Subgroups — Q₈×AGL(1,ℤ/8)×C₆₁ #29564799

Order 15616.

2010 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3386	C₂	1	6	yes	maximal, complemented
4	C₆₁×#3384	C₂	1	6	yes	maximal, complemented
5	Q₈×SD₁₆×C₆₁	C₂	1	2	yes	maximal, complemented
6	Q₈×D₈×C₆₁	C₂	1	2	yes	maximal, complemented
7	C₆₁×#3165	C₂	1	3	yes	maximal, complemented
8	C₆₁×#3079	C₂	1	3	yes	maximal, complemented
9	C₆₁×#3077	C₂	1	3	yes	maximal, complemented
10	Q₈×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×AGL(1,ℤ/8)×C₆₁	C₂	1	3	yes	maximal
12	C₂³×Q₈×C₆₁	—	2	1	no	—
13	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	1	yes	characteristic
14	Q₈²×C₆₁	C₂²	1	1	yes	characteristic, complemented
15	C₆₁×#557	C₂²	1	3	yes	complemented
16	C₆₁×#554	—	2	3	no	—
17	C₆₁×#554	C₂²	1	3	yes	complemented
18	D₄×Q₈×C₆₁	—	2	1	no	—
19	D₄×Q₈×C₆₁	—	2	3	no	—
20	D₄×Q₈×C₆₁	—	2	1	no	—
21	D₄×Q₈×C₆₁	C₂²	1	2	yes	—
22	D₄×Q₈×C₆₁	C₂²	1	6	yes	—
23	D₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
24	C₆₁×#533	—	2	3	no	—
25	C₆₁×#533	C₂²	1	3	yes	—
26	C₂×C₄⋊Q₈×C₆₁	C₂²	1	3	yes	—
27	C₂×C₂²⋊Q₈×C₆₁	—	2	3	no	—
28	C₄×#41×C₆₁	C₂²	1	3	yes	—
29	C₂×C₄×Q₈×C₆₁	—	2	1	no	—
30	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₆₁	C₂²	1	3	yes	—
32	C₆₁×#501	C₂²	1	12	yes	complemented
33	C₆₁×#500	C₂²	1	6	yes	complemented
34	C₆₁×#498	C₂²	1	6	yes	complemented
35	C₆₁×#478	C₂²	1	12	yes	—
36	C₆₁×#477	C₂²	1	6	yes	complemented
37	C₆₁×#476	C₂²	1	6	yes	—
38	C₆₁×#476	C₂²	1	6	yes	complemented
39	C₆₁×#475	C₂²	1	6	yes	complemented
40	C₆₁×#474	C₂²	1	6	yes	—
41	C₆₁×#474	C₂²	1	6	yes	complemented
42	C₆₁×#446	C₂²	1	6	yes	complemented
43	C₈×Q₈×C₆₁	C₂²	1	2	yes	complemented
44	C₆₁×#442	C₂²	1	6	yes	—
45	C₆₁×#440	C₂²	1	6	yes	—
46	C₄×SD₁₆×C₆₁	C₂²	1	6	yes	—
47	C₄×D₈×C₆₁	C₂²	1	6	yes	—
48	C₆₁×#428	C₂²	1	6	yes	—
49	C₆₁×#428	C₂²	1	3	yes	—
50	C₆₁×#423	C₂²	1	3	yes	—
51	C₆₁×#418	C₂²	1	3	yes	—
52	C₆₁×#417	C₂²	1	3	yes	—
53	C₄×M₁₆×C₆₁	C₂²	1	3	yes	—
54	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
55	C₂²×Q₈×C₆₁	—	2	6	no	—
56	C₂²×Q₈×C₆₁	—	2	2	no	—
57	C₂²×Q₈×C₆₁	—	2	1	no	—
58	C₂²×Q₈×C₆₁	—	2	3	no	—
59	C₂²×Q₈×C₆₁	—	4	8	no	—
60	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
61	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₆₁	—	2	3	no	—
63	AGL(1,ℤ/8)×C₆₁	Q₈	1	8	yes	direct product factor
64	C₂×SD₁₆×C₆₁	C₂³	1	2	yes	—
