Subgroups — C₂×C₃₁×#3081 #21517244

Order 7936.

2234 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×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#518	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3081	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#478	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#478	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#476	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#474	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#428	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#418	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
13	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#544	—	2	4	no	—
17	C₃₁×#544	C₂²	1	8	yes	complemented
18	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
19	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	complemented
20	C₂×C₃₁×#125	—	2	1	no	—
21	C₂×C₃₁×#125	C₂²	1	2	yes	complemented
22	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
23	C₃₁×#518	—	2	4	no	—
24	C₃₁×#518	C₂²	1	8	yes	—
25	C₂×C₄×D₄×C₃₁	—	2	1	no	—
26	C₂×C₄×D₄×C₃₁	—	2	1	no	—
27	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
29	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
32	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₂²×C₂²⋊C₄×C₃₁	—	2	2	no	—
34	C₃₁×#478	C₂²	1	16	yes	complemented
35	C₃₁×#478	C₂²	1	16	yes	complemented
36	C₃₁×#476	C₂²	1	16	yes	complemented
37	C₃₁×#474	C₂²	1	16	yes	complemented
38	C₃₁×#428	C₂²	1	8	yes	complemented
39	C₃₁×#427	C₂²	1	8	yes	complemented
40	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic, complemented
41	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic, complemented
42	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
43	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic, complemented
44	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic, complemented
45	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
46	C₃₁×#424	C₂²	1	8	yes	complemented
47	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
48	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
49	C₃₁×#418	C₂²	1	8	yes	—
50	C₂×C₃₁×#102	C₂²	1	8	yes	—
51	C₂×C₃₁×#102	C₂²	1	2	yes	—
52	C₂×C₃₁×#102	C₂²	1	2	yes	—
53	C₂×C₃₁×#102	C₂²	1	2	yes	—
54	C₂×C₃₁×#102	C₂²	1	2	yes	—
55	C₂⁵×C₃₁	—	2	1	no	—
56	C₂²×Q₈×C₃₁	—	2	2	no	—
57	C₂²×D₄×C₃₁	—	2	1	no	—
58	C₂²×D₄×C₃₁	—	2	2	no	—
59	C₂²×D₄×C₃₁	—	2	4	no	—
60	C₂²×D₄×C₃₁	Q₈	1	8	yes	complemented
61	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
62	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
63	C₂³×C₄×C₃₁	—	2	1	no	—
64	C₂³×C₄×C₃₁	—	2	2	no	—
65	C₂³×C₄×C₃₁	—	2	2	no	—
66	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₃₁	—	2	4	no	—
68	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
69	C₂²×C₈×C₃₁	—	2	4	no	—
70	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
72	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
74	C₃₁×C₄⋊Q₈	—	2	8	no	—
75	C₃₁×C₄⋊Q₈	—	2	4	no	—
76	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	complemented
77	C₃₁×#125	—	2	8	no	—
78	C₃₁×#125	—	2	4	no	—
79	C₃₁×#125	C₂³	1	8	yes	complemented
80	C₃₁×C₂²⋊Q₈	—	2	16	no	—
81	C₄×D₄×C₃₁	—	2	8	no	—
82	C₄×D₄×C₃₁	—	2	8	no	—
83	C₄×D₄×C₃₁	—	2	4	no	—
84	C₄×D₄×C₃₁	—	2	4	no	—
85	C₄×D₄×C₃₁	C₂³	1	8	yes	—
86	C₄×D₄×C₃₁	C₂³	1	8	yes	—
87	C₃₁×#117	—	2	2	no	—
88	C₃₁×#117	—	2	2	no	—
89	C₃₁×#117	D₄	1	4	yes	—
90	C₃₁×#117	C₂³	1	4	yes	—
91	C₃₁×#117	C₂³	1	4	yes	—
92	C₃₁×#117	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
94	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
