Subgroups — C₂×C₃₁×#3372 #21518064

Order 7936.

1854 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	C₂×C₃₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#554	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3372	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#491	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#479	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#461	C₂	1	2	yes	maximal
9	C₂×C₃₁×#461	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	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
15	C₃₁×#557	—	2	4	no	—
16	C₃₁×#557	C₂²	1	8	yes	complemented
17	C₃₁×#554	—	2	4	no	—
18	C₃₁×#554	C₂²	1	8	yes	—
19	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
21	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#125	—	2	1	no	—
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	2	yes	—
29	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
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	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
34	C₃₁×#491	C₂²	1	16	yes	complemented
35	C₃₁×#487	C₂²	1	8	yes	complemented
36	C₃₁×#479	C₂²	1	16	yes	complemented
37	C₃₁×#477	C₂²	1	8	yes	complemented
38	C₃₁×#461	C₂²	1	16	yes	—
39	C₃₁×#461	C₂²	1	8	yes	—
40	C₃₁×#446	C₂²	1	8	yes	complemented
41	C₃₁×#440	C₂²	1	16	yes	—
42	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	—
43	C₂×C₃₁×#107	C₂²	1	2	yes	—
44	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
45	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
46	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
47	C₂×C₃₁×#103	C₂²	1	2	yes	—
48	C₂×C₃₁×#103	C₂²	1	2	yes	—
49	C₂×C₃₁×#103	C₂²	1	2	yes	—
50	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
51	C₂×C₃₁×#103	C₂²	1	2	yes	—
52	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
53	C₂×C₃₁×#97	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
55	C₂²×Q₈×C₃₁	—	2	1	no	—
56	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
57	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
58	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₃₁	—	2	1	no	—
61	C₂³×C₄×C₃₁	—	2	2	no	—
62	C₂×SD₁₆×C₃₁	—	2	4	no	—
63	C₂×SD₁₆×C₃₁	—	2	8	no	—
64	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
65	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
66	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
69	C₃₁×C₄⋊Q₈	—	2	2	no	—
70	C₃₁×C₄⋊Q₈	—	2	4	no	—
71	C₃₁×C₄⋊Q₈	—	2	2	no	—
72	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
73	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	—
74	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
75	C₃₁×#125	—	2	8	no	—
76	C₃₁×#125	—	2	16	no	—
77	C₃₁×#125	—	2	4	no	—
78	C₃₁×#125	C₂³	1	8	yes	—
79	C₃₁×C₂²⋊Q₈	—	2	8	no	—
80	C₃₁×C₂²⋊Q₈	—	2	16	no	—
81	C₄×Q₈×C₃₁	—	2	8	no	—
82	C₄×Q₈×C₃₁	—	2	4	no	—
83	C₄×Q₈×C₃₁	—	2	2	no	—
84	C₄×Q₈×C₃₁	D₄	1	4	yes	—
85	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
86	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
87	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
88	C₄×D₄×C₃₁	—	2	4	no	—
89	C₄×D₄×C₃₁	—	2	2	no	—
90	C₄×D₄×C₃₁	C₂³	1	4	yes	—
91	C₄×D₄×C₃₁	C₂³	1	8	yes	—
92	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
94	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
96	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
97	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
100	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
102	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
107	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
108	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
109	C₂×C₄²×C₃₁	—	2	2	no	—
110	C₂×C₄²×C₃₁	—	2	1	no	—
111	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
113	C₃₁×#107	C₂³	1	8	yes	—
114	C₃₁×#106	C₂³	1	4	yes	—
115	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
116	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
117	C₃₁×#103	C₂³	1	4	yes	—
118	C₃₁×#103	C₂³	1	8	yes	—
119	C₃₁×#103	C₂³	1	8	yes	—
120	C₃₁×#103	C₂³	1	8	yes	—
121	C₃₁×#103	C₂³	1	8	yes	—
122	C₃₁×#102	C₂³	1	4	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	2	no	—