65	C₂×D₈×C₆₁	C₂³	1	2	yes	—
66	C₂×M₁₆×C₆₁	C₂³	1	3	yes	—
67	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₆₁	—	2	3	no	—
69	C₄⋊Q₈×C₆₁	—	2	3	no	—
70	C₄⋊Q₈×C₆₁	—	2	3	no	—
71	C₄⋊Q₈×C₆₁	—	2	3	no	—
72	C₄⋊Q₈×C₆₁	C₂³	1	6	yes	—
73	C₄⋊Q₈×C₆₁	C₂³	1	6	yes	—
74	C₄⋊Q₈×C₆₁	C₂³	1	3	yes	—
75	C₄⋊Q₈×C₆₁	C₂³	1	3	yes	—
76	#125×C₆₁	—	2	6	no	—
77	#125×C₆₁	C₂³	1	6	yes	—
78	C₂²⋊Q₈×C₆₁	—	2	3	no	—
79	C₂²⋊Q₈×C₆₁	—	2	3	no	—
80	C₂²⋊Q₈×C₆₁	—	2	3	no	—
81	C₂²⋊Q₈×C₆₁	—	2	3	no	—
82	C₂²⋊Q₈×C₆₁	—	2	6	no	—
83	C₂²⋊Q₈×C₆₁	—	2	6	no	—
84	C₂²⋊Q₈×C₆₁	—	2	12	no	—
85	C₂²⋊Q₈×C₆₁	—	2	6	no	—
86	C₄×Q₈×C₆₁	—	2	1	no	—
87	C₄×Q₈×C₆₁	—	2	3	no	—
88	C₄×Q₈×C₆₁	—	2	3	no	—
89	C₄×Q₈×C₆₁	—	2	3	no	—
90	C₄×Q₈×C₆₁	D₄	1	3	yes	—
91	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
92	C₄×Q₈×C₆₁	C₂³	1	3	yes	—
93	C₄×Q₈×C₆₁	C₂³	1	3	yes	—
94	C₄×Q₈×C₆₁	C₂³	1	1	yes	characteristic
95	C₄×Q₈×C₆₁	C₂³	1	3	yes	—
96	C₄×D₄×C₆₁	—	2	3	no	—
97	C₄×D₄×C₆₁	—	2	3	no	—
98	C₄×D₄×C₆₁	—	2	3	no	—
99	C₄×D₄×C₆₁	—	2	3	no	—
100	C₄×D₄×C₆₁	C₂³	1	6	yes	—
101	C₄×D₄×C₆₁	C₂³	1	6	yes	—
102	C₄×D₄×C₆₁	C₂³	1	3	yes	—
103	C₄×D₄×C₆₁	C₂³	1	3	yes	—
104	#117×C₆₁	—	2	3	no	—
105	#117×C₆₁	C₂³	1	3	yes	—
106	C₂×C₄⋊C₄×C₆₁	—	2	3	no	—
107	C₂×C₄⋊C₄×C₆₁	—	2	3	no	—
108	C₂×C₄⋊C₄×C₆₁	D₄	1	3	yes	—
109	C₂×C₄⋊C₄×C₆₁	C₂³	1	3	yes	—
110	C₂×C₂²⋊C₄×C₆₁	—	2	3	no	—
111	C₂×C₄²×C₆₁	—	2	3	no	—
112	C₂×C₄²×C₆₁	C₂³	1	3	yes	—
113	#107×C₆₁	C₂³	1	6	yes	—
114	#107×C₆₁	C₂³	1	12	yes	—
115	#106×C₆₁	C₂³	1	6	yes	—
116	#106×C₆₁	C₂³	1	12	yes	—
117	C₄⋊C₈×C₆₁	C₂³	1	6	yes	—
118	C₄⋊C₈×C₆₁	C₂³	1	6	yes	—
119	#103×C₆₁	C₂³	1	6	yes	—
120	#102×C₆₁	C₂³	1	6	yes	—
121	#102×C₆₁	C₂³	1	6	yes	—
122	#102×C₆₁	C₂³	1	6	yes	—
123	#97×C₆₁	C₂³	1	6	yes	—
124	C₄×C₈×C₆₁	C₂³	1	6	yes	—
125	C₂⁴×C₆₁	—	2	1	no	—
126	#41×C₆₁	—	2	2	no	—
127	#41×C₆₁	C₂×Q₈	1	4	yes	complemented
128	C₂×Q₈×C₆₁	—	2	4	no	—
129	C₂×Q₈×C₆₁	—	2	3	no	—
130	C₂×Q₈×C₆₁	—	2	6	no	—
131	C₂×Q₈×C₆₁	—	2	6	no	—
132	C₂×Q₈×C₆₁	—	2	3	no	—
133	C₂×Q₈×C₆₁	—	4	12	no	—
134	C₂×Q₈×C₆₁	—	4	24	no	—
135	C₂×Q₈×C₆₁	—	4	4	no	—
136	C₂×Q₈×C₆₁	—	4	8	no	—
137	C₂×Q₈×C₆₁	C₂×D₄	1	3	yes	—
138	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
139	C₂×Q₈×C₆₁	C₂⁴	1	1	yes	characteristic
140	C₂×D₄×C₆₁	—	2	1	no	—
141	C₂×D₄×C₆₁	—	2	1	no	—
142	C₂×D₄×C₆₁	—	2	2	no	—
143	C₂×D₄×C₆₁	C₂×Q₈	1	4	yes	complemented
144	C₂×D₄×C₆₁	C₂⁴	1	2	yes	—
