Subgroups — C₂²×C₃₁×#471 #21517222

Order 7936.

2758 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
3	C₂³×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#471	C₂	1	24	yes	maximal, direct product factor
6	C₂²×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₃₁×#101	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₃₁×#100	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
10	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#137	—	2	6	no	—
13	C₂×C₃₁×#137	C₂²	1	12	yes	complemented
14	C₂×C₃₁×AGL(1,ℤ/8)	—	2	6	no	—
15	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	12	yes	—
16	C₂²×Q₁₆×C₃₁	—	2	1	no	—
17	C₂²×SD₁₆×C₃₁	—	2	1	no	—
18	C₂²×SD₁₆×C₃₁	—	2	1	no	—
19	C₂²×SD₁₆×C₃₁	—	2	1	no	—
20	C₂²×SD₁₆×C₃₁	—	2	1	no	—
21	C₂²×SD₁₆×C₃₁	—	2	6	no	—
22	C₂²×SD₁₆×C₃₁	C₂²	1	12	yes	complemented
23	C₂²×D₈×C₃₁	—	2	1	no	—
24	C₂×C₃₁×#131	—	2	6	no	—
25	C₂×C₃₁×#131	C₂²	1	12	yes	complemented
26	C₂²×M₁₆×C₃₁	—	2	1	no	—
27	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
28	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
29	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
30	C₂³×C₈×C₃₁	—	2	1	no	—
31	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#471	C₂²	1	64	yes	direct product factor
33	C₂×C₃₁×#108	C₂²	1	12	yes	complemented
34	C₂×C₃₁×#101	C₂²	1	12	yes	complemented
35	C₂×C₃₁×#100	C₂²	1	12	yes	complemented
36	C₂⁵×C₃₁	—	2	1	no	—
37	C₂×#41×C₃₁	—	2	3	no	—
38	C₂×#41×C₃₁	—	2	3	no	—
39	C₂×#41×C₃₁	D₄	1	6	yes	—
40	C₂×#41×C₃₁	C₂³	1	6	yes	—
41	C₂²×Q₈×C₃₁	—	2	3	no	—
42	C₂²×Q₈×C₃₁	—	2	3	no	—
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₂²×Q₈×C₃₁	—	2	1	no	—
45	C₂²×Q₈×C₃₁	—	2	3	no	—
46	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
47	C₂²×Q₈×C₃₁	C₂³	1	6	yes	—
48	C₂²×D₄×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	—	2	3	no	—
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	—	2	3	no	—
53	C₂²×D₄×C₃₁	—	2	3	no	—
54	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
55	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
56	C₂³×C₄×C₃₁	—	2	1	no	—
57	C₂³×C₄×C₃₁	—	2	1	no	—
58	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×#137	—	2	24	no	—
60	C₃₁×#137	C₂³	1	16	yes	complemented
61	C₃₁×AGL(1,ℤ/8)	—	2	24	no	—
62	C₃₁×AGL(1,ℤ/8)	C₂³	1	16	yes	—
63	C₂×Q₁₆×C₃₁	—	2	12	no	—
64	C₂×SD₁₆×C₃₁	—	2	12	no	—
65	C₂×SD₁₆×C₃₁	—	2	12	no	—
66	C₂×SD₁₆×C₃₁	—	2	12	no	—
67	C₂×SD₁₆×C₃₁	—	2	12	no	—
68	C₂×SD₁₆×C₃₁	—	2	24	no	—
69	C₂×SD₁₆×C₃₁	C₂³	1	16	yes	complemented
70	C₂×D₈×C₃₁	—	2	12	no	—
71	C₃₁×#131	—	2	24	no	—
72	C₃₁×#131	C₂³	1	16	yes	complemented
73	C₂×M₁₆×C₃₁	—	2	3	no	—
74	C₂×M₁₆×C₃₁	—	2	3	no	—
75	C₂×M₁₆×C₃₁	—	2	3	no	—
76	C₂×M₁₆×C₃₁	—	2	12	no	—
77	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
