Subgroups — D₈×SD₁₆×C₃₁ #21476728

Order 7936.

798 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
3	D₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1673	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1668	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1665	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1639	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1627	C₂	1	2	yes	maximal, complemented
10	C₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	—	2	2	no	—
13	C₂²×D₈×C₃₁	—	2	1	no	—
14	D₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
15	D₄²×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#500	—	2	1	no	—
17	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
18	C₃₁×#493	C₂²	1	1	yes	characteristic
19	C₃₁×#492	C₂²	1	2	yes	complemented
20	C₃₁×#486	C₂²	1	2	yes	complemented
21	C₃₁×#474	—	2	2	no	—
22	C₃₁×#474	C₂²	1	2	yes	complemented
23	C₃₁×#466	—	2	1	no	—
24	C₃₁×#465	—	2	2	no	—
25	C₃₁×#461	C₂²	1	2	yes	—
26	C₃₁×#460	C₂²	1	2	yes	complemented
27	C₃₁×#459	C₂²	1	2	yes	—
28	C₃₁×#450	—	2	2	no	—
29	C₃₁×#448	—	2	2	no	—
30	C₃₁×#447	—	2	2	no	—
31	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
32	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
33	C₄×D₈×C₃₁	—	2	1	no	—
34	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
37	C₃₁×#335	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#334	C₂²	1	1	yes	characteristic, complemented
39	C₃₁×#332	C₂²	1	2	yes	complemented
40	C₃₁×#331	C₂²	1	2	yes	complemented
41	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
45	C₂²×Q₈×C₃₁	—	2	2	no	—
46	C₂²×D₄×C₃₁	—	2	2	no	—
47	C₂²×D₄×C₃₁	—	2	2	no	—
48	C₂×SD₁₆×C₃₁	—	2	2	no	—
49	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×SD₁₆×C₃₁	—	4	8	no	—
53	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
54	C₂×D₈×C₃₁	—	2	2	no	—
55	C₂×D₈×C₃₁	—	2	1	no	—
56	C₂×D₈×C₃₁	—	2	1	no	—
57	C₂×D₈×C₃₁	—	4	4	no	—
58	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₃₁	—	2	1	no	—
60	C₂²×C₈×C₃₁	—	2	2	no	—
61	C₃₁×C₄⋊Q₈	—	2	2	no	—
62	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
63	C₃₁×#127	C₂³	1	2	yes	—
64	C₃₁×C₂²⋊Q₈	—	2	2	no	—
65	C₃₁×C₂²⋊Q₈	—	4	2	no	—
66	C₃₁×#121	—	2	2	no	—
67	C₃₁×#121	—	2	2	no	—
68	C₃₁×#120	—	4	2	no	—
69	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₄×D₄×C₃₁	—	2	2	no	—
71	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₄×D₄×C₃₁	C₂³	1	2	yes	—
73	C₃₁×#107	—	2	1	no	—
74	C₃₁×#107	—	2	1	no	—
75	C₃₁×#107	—	2	2	no	—
76	C₃₁×#107	D₄	1	1	yes	characteristic
77	C₃₁×#106	D₄	1	1	yes	characteristic
78	C₃₁×C₄⋊C₈	—	2	1	no	—
79	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
81	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
82	C₃₁×#103	—	2	2	no	—
83	C₃₁×#103	D₄	1	1	yes	characteristic
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	—	4	2	no	—
88	C₃₁×#102	D₄	1	2	yes	—
89	C₃₁×#102	D₄	1	1	yes	characteristic
90	C₃₁×C₂²⋊C₈	—	2	1	no	—
91	C₃₁×C₂²⋊C₈	—	2	2	no	—
92	C₄×C₈×C₃₁	—	2	1	no	—
