Subgroups — C₂×C₃₁×#3216 #21517547

Order 7936.

2166 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#532	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3216	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#497	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#489	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#462	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#460	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#452	C₂	1	4	yes	maximal, complemented
9	C₂×C₃₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₃₁	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₃₁	C₂²	1	4	yes	—
12	C₂²×SD₁₆×C₃₁	C₂²	1	4	yes	—
13	C₃₁×#533	—	2	4	no	—
14	C₃₁×#533	C₂²	1	8	yes	complemented
15	C₃₁×#532	—	2	4	no	—
16	C₃₁×#532	C₂²	1	8	yes	—
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#125	—	2	1	no	—
22	C₂×C₃₁×#124	C₂²	1	2	yes	—
23	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
24	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₂×C₃₁×#121	—	2	1	no	—
26	C₂×C₃₁×#121	—	2	1	no	—
27	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
28	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₃₁	—	2	1	no	—
30	C₂×C₄×D₄×C₃₁	—	2	1	no	—
31	C₂×C₃₁×#117	—	2	1	no	—
32	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#497	C₂²	1	16	yes	complemented
34	C₃₁×#489	C₂²	1	16	yes	complemented
35	C₃₁×#462	C₂²	1	16	yes	complemented
36	C₃₁×#460	C₂²	1	16	yes	complemented
37	C₃₁×#452	C₂²	1	32	yes	complemented
38	C₃₁×#432	C₂²	1	8	yes	complemented
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
40	C₂×C₃₁×#103	C₂²	1	4	yes	—
41	C₂×C₃₁×#102	C₂²	1	4	yes	—
42	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
43	C₂×C₃₁×#97	C₂²	1	2	yes	—
44	C₂×#41×C₃₁	—	2	4	no	—
45	C₂×#41×C₃₁	—	4	8	no	—
46	C₂×#41×C₃₁	C₂³	1	8	yes	—
47	C₂²×Q₈×C₃₁	—	2	1	no	—
48	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
49	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×D₄×C₃₁	—	2	2	no	—
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	—	2	1	no	—
53	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
54	C₂³×C₄×C₃₁	—	2	2	no	—
55	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
56	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
57	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×Q₁₆×C₃₁	—	4	8	no	—
59	C₂×Q₁₆×C₃₁	C₂³	1	16	yes	—
60	C₂×SD₁₆×C₃₁	—	4	8	no	—
61	C₂×SD₁₆×C₃₁	C₂³	1	16	yes	—
62	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
63	C₃₁×C₄⋊Q₈	—	2	2	no	—
64	C₃₁×C₄⋊Q₈	—	2	2	no	—
65	C₃₁×C₄⋊Q₈	—	2	2	no	—
66	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
67	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
68	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
69	C₃₁×#127	—	2	2	no	—
70	C₃₁×#127	C₂³	1	4	yes	—
71	C₃₁×#125	—	2	8	no	—
72	C₃₁×#124	—	2	4	no	—
73	C₃₁×#124	C₂³	1	8	yes	—
74	C₃₁×C₂²⋊Q₈	—	2	8	no	—
75	C₃₁×C₂²⋊Q₈	—	2	4	no	—
76	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
77	C₃₁×#121	—	2	8	no	—
78	C₃₁×#121	—	2	8	no	—
79	C₄×Q₈×C₃₁	—	2	8	no	—
80	C₄×Q₈×C₃₁	—	2	4	no	—
81	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
82	C₄×D₄×C₃₁	—	2	8	no	—
83	C₄×D₄×C₃₁	—	2	8	no	—
84	C₃₁×#117	—	2	8	no	—
85	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
87	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
88	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
