Subgroups — C₂×C₃₁×#3368 #21518067

Order 7936.

2122 conjugacy classes of subgroups.

The subgroup lattice has 548 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×#550	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3368	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#490	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#478	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#474	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#460	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#460	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#440	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	—	2	1	no	—
14	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#557	—	2	4	no	—
17	C₃₁×#557	C₂²	1	8	yes	complemented
18	C₃₁×#550	—	2	4	no	—
19	C₃₁×#550	C₂²	1	8	yes	—
20	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
21	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#127	C₂²	1	2	yes	—
24	C₂×C₃₁×#125	—	2	2	no	—
25	C₂×C₃₁×#125	C₂²	1	2	yes	—
26	C₂×C₃₁×#121	—	2	1	no	—
27	C₂×C₃₁×#121	—	2	2	no	—
28	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
29	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
31	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
34	C₃₁×#490	C₂²	1	16	yes	complemented
35	C₃₁×#486	C₂²	1	8	yes	complemented
36	C₃₁×#478	C₂²	1	16	yes	complemented
37	C₃₁×#474	C₂²	1	8	yes	complemented
38	C₃₁×#460	C₂²	1	16	yes	complemented
39	C₃₁×#460	C₂²	1	8	yes	complemented
40	C₃₁×#446	C₂²	1	8	yes	complemented
41	C₃₁×#440	C₂²	1	16	yes	complemented
42	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	complemented
43	C₂×C₃₁×#107	C₂²	1	2	yes	—
44	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
45	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
47	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
48	C₂×C₃₁×#102	C₂²	1	2	yes	—
49	C₂×C₃₁×#102	C₂²	1	2	yes	—
50	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
51	C₂×C₃₁×#102	C₂²	1	2	yes	—
52	C₂×C₃₁×#102	C₂²	1	2	yes	—
53	C₂×C₃₁×#97	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
55	C₂×#41×C₃₁	—	2	8	no	—
56	C₂×#41×C₃₁	—	4	8	no	—
57	C₂²×Q₈×C₃₁	—	2	1	no	—
58	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
59	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₃₁	—	2	2	no	—
61	C₂²×D₄×C₃₁	—	2	1	no	—
62	C₂²×D₄×C₃₁	—	2	2	no	—
63	C₂²×D₄×C₃₁	—	2	1	no	—
64	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
65	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₃₁	—	2	2	no	—
67	C₂³×C₄×C₃₁	—	2	1	no	—
68	C₂×SD₁₆×C₃₁	—	2	4	no	—
69	C₂×SD₁₆×C₃₁	—	2	8	no	—
70	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
71	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
72	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
75	C₃₁×C₄⋊Q₈	—	2	8	no	—
76	C₃₁×C₄⋊Q₈	—	2	2	no	—
77	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
78	C₃₁×#127	—	2	4	no	—
79	C₃₁×#127	—	2	2	no	—
80	C₃₁×#127	C₂³	1	4	yes	—
81	C₃₁×#127	C₂³	1	8	yes	—
82	C₃₁×#125	—	2	16	no	—
83	C₃₁×#125	—	2	4	no	—
84	C₃₁×#125	C₂³	1	8	yes	—
85	C₃₁×#121	—	2	8	no	—
86	C₃₁×#121	—	2	16	no	—
87	C₄×Q₈×C₃₁	—	2	8	no	—
88	C₄×Q₈×C₃₁	—	2	4	no	—
89	C₄×Q₈×C₃₁	—	2	2	no	—
90	C₄×Q₈×C₃₁	D₄	1	4	yes	—
91	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
92	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
93	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