93	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
96	C₂⁴×C₃₁	—	4	2	no	—
97	C₂×Q₈×C₃₁	—	2	2	no	—
98	C₂×Q₈×C₃₁	—	4	2	no	—
99	C₂×Q₈×C₃₁	—	4	4	no	—
100	C₂×Q₈×C₃₁	—	8	2	no	—
101	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	4	2	no	—
106	C₂×D₄×C₃₁	—	4	2	no	—
107	C₂×D₄×C₃₁	—	4	4	no	—
108	C₂×D₄×C₃₁	—	4	4	no	—
109	C₂×D₄×C₃₁	—	8	2	no	—
110	C₂×D₄×C₃₁	—	8	2	no	—
111	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	—	4	2	no	—
116	SD₁₆×C₃₁	—	4	4	no	—
117	SD₁₆×C₃₁	—	4	4	no	—
118	SD₁₆×C₃₁	—	4	4	no	—
119	SD₁₆×C₃₁	—	8	2	no	—
120	SD₁₆×C₃₁	D₈	1	4	yes	direct product factor
121	D₈×C₃₁	—	4	4	no	—
122	D₈×C₃₁	—	4	2	no	—
123	D₈×C₃₁	—	8	1	no	—
124	D₈×C₃₁	SD₁₆	1	4	yes	direct product factor
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	1	no	—
129	C₂×C₈×C₃₁	—	4	4	no	—
130	C₂×C₈×C₃₁	—	4	2	no	—
131	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₃₁	—	2	1	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	4	2	no	—
139	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
141	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂²⋊C₄×C₃₁	—	2	1	no	—
143	C₂²⋊C₄×C₃₁	—	2	2	no	—
144	C₂²⋊C₄×C₃₁	—	4	2	no	—
145	C₂²⋊C₄×C₃₁	—	4	2	no	—
146	C₄²×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
148	D₈×SD₁₆	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂³×C₃₁	—	2	2	no	—
150	C₂³×C₃₁	—	2	1	no	—
151	C₂³×C₃₁	—	4	2	no	—
152	C₂³×C₃₁	—	8	4	no	—
153	C₂³×C₃₁	—	8	4	no	—
154	C₂³×C₃₁	—	16	2	no	—
155	Q₈×C₃₁	—	2	1	no	—
156	Q₈×C₃₁	—	4	2	no	—
157	Q₈×C₃₁	—	4	2	no	—
158	Q₈×C₃₁	—	8	4	no	—
159	Q₈×C₃₁	C₂×D₈	1	2	yes	complemented
160	D₄×C₃₁	—	2	1	no	—
161	D₄×C₃₁	—	2	2	no	—
162	D₄×C₃₁	—	4	2	no	—
163	D₄×C₃₁	—	4	2	no	—
164	D₄×C₃₁	—	4	2	no	—
165	D₄×C₃₁	—	4	2	no	—
166	D₄×C₃₁	—	8	2	no	—
167	D₄×C₃₁	—	8	4	no	—
168	D₄×C₃₁	—	8	4	no	—
169	D₄×C₃₁	C₂×D₈	1	2	yes	—
170	D₄×C₃₁	C₂×SD₁₆	1	4	yes	complemented
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	1	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	—	4	1	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	8	4	no	—
181	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
184	C₈×C₃₁	—	2	1	no	—
185	C₈×C₃₁	—	2	1	no	—
186	C₈×C₃₁	—	4	1	no	—
187	C₈×C₃₁	—	4	2	no	—
188	C₈×C₃₁	—	4	1	no	—
189	C₈×C₃₁	—	4	1	no	—
190	C₈×C₃₁	C₂×D₈	1	2	yes	complemented
191	C₈×C₃₁	C₂×SD₁₆	1	2	yes	complemented
192	Q₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
193	D₄×SD₁₆	C₂×C₃₁	1	2	yes	complemented
194	D₄×D₈	C₂×C₃₁	1	1	yes	characteristic
195	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#1673	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1668	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#1665	C₂×C₃₁	1	2	yes	complemented
199	#1639	C₂×C₃₁	1	2	yes	complemented
200	#1627	C₂×C₃₁	1	2	yes	complemented
201	C₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
202	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
203	C₂²×C₃₁	—	2	2	no	—
204	C₂²×C₃₁	—	2	4	no	—
205	C₂²×C₃₁	—	4	2	no	—
206	C₂²×C₃₁	—	4	1	no	—
207	C₂²×C₃₁	—	4	2	no	—
