yes	—
295	C₂×#95	C₂²×C₃₁	1	2	yes	—
296	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
297	C₂×#95	C₂²×C₃₁	1	2	yes	—
298	C₂×C₃₁	#2451	1	8	yes	direct product factor
299	C₂×C₃₁	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
300	C₂×C₃₁	C₂×#533	1	1	yes	characteristic
301	C₂×C₃₁	C₂×#542	1	1	yes	characteristic
302	C₂×C₃₁	C₂×#546	1	1	yes	characteristic
303	C₂×C₃₁	C₂×D₄×Q₈	1	1	yes	characteristic
304	C₂×C₃₁	C₂×#557	1	1	yes	characteristic
305	C₂×C₃₁	C₂×Q₈²	1	1	yes	characteristic
306	C₂²×Q₈	—	2	4	no	—
307	C₂²×Q₈	—	2	8	no	—
308	C₂²×Q₈	—	2	4	no	—
309	C₂²×Q₈	—	2	8	no	—
310	C₂²×Q₈	—	4	4	no	—
311	C₂²×Q₈	Q₈×C₃₁	1	8	yes	complemented
312	C₂²×Q₈	C₂³×C₃₁	1	8	yes	—
313	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
314	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
315	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
316	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
317	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
318	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
319	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
320	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
321	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
322	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
323	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
324	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
325	C₄⋊Q₈	—	2	32	no	—
326	C₄×Q₈	—	2	32	no	—
327	C₂×C₄⋊C₄	—	2	8	no	—
328	C₂×C₄⋊C₄	—	2	8	no	—
329	C₂×C₄⋊C₄	—	2	4	no	—
330	C₂×C₄⋊C₄	—	2	4	no	—
331	C₂×C₄⋊C₄	—	2	4	no	—
332	C₂×C₄⋊C₄	—	4	4	no	—
333	C₂×C₄⋊C₄	—	4	4	no	—
334	C₂×C₄⋊C₄	D₄×C₃₁	1	8	yes	—
335	C₂×C₄⋊C₄	Q₈×C₃₁	1	8	yes	complemented
336	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
337	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
338	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
339	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
340	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
341	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
342	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
343	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
344	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
345	C₂×C₄²	—	2	8	no	—
346	C₂×C₄²	—	2	4	no	—
347	C₂×C₄²	—	2	4	no	—
348	C₂×C₄²	—	2	4	no	—
349	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
350	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
351	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
352	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
353	#95	C₂³×C₃₁	1	4	yes	—
354	#95	C₂³×C₃₁	1	4	yes	—
355	#95	C₂³×C₃₁	1	8	yes	—
356	#95	C₂³×C₃₁	1	8	yes	—
357	#95	C₂³×C₃₁	1	8	yes	—
358	#95	C₂³×C₃₁	1	8	yes	—
359	#95	C₂³×C₃₁	1	8	yes	—
360	#95	C₂³×C₃₁	1	8	yes	—
361	C₃₁	C₂×#2451	1	1	yes	characteristic, direct product factor, Sylow
362	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
363	C₂×Q₈	—	2	32	no	—
364	C₂×Q₈	—	2	8	no	—
365	C₂×Q₈	—	2	16	no	—
366	C₂×Q₈	—	2	8	no	—
367	C₂×Q₈	—	2	16	no	—
368	C₂×Q₈	—	4	8	no	—
369	C₂×Q₈	—	4	32	no	—
370	C₂×Q₈	C₂×Q₈×C₃₁	1	16	yes	complemented
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	4	no	—
378	C₂²×C₄	—	2	4	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	4	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
396	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
397	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
398	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
399	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
400	C₂²×C₄	C₂×Q₈×C₃₁	1	8	yes	—
401	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
402	C₂²×C₄	#41×C₃₁	1	4	yes	—
403	C₂²×C₄	#41×C₃₁	1	4	yes	—
404	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
405	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
406	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