94	C₄×D₄×C₃₁	—	2	4	no	—
95	C₄×D₄×C₃₁	—	2	2	no	—
96	C₄×D₄×C₃₁	C₂³	1	4	yes	—
97	C₄×D₄×C₃₁	C₂³	1	8	yes	—
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	1	no	—
102	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
108	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
109	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
110	C₂×C₄²×C₃₁	—	2	2	no	—
111	C₂×C₄²×C₃₁	—	2	1	no	—
112	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
114	C₃₁×#107	C₂³	1	8	yes	—
115	C₃₁×#106	C₂³	1	4	yes	—
116	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
117	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
118	C₃₁×#103	C₂³	1	4	yes	—
119	C₃₁×#102	C₂³	1	4	yes	—
120	C₃₁×#102	C₂³	1	8	yes	—
121	C₃₁×#102	C₂³	1	8	yes	—
122	C₃₁×#102	C₂³	1	8	yes	—
123	C₃₁×#102	C₂³	1	8	yes	—
124	C₃₁×#97	C₂³	1	8	yes	—
125	C₄×C₈×C₃₁	C₂³	1	4	yes	—
126	C₂⁴×C₃₁	—	2	2	no	—
127	C₂⁴×C₃₁	—	2	1	no	—
128	#41×C₃₁	—	4	32	no	—
129	C₂×Q₈×C₃₁	—	2	4	no	—
130	C₂×Q₈×C₃₁	—	2	1	no	—
131	C₂×Q₈×C₃₁	—	2	4	no	—
132	C₂×Q₈×C₃₁	—	2	2	no	—
133	C₂×Q₈×C₃₁	—	4	4	no	—
134	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
135	C₂×Q₈×C₃₁	#41	1	4	yes	complemented
136	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
137	C₂×D₄×C₃₁	—	2	2	no	—
138	C₂×D₄×C₃₁	—	2	1	no	—
139	C₂×D₄×C₃₁	—	2	8	no	—
140	C₂×D₄×C₃₁	—	2	4	no	—
141	C₂×D₄×C₃₁	—	2	8	no	—
142	C₂×D₄×C₃₁	—	2	4	no	—
143	C₂×D₄×C₃₁	—	2	2	no	—
144	C₂×D₄×C₃₁	—	2	4	no	—
145	C₂×D₄×C₃₁	—	2	4	no	—
146	C₂×D₄×C₃₁	—	2	2	no	—
147	C₂×D₄×C₃₁	—	4	8	no	—
148	C₂×D₄×C₃₁	—	4	4	no	—
149	C₂×D₄×C₃₁	—	8	4	no	—
150	C₂×D₄×C₃₁	—	8	2	no	—
151	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
152	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₃₁	—	2	1	no	—
154	C₂²×C₄×C₃₁	—	2	1	no	—
155	C₂²×C₄×C₃₁	—	2	2	no	—
156	C₂²×C₄×C₃₁	—	2	1	no	—
157	C₂²×C₄×C₃₁	—	2	8	no	—
158	C₂²×C₄×C₃₁	—	2	4	no	—
159	C₂²×C₄×C₃₁	—	4	8	no	—
160	C₂²×C₄×C₃₁	—	4	4	no	—
161	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
166	SD₁₆×C₃₁	—	2	8	no	—
167	SD₁₆×C₃₁	—	2	16	no	—
168	C₂×C₈×C₃₁	—	2	2	no	—
169	C₂×C₈×C₃₁	—	2	4	no	—
170	C₂×C₈×C₃₁	—	2	2	no	—
171	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
172	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
173	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
174	C₄⋊C₄×C₃₁	—	2	8	no	—
175	C₄⋊C₄×C₃₁	—	2	4	no	—
176	C₄⋊C₄×C₃₁	—	2	8	no	—
177	C₄⋊C₄×C₃₁	—	2	4	no	—
178	C₄⋊C₄×C₃₁	—	2	8	no	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₄⋊C₄×C₃₁	—	2	1	no	—
181	C₄⋊C₄×C₃₁	—	2	2	no	—
182	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
183	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
184	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
185	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
186	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
187	C₂²⋊C₄×C₃₁	—	2	8	no	—
188	C₂²⋊C₄×C₃₁	—	2	4	no	—
189	C₄²×C₃₁	—	2	8	no	—
190	C₄²×C₃₁	—	2	4	no	—
191	C₄²×C₃₁	C₂×D₄	1	4	yes	—
192	C₄²×C₃₁	C₂×D₄	1	2	yes	—
193	C₄²×C₃₁	C₂⁴	1	2	yes	—
194	C₄²×C₃₁	C₂⁴	1	4	yes	—
195	C₂×#3368	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂³×C₃₁	—	2	4	no	—
197	C₂³×C₃₁	—	2	2	no	—
198	C₂³×C₃₁	—	4	4	no	—
199	C₂³×C₃₁	—	4	2	no	—
200	C₂³×C₃₁	—	4	4	no	—
201	C₂³×C₃₁	—	4	2	no	—
202	C₂³×C₃₁	—	4	4	no	—
203	C₂³×C₃₁	—	4	2	no	—
204	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