128	C₂×Q₈×C₃₁	—	2	1	no	—
129	C₂×Q₈×C₃₁	—	2	4	no	—
130	C₂×Q₈×C₃₁	—	2	2	no	—
131	C₂×Q₈×C₃₁	—	2	4	no	—
132	C₂×Q₈×C₃₁	—	2	4	no	—
133	C₂×Q₈×C₃₁	—	2	2	no	—
134	C₂×Q₈×C₃₁	—	8	2	no	—
135	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
136	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
137	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
138	C₂×D₄×C₃₁	—	2	1	no	—
139	C₂×D₄×C₃₁	—	2	2	no	—
140	C₂×D₄×C₃₁	#41	1	4	yes	—
141	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
142	C₂²×C₄×C₃₁	—	2	1	no	—
143	C₂²×C₄×C₃₁	—	2	2	no	—
144	C₂²×C₄×C₃₁	—	2	1	no	—
145	C₂²×C₄×C₃₁	—	2	1	no	—
146	C₂²×C₄×C₃₁	—	2	2	no	—
147	C₂²×C₄×C₃₁	—	2	4	no	—
148	C₂²×C₄×C₃₁	—	2	8	no	—
149	C₂²×C₄×C₃₁	—	4	4	no	—
150	C₂²×C₄×C₃₁	—	8	4	no	—
151	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
156	SD₁₆×C₃₁	—	2	8	no	—
157	SD₁₆×C₃₁	—	2	16	no	—
158	C₂×C₈×C₃₁	—	2	2	no	—
159	C₂×C₈×C₃₁	—	2	4	no	—
160	C₂×C₈×C₃₁	—	2	2	no	—
161	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
162	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
163	C₂×C₈×C₃₁	C₂⁴	1	4	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	4	no	—
168	C₄⋊C₄×C₃₁	—	2	4	no	—
169	C₄⋊C₄×C₃₁	—	2	8	no	—
170	C₄⋊C₄×C₃₁	—	2	8	no	—
171	C₄⋊C₄×C₃₁	—	2	4	no	—
172	C₄⋊C₄×C₃₁	—	2	8	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	2	no	—
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
177	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
178	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
179	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
180	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
181	C₂²⋊C₄×C₃₁	—	2	4	no	—
182	C₂²⋊C₄×C₃₁	—	2	8	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×#3372	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	8	no	—
196	Q₈×C₃₁	—	2	2	no	—
197	Q₈×C₃₁	—	2	4	no	—
198	Q₈×C₃₁	—	2	2	no	—
199	Q₈×C₃₁	—	2	4	no	—
200	Q₈×C₃₁	—	4	4	no	—
201	Q₈×C₃₁	—	4	2	no	—
202	Q₈×C₃₁	—	8	4	no	—
203	D₄×C₃₁	—	2	4	no	—
204	D₄×C₃₁	—	2	2	no	—
205	D₄×C₃₁	C₂×#41	1	4	yes	—
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	4	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	4	no	—
214	C₂×C₄×C₃₁	—	4	4	no	—
215	C₂×C₄×C₃₁	—	4	2	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	—	4	2	no	—
218	C₂×C₄×C₃₁	—	4	4	no	—
219	C₂×C₄×C₃₁	—	4	8	no	—
220	C₂×C₄×C₃₁	—	8	8	no	—
221	C₂×C₄×C₃₁	#137	1	4	yes	—
222	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
223	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
224	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
225	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
227	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
228	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
229	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
230	C₈×C₃₁	—	2	2	no	—
231	C₈×C₃₁	—	2	4	no	—
232	C₈×C₃₁	—	2	2	no	—
233	C₂×#557	C₂×C₃₁	1	1	yes	characteristic, complemented
234	C₂×#554	C₂×C₃₁	1	1	yes	characteristic
235	#3372	C₂×C₃₁	1	16	yes	direct product factor
236	C₂×#491	C₂×C₃₁	1	2	yes	complemented
237	C₂×#487	C₂×C₃₁	1	1	yes	characteristic, complemented
238	C₂×#479	C₂×C₃₁	1	2	yes	complemented
239	C₂×#477	C₂×C₃₁	1	1	yes	characteristic, complemented
240	C₂×#461	C₂×C₃₁	1	1	yes	characteristic
241	C₂×#461	C₂×C₃₁	1	2	yes	—
242	C₂×#446	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂×#440	C₂×C₃₁	1	2	yes	—
244	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
245	C₂²×C₃₁	—	2	4	no	—
246	C₂²×C₃₁	—	4	8	no	—
247	C₂²×C₃₁	—	4	2	no	—
248	C₂²×C₃₁	—	4	2	no	—
249	C₂²×C₃₁	#548	1	2	yes	—
250	C₂²×C₃₁	C₂×#137	1	2	yes	—
251	C₂²×C₃₁	#577	1	2	yes	—
252	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
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	2	no	—
259	C₄×C₃₁	—	4	2	no	—
260	C₄×C₃₁	—	4	4	no	—
261	C₄×C₃₁	#548	1	2	yes	—
262	C₄×C₃₁	C₂×#137	1	4	yes	—
263	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
264	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
265	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
266	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
267	#557	—	2	4	no	—
268	#557	C₂²×C₃₁	1	8	yes	complemented