78	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
79	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
80	C₂²×C₈×C₃₁	—	2	1	no	—
81	C₂²×C₈×C₃₁	—	2	1	no	—
82	C₂²×C₈×C₃₁	—	2	1	no	—
83	C₂²×C₈×C₃₁	—	2	12	no	—
84	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
85	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
86	C₂²×C₈×C₃₁	D₄	1	6	yes	—
87	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
88	C₃₁×#108	C₂³	1	16	yes	complemented
89	C₃₁×#101	C₂³	1	16	yes	complemented
90	C₃₁×#100	C₂³	1	16	yes	complemented
91	C₂⁴×C₃₁	—	2	1	no	—
92	C₂⁴×C₃₁	—	2	6	no	—
93	C₂⁴×C₃₁	—	4	3	no	—
94	C₂⁴×C₃₁	—	4	1	no	—
95	C₂⁴×C₃₁	—	8	4	no	—
96	C₂⁴×C₃₁	#41	1	1	yes	characteristic
97	#41×C₃₁	—	2	12	no	—
98	#41×C₃₁	—	2	6	no	—
99	#41×C₃₁	—	2	6	no	—
100	#41×C₃₁	C₂×D₄	1	12	yes	—
101	#41×C₃₁	C₂⁴	1	4	yes	—
102	C₂×Q₈×C₃₁	—	2	3	no	—
103	C₂×Q₈×C₃₁	—	2	6	no	—
104	C₂×Q₈×C₃₁	—	2	6	no	—
105	C₂×Q₈×C₃₁	—	2	6	no	—
106	C₂×Q₈×C₃₁	—	2	6	no	—
107	C₂×Q₈×C₃₁	—	2	6	no	—
108	C₂×Q₈×C₃₁	—	4	3	no	—
109	C₂×Q₈×C₃₁	—	4	3	no	—
110	C₂×Q₈×C₃₁	—	4	6	no	—
111	C₂×Q₈×C₃₁	C₂×D₄	1	6	yes	—
112	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
113	C₂×D₄×C₃₁	—	2	6	no	—
114	C₂×D₄×C₃₁	—	2	6	no	—
115	C₂×D₄×C₃₁	—	2	3	no	—
116	C₂×D₄×C₃₁	—	2	6	no	—
117	C₂×D₄×C₃₁	—	2	6	no	—
118	C₂×D₄×C₃₁	—	2	6	no	—
119	C₂×D₄×C₃₁	—	2	6	no	—
120	C₂×D₄×C₃₁	—	2	6	no	—
121	C₂×D₄×C₃₁	—	4	3	no	—
122	C₂×D₄×C₃₁	—	4	3	no	—
123	C₂×D₄×C₃₁	—	4	6	no	—
124	C₂×D₄×C₃₁	C₂×D₄	1	6	yes	—
125	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
126	C₂²×C₄×C₃₁	—	2	1	no	—
127	C₂²×C₄×C₃₁	—	2	6	no	—
128	C₂²×C₄×C₃₁	—	2	6	no	—
129	C₂²×C₄×C₃₁	—	2	6	no	—
130	C₂²×C₄×C₃₁	—	4	3	no	—
131	C₂²×C₄×C₃₁	—	4	1	no	—
132	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	#41	1	3	yes	—
137	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
138	Q₁₆×C₃₁	—	2	16	no	—
139	SD₁₆×C₃₁	—	2	16	no	—
140	SD₁₆×C₃₁	—	2	16	no	—
141	SD₁₆×C₃₁	—	2	16	no	—
142	SD₁₆×C₃₁	—	2	16	no	—
143	D₈×C₃₁	—	2	16	no	—
144	M₁₆×C₃₁	—	2	6	no	—
145	M₁₆×C₃₁	—	2	6	no	—
146	M₁₆×C₃₁	—	2	6	no	—
147	M₁₆×C₃₁	—	2	16	no	—
148	M₁₆×C₃₁	C₂⁴	1	4	yes	—
149	M₁₆×C₃₁	C₂⁴	1	4	yes	—
150	M₁₆×C₃₁	C₂⁴	1	4	yes	—
151	C₂×C₈×C₃₁	—	2	6	no	—
152	C₂×C₈×C₃₁	—	2	6	no	—
153	C₂×C₈×C₃₁	—	2	6	no	—
154	C₂×C₈×C₃₁	—	2	16	no	—
155	C₂×C₈×C₃₁	C₂×D₄	1	6	yes	—
156	C₂×C₈×C₃₁	C₂×D₄	1	6	yes	—
157	C₂×C₈×C₃₁	C₂×D₄	1	12	yes	—
158	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
159	C₂²×#471	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₃₁	—	2	4	no	—
161	C₂³×C₃₁	—	2	6	no	—
162	C₂³×C₃₁	—	2	4	no	—
163	C₂³×C₃₁	—	4	4	no	—
164	C₂³×C₃₁	—	4	6	no	—
165	C₂³×C₃₁	—	4	6	no	—
166	C₂³×C₃₁	—	8	12	no	—
167	C₂³×C₃₁	—	8	12	no	—
168	C₂³×C₃₁	—	8	4	no	—
169	C₂³×C₃₁	#121	1	3	yes	—
170	C₂³×C₃₁	#121	1	1	yes	characteristic