205	Q₈×C₃₁	—	2	4	no	—
206	Q₈×C₃₁	—	2	8	no	—
207	Q₈×C₃₁	—	2	2	no	—
208	Q₈×C₃₁	—	4	8	no	—
209	Q₈×C₃₁	C₂×#41	1	4	yes	complemented
210	D₄×C₃₁	—	2	2	no	—
211	D₄×C₃₁	—	2	4	no	—
212	D₄×C₃₁	—	2	2	no	—
213	D₄×C₃₁	—	2	4	no	—
214	D₄×C₃₁	—	4	16	no	—
215	D₄×C₃₁	—	4	4	no	—
216	D₄×C₃₁	—	4	2	no	—
217	D₄×C₃₁	—	4	8	no	—
218	D₄×C₃₁	—	8	8	no	—
219	D₄×C₃₁	—	8	4	no	—
220	C₂×C₄×C₃₁	—	2	4	no	—
221	C₂×C₄×C₃₁	—	2	4	no	—
222	C₂×C₄×C₃₁	—	2	4	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	2	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	4	16	no	—
229	C₂×C₄×C₃₁	—	4	2	no	—
230	C₂×C₄×C₃₁	—	4	4	no	—
231	C₂×C₄×C₃₁	—	4	2	no	—
232	C₂×C₄×C₃₁	—	4	8	no	—
233	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
234	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
235	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
236	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
237	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
239	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
240	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
241	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
242	C₈×C₃₁	—	2	2	no	—
243	C₈×C₃₁	—	2	4	no	—
244	C₈×C₃₁	—	2	2	no	—
245	C₂×#557	C₂×C₃₁	1	1	yes	characteristic, complemented
246	C₂×#550	C₂×C₃₁	1	1	yes	characteristic
247	#3368	C₂×C₃₁	1	16	yes	direct product factor
248	C₂×#490	C₂×C₃₁	1	2	yes	complemented
249	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
250	C₂×#478	C₂×C₃₁	1	2	yes	complemented
251	C₂×#474	C₂×C₃₁	1	1	yes	characteristic, complemented
252	C₂×#460	C₂×C₃₁	1	1	yes	characteristic, complemented
253	C₂×#460	C₂×C₃₁	1	2	yes	complemented
254	C₂×#446	C₂×C₃₁	1	1	yes	characteristic, complemented
255	C₂×#440	C₂×C₃₁	1	2	yes	complemented
256	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
257	C₂²×C₃₁	—	4	4	no	—
258	C₂²×C₃₁	—	4	2	no	—
259	C₂²×C₃₁	—	4	4	no	—
260	C₂²×C₃₁	—	4	2	no	—
261	C₂²×C₃₁	—	4	4	no	—
262	C₂²×C₃₁	—	4	2	no	—
263	C₂²×C₃₁	—	8	8	no	—
264	C₂²×C₃₁	—	8	4	no	—
265	C₂²×C₃₁	#548	1	2	yes	—
266	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
267	C₂²×C₃₁	#577	1	2	yes	—
268	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
269	C₄×C₃₁	—	2	4	no	—
270	C₄×C₃₁	—	2	4	no	—
271	C₄×C₃₁	—	2	4	no	—
272	C₄×C₃₁	—	2	2	no	—
273	C₄×C₃₁	—	4	2	no	—
274	C₄×C₃₁	—	4	4	no	—
275	C₄×C₃₁	—	4	2	no	—
276	C₄×C₃₁	#548	1	2	yes	—
277	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
278	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
279	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
280	C₂²×#41	—	2	1	no	—
281	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
282	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
283	#557	—	2	4	no	—
284	#557	C₂²×C₃₁	1	8	yes	complemented
285	#550	—	2	4	no	—
286	#550	C₂²×C₃₁	1	8	yes	—
287	C₂×C₄⋊Q₈	—	2	1	no	—
288	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×#127	C₂²×C₃₁	1	2	yes	—
291	C₂×#125	—	2	2	no	—
292	C₂×#125	C₂²×C₃₁	1	2	yes	—
293	C₂×#121	—	2	1	no	—
294	C₂×#121	—	2	2	no	—
295	C₂×C₄×Q₈	—	2	1	no	—
296	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
297	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
298	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
299	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
300	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