145	C₂×D₄×C₆₁	C₂⁴	1	1	yes	characteristic
146	C₂²×C₄×C₆₁	—	2	1	no	—
147	C₂²×C₄×C₆₁	—	2	3	no	—
148	C₂²×C₄×C₆₁	—	2	3	no	—
149	C₂²×C₄×C₆₁	—	2	6	no	—
150	C₂²×C₄×C₆₁	—	2	6	no	—
151	C₂²×C₄×C₆₁	—	2	3	no	—
152	C₂²×C₄×C₆₁	—	4	12	no	—
153	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
154	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
155	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
156	SD₁₆×C₆₁	C₂×Q₈	1	8	yes	complemented
157	D₈×C₆₁	C₂×Q₈	1	8	yes	complemented
158	M₁₆×C₆₁	—	2	6	no	—
159	M₁₆×C₆₁	C₂×Q₈	1	4	yes	complemented
160	C₂×C₈×C₆₁	C₂⁴	1	6	yes	—
161	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
162	C₄⋊C₄×C₆₁	—	2	6	no	—
163	C₄⋊C₄×C₆₁	—	2	3	no	—
164	C₄⋊C₄×C₆₁	—	2	6	no	—
165	C₄⋊C₄×C₆₁	—	2	6	no	—
166	C₄⋊C₄×C₆₁	—	2	3	no	—
167	C₄⋊C₄×C₆₁	—	2	3	no	—
168	C₄⋊C₄×C₆₁	—	2	3	no	—
169	C₄⋊C₄×C₆₁	—	2	3	no	—
170	C₄⋊C₄×C₆₁	—	2	6	no	—
171	C₄⋊C₄×C₆₁	C₂×D₄	1	3	yes	—
172	C₄⋊C₄×C₆₁	C₂×D₄	1	6	yes	—
173	C₄⋊C₄×C₆₁	C₂×D₄	1	3	yes	—
174	C₄⋊C₄×C₆₁	C₂⁴	1	3	yes	—
175	C₄⋊C₄×C₆₁	C₂⁴	1	3	yes	—
176	C₄⋊C₄×C₆₁	C₂⁴	1	6	yes	—
177	C₂²⋊C₄×C₆₁	—	2	3	no	—
178	C₂²⋊C₄×C₆₁	—	2	6	no	—
179	C₂²⋊C₄×C₆₁	—	2	6	no	—
180	C₂²⋊C₄×C₆₁	—	2	3	no	—
181	C₂²⋊C₄×C₆₁	—	2	3	no	—
182	C₄²×C₆₁	—	2	3	no	—
183	C₄²×C₆₁	—	2	3	no	—
184	C₄²×C₆₁	—	2	3	no	—
185	C₄²×C₆₁	C₂×D₄	1	3	yes	—
186	C₄²×C₆₁	C₂×D₄	1	3	yes	—
187	C₄²×C₆₁	C₂⁴	1	3	yes	—
188	C₄²×C₆₁	C₂⁴	1	3	yes	—
189	C₂³×C₆₁	—	2	4	no	—
190	C₂³×C₆₁	—	2	1	no	—
191	C₂³×C₆₁	—	2	2	no	—
192	C₂³×C₆₁	—	4	2	no	—
193	C₂³×C₆₁	—	4	2	no	—
194	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
195	Q₈×C₆₁	—	2	6	no	—
196	Q₈×C₆₁	—	2	1	no	—
197	Q₈×C₆₁	—	4	12	no	—
198	Q₈×C₆₁	—	4	6	no	—
199	Q₈×C₆₁	—	4	12	no	—
200	Q₈×C₆₁	—	8	3	no	—
201	Q₈×C₆₁	AGL(1,ℤ/8)	1	4	yes	direct product factor
202	Q₈×C₆₁	C₂²×Q₈	1	2	yes	complemented
203	D₄×C₆₁	—	2	4	no	—
204	D₄×C₆₁	—	2	4	no	—
205	D₄×C₆₁	—	2	2	no	—
206	D₄×C₆₁	—	2	1	no	—
207	D₄×C₆₁	C₂²×Q₈	1	4	yes	—
208	D₄×C₆₁	C₂²×Q₈	1	2	yes	complemented
209	C₂×C₄×C₆₁	—	2	1	no	—
210	C₂×C₄×C₆₁	—	2	4	no	—
211	C₂×C₄×C₆₁	—	2	6	no	—
212	C₂×C₄×C₆₁	—	2	3	no	—
213	C₂×C₄×C₆₁	—	2	3	no	—
214	C₂×C₄×C₆₁	—	2	6	no	—
215	C₂×C₄×C₆₁	—	4	3	no	—
216	C₂×C₄×C₆₁	—	4	12	no	—
217	C₂×C₄×C₆₁	—	4	6	no	—
218	C₂×C₄×C₆₁	—	4	12	no	—
219	C₂×C₄×C₆₁	—	4	6	no	—
220	C₂×C₄×C₆₁	—	8	3	no	—
221	C₂×C₄×C₆₁	C₂²×D₄	1	3	yes	—
222	C₂×C₄×C₆₁	C₂²×D₄	1	3	yes	—
223	C₂×C₄×C₆₁	C₂²×D₄	1	3	yes	—