107	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
108	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
109	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
111	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
112	C₂×C₄²×C₃₁	—	2	1	no	—
113	C₂×C₄²×C₃₁	—	2	1	no	—
114	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
115	C₃₁×#107	C₂³	1	4	yes	complemented
116	C₃₁×#107	C₂³	1	4	yes	complemented
117	C₃₁×#107	C₂³	1	8	yes	complemented
118	C₃₁×#106	C₂³	1	4	yes	complemented
119	C₃₁×#106	C₂³	1	4	yes	complemented
120	C₃₁×#106	C₂³	1	8	yes	complemented
121	C₃₁×C₄⋊C₈	C₂³	1	8	yes	complemented
122	C₃₁×C₄⋊C₈	C₂³	1	8	yes	complemented
123	C₃₁×#102	C₂³	1	8	yes	—
124	C₃₁×#102	C₂³	1	8	yes	—
125	C₃₁×#102	C₂³	1	8	yes	—
126	C₃₁×#102	C₂³	1	8	yes	—
127	C₂⁴×C₃₁	—	2	2	no	—
128	C₂⁴×C₃₁	—	2	8	no	—
129	C₂⁴×C₃₁	—	2	4	no	—
130	C₂⁴×C₃₁	—	8	2	no	—
131	C₂⁴×C₃₁	—	8	2	no	—
132	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂×Q₈×C₃₁	—	2	8	no	—
134	C₂×Q₈×C₃₁	—	8	4	no	—
135	C₂×D₄×C₃₁	—	2	16	no	—
136	C₂×D₄×C₃₁	—	2	8	no	—
137	C₂×D₄×C₃₁	—	2	4	no	—
138	C₂×D₄×C₃₁	—	2	2	no	—
139	C₂×D₄×C₃₁	—	2	4	no	—
140	C₂×D₄×C₃₁	—	2	8	no	—
141	C₂×D₄×C₃₁	C₂×Q₈	1	8	yes	—
142	C₂×D₄×C₃₁	C₂×Q₈	1	16	yes	complemented
143	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₃₁	—	2	1	no	—
145	C₂²×C₄×C₃₁	—	2	1	no	—
146	C₂²×C₄×C₃₁	—	2	1	no	—
147	C₂²×C₄×C₃₁	—	2	1	no	—
148	C₂²×C₄×C₃₁	—	2	2	no	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	2	8	no	—
151	C₂²×C₄×C₃₁	—	2	8	no	—
152	C₂²×C₄×C₃₁	—	8	4	no	—
153	C₂²×C₄×C₃₁	—	8	4	no	—
154	C₂²×C₄×C₃₁	—	8	2	no	—
155	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
157	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
158	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
160	C₂²×C₄×C₃₁	#41	1	4	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
163	M₁₆×C₃₁	—	2	8	no	—
164	C₂×C₈×C₃₁	—	2	2	no	—
165	C₂×C₈×C₃₁	—	2	2	no	—
166	C₂×C₈×C₃₁	—	2	8	no	—
167	C₂×C₈×C₃₁	—	2	4	no	—
168	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
169	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
170	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
171	C₄⋊C₄×C₃₁	—	2	4	no	—
172	C₄⋊C₄×C₃₁	—	2	2	no	—
173	C₄⋊C₄×C₃₁	—	2	8	no	—
174	C₄⋊C₄×C₃₁	—	2	8	no	—
175	C₄⋊C₄×C₃₁	—	2	8	no	—
176	C₄⋊C₄×C₃₁	—	2	1	no	—
177	C₄⋊C₄×C₃₁	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	2	1	no	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
181	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
182	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
183	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
184	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
185	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
186	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
187	C₂²⋊C₄×C₃₁	—	2	4	no	—
188	C₂²⋊C₄×C₃₁	—	2	4	no	—
189	C₂²⋊C₄×C₃₁	—	2	8	no	—
190	C₂²⋊C₄×C₃₁	—	2	8	no	—
191	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
192	C₄²×C₃₁	—	2	4	no	—
193	C₄²×C₃₁	—	2	4	no	—
194	C₄²×C₃₁	C₂×D₄	1	4	yes	—
195	C₄²×C₃₁	C₂⁴	1	4	yes	—
196	C₂×#3081	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂³×C₃₁	—	2	2	no	—
198	C₂³×C₃₁	—	2	2	no	—
199	C₂³×C₃₁	—	2	8	no	—
200	C₂³×C₃₁	—	2	4	no	—
201	C₂³×C₃₁	—	2	16	no	—
202	C₂³×C₃₁	—	4	4	no	—
203	C₂³×C₃₁	—	4	4	no	—
204	C₂³×C₃₁	—	8	16	no	—
205	C₂³×C₃₁	—	8	4	no	—
206	C₂³×C₃₁	—	8	4	no	—
207	C₂³×C₃₁	C₂²⋊Q₈	1	4	yes	—
208	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
209	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
210	Q₈×C₃₁	—	8	8	no	—
211	D₄×C₃₁	—	2	8	no	—
212	D₄×C₃₁	—	2	16	no	—
213	D₄×C₃₁	—	2	4	no	—
