Subgroups — C₂×C₃₁×#3079 #21517241

Order 7936.

2358 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3079	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#477	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#475	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	2	yes	maximal, complemented
9	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#417	C₂	1	2	yes	maximal
11	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×Q₈×C₃₁	—	2	1	no	—
13	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
15	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
16	C₂×C₃₁×C₄⋊Q₈	—	2	4	no	—
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊Q₈	C₂²	1	8	yes	complemented
20	C₄×#41×C₃₁	—	2	4	no	—
21	C₄×#41×C₃₁	C₂²	1	8	yes	—
22	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₃₁	—	2	1	no	—
25	C₂×C₄×D₄×C₃₁	—	2	1	no	—
26	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#117	—	2	1	no	—
29	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
30	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
31	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₂²×C₄²×C₃₁	—	2	1	no	—
33	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#477	C₂²	1	16	yes	complemented
35	C₃₁×#476	C₂²	1	16	yes	complemented
36	C₃₁×#475	C₂²	1	16	yes	complemented
37	C₃₁×#474	C₂²	1	16	yes	complemented
38	C₃₁×#428	C₂²	1	16	yes	complemented
39	C₂×C₃₁×#107	C₂²	1	4	yes	—
40	C₂×C₃₁×#106	C₂²	1	4	yes	—
41	C₃₁×#423	C₂²	1	8	yes	complemented
42	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
43	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
44	C₃₁×#417	C₂²	1	16	yes	—
45	C₂×C₃₁×#103	C₂²	1	4	yes	—
46	C₂×C₃₁×#102	C₂²	1	4	yes	—
47	C₂×#41×C₃₁	—	2	2	no	—
48	C₂×#41×C₃₁	—	2	4	no	—
49	C₂×#41×C₃₁	Q₈	1	8	yes	complemented
50	C₂×#41×C₃₁	C₂³	1	4	yes	—
51	C₂²×Q₈×C₃₁	—	2	2	no	—
52	C₂²×Q₈×C₃₁	—	2	2	no	—
53	C₂²×Q₈×C₃₁	—	2	8	no	—
54	C₂²×Q₈×C₃₁	—	4	8	no	—
55	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×D₄×C₃₁	—	2	1	no	—
57	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₃₁	—	2	1	no	—
59	C₂³×C₄×C₃₁	—	2	2	no	—
60	C₂³×C₄×C₃₁	—	2	1	no	—
61	C₂³×C₄×C₃₁	—	2	1	no	—
62	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
64	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×M₁₆×C₃₁	—	2	8	no	—
66	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
67	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
68	C₃₁×C₄⋊Q₈	—	2	4	no	—
69	C₃₁×C₄⋊Q₈	—	2	4	no	—
70	C₃₁×C₄⋊Q₈	—	2	8	no	—
71	C₃₁×C₄⋊Q₈	—	2	8	no	—
72	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	—
73	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	—
74	C₄×Q₈×C₃₁	—	2	2	no	—
75	C₄×Q₈×C₃₁	—	2	2	no	—
76	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
77	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
78	C₄×D₄×C₃₁	—	2	8	no	—
79	C₄×D₄×C₃₁	—	2	8	no	—
80	C₄×D₄×C₃₁	—	2	2	no	—
81	C₄×D₄×C₃₁	—	2	2	no	—
82	C₄×D₄×C₃₁	C₂³	1	4	yes	—
83	C₄×D₄×C₃₁	C₂³	1	4	yes	—
84	C₃₁×#117	—	2	2	no	—
85	C₃₁×#117	—	2	8	no	—
86	C₃₁×#117	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
89	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
90	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
91	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
93	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
98	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
101	C₂×C₄²×C₃₁	—	2	1	no	—