171	C₂³×C₃₁	C₂×#41	1	3	yes	—
172	Q₈×C₃₁	—	2	6	no	—
173	Q₈×C₃₁	—	2	4	no	—
174	Q₈×C₃₁	—	2	4	no	—
175	Q₈×C₃₁	—	4	6	no	—
176	Q₈×C₃₁	—	4	6	no	—
177	Q₈×C₃₁	C₂²×D₄	1	4	yes	—
178	D₄×C₃₁	—	2	12	no	—
179	D₄×C₃₁	—	2	4	no	—
180	D₄×C₃₁	—	2	6	no	—
181	D₄×C₃₁	—	2	4	no	—
182	D₄×C₃₁	—	2	4	no	—
183	D₄×C₃₁	—	4	6	no	—
184	D₄×C₃₁	—	4	6	no	—
185	D₄×C₃₁	C₂²×D₄	1	4	yes	—
186	C₂×C₄×C₃₁	—	2	6	no	—
187	C₂×C₄×C₃₁	—	2	12	no	—
188	C₂×C₄×C₃₁	—	2	4	no	—
189	C₂×C₄×C₃₁	—	2	4	no	—
190	C₂×C₄×C₃₁	—	4	6	no	—
191	C₂×C₄×C₃₁	—	4	6	no	—
192	C₂×C₄×C₃₁	#121	1	3	yes	—
193	C₂×C₄×C₃₁	#121	1	3	yes	—
194	C₂×C₄×C₃₁	#121	1	6	yes	—
195	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
196	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
197	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
198	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
199	C₂×C₄×C₃₁	C₂×#41	1	3	yes	—
200	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
201	C₈×C₃₁	—	2	4	no	—
202	C₈×C₃₁	—	2	4	no	—
203	C₈×C₃₁	—	2	4	no	—
204	C₈×C₃₁	C₂²×D₄	1	4	yes	—
205	C₈×C₃₁	C₂²×D₄	1	4	yes	—
206	C₂²×#137	C₂×C₃₁	1	1	yes	characteristic, complemented
207	C₂²×AGL(1,ℤ/8)	C₂×C₃₁	1	1	yes	characteristic
208	C₂³×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
209	C₂²×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
210	C₂×#471	C₂×C₃₁	1	24	yes	direct product factor
211	C₂²×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
212	C₂²×#101	C₂×C₃₁	1	1	yes	characteristic, complemented
213	C₂²×#100	C₂×C₃₁	1	1	yes	characteristic, complemented
214	C₂²×C₃₁	—	2	6	no	—
215	C₂²×C₃₁	—	2	6	no	—
216	C₂²×C₃₁	—	2	4	no	—
217	C₂²×C₃₁	—	4	12	no	—
218	C₂²×C₃₁	—	4	4	no	—
219	C₂²×C₃₁	—	8	4	no	—
220	C₂²×C₃₁	—	8	12	no	—
221	C₂²×C₃₁	—	8	12	no	—
222	C₂²×C₃₁	#471	1	4	yes	direct product factor
223	C₂²×C₃₁	C₂×#121	1	3	yes	—
224	C₂²×C₃₁	C₂×#121	1	3	yes	—
225	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
226	C₄×C₃₁	—	2	4	no	—
227	C₄×C₃₁	—	4	4	no	—
228	C₄×C₃₁	C₂×#121	1	3	yes	—
229	C₄×C₃₁	C₂×#121	1	3	yes	—
230	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
231	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
232	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
233	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
234	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×#137	—	2	6	no	—
236	C₂×#137	C₂²×C₃₁	1	12	yes	complemented
237	C₂×AGL(1,ℤ/8)	—	2	6	no	—
238	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	12	yes	—
239	C₂²×Q₁₆	—	2	1	no	—
240	C₂²×SD₁₆	—	2	1	no	—
241	C₂²×SD₁₆	—	2	1	no	—
242	C₂²×SD₁₆	—	2	1	no	—
243	C₂²×SD₁₆	—	2	1	no	—
244	C₂²×SD₁₆	—	2	6	no	—
245	C₂²×SD₁₆	C₂²×C₃₁	1	12	yes	complemented
246	C₂²×D₈	—	2	1	no	—
247	C₂×#131	—	2	6	no	—