89	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
90	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
91	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
93	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
96	C₂×C₄²×C₃₁	—	2	2	no	—
97	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₃₁	D₄	1	4	yes	—
100	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
103	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
104	C₃₁×#103	C₂³	1	16	yes	—
105	C₃₁×#102	C₂³	1	16	yes	—
106	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
107	C₃₁×#97	C₂³	1	8	yes	—
108	C₂⁴×C₃₁	—	2	2	no	—
109	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
110	#41×C₃₁	—	4	16	no	—
111	#41×C₃₁	—	4	16	no	—
112	C₂×Q₈×C₃₁	—	2	2	no	—
113	C₂×Q₈×C₃₁	—	2	4	no	—
114	C₂×Q₈×C₃₁	—	2	2	no	—
115	C₂×Q₈×C₃₁	—	2	4	no	—
116	C₂×Q₈×C₃₁	—	2	4	no	—
117	C₂×Q₈×C₃₁	—	2	8	no	—
118	C₂×Q₈×C₃₁	—	4	4	no	—
119	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
120	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
121	C₂×D₄×C₃₁	—	2	8	no	—
122	C₂×D₄×C₃₁	—	2	4	no	—
123	C₂×D₄×C₃₁	—	2	4	no	—
124	C₂×D₄×C₃₁	—	2	2	no	—
125	C₂×D₄×C₃₁	—	2	8	no	—
126	C₂×D₄×C₃₁	—	4	4	no	—
127	C₂×D₄×C₃₁	—	4	8	no	—
128	C₂×D₄×C₃₁	—	4	4	no	—
129	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₃₁	—	2	4	no	—
131	C₂²×C₄×C₃₁	—	2	4	no	—
132	C₂²×C₄×C₃₁	—	2	2	no	—
133	C₂²×C₄×C₃₁	—	2	2	no	—
134	C₂²×C₄×C₃₁	—	2	2	no	—
135	C₂²×C₄×C₃₁	—	2	8	no	—
136	C₂²×C₄×C₃₁	—	4	2	no	—
137	C₂²×C₄×C₃₁	—	4	8	no	—
138	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
139	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
145	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
148	Q₁₆×C₃₁	—	4	16	no	—
149	SD₁₆×C₃₁	—	4	16	no	—
150	C₂×C₈×C₃₁	—	2	8	no	—
151	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
152	C₄⋊C₄×C₃₁	—	2	4	no	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₄⋊C₄×C₃₁	—	2	4	no	—
155	C₄⋊C₄×C₃₁	—	2	4	no	—
156	C₄⋊C₄×C₃₁	—	2	4	no	—
157	C₄⋊C₄×C₃₁	—	2	8	no	—
158	C₄⋊C₄×C₃₁	—	2	8	no	—
159	C₄⋊C₄×C₃₁	—	2	2	no	—
160	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
161	C₂²⋊C₄×C₃₁	—	2	8	no	—
162	C₂²⋊C₄×C₃₁	—	2	8	no	—
163	C₂²⋊C₄×C₃₁	—	2	8	no	—
164	C₄²×C₃₁	—	2	8	no	—
165	C₄²×C₃₁	C₂×D₄	1	4	yes	—
166	C₄²×C₃₁	C₂×D₄	1	4	yes	—
167	C₄²×C₃₁	C₂×D₄	1	2	yes	—
168	C₄²×C₃₁	C₂×D₄	1	2	yes	—
169	C₄²×C₃₁	C₂⁴	1	2	yes	—
170	C₄²×C₃₁	C₂⁴	1	2	yes	—
171	C₂×#3216	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂³×C₃₁	—	2	2	no	—
173	C₂³×C₃₁	—	2	2	no	—
174	C₂³×C₃₁	—	2	2	no	—
175	C₂³×C₃₁	—	2	4	no	—
176	C₂³×C₃₁	—	4	4	no	—
177	C₂³×C₃₁	—	4	4	no	—
178	C₂³×C₃₁	—	4	4	no	—
179	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
180	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
181	Q₈×C₃₁	—	2	4	no	—
182	Q₈×C₃₁	—	2	4	no	—
183	Q₈×C₃₁	—	2	8	no	—
184	Q₈×C₃₁	—	4	8	no	—
185	Q₈×C₃₁	—	4	4	no	—
186	Q₈×C₃₁	—	4	4	no	—
187	D₄×C₃₁	—	2	8	no	—
188	D₄×C₃₁	—	4	8	no	—
189	D₄×C₃₁	—	4	16	no	—
190	D₄×C₃₁	—	4	4	no	—
191	D₄×C₃₁	—	4	8	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	4	no	—
197	C₂×C₄×C₃₁	—	2	8	no	—
198	C₂×C₄×C₃₁	—	2	4	no	—
199	C₂×C₄×C₃₁	—	2	4	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	2	4	no	—
202	C₂×C₄×C₃₁	—	4	4	no	—
203	C₂×C₄×C₃₁	—	4	16	no	—
204	C₂×C₄×C₃₁	—	4	4	no	—
205	C₂×C₄×C₃₁	—	4	4	no	—
206	C₂×C₄×C₃₁	—	4	4	no	—
207	C₂×C₄×C₃₁	#137	1	4	yes	—
208	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