102	C₂×C₄²×C₃₁	—	2	1	no	—
103	C₂×C₄²×C₃₁	—	2	1	no	—
104	C₂×C₄²×C₃₁	—	2	8	no	—
105	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
106	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
107	C₂×C₄²×C₃₁	D₄	1	4	yes	—
108	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
111	C₃₁×#107	C₂³	1	16	yes	—
112	C₃₁×#106	C₂³	1	16	yes	—
113	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
114	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
115	C₃₁×#103	C₂³	1	16	yes	—
116	C₃₁×#102	C₂³	1	16	yes	—
117	C₂⁴×C₃₁	—	2	1	no	—
118	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
119	#41×C₃₁	—	2	16	no	—
120	#41×C₃₁	—	2	8	no	—
121	#41×C₃₁	C₂×Q₈	1	16	yes	complemented
122	C₂×Q₈×C₃₁	—	2	1	no	—
123	C₂×Q₈×C₃₁	—	2	8	no	—
124	C₂×Q₈×C₃₁	—	2	8	no	—
125	C₂×Q₈×C₃₁	—	2	2	no	—
126	C₂×Q₈×C₃₁	—	4	8	no	—
127	C₂×Q₈×C₃₁	—	4	32	no	—
128	C₂×Q₈×C₃₁	—	4	8	no	—
129	C₂×Q₈×C₃₁	C₂×Q₈	1	4	yes	complemented
130	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
131	C₂×D₄×C₃₁	—	2	8	no	—
132	C₂×D₄×C₃₁	—	2	4	no	—
133	C₂×D₄×C₃₁	—	2	1	no	—
134	C₂×D₄×C₃₁	—	2	2	no	—
135	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	complemented
136	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
137	C₂²×C₄×C₃₁	—	2	1	no	—
138	C₂²×C₄×C₃₁	—	2	8	no	—
139	C₂²×C₄×C₃₁	—	2	4	no	—
140	C₂²×C₄×C₃₁	—	2	4	no	—
141	C₂²×C₄×C₃₁	—	2	4	no	—
142	C₂²×C₄×C₃₁	—	2	8	no	—
143	C₂²×C₄×C₃₁	—	2	1	no	—
144	C₂²×C₄×C₃₁	—	2	1	no	—
145	C₂²×C₄×C₃₁	—	2	4	no	—
146	C₂²×C₄×C₃₁	—	2	4	no	—
147	C₂²×C₄×C₃₁	—	2	4	no	—
148	C₂²×C₄×C₃₁	—	4	8	no	—
149	C₂²×C₄×C₃₁	—	4	4	no	—
150	C₂²×C₄×C₃₁	—	4	4	no	—
151	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
152	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
153	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
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	16	no	—
164	C₂×C₈×C₃₁	—	2	8	no	—
165	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
166	C₄⋊C₄×C₃₁	—	2	8	no	—
167	C₄⋊C₄×C₃₁	—	2	8	no	—
168	C₄⋊C₄×C₃₁	—	2	8	no	—
169	C₄⋊C₄×C₃₁	—	2	4	no	—
170	C₄⋊C₄×C₃₁	—	2	4	no	—
171	C₄⋊C₄×C₃₁	—	2	4	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
175	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
176	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
177	C₂²⋊C₄×C₃₁	—	2	4	no	—
178	C₂²⋊C₄×C₃₁	—	2	4	no	—
179	C₂²⋊C₄×C₃₁	—	2	4	no	—
180	C₄²×C₃₁	—	2	4	no	—
181	C₄²×C₃₁	—	2	4	no	—
182	C₄²×C₃₁	—	2	4	no	—
183	C₄²×C₃₁	C₂×D₄	1	4	yes	—
184	C₄²×C₃₁	C₂×D₄	1	4	yes	—
185	C₄²×C₃₁	C₂×D₄	1	2	yes	—
186	C₄²×C₃₁	C₂×D₄	1	2	yes	—
187	C₄²×C₃₁	C₂⁴	1	2	yes	—
188	C₄²×C₃₁	C₂⁴	1	2	yes	—
189	C₂×#3079	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂³×C₃₁	—	2	2	no	—
191	C₂³×C₃₁	—	2	2	no	—
192	C₂³×C₃₁	—	2	4	no	—
193	C₂³×C₃₁	—	2	2	no	—
194	C₂³×C₃₁	—	4	2	no	—
195	C₂³×C₃₁	—	4	2	no	—
196	C₂³×C₃₁	C₂²⋊Q₈	1	4	yes	—
197	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
198	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
199	Q₈×C₃₁	—	2	4	no	—
200	Q₈×C₃₁	—	2	2	no	—
201	Q₈×C₃₁	—	4	16	no	—
202	Q₈×C₃₁	—	4	16	no	—
203	Q₈×C₃₁	C₂²×Q₈	1	4	yes	complemented
204	D₄×C₃₁	—	2	16	no	—
205	D₄×C₃₁	—	2	4	no	—
206	D₄×C₃₁	—	2	2	no	—
207	D₄×C₃₁	C₂²×Q₈	1	4	yes	complemented
208	C₂×C₄×C₃₁	—	2	4	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	16	no	—
213	C₂×C₄×C₃₁	—	2	8	no	—
214	C₂×C₄×C₃₁	—	2	8	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	8	no	—
218	C₂×C₄×C₃₁	—	4	16	no	—