269	#554	—	2	4	no	—
270	#554	C₂²×C₃₁	1	8	yes	—
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	1	no	—
275	C₂×#125	—	2	2	no	—
276	C₂×#125	C₂²×C₃₁	1	2	yes	—
277	C₂×C₂²⋊Q₈	—	2	1	no	—
278	C₂×C₂²⋊Q₈	—	2	2	no	—
279	C₂×C₄×Q₈	—	2	1	no	—
280	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
282	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
283	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
284	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
285	C₂²×C₄⋊C₄	—	2	1	no	—
286	#491	C₂²×C₃₁	1	16	yes	complemented
287	#487	C₂²×C₃₁	1	8	yes	complemented
288	#479	C₂²×C₃₁	1	16	yes	complemented
289	#477	C₂²×C₃₁	1	8	yes	complemented
290	#461	C₂²×C₃₁	1	16	yes	—
291	#461	C₂²×C₃₁	1	8	yes	—
292	#446	C₂²×C₃₁	1	8	yes	complemented
293	#440	C₂²×C₃₁	1	16	yes	—
294	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	—
295	C₂×#107	C₂²×C₃₁	1	2	yes	—
296	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
297	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
298	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
299	C₂×#103	C₂²×C₃₁	1	2	yes	—
300	C₂×#103	C₂²×C₃₁	1	2	yes	—
301	C₂×#103	C₂²×C₃₁	1	2	yes	—
302	C₂×#103	C₂²×C₃₁	1	2	yes	—
303	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
304	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
305	C₂×#97	C₂²×C₃₁	1	2	yes	—
306	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
307	C₂×C₃₁	—	4	4	no	—
308	C₂×C₃₁	#3372	1	4	yes	direct product factor
309	C₂×C₃₁	C₂×#548	1	1	yes	characteristic
310	C₂×C₃₁	C₂²×#137	1	1	yes	characteristic
311	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
312	C₂²×Q₈	—	2	1	no	—
313	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
314	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
315	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
316	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
317	C₂³×C₄	—	2	1	no	—
318	C₂³×C₄	—	2	2	no	—
319	C₂×SD₁₆	—	2	4	no	—
320	C₂×SD₁₆	—	2	8	no	—
321	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
322	C₂×SD₁₆	C₂³×C₃₁	1	8	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	2	no	—
327	C₄⋊Q₈	—	2	4	no	—
328	C₄⋊Q₈	—	2	2	no	—
329	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
330	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	—
331	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
332	#125	—	2	8	no	—
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	1	no	—
353	C₂×C₄⋊C₄	—	2	1	no	—
354	C₂×C₄⋊C₄	—	2	2	no	—
355	C₂×C₄⋊C₄	—	2	2	no	—
356	C₂×C₄⋊C₄	—	2	1	no	—
357	C₂×C₄⋊C₄	—	2	2	no	—
358	C₂×C₄⋊C₄	—	2	8	no	—
359	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
360	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
361	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
362	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
363	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
364	C₂×C₂²⋊C₄	—	2	1	no	—
365	C₂×C₂²⋊C₄	—	2	2	no	—
366	C₂×C₄²	—	2	2	no	—
367	C₂×C₄²	—	2	1	no	—
368	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
369	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
370	#107	C₂³×C₃₁	1	8	yes	—
371	#106	C₂³×C₃₁	1	4	yes	—
372	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
373	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
374	#103	C₂³×C₃₁	1	4	yes	—
375	#103	C₂³×C₃₁	1	8	yes	—
376	#103	C₂³×C₃₁	1	8	yes	—
377	#103	C₂³×C₃₁	1	8	yes	—
378	#103	C₂³×C₃₁	1	8	yes	—
379	#102	C₂³×C₃₁	1	4	yes	—
380	#97	C₂³×C₃₁	1	8	yes	—
381	C₄×C₈	C₂³×C₃₁	1	4	yes	—
382	C₃₁	C₂×#3372	1	1	yes	characteristic, direct product factor, Sylow
383	C₂⁴	—	2	1	no	—
384	C₂×Q₈	—	2	4	no	—
385	C₂×Q₈	—	2	2	no	—
386	C₂×Q₈	—	2	1	no	—
387	C₂×Q₈	—	2	4	no	—
388	C₂×Q₈	—	2	2	no	—
389	C₂×Q₈	—	2	4	no	—
390	C₂×Q₈	—	2	2	no	—
391	C₂×Q₈	—	2	4	no	—
392	C₂×Q₈	—	8	2	no	—
393	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
394	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
395	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
396	C₂×D₄	—	2	1	no	—
397	C₂×D₄	—	2	2	no	—
398	C₂×D₄	#41×C₃₁	1	4	yes	—
399	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	1	no	—
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	2	4	no	—