214	D₄×C₃₁	C₂²×Q₈	1	8	yes	—
215	C₂×C₄×C₃₁	—	2	4	no	—
216	C₂×C₄×C₃₁	—	2	4	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	4	no	—
222	C₂×C₄×C₃₁	—	4	2	no	—
223	C₂×C₄×C₃₁	—	4	4	no	—
224	C₂×C₄×C₃₁	—	4	2	no	—
225	C₂×C₄×C₃₁	—	4	2	no	—
226	C₂×C₄×C₃₁	—	4	2	no	—
227	C₂×C₄×C₃₁	—	8	8	no	—
228	C₂×C₄×C₃₁	—	8	8	no	—
229	C₂×C₄×C₃₁	—	8	4	no	—
230	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
231	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
232	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
233	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
234	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
235	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
237	C₂×C₄×C₃₁	C₂²×Q₈	1	4	yes	—
238	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
239	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
240	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
241	C₈×C₃₁	—	2	2	no	—
242	C₈×C₃₁	—	2	2	no	—
243	C₈×C₃₁	—	2	4	no	—
244	C₂×#544	C₂×C₃₁	1	1	yes	characteristic, complemented
245	C₂×#518	C₂×C₃₁	1	1	yes	characteristic
246	#3081	C₂×C₃₁	1	16	yes	direct product factor
247	C₂×#478	C₂×C₃₁	1	2	yes	complemented
248	C₂×#478	C₂×C₃₁	1	2	yes	complemented
249	C₂×#476	C₂×C₃₁	1	2	yes	complemented
250	C₂×#474	C₂×C₃₁	1	2	yes	complemented
251	C₂×#428	C₂×C₃₁	1	1	yes	characteristic, complemented
252	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
253	C₂×#424	C₂×C₃₁	1	1	yes	characteristic, complemented
254	C₂×#418	C₂×C₃₁	1	1	yes	characteristic
255	C₂²×#102	C₂×C₃₁	1	1	yes	characteristic
256	C₂²×C₃₁	—	2	8	no	—
257	C₂²×C₃₁	—	2	2	no	—
258	C₂²×C₃₁	—	2	2	no	—
259	C₂²×C₃₁	—	2	8	no	—
260	C₂²×C₃₁	—	4	16	no	—
261	C₂²×C₃₁	—	4	4	no	—
262	C₂²×C₃₁	—	4	4	no	—
263	C₂²×C₃₁	—	8	16	no	—
264	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	4	yes	—
265	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
266	C₂²×C₃₁	#576	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	—	4	2	no	—
272	C₄×C₃₁	—	4	4	no	—
273	C₄×C₃₁	—	4	2	no	—
274	C₄×C₃₁	—	4	2	no	—
275	C₄×C₃₁	—	4	2	no	—
276	C₄×C₃₁	C₂×C₂²⋊Q₈	1	4	yes	—
277	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
278	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
279	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
280	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
281	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
282	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
283	#544	—	2	4	no	—
284	#544	C₂²×C₃₁	1	8	yes	complemented
285	C₂×C₄⋊Q₈	—	2	1	no	—
286	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
287	C₂×#125	—	2	1	no	—
288	C₂×#125	C₂²×C₃₁	1	2	yes	complemented
289	C₂×C₂²⋊Q₈	—	2	2	no	—
290	#518	—	2	4	no	—
291	#518	C₂²×C₃₁	1	8	yes	—
292	C₂×C₄×D₄	—	2	1	no	—
293	C₂×C₄×D₄	—	2	1	no	—
294	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
295	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
296	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
297	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
298	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
299	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
300	C₂²×C₂²⋊C₄	—	2	2	no	—
301	#478	C₂²×C₃₁	1	16	yes	complemented
302	#478	C₂²×C₃₁	1	16	yes	complemented
303	#476	C₂²×C₃₁	1	16	yes	complemented
304	#474	C₂²×C₃₁	1	16	yes	complemented
305	#428	C₂²×C₃₁	1	8	yes	complemented
306	#427	C₂²×C₃₁	1	8	yes	complemented
307	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic, complemented
308	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
309	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic, complemented