219	C₂×C₄×C₃₁	—	4	2	no	—
220	C₂×C₄×C₃₁	—	4	8	no	—
221	C₂×C₄×C₃₁	—	4	2	no	—
222	C₂×C₄×C₃₁	—	4	8	no	—
223	C₂×C₄×C₃₁	—	4	8	no	—
224	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
225	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
226	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
227	C₂×C₄×C₃₁	#137	1	4	yes	—
228	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
229	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
230	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
231	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
233	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
234	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
235	C₂×C₄×C₃₁	C₂²×Q₈	1	4	yes	—
236	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
237	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
238	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
239	C₈×C₃₁	—	2	8	no	—
240	C₂²×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
241	C₂×C₄×#41	C₂×C₃₁	1	1	yes	characteristic
242	#3079	C₂×C₃₁	1	16	yes	direct product factor
243	C₂×#477	C₂×C₃₁	1	2	yes	complemented
244	C₂×#475	C₂×C₃₁	1	2	yes	complemented
245	C₂×#476	C₂×C₃₁	1	2	yes	complemented
246	C₂×#474	C₂×C₃₁	1	2	yes	complemented
247	C₂×#428	C₂×C₃₁	1	2	yes	complemented
248	C₂×#423	C₂×C₃₁	1	1	yes	characteristic, complemented
249	C₂×#417	C₂×C₃₁	1	2	yes	—
250	C₂²×C₃₁	—	2	8	no	—
251	C₂²×C₃₁	—	2	2	no	—
252	C₂²×C₃₁	—	2	2	no	—
253	C₂²×C₃₁	—	2	4	no	—
254	C₂²×C₃₁	—	4	8	no	—
255	C₂²×C₃₁	—	4	2	no	—
256	C₂²×C₃₁	—	4	2	no	—
257	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	4	yes	—
258	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
259	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
260	C₂²×C₃₁	C₂×#137	1	2	yes	—
261	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
262	C₄×C₃₁	—	2	4	no	—
263	C₄×C₃₁	—	2	2	no	—
264	C₄×C₃₁	—	2	2	no	—
265	C₄×C₃₁	—	4	2	no	—
266	C₄×C₃₁	—	4	8	no	—
267	C₄×C₃₁	—	4	2	no	—
268	C₄×C₃₁	C₂×C₂²⋊Q₈	1	4	yes	—
269	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
270	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
271	C₄×C₃₁	C₂×#137	1	4	yes	—
272	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
273	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
274	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
275	C₂³×Q₈	—	2	1	no	—
276	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
277	C₂×C₄⋊Q₈	—	2	1	no	—
278	C₂×C₄⋊Q₈	—	2	4	no	—
279	C₂×C₄⋊Q₈	—	2	1	no	—
280	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
281	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
282	C₂×C₄⋊Q₈	C₂²×C₃₁	1	8	yes	complemented
283	C₄×#41	—	2	4	no	—
284	C₄×#41	C₂²×C₃₁	1	8	yes	—
285	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
286	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×C₄×D₄	—	2	1	no	—
288	C₂×C₄×D₄	—	2	1	no	—
289	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×#117	—	2	1	no	—
292	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
293	C₂²×C₄⋊C₄	—	2	1	no	—
294	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
295	C₂²×C₄²	—	2	1	no	—
296	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
297	#477	C₂²×C₃₁	1	16	yes	complemented
298	#476	C₂²×C₃₁	1	16	yes	complemented
299	#475	C₂²×C₃₁	1	16	yes	complemented
300	#474	C₂²×C₃₁	1	16	yes	complemented
301	#428	C₂²×C₃₁	1	16	yes	complemented