224	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₆₁	C₂²×D₄	1	3	yes	—
226	C₂×C₄×C₆₁	C₂²×Q₈	1	2	yes	—
227	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
228	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
229	C₈×C₆₁	—	2	6	no	—
230	C₈×C₆₁	C₂²×Q₈	1	4	yes	complemented
231	Q₈×AGL(1,ℤ/8)	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂²×C₆₁	—	2	1	no	—
233	C₂²×C₆₁	—	2	1	no	—
234	C₂²×C₆₁	—	2	2	no	—
235	C₂²×C₆₁	—	2	4	no	—
236	C₂²×C₆₁	—	4	8	no	—
237	C₂²×C₆₁	—	4	1	no	—
238	C₂²×C₆₁	—	4	2	no	—
239	C₂²×C₆₁	—	4	2	no	—
240	C₂²×C₆₁	—	4	1	no	—
241	C₂²×C₆₁	D₄×Q₈	1	2	yes	—
242	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
243	C₄×C₆₁	—	2	3	no	—
244	C₄×C₆₁	—	2	3	no	—
245	C₄×C₆₁	—	2	2	no	—
246	C₄×C₆₁	—	2	3	no	—
247	C₄×C₆₁	—	4	3	no	—
248	C₄×C₆₁	—	4	6	no	—
249	C₄×C₆₁	—	4	3	no	—
250	C₄×C₆₁	D₄×Q₈	1	2	yes	—
251	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	6	yes	—
252	C₄×C₆₁	C₂³×Q₈	1	1	yes	characteristic
253	C₄×C₆₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
254	Q₈×#41	C₂×C₆₁	1	1	yes	characteristic, complemented
255	C₂×D₄×Q₈	C₂×C₆₁	1	1	yes	characteristic, complemented
256	#3386	C₂×C₆₁	1	6	yes	complemented
257	#3384	C₂×C₆₁	1	6	yes	complemented
258	Q₈×SD₁₆	C₂×C₆₁	1	2	yes	complemented
259	Q₈×D₈	C₂×C₆₁	1	2	yes	complemented
260	#3165	C₂×C₆₁	1	3	yes	complemented
261	#3079	C₂×C₆₁	1	3	yes	complemented
262	#3077	C₂×C₆₁	1	3	yes	complemented
263	Q₈×M₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
264	C₄×AGL(1,ℤ/8)	C₂×C₆₁	1	3	yes	—
265	C₂×C₆₁	—	2	2	no	—
266	C₂×C₆₁	—	4	2	no	—
267	C₂×C₆₁	—	4	4	no	—
268	C₂×C₆₁	C₂×D₄×Q₈	1	1	yes	characteristic
269	C₂×C₆₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
270	C₂×C₆₁	#3586	1	1	yes	characteristic
271	C₂³×Q₈	—	2	1	no	—
272	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	1	yes	characteristic
273	Q₈²	C₂²×C₆₁	1	1	yes	characteristic, complemented
274	#557	C₂²×C₆₁	1	3	yes	complemented
275	#554	—	2	3	no	—
276	#554	C₂²×C₆₁	1	3	yes	complemented
277	D₄×Q₈	—	2	1	no	—
278	D₄×Q₈	—	2	1	no	—
279	D₄×Q₈	—	2	3	no	—
280	D₄×Q₈	C₂²×C₆₁	1	2	yes	—
281	D₄×Q₈	C₂²×C₆₁	1	6	yes	—
282	D₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic, complemented
283	#533	—	2	3	no	—
284	#533	C₂²×C₆₁	1	3	yes	—
285	C₂×C₄⋊Q₈	C₂²×C₆₁	1	3	yes	—
286	C₂×C₂²⋊Q₈	—	2	3	no	—