310	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic, complemented
311	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
312	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic, complemented
313	#424	C₂²×C₃₁	1	8	yes	complemented
314	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
315	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
316	#418	C₂²×C₃₁	1	8	yes	—
317	C₂×#102	C₂²×C₃₁	1	2	yes	—
318	C₂×#102	C₂²×C₃₁	1	2	yes	—
319	C₂×#102	C₂²×C₃₁	1	2	yes	—
320	C₂×#102	C₂²×C₃₁	1	8	yes	—
321	C₂×#102	C₂²×C₃₁	1	2	yes	—
322	C₂×C₃₁	—	2	4	no	—
323	C₂×C₃₁	—	4	8	no	—
324	C₂×C₃₁	#3081	1	4	yes	direct product factor
325	C₂×C₃₁	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
326	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
327	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
328	C₂⁵	—	2	1	no	—
329	C₂²×Q₈	—	2	2	no	—
330	C₂²×D₄	—	2	1	no	—
331	C₂²×D₄	—	2	2	no	—
332	C₂²×D₄	—	2	4	no	—
333	C₂²×D₄	Q₈×C₃₁	1	8	yes	complemented
334	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
335	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
336	C₂³×C₄	—	2	1	no	—
337	C₂³×C₄	—	2	2	no	—
338	C₂³×C₄	—	2	2	no	—
339	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
340	C₂×M₁₆	—	2	4	no	—
341	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
342	C₂²×C₈	—	2	4	no	—
343	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
344	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
345	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
346	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
347	C₄⋊Q₈	—	2	8	no	—
348	C₄⋊Q₈	—	2	4	no	—
349	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	complemented
350	#125	—	2	8	no	—
351	#125	—	2	4	no	—
352	#125	C₂³×C₃₁	1	8	yes	complemented
353	C₂²⋊Q₈	—	2	16	no	—
354	C₄×D₄	—	2	8	no	—
355	C₄×D₄	—	2	8	no	—
356	C₄×D₄	—	2	4	no	—
357	C₄×D₄	—	2	4	no	—
358	C₄×D₄	C₂³×C₃₁	1	8	yes	—
359	C₄×D₄	C₂³×C₃₁	1	8	yes	—
360	#117	—	2	2	no	—
361	#117	—	2	2	no	—
362	#117	D₄×C₃₁	1	4	yes	—
363	#117	C₂³×C₃₁	1	4	yes	—
364	#117	C₂³×C₃₁	1	4	yes	—
365	#117	C₂³×C₃₁	1	4	yes	—
366	C₂×C₄⋊C₄	—	2	1	no	—
367	C₂×C₄⋊C₄	—	2	2	no	—
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	—	2	2	no	—
370	C₂×C₄⋊C₄	—	2	2	no	—
371	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
372	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
373	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
374	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
375	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
376	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
377	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
378	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
379	C₂×C₂²⋊C₄	—	2	1	no	—
380	C₂×C₂²⋊C₄	—	2	1	no	—
381	C₂×C₂²⋊C₄	—	2	2	no	—
382	C₂×C₂²⋊C₄	—	2	2	no	—
383	C₂×C₂²⋊C₄	—	2	16	no	—
384	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
385	C₂×C₄²	—	2	1	no	—
386	C₂×C₄²	—	2	1	no	—
387	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
388	#107	C₂³×C₃₁	1	4	yes	complemented
389	#107	C₂³×C₃₁	1	4	yes	complemented
390	#107	C₂³×C₃₁	1	8	yes	complemented
391	#106	C₂³×C₃₁	1	4	yes	complemented
392	#106	C₂³×C₃₁	1	4	yes	complemented
393	#106	C₂³×C₃₁	1	8	yes	complemented
394	C₄⋊C₈	C₂³×C₃₁	1	8	yes	complemented
395	C₄⋊C₈	C₂³×C₃₁	1	8	yes	complemented
396	#102	C₂³×C₃₁	1	8	yes	—
397	#102	C₂³×C₃₁	1	8	yes	—
398	#102	C₂³×C₃₁	1	8	yes	—
399	#102	C₂³×C₃₁	1	8	yes	—
400	C₃₁	C₂×#3081	1	1	yes	characteristic, direct product factor, Sylow
401	C₂⁴	—	2	2	no	—
402	C₂⁴	—	2	8	no	—
403	C₂⁴	—	2	4	no	—
404	C₂⁴	—	8	2	no	—