301	#490	C₂²×C₃₁	1	16	yes	complemented
302	#486	C₂²×C₃₁	1	8	yes	complemented
303	#478	C₂²×C₃₁	1	16	yes	complemented
304	#474	C₂²×C₃₁	1	8	yes	complemented
305	#460	C₂²×C₃₁	1	16	yes	complemented
306	#460	C₂²×C₃₁	1	8	yes	complemented
307	#446	C₂²×C₃₁	1	8	yes	complemented
308	#440	C₂²×C₃₁	1	16	yes	complemented
309	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	complemented
310	C₂×#107	C₂²×C₃₁	1	2	yes	—
311	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
312	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
313	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
314	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
315	C₂×#102	C₂²×C₃₁	1	2	yes	—
316	C₂×#102	C₂²×C₃₁	1	2	yes	—
317	C₂×#102	C₂²×C₃₁	1	2	yes	—
318	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
319	C₂×#102	C₂²×C₃₁	1	2	yes	—
320	C₂×#97	C₂²×C₃₁	1	2	yes	—
321	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
322	C₂×C₃₁	—	8	4	no	—
323	C₂×C₃₁	—	8	2	no	—
324	C₂×C₃₁	#3368	1	4	yes	direct product factor
325	C₂×C₃₁	C₂×#548	1	1	yes	characteristic
326	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
327	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
328	C₂×#41	—	2	8	no	—
329	C₂×#41	—	4	8	no	—
330	C₂²×Q₈	—	2	1	no	—
331	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
332	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
333	C₂²×D₄	—	2	2	no	—
334	C₂²×D₄	—	2	1	no	—
335	C₂²×D₄	—	2	2	no	—
336	C₂²×D₄	—	2	1	no	—
337	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
338	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
339	C₂³×C₄	—	2	2	no	—
340	C₂³×C₄	—	2	1	no	—
341	C₂×SD₁₆	—	2	4	no	—
342	C₂×SD₁₆	—	2	8	no	—
343	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
344	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
345	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
346	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
347	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
348	C₄⋊Q₈	—	2	8	no	—
349	C₄⋊Q₈	—	2	2	no	—
350	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
351	#127	—	2	4	no	—
352	#127	—	2	2	no	—
353	#127	C₂³×C₃₁	1	4	yes	—
354	#127	C₂³×C₃₁	1	8	yes	—
355	#125	—	2	16	no	—
356	#125	—	2	4	no	—
357	#125	C₂³×C₃₁	1	8	yes	—
358	#121	—	2	8	no	—
359	#121	—	2	16	no	—
360	C₄×Q₈	—	2	8	no	—
361	C₄×Q₈	—	2	4	no	—
362	C₄×Q₈	—	2	2	no	—
363	C₄×Q₈	D₄×C₃₁	1	4	yes	—
364	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
365	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
366	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
367	C₄×D₄	—	2	4	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	C₂³×C₃₁	1	4	yes	—
370	C₄×D₄	C₂³×C₃₁	1	8	yes	—
371	C₂×C₄⋊C₄	—	2	2	no	—
372	C₂×C₄⋊C₄	—	2	1	no	—
373	C₂×C₄⋊C₄	—	2	2	no	—
374	C₂×C₄⋊C₄	—	2	1	no	—
375	C₂×C₄⋊C₄	—	2	2	no	—
376	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
377	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
378	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
379	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
380	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
381	C₂×C₂²⋊C₄	—	2	2	no	—
382	C₂×C₂²⋊C₄	—	2	1	no	—
383	C₂×C₄²	—	2	2	no	—
384	C₂×C₄²	—	2	1	no	—
385	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
386	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
387	#107	C₂³×C₃₁	1	8	yes	—