287	C₄×#41	C₂²×C₆₁	1	3	yes	—
288	C₂×C₄×Q₈	—	2	1	no	—
289	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
290	C₂×C₄×D₄	C₂²×C₆₁	1	3	yes	—
291	#501	C₂²×C₆₁	1	12	yes	complemented
292	#500	C₂²×C₆₁	1	6	yes	complemented
293	#498	C₂²×C₆₁	1	6	yes	complemented
294	#478	C₂²×C₆₁	1	12	yes	—
295	#477	C₂²×C₆₁	1	6	yes	complemented
296	#476	C₂²×C₆₁	1	6	yes	—
297	#476	C₂²×C₆₁	1	6	yes	complemented
298	#475	C₂²×C₆₁	1	6	yes	complemented
299	#474	C₂²×C₆₁	1	6	yes	—
300	#474	C₂²×C₆₁	1	6	yes	complemented
301	#446	C₂²×C₆₁	1	6	yes	complemented
302	C₈×Q₈	C₂²×C₆₁	1	2	yes	complemented
303	#442	C₂²×C₆₁	1	6	yes	—
304	#440	C₂²×C₆₁	1	6	yes	—
305	C₄×SD₁₆	C₂²×C₆₁	1	6	yes	—
306	C₄×D₈	C₂²×C₆₁	1	6	yes	—
307	#428	C₂²×C₆₁	1	6	yes	—
308	#428	C₂²×C₆₁	1	3	yes	—
309	#423	C₂²×C₆₁	1	3	yes	—
310	#418	C₂²×C₆₁	1	3	yes	—
311	#417	C₂²×C₆₁	1	3	yes	—
312	C₄×M₁₆	C₂²×C₆₁	1	3	yes	—
313	C₆₁	Q₈×AGL(1,ℤ/8)	1	1	yes	characteristic, direct product factor, Sylow
314	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
315	C₂²×Q₈	—	2	1	no	—
316	C₂²×Q₈	—	2	6	no	—
317	C₂²×Q₈	—	2	2	no	—
318	C₂²×Q₈	—	2	3	no	—
319	C₂²×Q₈	—	4	8	no	—
320	C₂²×Q₈	D₄×C₆₁	1	1	yes	characteristic
321	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
322	C₂³×C₄	—	2	3	no	—
323	AGL(1,ℤ/8)	Q₈×C₆₁	1	8	yes	direct product factor
324	C₂×SD₁₆	C₂³×C₆₁	1	2	yes	—
325	C₂×D₈	C₂³×C₆₁	1	2	yes	—
326	C₂×M₁₆	C₂³×C₆₁	1	3	yes	—
327	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
328	C₄⋊Q₈	—	2	3	no	—
329	C₄⋊Q₈	—	2	3	no	—
330	C₄⋊Q₈	—	2	3	no	—
331	C₄⋊Q₈	—	2	3	no	—
332	C₄⋊Q₈	C₂³×C₆₁	1	6	yes	—
333	C₄⋊Q₈	C₂³×C₆₁	1	6	yes	—
334	C₄⋊Q₈	C₂³×C₆₁	1	3	yes	—
335	C₄⋊Q₈	C₂³×C₆₁	1	3	yes	—
336	#125	—	2	6	no	—
337	#125	C₂³×C₆₁	1	6	yes	—
338	C₂²⋊Q₈	—	2	3	no	—
339	C₂²⋊Q₈	—	2	3	no	—
340	C₂²⋊Q₈	—	2	3	no	—
341	C₂²⋊Q₈	—	2	6	no	—
342	C₂²⋊Q₈	—	2	12	no	—
343	C₂²⋊Q₈	—	2	6	no	—
344	C₂²⋊Q₈	—	2	3	no	—
345	C₂²⋊Q₈	—	2	6	no	—
346	C₄×Q₈	—	2	1	no	—
347	C₄×Q₈	—	2	3	no	—
348	C₄×Q₈	—	2	3	no	—
349	C₄×Q₈	—	2	3	no	—
350	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
351	C₄×Q₈	D₄×C₆₁	1	3	yes	—
352	C₄×Q₈	C₂³×C₆₁	1	3	yes	—
353	C₄×Q₈	C₂³×C₆₁	1	3	yes	—
354	C₄×Q₈	C₂³×C₆₁	1	1	yes	characteristic
355	C₄×Q₈	C₂³×C₆₁	1	3	yes	—
356	C₄×D₄	—	2	3	no	—
357	C₄×D₄	—	2	3	no	—
358	C₄×D₄	—	2	3	no	—
359	C₄×D₄	—	2	3	no	—
360	C₄×D₄	C₂³×C₆₁	1	6	yes	—
361	C₄×D₄	C₂³×C₆₁	1	6	yes	—