406	C₂²×C₄	—	2	8	no	—
407	C₂²×C₄	—	4	4	no	—
408	C₂²×C₄	—	8	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	—	2	8	no	—
415	SD₁₆	—	2	16	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	4	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
420	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
421	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
422	C₄⋊C₄	—	2	8	no	—
423	C₄⋊C₄	—	2	4	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	8	no	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	8	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
435	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
436	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
437	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
438	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
439	C₂²⋊C₄	—	2	4	no	—
440	C₂²⋊C₄	—	2	8	no	—
441	C₄²	—	2	8	no	—
442	C₄²	—	2	4	no	—
443	C₄²	C₂×D₄×C₃₁	1	4	yes	—
444	C₄²	C₂×D₄×C₃₁	1	2	yes	—
445	C₄²	C₂⁴×C₃₁	1	2	yes	—
446	C₄²	C₂⁴×C₃₁	1	4	yes	—
447	C₂³	—	2	4	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	4	2	no	—
450	C₂³	—	4	2	no	—
451	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
452	Q₈	—	2	8	no	—
453	Q₈	—	2	2	no	—
454	Q₈	—	2	4	no	—
455	Q₈	—	2	2	no	—
456	Q₈	—	2	4	no	—
457	Q₈	—	4	4	no	—
458	Q₈	—	4	2	no	—
459	Q₈	—	8	4	no	—
460	D₄	—	2	4	no	—
461	D₄	—	2	2	no	—
462	D₄	C₂×#41×C₃₁	1	4	yes	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	4	4	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	4	no	—
476	C₂×C₄	—	4	8	no	—
477	C₂×C₄	—	8	8	no	—
478	C₂×C₄	C₃₁×#137	1	4	yes	—
479	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
480	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
481	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
482	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
483	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
484	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
485	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
486	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
487	C₈	—	2	2	no	—
488	C₈	—	2	4	no	—
489	C₈	—	2	2	no	—
490	C₂²	—	2	4	no	—
491	C₂²	—	4	8	no	—
492	C₂²	—	4	2	no	—
493	C₂²	—	4	2	no	—
494	C₂²	C₃₁×#548	1	2	yes	—
495	C₂²	C₂×C₃₁×#137	1	2	yes	—
496	C₂²	C₃₁×#577	1	2	yes	—
497	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
498	C₄	—	2	4	no	—
499	C₄	—	2	4	no	—
500	C₄	—	2	2	no	—
501	C₄	—	4	4	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	4	no	—
505	C₄	—	4	2	no	—
506	C₄	C₃₁×#548	1	2	yes	—
507	C₄	C₂×C₃₁×#137	1	4	yes	—
508	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
509	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
510	C₂	—	4	4	no	—
511	C₂	C₃₁×#3372	1	4	yes	direct product factor
512	C₂	C₂×C₃₁×#548	1	1	yes	characteristic
513	C₂	C₂²×C₃₁×#137	1	1	yes	characteristic
514	C₂	C₂×C₃₁×#577	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×#3372	C₂	direct (N × H)
4. C₂×C₃₁×#491	C₂	semidirect
5. C₂×C₃₁×#487	C₂	semidirect
6. C₂×C₃₁×#479	C₂	semidirect
7. C₂×C₃₁×#477	C₂	semidirect
10. C₂×C₃₁×#446	C₂	semidirect
16. C₃₁×#557	C₂²	semidirect
34. C₃₁×#491	C₂²	semidirect
35. C₃₁×#487	C₂²	semidirect
36. C₃₁×#479	C₂²	semidirect
37. C₃₁×#477	C₂²	semidirect
40. C₃₁×#446	C₂²	semidirect
189. C₂×#3372	C₃₁	direct (N × H)
233. C₂×#557	C₂×C₃₁	semidirect
235. #3372	C₂×C₃₁	direct (N × H)
236. C₂×#491	C₂×C₃₁	semidirect
237. C₂×#487	C₂×C₃₁	semidirect
238. C₂×#479	C₂×C₃₁	semidirect
239. C₂×#477	C₂×C₃₁	semidirect
242. C₂×#446	C₂×C₃₁	semidirect
268. #557	C₂²×C₃₁	semidirect
286. #491	C₂²×C₃₁	semidirect
287. #487	C₂²×C₃₁	semidirect
288. #479	C₂²×C₃₁	semidirect
289. #477	C₂²×C₃₁	semidirect
292. #446	C₂²×C₃₁	semidirect
308. C₂×C₃₁	#3372	direct (N × H)
382. C₃₁	C₂×#3372	direct (N × H)
511. C₂	C₃₁×#3372	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×C₃₁×#554	2
C₃₁×#3372	2
C₂×C₃₁×#491	2
C₂×C₃₁×#487	2
C₂×C₃₁×#479	2
C₂×C₃₁×#477	2
C₂×C₃₁×#461	2
C₂×C₃₁×#446	2
C₂×C₃₁×#440	2
C₂×C₄×SD₁₆×C₃₁	2
C₂×#3372	31

Subgroups — C2×C31×#3372 #21518064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3372 #21518064