302	C₂×#107	C₂²×C₃₁	1	4	yes	—
303	C₂×#106	C₂²×C₃₁	1	4	yes	—
304	#423	C₂²×C₃₁	1	8	yes	complemented
305	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
306	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
307	#417	C₂²×C₃₁	1	16	yes	—
308	C₂×#103	C₂²×C₃₁	1	4	yes	—
309	C₂×#102	C₂²×C₃₁	1	4	yes	—
310	C₂×C₃₁	—	2	4	no	—
311	C₂×C₃₁	—	4	4	no	—
312	C₂×C₃₁	#3079	1	4	yes	direct product factor
313	C₂×C₃₁	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
314	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
315	C₂×C₃₁	C₂²×#137	1	1	yes	characteristic
316	C₂×#41	—	2	2	no	—
317	C₂×#41	—	2	4	no	—
318	C₂×#41	Q₈×C₃₁	1	8	yes	complemented
319	C₂×#41	C₂³×C₃₁	1	4	yes	—
320	C₂²×Q₈	—	2	2	no	—
321	C₂²×Q₈	—	2	2	no	—
322	C₂²×Q₈	—	2	8	no	—
323	C₂²×Q₈	—	4	8	no	—
324	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
325	C₂²×D₄	—	2	1	no	—
326	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
327	C₂³×C₄	—	2	1	no	—
328	C₂³×C₄	—	2	2	no	—
329	C₂³×C₄	—	2	1	no	—
330	C₂³×C₄	—	2	1	no	—
331	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
332	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
333	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
334	C₂×M₁₆	—	2	8	no	—
335	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
336	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
337	C₄⋊Q₈	—	2	4	no	—
338	C₄⋊Q₈	—	2	4	no	—
339	C₄⋊Q₈	—	2	8	no	—
340	C₄⋊Q₈	—	2	8	no	—
341	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	—
342	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	—
343	C₄×Q₈	—	2	2	no	—
344	C₄×Q₈	—	2	2	no	—
345	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
346	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
347	C₄×D₄	—	2	8	no	—
348	C₄×D₄	—	2	8	no	—
349	C₄×D₄	—	2	2	no	—
350	C₄×D₄	—	2	2	no	—
351	C₄×D₄	C₂³×C₃₁	1	4	yes	—
352	C₄×D₄	C₂³×C₃₁	1	4	yes	—
353	#117	—	2	2	no	—
354	#117	—	2	8	no	—
355	#117	C₂³×C₃₁	1	4	yes	—
356	C₂×C₄⋊C₄	—	2	2	no	—
357	C₂×C₄⋊C₄	—	2	2	no	—
358	C₂×C₄⋊C₄	—	2	8	no	—
359	C₂×C₄⋊C₄	—	2	2	no	—
360	C₂×C₄⋊C₄	—	2	4	no	—
361	C₂×C₄⋊C₄	—	2	1	no	—
362	C₂×C₄⋊C₄	—	2	1	no	—
363	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
364	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
365	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
366	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
367	C₂×C₂²⋊C₄	—	2	1	no	—
368	C₂×C₂²⋊C₄	—	2	1	no	—
369	C₂×C₂²⋊C₄	—	2	1	no	—
370	C₂×C₄²	—	2	1	no	—
371	C₂×C₄²	—	2	1	no	—
372	C₂×C₄²	—	2	1	no	—
373	C₂×C₄²	—	2	8	no	—
374	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
375	C₂×C₄²	D₄×C₃₁	1	4	yes	—
376	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
377	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
378	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
379	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
380	#107	C₂³×C₃₁	1	16	yes	—
381	#106	C₂³×C₃₁	1	16	yes	—
382	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
383	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
384	#103	C₂³×C₃₁	1	16	yes	—
385	#102	C₂³×C₃₁	1	16	yes	—
386	C₃₁	C₂×#3079	1	1	yes	characteristic, direct product factor, Sylow
387	C₂⁴	—	2	1	no	—
388	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
389	#41	—	2	16	no	—
390	#41	—	2	8	no	—
391	#41	C₂×Q₈×C₃₁	1	16	yes	complemented
392	C₂×Q₈	—	2	1	no	—
393	C₂×Q₈	—	2	8	no	—
394	C₂×Q₈	—	2	8	no	—
395	C₂×Q₈	—	2	2	no	—