362	C₄×D₄	C₂³×C₆₁	1	3	yes	—
363	C₄×D₄	C₂³×C₆₁	1	3	yes	—
364	#117	—	2	3	no	—
365	#117	C₂³×C₆₁	1	3	yes	—
366	C₂×C₄⋊C₄	—	2	3	no	—
367	C₂×C₄⋊C₄	—	2	3	no	—
368	C₂×C₄⋊C₄	D₄×C₆₁	1	3	yes	—
369	C₂×C₄⋊C₄	C₂³×C₆₁	1	3	yes	—
370	C₂×C₂²⋊C₄	—	2	3	no	—
371	C₂×C₄²	—	2	3	no	—
372	C₂×C₄²	C₂³×C₆₁	1	3	yes	—
373	#107	C₂³×C₆₁	1	6	yes	—
374	#107	C₂³×C₆₁	1	12	yes	—
375	#106	C₂³×C₆₁	1	6	yes	—
376	#106	C₂³×C₆₁	1	12	yes	—
377	C₄⋊C₈	C₂³×C₆₁	1	6	yes	—
378	C₄⋊C₈	C₂³×C₆₁	1	6	yes	—
379	#103	C₂³×C₆₁	1	6	yes	—
380	#102	C₂³×C₆₁	1	6	yes	—
381	#102	C₂³×C₆₁	1	6	yes	—
382	#102	C₂³×C₆₁	1	6	yes	—
383	#97	C₂³×C₆₁	1	6	yes	—
384	C₄×C₈	C₂³×C₆₁	1	6	yes	—
385	C₂⁴	—	2	1	no	—
386	#41	—	2	2	no	—
387	#41	C₂×Q₈×C₆₁	1	4	yes	complemented
388	C₂×Q₈	—	2	4	no	—
389	C₂×Q₈	—	2	3	no	—
390	C₂×Q₈	—	2	6	no	—
391	C₂×Q₈	—	2	6	no	—
392	C₂×Q₈	—	2	3	no	—
393	C₂×Q₈	—	4	12	no	—
394	C₂×Q₈	—	4	24	no	—
395	C₂×Q₈	—	4	8	no	—
396	C₂×Q₈	—	4	4	no	—
397	C₂×Q₈	C₂×D₄×C₆₁	1	3	yes	—
398	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
399	C₂×Q₈	C₂⁴×C₆₁	1	1	yes	characteristic
400	C₂×D₄	—	2	1	no	—
401	C₂×D₄	—	2	1	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	C₂×Q₈×C₆₁	1	4	yes	complemented
404	C₂×D₄	C₂⁴×C₆₁	1	2	yes	—
405	C₂×D₄	C₂⁴×C₆₁	1	1	yes	characteristic
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	3	no	—
408	C₂²×C₄	—	2	3	no	—
409	C₂²×C₄	—	2	6	no	—
410	C₂²×C₄	—	2	6	no	—
411	C₂²×C₄	—	2	3	no	—
412	C₂²×C₄	—	4	12	no	—
413	C₂²×C₄	C₂×D₄×C₆₁	1	3	yes	—
414	C₂²×C₄	C₂×D₄×C₆₁	1	3	yes	—
415	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
416	SD₁₆	C₂×Q₈×C₆₁	1	8	yes	complemented
417	D₈	C₂×Q₈×C₆₁	1	8	yes	complemented
418	M₁₆	—	2	6	no	—
419	M₁₆	C₂×Q₈×C₆₁	1	4	yes	complemented
420	C₂×C₈	C₂⁴×C₆₁	1	6	yes	—
421	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
422	C₄⋊C₄	—	2	6	no	—
423	C₄⋊C₄	—	2	3	no	—
424	C₄⋊C₄	—	2	6	no	—
425	C₄⋊C₄	—	2	6	no	—
426	C₄⋊C₄	—	2	3	no	—
427	C₄⋊C₄	—	2	3	no	—
428	C₄⋊C₄	—	2	3	no	—
429	C₄⋊C₄	—	2	3	no	—
430	C₄⋊C₄	—	2	6	no	—
431	C₄⋊C₄	C₂×D₄×C₆₁	1	3	yes	—
432	C₄⋊C₄	C₂×D₄×C₆₁	1	6	yes	—
433	C₄⋊C₄	C₂×D₄×C₆₁	1	3	yes	—
434	C₄⋊C₄	C₂⁴×C₆₁	1	3	yes	—
435	C₄⋊C₄	C₂⁴×C₆₁	1	3	yes	—
436	C₄⋊C₄	C₂⁴×C₆₁	1	6	yes	—
437	C₂²⋊C₄	—	2	3	no	—
438	C₂²⋊C₄	—	2	6	no	—
439	C₂²⋊C₄	—	2	6	no	—
440	C₂²⋊C₄	—	2	3	no	—
441	C₂²⋊C₄	—	2	3	no	—
442	C₄²	—	2	3	no	—
443	C₄²	—	2	3	no	—
444	C₄²	—	2	3	no	—