208	C₂²×C₃₁	—	4	1	no	—
209	C₂²×C₃₁	—	8	2	no	—
210	C₂²×C₃₁	—	8	2	no	—
211	C₂²×C₃₁	—	8	2	no	—
212	C₂²×C₃₁	—	16	8	no	—
213	C₂²×C₃₁	D₄²	1	1	yes	characteristic
214	C₄×C₃₁	—	2	2	no	—
215	C₄×C₃₁	—	2	1	no	—
216	C₄×C₃₁	—	4	2	no	—
217	C₄×C₃₁	—	4	1	no	—
218	C₄×C₃₁	—	4	1	no	—
219	C₄×C₃₁	—	8	2	no	—
220	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
221	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
222	C₄×C₃₁	#576	1	1	yes	characteristic
223	C₄×C₃₁	#577	1	1	yes	characteristic
224	C₂²×SD₁₆	—	2	2	no	—
225	C₂²×D₈	—	2	1	no	—
226	D₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
227	D₄²	C₂²×C₃₁	1	2	yes	—
228	#500	—	2	1	no	—
229	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
230	#493	C₂²×C₃₁	1	1	yes	characteristic
231	#492	C₂²×C₃₁	1	2	yes	complemented
232	#486	C₂²×C₃₁	1	2	yes	complemented
233	#474	—	2	2	no	—
234	#474	C₂²×C₃₁	1	2	yes	complemented
235	#466	—	2	1	no	—
236	#465	—	2	2	no	—
237	#461	C₂²×C₃₁	1	2	yes	—
238	#460	C₂²×C₃₁	1	2	yes	complemented
239	#459	C₂²×C₃₁	1	2	yes	—
240	#450	—	2	2	no	—
241	#448	—	2	2	no	—
242	#447	—	2	2	no	—
243	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
244	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
245	C₄×D₈	—	2	1	no	—
246	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
247	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
248	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
249	#335	C₂²×C₃₁	1	1	yes	characteristic, complemented
250	#334	C₂²×C₃₁	1	1	yes	characteristic, complemented
251	#332	C₂²×C₃₁	1	2	yes	complemented
252	#331	C₂²×C₃₁	1	2	yes	complemented
253	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
254	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
255	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
256	C₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
257	C₂×C₃₁	—	4	4	no	—
258	C₂×C₃₁	—	4	2	no	—
259	C₂×C₃₁	—	16	2	no	—
260	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
261	C₂×C₃₁	D₄×SD₁₆	1	1	yes	characteristic
262	C₂×C₃₁	#3290	1	1	yes	characteristic
263	C₂²×Q₈	—	2	2	no	—
264	C₂²×D₄	—	2	2	no	—
265	C₂²×D₄	—	2	2	no	—
266	C₂×SD₁₆	—	2	2	no	—
267	C₂×SD₁₆	—	2	2	no	—
268	C₂×SD₁₆	—	2	2	no	—
269	C₂×SD₁₆	—	2	2	no	—
270	C₂×SD₁₆	—	4	8	no	—
271	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
272	C₂×D₈	—	2	2	no	—
273	C₂×D₈	—	2	1	no	—
274	C₂×D₈	—	2	1	no	—
275	C₂×D₈	—	4	4	no	—
276	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
277	C₂²×C₈	—	2	1	no	—
278	C₂²×C₈	—	2	2	no	—
279	C₄⋊Q₈	—	2	2	no	—
280	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
281	#127	C₂³×C₃₁	1	2	yes	—
282	C₂²⋊Q₈	—	2	2	no	—
283	C₂²⋊Q₈	—	4	2	no	—
284	#121	—	2	2	no	—
285	#121	—	2	2	no	—
286	#120	—	4	2	no	—
287	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
288	C₄×D₄	—	2	2	no	—
289	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
290	C₄×D₄	C₂³×C₃₁	1	2	yes	—
291	#107	—	2	1	no	—
292	#107	—	2	1	no	—
293	#107	—	2	2	no	—
294	#107	D₄×C₃₁	1	1	yes	characteristic