396	C₂×Q₈	—	4	8	no	—
397	C₂×Q₈	—	4	32	no	—
398	C₂×Q₈	—	4	8	no	—
399	C₂×Q₈	C₂×Q₈×C₃₁	1	4	yes	complemented
400	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
401	C₂×D₄	—	2	8	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	2	1	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	complemented
406	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
407	C₂²×C₄	—	2	1	no	—
408	C₂²×C₄	—	2	8	no	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	—	2	4	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	8	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	4	8	no	—
419	C₂²×C₄	—	4	4	no	—
420	C₂²×C₄	—	4	4	no	—
421	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
422	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
423	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
424	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
425	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
426	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
427	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
428	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
429	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
430	C₂²×C₄	#41×C₃₁	1	4	yes	—
431	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
432	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
433	M₁₆	—	2	16	no	—
434	C₂×C₈	—	2	8	no	—
435	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
436	C₄⋊C₄	—	2	8	no	—
437	C₄⋊C₄	—	2	8	no	—
438	C₄⋊C₄	—	2	8	no	—
439	C₄⋊C₄	—	2	4	no	—
440	C₄⋊C₄	—	2	4	no	—
441	C₄⋊C₄	—	2	4	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
445	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
446	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
447	C₂²⋊C₄	—	2	4	no	—
448	C₂²⋊C₄	—	2	4	no	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₄²	—	2	4	no	—
451	C₄²	—	2	4	no	—
452	C₄²	—	2	4	no	—
453	C₄²	C₂×D₄×C₃₁	1	4	yes	—
454	C₄²	C₂×D₄×C₃₁	1	4	yes	—
455	C₄²	C₂×D₄×C₃₁	1	2	yes	—
456	C₄²	C₂×D₄×C₃₁	1	2	yes	—
457	C₄²	C₂⁴×C₃₁	1	2	yes	—
458	C₄²	C₂⁴×C₃₁	1	2	yes	—
459	C₂³	—	2	2	no	—
460	C₂³	—	2	2	no	—
461	C₂³	—	2	4	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	4	2	no	—
464	C₂³	—	4	2	no	—
465	C₂³	C₃₁×C₂²⋊Q₈	1	4	yes	—
466	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
467	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
468	Q₈	—	2	4	no	—
469	Q₈	—	2	2	no	—
470	Q₈	—	4	16	no	—
471	Q₈	—	4	16	no	—
472	Q₈	C₂²×Q₈×C₃₁	1	4	yes	complemented
473	D₄	—	2	16	no	—
474	D₄	—	2	4	no	—
475	D₄	—	2	2	no	—
476	D₄	C₂²×Q₈×C₃₁	1	4	yes	complemented
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	16	no	—
482	C₂×C₄	—	2	8	no	—
483	C₂×C₄	—	2	8	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	8	no	—
487	C₂×C₄	—	4	16	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	8	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	8	no	—
492	C₂×C₄	—	4	8	no	—
493	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
494	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
495	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
496	C₂×C₄	C₃₁×#137	1	4	yes	—
497	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
498	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
499	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
500	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
501	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