445	C₄²	C₂×D₄×C₆₁	1	3	yes	—
446	C₄²	C₂×D₄×C₆₁	1	3	yes	—
447	C₄²	C₂⁴×C₆₁	1	3	yes	—
448	C₄²	C₂⁴×C₆₁	1	3	yes	—
449	C₂³	—	2	4	no	—
450	C₂³	—	2	1	no	—
451	C₂³	—	2	2	no	—
452	C₂³	—	4	2	no	—
453	C₂³	—	4	2	no	—
454	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
455	Q₈	—	2	6	no	—
456	Q₈	—	2	1	no	—
457	Q₈	—	4	12	no	—
458	Q₈	—	4	6	no	—
459	Q₈	—	4	12	no	—
460	Q₈	—	8	3	no	—
461	Q₈	AGL(1,ℤ/8)×C₆₁	1	4	yes	direct product factor
462	Q₈	C₂²×Q₈×C₆₁	1	2	yes	complemented
463	D₄	—	2	4	no	—
464	D₄	—	2	4	no	—
465	D₄	—	2	2	no	—
466	D₄	—	2	1	no	—
467	D₄	C₂²×Q₈×C₆₁	1	4	yes	—
468	D₄	C₂²×Q₈×C₆₁	1	2	yes	complemented
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	6	no	—
472	C₂×C₄	—	2	3	no	—
473	C₂×C₄	—	2	3	no	—
474	C₂×C₄	—	2	6	no	—
475	C₂×C₄	—	4	3	no	—
476	C₂×C₄	—	4	12	no	—
477	C₂×C₄	—	4	6	no	—
478	C₂×C₄	—	4	12	no	—
479	C₂×C₄	—	4	6	no	—
480	C₂×C₄	—	8	3	no	—
481	C₂×C₄	C₂²×D₄×C₆₁	1	3	yes	—
482	C₂×C₄	C₂²×D₄×C₆₁	1	3	yes	—
483	C₂×C₄	C₂²×D₄×C₆₁	1	3	yes	—
484	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₆₁	1	3	yes	—
486	C₂×C₄	C₂²×Q₈×C₆₁	1	2	yes	—
487	C₂×C₄	2₋¹⁺⁴×C₆₁	1	2	yes	—
488	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
489	C₈	—	2	6	no	—
490	C₈	C₂²×Q₈×C₆₁	1	4	yes	complemented
491	C₂²	—	2	1	no	—
492	C₂²	—	2	1	no	—
493	C₂²	—	2	2	no	—
494	C₂²	—	2	4	no	—
495	C₂²	—	4	8	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	4	1	no	—
498	C₂²	—	4	1	no	—
499	C₂²	—	4	2	no	—
500	C₂²	D₄×Q₈×C₆₁	1	2	yes	—
501	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
502	C₄	—	2	3	no	—
503	C₄	—	2	3	no	—
504	C₄	—	2	3	no	—
505	C₄	—	2	2	no	—
506	C₄	—	4	6	no	—
507	C₄	—	4	3	no	—
508	C₄	—	4	3	no	—
509	C₄	D₄×Q₈×C₆₁	1	2	yes	—
510	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	6	yes	—
511	C₄	C₂³×Q₈×C₆₁	1	1	yes	characteristic
512	C₄	C₂×2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
513	C₂	—	2	2	no	—
514	C₂	—	4	4	no	—
515	C₂	—	4	2	no	—
516	C₂	C₂×D₄×Q₈×C₆₁	1	1	yes	characteristic
517	C₂	C₂²×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
518	C₂	C₆₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
prime radical	C₂×AGL(1,ℤ/8)×C₆₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄×Q₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×#41×C₆₁	C₂	semidirect
2. C₂×D₄×Q₈×C₆₁	C₂	semidirect
3. C₆₁×#3386	C₂	semidirect