295	#106	D₄×C₃₁	1	1	yes	characteristic
296	C₄⋊C₈	—	2	1	no	—
297	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
299	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
300	#103	—	2	2	no	—
301	#103	D₄×C₃₁	1	1	yes	characteristic
302	#102	—	2	2	no	—
303	#102	—	2	2	no	—
304	#102	—	2	2	no	—
305	#102	—	4	2	no	—
306	#102	D₄×C₃₁	1	2	yes	—
307	#102	D₄×C₃₁	1	1	yes	characteristic
308	C₂²⋊C₈	—	2	1	no	—
309	C₂²⋊C₈	—	2	2	no	—
310	C₄×C₈	—	2	1	no	—
311	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
312	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
313	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
314	C₃₁	D₈×SD₁₆	1	1	yes	characteristic, direct product factor, Sylow
315	C₂⁴	—	4	2	no	—
316	C₂×Q₈	—	2	2	no	—
317	C₂×Q₈	—	4	2	no	—
318	C₂×Q₈	—	4	4	no	—
319	C₂×Q₈	—	8	2	no	—
320	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	2	2	no	—
323	C₂×D₄	—	2	2	no	—
324	C₂×D₄	—	4	2	no	—
325	C₂×D₄	—	4	2	no	—
326	C₂×D₄	—	4	4	no	—
327	C₂×D₄	—	4	4	no	—
328	C₂×D₄	—	8	2	no	—
329	C₂×D₄	—	8	2	no	—
330	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
331	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	4	2	no	—
335	SD₁₆	—	4	4	no	—
336	SD₁₆	—	4	4	no	—
337	SD₁₆	—	4	4	no	—
338	SD₁₆	—	8	2	no	—
339	SD₁₆	D₈×C₃₁	1	4	yes	direct product factor
340	D₈	—	4	4	no	—
341	D₈	—	4	2	no	—
342	D₈	—	8	1	no	—
343	D₈	SD₁₆×C₃₁	1	4	yes	direct product factor
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	4	4	no	—
349	C₂×C₈	—	4	2	no	—
350	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
351	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
352	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
353	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	4	2	no	—
358	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
360	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
361	C₂²⋊C₄	—	2	1	no	—
362	C₂²⋊C₄	—	2	2	no	—
363	C₂²⋊C₄	—	4	2	no	—
364	C₂²⋊C₄	—	4	2	no	—
365	C₄²	—	2	1	no	—
366	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
367	C₂³	—	2	2	no	—
368	C₂³	—	2	1	no	—
369	C₂³	—	4	2	no	—
370	C₂³	—	8	4	no	—
371	C₂³	—	8	4	no	—
372	C₂³	—	16	2	no	—
373	Q₈	—	2	1	no	—
374	Q₈	—	4	2	no	—
375	Q₈	—	4	2	no	—
376	Q₈	—	8	4	no	—
377	Q₈	C₂×D₈×C₃₁	1	2	yes	complemented
378	D₄	—	2	1	no	—
379	D₄	—	2	2	no	—
380	D₄	—	4	2	no	—
381	D₄	—	4	2	no	—
382	D₄	—	4	2	no	—
383	D₄	—	4	2	no	—
384	D₄	—	8	2	no	—
385	D₄	—	8	4	no	—
386	D₄	—	8	4	no	—
387	D₄	C₂×D₈×C₃₁	1	2	yes	—
388	D₄	C₂×SD₁₆×C₃₁	1	4	yes	complemented
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	8	4	no	—
399	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
400	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
401	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
402	C₈	—	2	1	no	—
403	C₈	—	2	1	no	—
404	C₈	—	4	1	no	—
405	C₈	—	4	2	no	—
406	C₈	—	4	1	no	—
407	C₈	—	4	1	no	—
408	C₈	C₂×D₈×C₃₁	1	2	yes	complemented
409	C₈	C₂×SD₁₆×C₃₁	1	2	yes	complemented
410	C₂²	—	2	2	no	—
411	C₂²	—	2	4	no	—
412	C₂²	—	4	2	no	—