502	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
503	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
504	C₂×C₄	C₂²×Q₈×C₃₁	1	4	yes	—
505	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
506	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
507	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
508	C₈	—	2	8	no	—
509	C₂²	—	2	8	no	—
510	C₂²	—	2	2	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	2	4	no	—
513	C₂²	—	4	8	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	2	no	—
516	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	4	yes	—
517	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
518	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
519	C₂²	C₂×C₃₁×#137	1	2	yes	—
520	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
521	C₄	—	2	4	no	—
522	C₄	—	2	2	no	—
523	C₄	—	2	2	no	—
524	C₄	—	4	8	no	—
525	C₄	—	4	2	no	—
526	C₄	—	4	2	no	—
527	C₄	C₂×C₃₁×C₂²⋊Q₈	1	4	yes	—
528	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
529	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
530	C₄	C₂×C₃₁×#137	1	4	yes	—
531	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
532	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
533	C₂	—	2	4	no	—
534	C₂	—	4	4	no	—
535	C₂	C₃₁×#3079	1	4	yes	direct product factor
536	C₂	C₂²×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
537	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
538	C₂	C₂²×C₃₁×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂²×#41×C₃₁	C₂²
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. C₂²×C₃₁×C₄⋊Q₈	C₂	semidirect
3. C₃₁×#3079	C₂	direct (N × H)
4. C₂×C₃₁×#477	C₂	semidirect
5. C₂×C₃₁×#475	C₂	semidirect
6. C₂×C₃₁×#476	C₂	semidirect
7. C₂×C₃₁×#474	C₂	semidirect
8. C₂×C₃₁×#428	C₂	semidirect
9. C₂×C₃₁×#423	C₂	semidirect
19. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
34. C₃₁×#477	C₂²	semidirect
35. C₃₁×#476	C₂²	semidirect
36. C₃₁×#475	C₂²	semidirect
37. C₃₁×#474	C₂²	semidirect
38. C₃₁×#428	C₂²	semidirect
41. C₃₁×#423	C₂²	semidirect
49. C₂×#41×C₃₁	Q₈	semidirect
121. #41×C₃₁	C₂×Q₈	semidirect
129. C₂×Q₈×C₃₁	C₂×Q₈	semidirect
135. C₂×D₄×C₃₁	C₂×Q₈	semidirect
189. C₂×#3079	C₃₁	direct (N × H)
203. Q₈×C₃₁	C₂²×Q₈	semidirect
207. D₄×C₃₁	C₂²×Q₈	semidirect
240. C₂²×C₄⋊Q₈	C₂×C₃₁	semidirect
242. #3079	C₂×C₃₁	direct (N × H)
243. C₂×#477	C₂×C₃₁	semidirect
244. C₂×#475	C₂×C₃₁	semidirect
245. C₂×#476	C₂×C₃₁	semidirect
246. C₂×#474	C₂×C₃₁	semidirect
247. C₂×#428	C₂×C₃₁	semidirect
248. C₂×#423	C₂×C₃₁	semidirect
282. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
297. #477	C₂²×C₃₁	semidirect
298. #476	C₂²×C₃₁	semidirect
299. #475	C₂²×C₃₁	semidirect
300. #474	C₂²×C₃₁	semidirect
301. #428	C₂²×C₃₁	semidirect
304. #423	C₂²×C₃₁	semidirect
312. C₂×C₃₁	#3079	direct (N × H)
318. C₂×#41	Q₈×C₃₁	semidirect
386. C₃₁	C₂×#3079	direct (N × H)
391. #41	C₂×Q₈×C₃₁	semidirect
399. C₂×Q₈	C₂×Q₈×C₃₁	semidirect
405. C₂×D₄	C₂×Q₈×C₃₁	semidirect
472. Q₈	C₂²×Q₈×C₃₁	semidirect
476. D₄	C₂²×Q₈×C₃₁	semidirect
535. C₂	C₃₁×#3079	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×C₄⋊Q₈	2
C₂×C₄×#41×C₃₁	2
C₃₁×#3079	2
C₂×C₃₁×#477	2
C₂×C₃₁×#475	2
C₂×C₃₁×#476	2
C₂×C₃₁×#474	2
C₂×C₃₁×#428	2
C₂×C₃₁×#423	2
C₂×C₃₁×#417	2
C₂×#3079	31

Subgroups — C2×C31×#3079 #21517241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3079 #21517241