4. C₆₁×#3384	C₂	semidirect
5. Q₈×SD₁₆×C₆₁	C₂	semidirect
6. Q₈×D₈×C₆₁	C₂	semidirect
7. C₆₁×#3165	C₂	semidirect
8. C₆₁×#3079	C₂	semidirect
9. C₆₁×#3077	C₂	semidirect
10. Q₈×M₁₆×C₆₁	C₂	semidirect
14. Q₈²×C₆₁	C₂²	semidirect
15. C₆₁×#557	C₂²	semidirect
17. C₆₁×#554	C₂²	semidirect
23. D₄×Q₈×C₆₁	C₂²	semidirect
32. C₆₁×#501	C₂²	semidirect
33. C₆₁×#500	C₂²	semidirect
34. C₆₁×#498	C₂²	semidirect
36. C₆₁×#477	C₂²	semidirect
38. C₆₁×#476	C₂²	semidirect
39. C₆₁×#475	C₂²	semidirect
41. C₆₁×#474	C₂²	semidirect
42. C₆₁×#446	C₂²	semidirect
43. C₈×Q₈×C₆₁	C₂²	semidirect
63. AGL(1,ℤ/8)×C₆₁	Q₈	direct (N × H)
127. #41×C₆₁	C₂×Q₈	semidirect
143. C₂×D₄×C₆₁	C₂×Q₈	semidirect
156. SD₁₆×C₆₁	C₂×Q₈	semidirect
157. D₈×C₆₁	C₂×Q₈	semidirect
159. M₁₆×C₆₁	C₂×Q₈	semidirect
201. Q₈×C₆₁	AGL(1,ℤ/8)	direct (N × H)
202. Q₈×C₆₁	C₂²×Q₈	semidirect
208. D₄×C₆₁	C₂²×Q₈	semidirect
230. C₈×C₆₁	C₂²×Q₈	semidirect
231. Q₈×AGL(1,ℤ/8)	C₆₁	direct (N × H)
254. Q₈×#41	C₂×C₆₁	semidirect
255. C₂×D₄×Q₈	C₂×C₆₁	semidirect
256. #3386	C₂×C₆₁	semidirect
257. #3384	C₂×C₆₁	semidirect
258. Q₈×SD₁₆	C₂×C₆₁	semidirect
259. Q₈×D₈	C₂×C₆₁	semidirect
260. #3165	C₂×C₆₁	semidirect
261. #3079	C₂×C₆₁	semidirect
262. #3077	C₂×C₆₁	semidirect
263. Q₈×M₁₆	C₂×C₆₁	semidirect
273. Q₈²	C₂²×C₆₁	semidirect
274. #557	C₂²×C₆₁	semidirect
276. #554	C₂²×C₆₁	semidirect
282. D₄×Q₈	C₂²×C₆₁	semidirect
291. #501	C₂²×C₆₁	semidirect
292. #500	C₂²×C₆₁	semidirect
293. #498	C₂²×C₆₁	semidirect
295. #477	C₂²×C₆₁	semidirect
297. #476	C₂²×C₆₁	semidirect
298. #475	C₂²×C₆₁	semidirect
300. #474	C₂²×C₆₁	semidirect
301. #446	C₂²×C₆₁	semidirect
302. C₈×Q₈	C₂²×C₆₁	semidirect
313. C₆₁	Q₈×AGL(1,ℤ/8)	direct (N × H)
323. AGL(1,ℤ/8)	Q₈×C₆₁	direct (N × H)
387. #41	C₂×Q₈×C₆₁	semidirect
403. C₂×D₄	C₂×Q₈×C₆₁	semidirect
416. SD₁₆	C₂×Q₈×C₆₁	semidirect
417. D₈	C₂×Q₈×C₆₁	semidirect
419. M₁₆	C₂×Q₈×C₆₁	semidirect
461. Q₈	AGL(1,ℤ/8)×C₆₁	direct (N × H)
462. Q₈	C₂²×Q₈×C₆₁	semidirect
468. D₄	C₂²×Q₈×C₆₁	semidirect
490. C₈	C₂²×Q₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#41×C₆₁	2
C₂×D₄×Q₈×C₆₁	2
C₆₁×#3386	2
C₆₁×#3384	2
Q₈×SD₁₆×C₆₁	2
Q₈×D₈×C₆₁	2
C₆₁×#3165	2
C₆₁×#3079	2
C₆₁×#3077	2
Q₈×M₁₆×C₆₁	2
C₄×AGL(1,ℤ/8)×C₆₁	2
Q₈×AGL(1,ℤ/8)	61

Subgroups — Q8×AGL(1,ℤ/8)×C61 #29564799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×AGL(1,ℤ/8)×C₆₁ #29564799