248	C₂×#131	C₂²×C₃₁	1	12	yes	complemented
249	C₂²×M₁₆	—	2	1	no	—
250	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
251	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
252	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
253	C₂³×C₈	—	2	1	no	—
254	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
255	#471	C₂²×C₃₁	1	64	yes	direct product factor
256	C₂×#108	C₂²×C₃₁	1	12	yes	complemented
257	C₂×#101	C₂²×C₃₁	1	12	yes	complemented
258	C₂×#100	C₂²×C₃₁	1	12	yes	complemented
259	C₂×C₃₁	—	2	3	no	—
260	C₂×C₃₁	—	2	1	no	—
261	C₂×C₃₁	—	4	4	no	—
262	C₂×C₃₁	—	8	4	no	—
263	C₂×C₃₁	C₂×#471	1	6	yes	direct product factor
264	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
265	C₂⁵	—	2	1	no	—
266	C₂×#41	—	2	3	no	—
267	C₂×#41	—	2	3	no	—
268	C₂×#41	D₄×C₃₁	1	6	yes	—
269	C₂×#41	C₂³×C₃₁	1	6	yes	—
270	C₂²×Q₈	—	2	3	no	—
271	C₂²×Q₈	—	2	1	no	—
272	C₂²×Q₈	—	2	3	no	—
273	C₂²×Q₈	—	2	1	no	—
274	C₂²×Q₈	—	2	3	no	—
275	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
276	C₂²×Q₈	C₂³×C₃₁	1	6	yes	—
277	C₂²×D₄	—	2	1	no	—
278	C₂²×D₄	—	2	1	no	—
279	C₂²×D₄	—	2	1	no	—
280	C₂²×D₄	—	2	3	no	—
281	C₂²×D₄	—	2	3	no	—
282	C₂²×D₄	—	2	3	no	—
283	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
284	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
285	C₂³×C₄	—	2	1	no	—
286	C₂³×C₄	—	2	1	no	—
287	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
288	#137	—	2	24	no	—
289	#137	C₂³×C₃₁	1	16	yes	complemented
290	AGL(1,ℤ/8)	—	2	24	no	—
291	AGL(1,ℤ/8)	C₂³×C₃₁	1	16	yes	—
292	C₂×Q₁₆	—	2	12	no	—
293	C₂×SD₁₆	—	2	12	no	—
294	C₂×SD₁₆	—	2	12	no	—
295	C₂×SD₁₆	—	2	12	no	—
296	C₂×SD₁₆	—	2	12	no	—
297	C₂×SD₁₆	—	2	24	no	—
298	C₂×SD₁₆	C₂³×C₃₁	1	16	yes	complemented
299	C₂×D₈	—	2	12	no	—
300	#131	—	2	24	no	—
301	#131	C₂³×C₃₁	1	16	yes	complemented
302	C₂×M₁₆	—	2	3	no	—
303	C₂×M₁₆	—	2	3	no	—
304	C₂×M₁₆	—	2	3	no	—
305	C₂×M₁₆	—	2	12	no	—
306	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
307	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
308	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
309	C₂²×C₈	—	2	1	no	—
310	C₂²×C₈	—	2	1	no	—
311	C₂²×C₈	—	2	1	no	—
312	C₂²×C₈	—	2	12	no	—
313	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
314	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
315	C₂²×C₈	D₄×C₃₁	1	6	yes	—
316	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
317	#108	C₂³×C₃₁	1	16	yes	complemented
318	#101	C₂³×C₃₁	1	16	yes	complemented
319	#100	C₂³×C₃₁	1	16	yes	complemented
320	C₃₁	C₂²×#471	1	1	yes	characteristic, direct product factor, Sylow
321	C₂⁴	—	2	1	no	—
322	C₂⁴	—	2	6	no	—
323	C₂⁴	—	4	3	no	—
324	C₂⁴	—	4	1	no	—
325	C₂⁴	—	8	4	no	—
326	C₂⁴	#41×C₃₁	1	1	yes	characteristic
327	#41	—	2	12	no	—
328	#41	—	2	6	no	—
329	#41	—	2	6	no	—