407	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
408	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
409	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
410	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
411	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
412	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
413	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
414	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
415	C₄⋊C₄	—	2	16	no	—
416	C₄⋊C₄	—	2	16	no	—
417	C₄⋊C₄	—	2	8	no	—
418	C₄⋊C₄	—	2	8	no	—
419	C₄⋊C₄	—	2	8	no	—
420	C₄⋊C₄	—	4	8	no	—
421	C₄⋊C₄	—	4	8	no	—
422	C₄⋊C₄	C₂×D₄×C₃₁	1	16	yes	—
423	C₄⋊C₄	C₂×Q₈×C₃₁	1	16	yes	complemented
424	C₄²	—	2	16	no	—
425	C₄²	—	2	8	no	—
426	C₄²	—	2	8	no	—
427	C₄²	—	2	8	no	—
428	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
429	C₂³	C₂²×Q₈×C₃₁	1	2	yes	—
430	C₂³	C₂²×Q₈×C₃₁	1	2	yes	—
431	C₂³	C₂×#41×C₃₁	1	2	yes	—
432	C₂³	C₂×#41×C₃₁	1	2	yes	—
433	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
434	C₂³	C₃₁×2₋¹⁺⁴	1	2	yes	—
435	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
436	Q₈	—	2	32	no	—
437	Q₈	—	4	32	no	—
438	C₂×C₄	—	2	16	no	—
439	C₂×C₄	—	2	8	no	—
440	C₂×C₄	—	2	16	no	—
441	C₂×C₄	—	2	8	no	—
442	C₂×C₄	—	2	8	no	—
443	C₂×C₄	—	2	8	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	4	4	no	—
469	C₂×C₄	—	4	8	no	—
470	C₂×C₄	—	4	4	no	—
471	C₂×C₄	—	4	8	no	—
472	C₂×C₄	C₃₁×C₂²⋊Q₈	1	16	yes	—
473	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
474	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
475	C₂×C₄	C₂²×Q₈×C₃₁	1	4	yes	—
476	C₂×C₄	C₂²×Q₈×C₃₁	1	4	yes	—
477	C₂×C₄	C₂²×Q₈×C₃₁	1	4	yes	—
478	C₂×C₄	C₂²×Q₈×C₃₁	1	8	yes	—
479	C₂×C₄	C₂²×Q₈×C₃₁	1	4	yes	—
480	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
481	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
482	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	4	yes	—
483	C₂²	C₃₁×#533	1	4	yes	—
484	C₂²	C₃₁×#542	1	4	yes	—
485	C₂²	C₃₁×#546	1	4	yes	—
486	C₂²	D₄×Q₈×C₃₁	1	4	yes	—
487	C₂²	C₃₁×#557	1	4	yes	—
488	C₂²	Q₈²×C₃₁	1	4	yes	—
489	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
490	C₂²	C₂³×Q₈×C₃₁	1	1	yes	characteristic
491	C₂²	C₂³×Q₈×C₃₁	1	1	yes	characteristic
492	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
493	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
494	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
495	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
496	C₄	—	2	8	no	—
497	C₄	—	2	4	no	—
498	C₄	—	2	8	no	—
499	C₄	—	2	4	no	—
500	C₄	—	2	4	no	—
501	C₄	—	2	4	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	4	no	—
504	C₄	—	4	4	no	—
505	C₄	—	4	2	no	—
506	C₄	C₂×C₃₁×C₂²⋊Q₈	1	8	yes	—
507	C₂	C₃₁×#2451	1	8	yes	direct product factor
508	C₂	C₂²×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
509	C₂	C₂×C₃₁×#533	1	1	yes	characteristic
510	C₂	C₂×C₃₁×#542	1	1	yes	characteristic
511	C₂	C₂×C₃₁×#546	1	1	yes	characteristic
512	C₂	C₂×D₄×Q₈×C₃₁	1	1	yes	characteristic
513	C₂	C₂×C₃₁×#557	1	1	yes	characteristic
514	C₂	C₂×Q₈²×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2451	C₂	direct (N × H)
54. C₂²×Q₈×C₃₁	Q₈	semidirect
78. C₂×C₄⋊C₄×C₃₁	Q₈	semidirect
112. C₂×Q₈×C₃₁	C₂×Q₈	semidirect
165. C₄⋊C₄×C₃₁	C₂×Q₈	semidirect
170. C₂×#2451	C₃₁	direct (N × H)
227. #2451	C₂×C₃₁	direct (N × H)
298. C₂×C₃₁	#2451	direct (N × H)
311. C₂²×Q₈	Q₈×C₃₁	semidirect
335. C₂×C₄⋊C₄	Q₈×C₃₁	semidirect
361. C₃₁	C₂×#2451	direct (N × H)
370. C₂×Q₈	C₂×Q₈×C₃₁	semidirect
423. C₄⋊C₄	C₂×Q₈×C₃₁	semidirect
507. C₂	C₃₁×#2451	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×C₄⋊Q₈	2
C₂²×C₄×Q₈×C₃₁	2
C₃₁×#2451	2
C₂×C₃₁×#398	2
C₂×C₃₁×#395	2
C₂×C₃₁×#391	2
C₂×C₃₁×#389	2
C₂×C₃₁×#387	2
C₂×C₄×C₄⋊C₄×C₃₁	2
C₂×#2451	31

Subgroups — C2×C31×#2451 #21491449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2451 #21491449