405	C₂⁴	—	8	2	no	—
406	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
407	C₂×Q₈	—	2	8	no	—
408	C₂×Q₈	—	8	4	no	—
409	C₂×D₄	—	2	16	no	—
410	C₂×D₄	—	2	8	no	—
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	2	2	no	—
413	C₂×D₄	—	2	4	no	—
414	C₂×D₄	—	2	8	no	—
415	C₂×D₄	C₂×Q₈×C₃₁	1	8	yes	—
416	C₂×D₄	C₂×Q₈×C₃₁	1	16	yes	complemented
417	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	8	no	—
425	C₂²×C₄	—	2	8	no	—
426	C₂²×C₄	—	8	4	no	—
427	C₂²×C₄	—	8	4	no	—
428	C₂²×C₄	—	8	2	no	—
429	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
430	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
431	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
433	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
434	C₂²×C₄	#41×C₃₁	1	4	yes	—
435	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
436	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
437	M₁₆	—	2	8	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	—	2	8	no	—
441	C₂×C₈	—	2	4	no	—
442	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
443	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
444	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	—	2	8	no	—
448	C₄⋊C₄	—	2	8	no	—
449	C₄⋊C₄	—	2	8	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
455	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
456	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
457	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
458	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
459	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
460	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
461	C₂²⋊C₄	—	2	4	no	—
462	C₂²⋊C₄	—	2	4	no	—
463	C₂²⋊C₄	—	2	8	no	—
464	C₂²⋊C₄	—	2	8	no	—
465	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
466	C₄²	—	2	4	no	—
467	C₄²	—	2	4	no	—
468	C₄²	C₂×D₄×C₃₁	1	4	yes	—
469	C₄²	C₂⁴×C₃₁	1	4	yes	—
470	C₂³	—	2	2	no	—
471	C₂³	—	2	2	no	—
472	C₂³	—	2	8	no	—
473	C₂³	—	2	4	no	—
474	C₂³	—	2	16	no	—
475	C₂³	—	4	4	no	—
476	C₂³	—	4	4	no	—
477	C₂³	—	8	16	no	—
478	C₂³	—	8	4	no	—
479	C₂³	—	8	4	no	—
480	C₂³	C₃₁×C₂²⋊Q₈	1	4	yes	—
481	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
482	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
483	Q₈	—	8	8	no	—
484	D₄	—	2	8	no	—
485	D₄	—	2	16	no	—
486	D₄	—	2	4	no	—
487	D₄	C₂²×Q₈×C₃₁	1	8	yes	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	4	4	no	—
497	C₂×C₄	—	4	2	no	—
498	C₂×C₄	—	4	2	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	—	8	8	no	—
501	C₂×C₄	—	8	8	no	—
502	C₂×C₄	—	8	4	no	—
503	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
504	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
505	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
506	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
507	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
508	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
509	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
510	C₂×C₄	C₂²×Q₈×C₃₁	1	4	yes	—
511	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
512	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
513	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
514	C₈	—	2	2	no	—
515	C₈	—	2	2	no	—
516	C₈	—	2	4	no	—
517	C₂²	—	2	8	no	—
518	C₂²	—	2	2	no	—
519	C₂²	—	2	2	no	—
520	C₂²	—	2	8	no	—
521	C₂²	—	4	16	no	—
522	C₂²	—	4	4	no	—
523	C₂²	—	4	4	no	—
524	C₂²	—	8	16	no	—
525	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	4	yes	—
526	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
527	C₂²	C₃₁×#576	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	—	4	2	no	—