209	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
210	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
211	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
215	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
216	C₈×C₃₁	—	2	8	no	—
217	C₂×#533	C₂×C₃₁	1	1	yes	characteristic, complemented
218	C₂×#532	C₂×C₃₁	1	1	yes	characteristic
219	#3216	C₂×C₃₁	1	16	yes	direct product factor
220	C₂×#497	C₂×C₃₁	1	2	yes	complemented
221	C₂×#489	C₂×C₃₁	1	2	yes	complemented
222	C₂×#462	C₂×C₃₁	1	2	yes	complemented
223	C₂×#460	C₂×C₃₁	1	2	yes	complemented
224	C₂×#452	C₂×C₃₁	1	4	yes	complemented
225	C₂×#432	C₂×C₃₁	1	1	yes	characteristic, complemented
226	C₂²×C₃₁	—	2	2	no	—
227	C₂²×C₃₁	—	2	2	no	—
228	C₂²×C₃₁	—	2	2	no	—
229	C₂²×C₃₁	—	4	4	no	—
230	C₂²×C₃₁	—	4	4	no	—
231	C₂²×C₃₁	—	4	4	no	—
232	C₂²×C₃₁	—	4	4	no	—
233	C₂²×C₃₁	—	8	8	no	—
234	C₂²×C₃₁	#535	1	2	yes	—
235	C₂²×C₃₁	C₂×#137	1	2	yes	—
236	C₂²×C₃₁	#575	1	2	yes	—
237	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
238	C₄×C₃₁	—	2	2	no	—
239	C₄×C₃₁	—	2	2	no	—
240	C₄×C₃₁	—	2	2	no	—
241	C₄×C₃₁	—	2	2	no	—
242	C₄×C₃₁	—	4	4	no	—
243	C₄×C₃₁	—	4	4	no	—
244	C₄×C₃₁	—	4	4	no	—
245	C₄×C₃₁	#535	1	2	yes	—
246	C₄×C₃₁	C₂×#137	1	4	yes	—
247	C₄×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
248	C₂²×#41	C₂²×C₃₁	1	2	yes	—
249	C₂²×Q₁₆	C₂²×C₃₁	1	4	yes	—
250	C₂²×SD₁₆	C₂²×C₃₁	1	4	yes	—
251	#533	—	2	4	no	—
252	#533	C₂²×C₃₁	1	8	yes	complemented
253	#532	—	2	4	no	—
254	#532	C₂²×C₃₁	1	8	yes	—
255	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×#125	—	2	1	no	—
260	C₂×#124	C₂²×C₃₁	1	2	yes	—
261	C₂×C₂²⋊Q₈	—	2	1	no	—
262	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
263	C₂×#121	—	2	1	no	—
264	C₂×#121	—	2	1	no	—
265	C₂×C₄×Q₈	—	2	1	no	—
266	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
267	C₂×C₄×D₄	—	2	1	no	—
268	C₂×C₄×D₄	—	2	1	no	—
269	C₂×#117	—	2	1	no	—
270	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
271	#497	C₂²×C₃₁	1	16	yes	complemented
272	#489	C₂²×C₃₁	1	16	yes	complemented
273	#462	C₂²×C₃₁	1	16	yes	complemented
274	#460	C₂²×C₃₁	1	16	yes	complemented
275	#452	C₂²×C₃₁	1	32	yes	complemented
276	#432	C₂²×C₃₁	1	8	yes	complemented
277	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
278	C₂×#103	C₂²×C₃₁	1	4	yes	—
279	C₂×#102	C₂²×C₃₁	1	4	yes	—
280	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
281	C₂×#97	C₂²×C₃₁	1	2	yes	—
282	C₂×C₃₁	—	4	2	no	—
283	C₂×C₃₁	—	8	4	no	—
284	C₂×C₃₁	#3216	1	4	yes	direct product factor
285	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
286	C₂×C₃₁	C₂²×#137	1	1	yes	characteristic
287	C₂×C₃₁	C₂×#575	1	1	yes	characteristic
288	C₂×#41	—	2	4	no	—
289	C₂×#41	—	4	8	no	—
290	C₂×#41	C₂³×C₃₁	1	8	yes	—
291	C₂²×Q₈	—	2	1	no	—
292	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
293	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
294	C₂²×D₄	—	2	2	no	—
295	C₂²×D₄	—	2	1	no	—
296	C₂²×D₄	—	2	1	no	—
297	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
298	C₂³×C₄	—	2	2	no	—
299	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
300	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
301	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
302	C₂×Q₁₆	—	4	8	no	—
303	C₂×Q₁₆	C₂³×C₃₁	1	16	yes	—
304	C₂×SD₁₆	—	4	8	no	—
305	C₂×SD₁₆	C₂³×C₃₁	1	16	yes	—
306	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