413	C₂²	—	4	1	no	—
414	C₂²	—	4	1	no	—
415	C₂²	—	4	2	no	—
416	C₂²	—	8	2	no	—
417	C₂²	—	8	2	no	—
418	C₂²	—	8	2	no	—
419	C₂²	—	16	8	no	—
420	C₂²	D₄²×C₃₁	1	1	yes	characteristic
421	C₄	—	2	2	no	—
422	C₄	—	2	1	no	—
423	C₄	—	4	2	no	—
424	C₄	—	4	1	no	—
425	C₄	—	4	1	no	—
426	C₄	—	8	2	no	—
427	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
428	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
429	C₄	C₃₁×#576	1	1	yes	characteristic
430	C₄	C₃₁×#577	1	1	yes	characteristic
431	C₂	—	4	2	no	—
432	C₂	—	4	4	no	—
433	C₂	—	16	2	no	—
434	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
435	C₂	D₄×SD₁₆×C₃₁	1	1	yes	characteristic
436	C₂	C₃₁×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	D₄×D₈×C₃₁	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×D₈×C₃₁	C₂	semidirect
2. D₄×SD₁₆×C₃₁	C₂	semidirect
4. C₃₁×#1706	C₂	semidirect
5. C₃₁×#1673	C₂	semidirect
6. C₃₁×#1668	C₂	semidirect
7. C₃₁×#1665	C₂	semidirect
8. C₃₁×#1639	C₂	semidirect
9. C₃₁×#1627	C₂	semidirect
10. C₈×SD₁₆×C₃₁	C₂	semidirect
11. C₈×D₈×C₃₁	C₂	semidirect
14. D₄×Q₈×C₃₁	C₂²	semidirect
17. C₃₁×#500	C₂²	semidirect
19. C₃₁×#492	C₂²	semidirect
20. C₃₁×#486	C₂²	semidirect
22. C₃₁×#474	C₂²	semidirect
26. C₃₁×#460	C₂²	semidirect
31. C₈×Q₈×C₃₁	C₂²	semidirect
36. C₈×D₄×C₃₁	C₂²	semidirect
37. C₃₁×#335	C₂²	semidirect
38. C₃₁×#334	C₂²	semidirect
39. C₃₁×#332	C₂²	semidirect
40. C₃₁×#331	C₂²	semidirect
41. C₃₁×Q₈⋊C₈	C₂²	semidirect
43. C₃₁×D₄⋊C₈	C₂²	semidirect
44. C₈²×C₃₁	C₂²	semidirect
120. SD₁₆×C₃₁	D₈	direct (N × H)
124. D₈×C₃₁	SD₁₆	direct (N × H)
148. D₈×SD₁₆	C₃₁	direct (N × H)
159. Q₈×C₃₁	C₂×D₈	semidirect
170. D₄×C₃₁	C₂×SD₁₆	semidirect
190. C₈×C₃₁	C₂×D₈	semidirect
191. C₈×C₃₁	C₂×SD₁₆	semidirect
192. Q₈×D₈	C₂×C₃₁	semidirect
193. D₄×SD₁₆	C₂×C₃₁	semidirect
195. #1706	C₂×C₃₁	semidirect
196. #1673	C₂×C₃₁	semidirect
197. #1668	C₂×C₃₁	semidirect
198. #1665	C₂×C₃₁	semidirect
199. #1639	C₂×C₃₁	semidirect
200. #1627	C₂×C₃₁	semidirect
201. C₈×SD₁₆	C₂×C₃₁	semidirect
202. C₈×D₈	C₂×C₃₁	semidirect
226. D₄×Q₈	C₂²×C₃₁	semidirect
229. #500	C₂²×C₃₁	semidirect
231. #492	C₂²×C₃₁	semidirect
232. #486	C₂²×C₃₁	semidirect
234. #474	C₂²×C₃₁	semidirect
238. #460	C₂²×C₃₁	semidirect
243. C₈×Q₈	C₂²×C₃₁	semidirect
248. C₈×D₄	C₂²×C₃₁	semidirect
249. #335	C₂²×C₃₁	semidirect
250. #334	C₂²×C₃₁	semidirect
251. #332	C₂²×C₃₁	semidirect
252. #331	C₂²×C₃₁	semidirect
253. Q₈⋊C₈	C₂²×C₃₁	semidirect
255. D₄⋊C₈	C₂²×C₃₁	semidirect
256. C₈²	C₂²×C₃₁	semidirect
314. C₃₁	D₈×SD₁₆	direct (N × H)
339. SD₁₆	D₈×C₃₁	direct (N × H)
343. D₈	SD₁₆×C₃₁	direct (N × H)
377. Q₈	C₂×D₈×C₃₁	semidirect
388. D₄	C₂×SD₁₆×C₃₁	semidirect
408. C₈	C₂×D₈×C₃₁	semidirect
409. C₈	C₂×SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₈×C₃₁	2
D₄×SD₁₆×C₃₁	2
D₄×D₈×C₃₁	2
C₃₁×#1706	2
C₃₁×#1673	2
C₃₁×#1668	2
C₃₁×#1665	2
C₃₁×#1639	2
C₃₁×#1627	2
C₈×SD₁₆×C₃₁	2
C₈×D₈×C₃₁	2
D₈×SD₁₆	31

Subgroups — D8×SD16×C31 #21476728

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₈×SD₁₆×C₃₁ #21476728