533	C₄	—	4	4	no	—
534	C₄	—	4	2	no	—
535	C₄	—	4	2	no	—
536	C₄	—	4	2	no	—
537	C₄	C₂×C₃₁×C₂²⋊Q₈	1	4	yes	—
538	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
539	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
540	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
541	C₂	—	2	4	no	—
542	C₂	—	4	8	no	—
543	C₂	C₃₁×#3081	1	4	yes	direct product factor
544	C₂	C₂²×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
545	C₂	C₂×C₃₁×#576	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	C₂³×D₄×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#544	C₂	semidirect
3. C₃₁×#3081	C₂	direct (N × H)
4. C₂×C₃₁×#478	C₂	semidirect
5. C₂×C₃₁×#478	C₂	semidirect
6. C₂×C₃₁×#476	C₂	semidirect
7. C₂×C₃₁×#474	C₂	semidirect
8. C₂×C₃₁×#428	C₂	semidirect
9. C₂×C₃₁×#427	C₂	semidirect
10. C₂×C₃₁×#424	C₂	semidirect
17. C₃₁×#544	C₂²	semidirect
19. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
21. C₂×C₃₁×#125	C₂²	semidirect
34. C₃₁×#478	C₂²	semidirect
35. C₃₁×#478	C₂²	semidirect
36. C₃₁×#476	C₂²	semidirect
37. C₃₁×#474	C₂²	semidirect
38. C₃₁×#428	C₂²	semidirect
39. C₃₁×#427	C₂²	semidirect
40. C₂×C₃₁×#107	C₂²	semidirect
41. C₂×C₃₁×#107	C₂²	semidirect
42. C₂×C₃₁×#107	C₂²	semidirect
43. C₂×C₃₁×#106	C₂²	semidirect
44. C₂×C₃₁×#106	C₂²	semidirect
45. C₂×C₃₁×#106	C₂²	semidirect
46. C₃₁×#424	C₂²	semidirect
47. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
48. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
60. C₂²×D₄×C₃₁	Q₈	semidirect
76. C₃₁×C₄⋊Q₈	C₂³	semidirect
79. C₃₁×#125	C₂³	semidirect
115. C₃₁×#107	C₂³	semidirect
116. C₃₁×#107	C₂³	semidirect
117. C₃₁×#107	C₂³	semidirect
118. C₃₁×#106	C₂³	semidirect
119. C₃₁×#106	C₂³	semidirect
120. C₃₁×#106	C₂³	semidirect
121. C₃₁×C₄⋊C₈	C₂³	semidirect
122. C₃₁×C₄⋊C₈	C₂³	semidirect
142. C₂×D₄×C₃₁	C₂×Q₈	semidirect
196. C₂×#3081	C₃₁	direct (N × H)
244. C₂×#544	C₂×C₃₁	semidirect
246. #3081	C₂×C₃₁	direct (N × H)
247. C₂×#478	C₂×C₃₁	semidirect
248. C₂×#478	C₂×C₃₁	semidirect
249. C₂×#476	C₂×C₃₁	semidirect
250. C₂×#474	C₂×C₃₁	semidirect
251. C₂×#428	C₂×C₃₁	semidirect
252. C₂×#427	C₂×C₃₁	semidirect
253. C₂×#424	C₂×C₃₁	semidirect
284. #544	C₂²×C₃₁	semidirect
286. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
288. C₂×#125	C₂²×C₃₁	semidirect
301. #478	C₂²×C₃₁	semidirect
302. #478	C₂²×C₃₁	semidirect
303. #476	C₂²×C₃₁	semidirect
304. #474	C₂²×C₃₁	semidirect
305. #428	C₂²×C₃₁	semidirect
306. #427	C₂²×C₃₁	semidirect
307. C₂×#107	C₂²×C₃₁	semidirect
308. C₂×#107	C₂²×C₃₁	semidirect
309. C₂×#107	C₂²×C₃₁	semidirect
310. C₂×#106	C₂²×C₃₁	semidirect
311. C₂×#106	C₂²×C₃₁	semidirect
312. C₂×#106	C₂²×C₃₁	semidirect
313. #424	C₂²×C₃₁	semidirect
314. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
315. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
324. C₂×C₃₁	#3081	direct (N × H)
333. C₂²×D₄	Q₈×C₃₁	semidirect
349. C₄⋊Q₈	C₂³×C₃₁	semidirect
352. #125	C₂³×C₃₁	semidirect
388. #107	C₂³×C₃₁	semidirect
389. #107	C₂³×C₃₁	semidirect
390. #107	C₂³×C₃₁	semidirect
391. #106	C₂³×C₃₁	semidirect
392. #106	C₂³×C₃₁	semidirect
393. #106	C₂³×C₃₁	semidirect
394. C₄⋊C₈	C₂³×C₃₁	semidirect
395. C₄⋊C₈	C₂³×C₃₁	semidirect
400. C₃₁	C₂×#3081	direct (N × H)
416. C₂×D₄	C₂×Q₈×C₃₁	semidirect
543. C₂	C₃₁×#3081	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×#544	2
C₂×C₃₁×#518	2
C₃₁×#3081	2
C₂×C₃₁×#478	2
C₂×C₃₁×#476	2
C₂×C₃₁×#474	2
C₂×C₃₁×#428	2
C₂×C₃₁×#427	2
C₂×C₃₁×#424	2
C₂×C₃₁×#418	2
C₂²×C₃₁×#102	2
C₂×#3081	31

Subgroups — C2×C31×#3081 #21517244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3081 #21517244