307	C₄⋊Q₈	—	2	2	no	—
308	C₄⋊Q₈	—	2	2	no	—
309	C₄⋊Q₈	—	2	2	no	—
310	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
311	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
312	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
313	#127	—	2	2	no	—
314	#127	C₂³×C₃₁	1	4	yes	—
315	#125	—	2	8	no	—
316	#124	—	2	4	no	—
317	#124	C₂³×C₃₁	1	8	yes	—
318	C₂²⋊Q₈	—	2	8	no	—
319	C₂²⋊Q₈	—	2	4	no	—
320	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
321	#121	—	2	8	no	—
322	#121	—	2	8	no	—
323	C₄×Q₈	—	2	8	no	—
324	C₄×Q₈	—	2	4	no	—
325	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
326	C₄×D₄	—	2	8	no	—
327	C₄×D₄	—	2	8	no	—
328	#117	—	2	8	no	—
329	C₂×C₄⋊C₄	—	2	1	no	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₄⋊C₄	—	2	1	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	2	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
337	C₂×C₂²⋊C₄	—	2	2	no	—
338	C₂×C₂²⋊C₄	—	2	2	no	—
339	C₂×C₂²⋊C₄	—	2	2	no	—
340	C₂×C₄²	—	2	2	no	—
341	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
342	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
343	C₂×C₄²	D₄×C₃₁	1	4	yes	—
344	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
345	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
346	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
347	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
348	#103	C₂³×C₃₁	1	16	yes	—
349	#102	C₂³×C₃₁	1	16	yes	—
350	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
351	#97	C₂³×C₃₁	1	8	yes	—
352	C₃₁	C₂×#3216	1	1	yes	characteristic, direct product factor, Sylow
353	C₂⁴	—	2	2	no	—
354	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
355	#41	—	4	16	no	—
356	#41	—	4	16	no	—
357	C₂×Q₈	—	2	2	no	—
358	C₂×Q₈	—	2	4	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	4	no	—
361	C₂×Q₈	—	2	4	no	—
362	C₂×Q₈	—	2	8	no	—
363	C₂×Q₈	—	4	4	no	—
364	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
365	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
366	C₂×D₄	—	2	8	no	—
367	C₂×D₄	—	2	4	no	—
368	C₂×D₄	—	2	4	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	8	no	—
371	C₂×D₄	—	4	4	no	—
372	C₂×D₄	—	4	8	no	—
373	C₂×D₄	—	4	4	no	—
374	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	4	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	8	no	—
381	C₂²×C₄	—	4	2	no	—
382	C₂²×C₄	—	4	8	no	—
383	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
385	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
386	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
387	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
389	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
390	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
391	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
392	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
393	Q₁₆	—	4	16	no	—
394	SD₁₆	—	4	16	no	—
395	C₂×C₈	—	2	8	no	—
396	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	4	no	—
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	—	2	8	no	—
403	C₄⋊C₄	—	2	8	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
406	C₂²⋊C₄	—	2	8	no	—
407	C₂²⋊C₄	—	2	8	no	—
408	C₂²⋊C₄	—	2	8	no	—
409	C₄²	—	2	8	no	—
410	C₄²	C₂×D₄×C₃₁	1	4	yes	—
411	C₄²	C₂×D₄×C₃₁	1	4	yes	—
412	C₄²	C₂×D₄×C₃₁	1	2	yes	—
413	C₄²	C₂×D₄×C₃₁	1	2	yes	—
414	C₄²	C₂⁴×C₃₁	1	2	yes	—
415	C₄²	C₂⁴×C₃₁	1	2	yes	—