388	#106	C₂³×C₃₁	1	4	yes	—
389	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
390	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
391	#103	C₂³×C₃₁	1	4	yes	—
392	#102	C₂³×C₃₁	1	4	yes	—
393	#102	C₂³×C₃₁	1	8	yes	—
394	#102	C₂³×C₃₁	1	8	yes	—
395	#102	C₂³×C₃₁	1	8	yes	—
396	#102	C₂³×C₃₁	1	8	yes	—
397	#97	C₂³×C₃₁	1	8	yes	—
398	C₄×C₈	C₂³×C₃₁	1	4	yes	—
399	C₃₁	C₂×#3368	1	1	yes	characteristic, direct product factor, Sylow
400	C₂⁴	—	2	2	no	—
401	C₂⁴	—	2	1	no	—
402	#41	—	4	32	no	—
403	C₂×Q₈	—	2	4	no	—
404	C₂×Q₈	—	2	1	no	—
405	C₂×Q₈	—	2	4	no	—
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	—	4	4	no	—
408	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
409	C₂×Q₈	#41×C₃₁	1	4	yes	complemented
410	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
411	C₂×D₄	—	2	2	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	2	8	no	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	—	2	8	no	—
416	C₂×D₄	—	2	4	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	4	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	4	8	no	—
422	C₂×D₄	—	4	4	no	—
423	C₂×D₄	—	8	4	no	—
424	C₂×D₄	—	8	2	no	—
425	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
426	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	8	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	4	8	no	—
434	C₂²×C₄	—	4	4	no	—
435	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
436	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
437	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
438	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
439	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
440	SD₁₆	—	2	8	no	—
441	SD₁₆	—	2	16	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	—	2	4	no	—
444	C₂×C₈	—	2	2	no	—
445	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
446	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
447	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
448	C₄⋊C₄	—	2	8	no	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	8	no	—
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	2	8	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
457	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
458	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
459	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
460	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
461	C₂²⋊C₄	—	2	8	no	—
462	C₂²⋊C₄	—	2	4	no	—
463	C₄²	—	2	8	no	—
464	C₄²	—	2	4	no	—
465	C₄²	C₂×D₄×C₃₁	1	4	yes	—
466	C₄²	C₂×D₄×C₃₁	1	2	yes	—
467	C₄²	C₂⁴×C₃₁	1	2	yes	—
468	C₄²	C₂⁴×C₃₁	1	4	yes	—
469	C₂³	—	2	4	no	—
470	C₂³	—	2	2	no	—
471	C₂³	—	4	4	no	—
472	C₂³	—	4	2	no	—
473	C₂³	—	4	4	no	—
474	C₂³	—	4	2	no	—
475	C₂³	—	4	4	no	—
476	C₂³	—	4	2	no	—
477	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
478	Q₈	—	2	4	no	—
479	Q₈	—	2	8	no	—
480	Q₈	—	2	2	no	—
481	Q₈	—	4	8	no	—
482	Q₈	C₂×#41×C₃₁	1	4	yes	complemented
483	D₄	—	2	2	no	—
484	D₄	—	2	4	no	—
485	D₄	—	2	2	no	—
486	D₄	—	2	4	no	—
487	D₄	—	4	16	no	—
488	D₄	—	4	4	no	—
489	D₄	—	4	2	no	—
490	D₄	—	4	8	no	—
491	D₄	—	8	8	no	—
492	D₄	—	8	4	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	4	16	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	4	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	8	no	—
506	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