330	#41	C₂×D₄×C₃₁	1	12	yes	—
331	#41	C₂⁴×C₃₁	1	4	yes	—
332	C₂×Q₈	—	2	3	no	—
333	C₂×Q₈	—	2	6	no	—
334	C₂×Q₈	—	2	6	no	—
335	C₂×Q₈	—	2	6	no	—
336	C₂×Q₈	—	2	6	no	—
337	C₂×Q₈	—	2	6	no	—
338	C₂×Q₈	—	4	3	no	—
339	C₂×Q₈	—	4	3	no	—
340	C₂×Q₈	—	4	6	no	—
341	C₂×Q₈	C₂×D₄×C₃₁	1	6	yes	—
342	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
343	C₂×D₄	—	2	6	no	—
344	C₂×D₄	—	2	6	no	—
345	C₂×D₄	—	2	3	no	—
346	C₂×D₄	—	2	6	no	—
347	C₂×D₄	—	2	6	no	—
348	C₂×D₄	—	2	6	no	—
349	C₂×D₄	—	2	6	no	—
350	C₂×D₄	—	2	6	no	—
351	C₂×D₄	—	4	3	no	—
352	C₂×D₄	—	4	3	no	—
353	C₂×D₄	—	4	6	no	—
354	C₂×D₄	C₂×D₄×C₃₁	1	6	yes	—
355	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	6	no	—
358	C₂²×C₄	—	2	6	no	—
359	C₂²×C₄	—	2	6	no	—
360	C₂²×C₄	—	4	3	no	—
361	C₂²×C₄	—	4	1	no	—
362	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
363	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
364	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
365	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
366	C₂²×C₄	#41×C₃₁	1	3	yes	—
367	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
368	Q₁₆	—	2	16	no	—
369	SD₁₆	—	2	16	no	—
370	SD₁₆	—	2	16	no	—
371	SD₁₆	—	2	16	no	—
372	SD₁₆	—	2	16	no	—
373	D₈	—	2	16	no	—
374	M₁₆	—	2	6	no	—
375	M₁₆	—	2	6	no	—
376	M₁₆	—	2	6	no	—
377	M₁₆	—	2	16	no	—
378	M₁₆	C₂⁴×C₃₁	1	4	yes	—
379	M₁₆	C₂⁴×C₃₁	1	4	yes	—
380	M₁₆	C₂⁴×C₃₁	1	4	yes	—
381	C₂×C₈	—	2	6	no	—
382	C₂×C₈	—	2	6	no	—
383	C₂×C₈	—	2	6	no	—
384	C₂×C₈	—	2	16	no	—
385	C₂×C₈	C₂×D₄×C₃₁	1	6	yes	—
386	C₂×C₈	C₂×D₄×C₃₁	1	6	yes	—
387	C₂×C₈	C₂×D₄×C₃₁	1	12	yes	—
388	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
389	C₂³	—	2	4	no	—
390	C₂³	—	2	6	no	—
391	C₂³	—	2	4	no	—
392	C₂³	—	4	4	no	—
393	C₂³	—	4	6	no	—
394	C₂³	—	4	6	no	—
395	C₂³	—	8	12	no	—
396	C₂³	—	8	12	no	—
397	C₂³	—	8	4	no	—
398	C₂³	C₃₁×#121	1	3	yes	—
399	C₂³	C₃₁×#121	1	1	yes	characteristic
400	C₂³	C₂×#41×C₃₁	1	3	yes	—
401	Q₈	—	2	6	no	—
402	Q₈	—	2	4	no	—
403	Q₈	—	2	4	no	—
404	Q₈	—	4	6	no	—
405	Q₈	—	4	6	no	—
406	Q₈	C₂²×D₄×C₃₁	1	4	yes	—
407	D₄	—	2	12	no	—
408	D₄	—	2	4	no	—
409	D₄	—	2	6	no	—
410	D₄	—	2	4	no	—
411	D₄	—	2	4	no	—
412	D₄	—	4	6	no	—
413	D₄	—	4	6	no	—
414	D₄	C₂²×D₄×C₃₁	1	4	yes	—
415	C₂×C₄	—	2	6	no	—
416	C₂×C₄	—	2	12	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	4	6	no	—
420	C₂×C₄	—	4	6	no	—
421	C₂×C₄	C₃₁×#121	1	3	yes	—
422	C₂×C₄	C₃₁×#121	1	3	yes	—
423	C₂×C₄	C₃₁×#121	1	6	yes	—
424	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
425	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
426	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
427	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
428	C₂×C₄	C₂×#41×C₃₁	1	3	yes	—
429	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