416	C₂³	—	2	2	no	—
417	C₂³	—	2	2	no	—
418	C₂³	—	2	2	no	—
419	C₂³	—	2	4	no	—
420	C₂³	—	4	4	no	—
421	C₂³	—	4	4	no	—
422	C₂³	—	4	4	no	—
423	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
424	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
425	Q₈	—	2	4	no	—
426	Q₈	—	2	4	no	—
427	Q₈	—	2	8	no	—
428	Q₈	—	4	8	no	—
429	Q₈	—	4	4	no	—
430	Q₈	—	4	4	no	—
431	D₄	—	2	8	no	—
432	D₄	—	4	8	no	—
433	D₄	—	4	16	no	—
434	D₄	—	4	4	no	—
435	D₄	—	4	8	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	8	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	4	4	no	—
447	C₂×C₄	—	4	16	no	—
448	C₂×C₄	—	4	4	no	—
449	C₂×C₄	—	4	4	no	—
450	C₂×C₄	—	4	4	no	—
451	C₂×C₄	C₃₁×#137	1	4	yes	—
452	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
453	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
454	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
455	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
456	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
457	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
458	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
459	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
460	C₈	—	2	8	no	—
461	C₂²	—	2	2	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	2	2	no	—
464	C₂²	—	4	4	no	—
465	C₂²	—	4	4	no	—
466	C₂²	—	4	4	no	—
467	C₂²	—	4	4	no	—
468	C₂²	—	8	8	no	—
469	C₂²	C₃₁×#535	1	2	yes	—
470	C₂²	C₂×C₃₁×#137	1	2	yes	—
471	C₂²	C₃₁×#575	1	2	yes	—
472	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
473	C₄	—	2	2	no	—
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	2	2	no	—
477	C₄	—	4	4	no	—
478	C₄	—	4	4	no	—
479	C₄	—	4	4	no	—
480	C₄	C₃₁×#535	1	2	yes	—
481	C₄	C₂×C₃₁×#137	1	4	yes	—
482	C₄	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
483	C₂	—	4	2	no	—
484	C₂	—	8	4	no	—
485	C₂	C₃₁×#3216	1	4	yes	direct product factor
486	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
487	C₂	C₂²×C₃₁×#137	1	1	yes	characteristic
488	C₂	C₂×C₃₁×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#533	C₂	semidirect
3. C₃₁×#3216	C₂	direct (N × H)
4. C₂×C₃₁×#497	C₂	semidirect
5. C₂×C₃₁×#489	C₂	semidirect
6. C₂×C₃₁×#462	C₂	semidirect
7. C₂×C₃₁×#460	C₂	semidirect
8. C₂×C₃₁×#452	C₂	semidirect
9. C₂×C₃₁×#432	C₂	semidirect
14. C₃₁×#533	C₂²	semidirect
33. C₃₁×#497	C₂²	semidirect
34. C₃₁×#489	C₂²	semidirect
35. C₃₁×#462	C₂²	semidirect
36. C₃₁×#460	C₂²	semidirect
37. C₃₁×#452	C₂²	semidirect
38. C₃₁×#432	C₂²	semidirect
171. C₂×#3216	C₃₁	direct (N × H)
217. C₂×#533	C₂×C₃₁	semidirect
219. #3216	C₂×C₃₁	direct (N × H)
220. C₂×#497	C₂×C₃₁	semidirect
221. C₂×#489	C₂×C₃₁	semidirect
222. C₂×#462	C₂×C₃₁	semidirect
223. C₂×#460	C₂×C₃₁	semidirect
224. C₂×#452	C₂×C₃₁	semidirect
225. C₂×#432	C₂×C₃₁	semidirect
252. #533	C₂²×C₃₁	semidirect
271. #497	C₂²×C₃₁	semidirect
272. #489	C₂²×C₃₁	semidirect
273. #462	C₂²×C₃₁	semidirect
274. #460	C₂²×C₃₁	semidirect
275. #452	C₂²×C₃₁	semidirect
276. #432	C₂²×C₃₁	semidirect
284. C₂×C₃₁	#3216	direct (N × H)
352. C₃₁	C₂×#3216	direct (N × H)
485. C₂	C₃₁×#3216	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×#533	2
C₂×C₃₁×#532	2
C₃₁×#3216	2
C₂×C₃₁×#497	2
C₂×C₃₁×#489	2
C₂×C₃₁×#462	2
C₂×C₃₁×#460	2
C₂×C₃₁×#452	2
C₂×C₃₁×#432	2
C₂×#3216	31

Subgroups — C2×C31×#3216 #21517547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3216 #21517547