507	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
508	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
509	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
510	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
511	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
512	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
513	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
514	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
515	C₈	—	2	2	no	—
516	C₈	—	2	4	no	—
517	C₈	—	2	2	no	—
518	C₂²	—	4	4	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	4	no	—
521	C₂²	—	4	2	no	—
522	C₂²	—	4	4	no	—
523	C₂²	—	4	2	no	—
524	C₂²	—	8	8	no	—
525	C₂²	—	8	4	no	—
526	C₂²	C₃₁×#548	1	2	yes	—
527	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
528	C₂²	C₃₁×#577	1	2	yes	—
529	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
530	C₄	—	2	4	no	—
531	C₄	—	2	4	no	—
532	C₄	—	2	4	no	—
533	C₄	—	2	2	no	—
534	C₄	—	4	2	no	—
535	C₄	—	4	4	no	—
536	C₄	—	4	2	no	—
537	C₄	C₃₁×#548	1	2	yes	—
538	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
539	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
540	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
541	C₂	—	8	4	no	—
542	C₂	—	8	2	no	—
543	C₂	C₃₁×#3368	1	4	yes	direct product factor
544	C₂	C₂×C₃₁×#548	1	1	yes	characteristic
545	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
546	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	C₂×C₃₁×#550	C₂
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×#3368	C₂	direct (N × H)
4. C₂×C₃₁×#490	C₂	semidirect
5. C₂×C₃₁×#486	C₂	semidirect
6. C₂×C₃₁×#478	C₂	semidirect
7. C₂×C₃₁×#474	C₂	semidirect
8. C₂×C₃₁×#460	C₂	semidirect
9. C₂×C₃₁×#460	C₂	semidirect
10. C₂×C₃₁×#446	C₂	semidirect
11. C₂×C₃₁×#440	C₂	semidirect
12. C₂×C₄×SD₁₆×C₃₁	C₂	semidirect
17. C₃₁×#557	C₂²	semidirect
34. C₃₁×#490	C₂²	semidirect
35. C₃₁×#486	C₂²	semidirect
36. C₃₁×#478	C₂²	semidirect
37. C₃₁×#474	C₂²	semidirect
38. C₃₁×#460	C₂²	semidirect
39. C₃₁×#460	C₂²	semidirect
40. C₃₁×#446	C₂²	semidirect
41. C₃₁×#440	C₂²	semidirect
42. C₄×SD₁₆×C₃₁	C₂²	semidirect
135. C₂×Q₈×C₃₁	#41	semidirect
195. C₂×#3368	C₃₁	direct (N × H)
209. Q₈×C₃₁	C₂×#41	semidirect
245. C₂×#557	C₂×C₃₁	semidirect
247. #3368	C₂×C₃₁	direct (N × H)
248. C₂×#490	C₂×C₃₁	semidirect
249. C₂×#486	C₂×C₃₁	semidirect
250. C₂×#478	C₂×C₃₁	semidirect
251. C₂×#474	C₂×C₃₁	semidirect
252. C₂×#460	C₂×C₃₁	semidirect
253. C₂×#460	C₂×C₃₁	semidirect
254. C₂×#446	C₂×C₃₁	semidirect
255. C₂×#440	C₂×C₃₁	semidirect
256. C₂×C₄×SD₁₆	C₂×C₃₁	semidirect
284. #557	C₂²×C₃₁	semidirect
301. #490	C₂²×C₃₁	semidirect
302. #486	C₂²×C₃₁	semidirect
303. #478	C₂²×C₃₁	semidirect
304. #474	C₂²×C₃₁	semidirect
305. #460	C₂²×C₃₁	semidirect
306. #460	C₂²×C₃₁	semidirect
307. #446	C₂²×C₃₁	semidirect
308. #440	C₂²×C₃₁	semidirect
309. C₄×SD₁₆	C₂²×C₃₁	semidirect
324. C₂×C₃₁	#3368	direct (N × H)
399. C₃₁	C₂×#3368	direct (N × H)
409. C₂×Q₈	#41×C₃₁	semidirect
482. Q₈	C₂×#41×C₃₁	semidirect
543. C₂	C₃₁×#3368	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×#550	2
C₃₁×#3368	2
C₂×C₃₁×#490	2
C₂×C₃₁×#486	2
C₂×C₃₁×#478	2
C₂×C₃₁×#474	2
C₂×C₃₁×#460	2
C₂×C₃₁×#446	2
C₂×C₃₁×#440	2
C₂×C₄×SD₁₆×C₃₁	2
C₂×#3368	31

Subgroups — C2×C31×#3368 #21518067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3368 #21518067