430	C₈	—	2	4	no	—
431	C₈	—	2	4	no	—
432	C₈	—	2	4	no	—
433	C₈	C₂²×D₄×C₃₁	1	4	yes	—
434	C₈	C₂²×D₄×C₃₁	1	4	yes	—
435	C₂²	—	2	6	no	—
436	C₂²	—	2	6	no	—
437	C₂²	—	2	4	no	—
438	C₂²	—	4	12	no	—
439	C₂²	—	4	4	no	—
440	C₂²	—	8	4	no	—
441	C₂²	—	8	12	no	—
442	C₂²	—	8	12	no	—
443	C₂²	C₃₁×#471	1	4	yes	direct product factor
444	C₂²	C₂×C₃₁×#121	1	3	yes	—
445	C₂²	C₂×C₃₁×#121	1	3	yes	—
446	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
447	C₄	—	2	4	no	—
448	C₄	—	4	4	no	—
449	C₄	C₂×C₃₁×#121	1	3	yes	—
450	C₄	C₂×C₃₁×#121	1	3	yes	—
451	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
452	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
453	C₂	—	2	3	no	—
454	C₂	—	2	1	no	—
455	C₂	—	4	4	no	—
456	C₂	—	8	4	no	—
457	C₂	C₂×C₃₁×#471	1	6	yes	direct product factor
458	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#121
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂²×C₃₁×AGL(1,ℤ/8)	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×#137	C₂	semidirect
3. C₂³×SD₁₆×C₃₁	C₂	semidirect
4. C₂²×C₃₁×#131	C₂	semidirect
5. C₂×C₃₁×#471	C₂	direct (N × H)
6. C₂²×C₃₁×#108	C₂	semidirect
7. C₂²×C₃₁×#101	C₂	semidirect
8. C₂²×C₃₁×#100	C₂	semidirect
13. C₂×C₃₁×#137	C₂²	semidirect
22. C₂²×SD₁₆×C₃₁	C₂²	semidirect
25. C₂×C₃₁×#131	C₂²	semidirect
32. C₃₁×#471	C₂²	direct (N × H)
33. C₂×C₃₁×#108	C₂²	semidirect
34. C₂×C₃₁×#101	C₂²	semidirect
35. C₂×C₃₁×#100	C₂²	semidirect
60. C₃₁×#137	C₂³	semidirect
69. C₂×SD₁₆×C₃₁	C₂³	semidirect
72. C₃₁×#131	C₂³	semidirect
88. C₃₁×#108	C₂³	semidirect
89. C₃₁×#101	C₂³	semidirect
90. C₃₁×#100	C₂³	semidirect
159. C₂²×#471	C₃₁	direct (N × H)
206. C₂²×#137	C₂×C₃₁	semidirect
208. C₂³×SD₁₆	C₂×C₃₁	semidirect
209. C₂²×#131	C₂×C₃₁	semidirect
210. C₂×#471	C₂×C₃₁	direct (N × H)
211. C₂²×#108	C₂×C₃₁	semidirect
212. C₂²×#101	C₂×C₃₁	semidirect
213. C₂²×#100	C₂×C₃₁	semidirect
222. C₂²×C₃₁	#471	direct (N × H)
236. C₂×#137	C₂²×C₃₁	semidirect
245. C₂²×SD₁₆	C₂²×C₃₁	semidirect
248. C₂×#131	C₂²×C₃₁	semidirect
255. #471	C₂²×C₃₁	direct (N × H)
256. C₂×#108	C₂²×C₃₁	semidirect
257. C₂×#101	C₂²×C₃₁	semidirect
258. C₂×#100	C₂²×C₃₁	semidirect
263. C₂×C₃₁	C₂×#471	direct (N × H)
289. #137	C₂³×C₃₁	semidirect
298. C₂×SD₁₆	C₂³×C₃₁	semidirect
301. #131	C₂³×C₃₁	semidirect
317. #108	C₂³×C₃₁	semidirect
318. #101	C₂³×C₃₁	semidirect
319. #100	C₂³×C₃₁	semidirect
320. C₃₁	C₂²×#471	direct (N × H)
443. C₂²	C₃₁×#471	direct (N × H)
457. C₂	C₂×C₃₁×#471	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×#137	2
C₂²×C₃₁×AGL(1,ℤ/8)	2
C₂³×SD₁₆×C₃₁	2
C₂²×C₃₁×#131	2
C₂×C₃₁×#471	2
C₂²×C₃₁×#108	2
C₂²×C₃₁×#101	2
C₂²×C₃₁×#100	2
C₂²×#471	31

Subgroups — C22×C31×#471 #21517222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#471 #21517222