Subgroups — C₆₁×#24329 #29527762

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#476	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×SD₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2538	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2058	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2023	C₂	1	2	yes	maximal
7	C₆₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1880	C₂	1	2	yes	maximal
10	C₆₁×#1865	C₂	1	2	yes	maximal, complemented
11	C₆₁×#1803	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₆₁	C₂²	1	1	yes	characteristic
13	Q₈²×C₆₁	—	2	4	no	—
14	Q₈²×C₆₁	—	2	4	no	—
15	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
16	C₂×C₄⋊Q₈×C₆₁	—	2	2	no	—
17	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
18	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
20	C₂×C₄×Q₈×C₆₁	—	2	1	no	—
21	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
22	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
25	C₆₁×#476	—	2	4	no	—
26	C₄×SD₁₆×C₆₁	—	2	4	no	—
27	C₂×#106×C₆₁	C₂²	1	2	yes	—
28	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×#103×C₆₁	C₂²	1	2	yes	—
31	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
32	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
33	C₂×#102×C₆₁	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
35	C₆₁×#396	—	2	2	no	—
36	C₆₁×#390	C₂²	1	2	yes	—
37	C₆₁×#389	C₂²	1	2	yes	—
38	C₆₁×#384	C₂²	1	1	yes	characteristic
39	C₆₁×#380	—	2	1	no	—
40	C₆₁×#340	C₂²	1	2	yes	—
41	C₆₁×#340	C₂²	1	2	yes	—
42	C₆₁×#340	C₂²	1	2	yes	—
43	C₆₁×#336	C₂²	1	2	yes	—
44	C₆₁×#336	C₂²	1	2	yes	—
45	C₂²×Q₈×C₆₁	—	2	2	no	—
46	C₂²×Q₈×C₆₁	—	2	1	no	—
47	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
48	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₆₁	—	2	1	no	—
51	C₂×SD₁₆×C₆₁	—	2	2	no	—
52	C₂×SD₁₆×C₆₁	—	2	2	no	—
53	C₂×SD₁₆×C₆₁	Q₈	1	4	yes	complemented
54	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈×C₆₁	—	2	16	no	—
58	C₄⋊Q₈×C₆₁	—	2	8	no	—
59	C₄⋊Q₈×C₆₁	—	2	4	no	—
60	C₄⋊Q₈×C₆₁	—	2	4	no	—
61	C₄⋊Q₈×C₆₁	—	2	2	no	—
62	C₄⋊Q₈×C₆₁	D₄	1	4	yes	—
63	C₂²⋊Q₈×C₆₁	—	4	4	no	—
64	C₄×Q₈×C₆₁	—	2	8	no	—
65	C₄×Q₈×C₆₁	—	2	4	no	—
66	C₄×Q₈×C₆₁	—	2	4	no	—
67	C₄×Q₈×C₆₁	—	2	2	no	—
68	C₄×Q₈×C₆₁	—	2	2	no	—
69	C₄×Q₈×C₆₁	—	2	2	no	—
70	C₄×Q₈×C₆₁	D₄	1	4	yes	—
71	C₄×D₄×C₆₁	—	2	2	no	—
72	C₄×D₄×C₆₁	—	2	2	no	—
73	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
74	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
75	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
76	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
79	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
82	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
86	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
89	C₂×C₄²×C₆₁	—	2	2	no	—
90	C₂×C₄²×C₆₁	—	2	1	no	—
91	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
92	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
93	#106×C₆₁	—	2	2	no	—
94	#106×C₆₁	—	2	4	no	—
95	C₄⋊C₈×C₆₁	—	2	2	no	—
96	#103×C₆₁	—	2	2	no	—
97	#103×C₆₁	—	2	4	no	—
98	#102×C₆₁	—	2	2	no	—
99	#102×C₆₁	—	2	4	no	—
100	C₄×C₈×C₆₁	—	2	2	no	—
101	#95×C₆₁	—	2	2	no	—
102	C₂⁴×C₆₁	—	2	1	no	—
103	C₂×Q₈×C₆₁	—	2	8	no	—
104	C₂×Q₈×C₆₁	—	2	2	no	—
105	C₂×Q₈×C₆₁	—	2	4	no	—
106	C₂×Q₈×C₆₁	—	2	2	no	—
107	C₂×Q₈×C₆₁	—	2	1	no	—
108	C₂×Q₈×C₆₁	—	2	1	no	—
109	C₂×Q₈×C₆₁	—	2	1	no	—
110	C₂×Q₈×C₆₁	—	8	1	no	—
111	C₂×Q₈×C₆₁	—	8	1	no	—
112	C₂×Q₈×C₆₁	—	8	2	no	—
113	C₂×Q₈×C₆₁	—	8	2	no	—
114	C₂×Q₈×C₆₁	SD₁₆	1	4	yes	complemented
115	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
116	C₂×Q₈×C₆₁	C₂×Q₈	1	2	yes	—
117	C₂×Q₈×C₆₁	#41	1	2	yes	—
118	C₂×D₄×C₆₁	—	2	1	no	—
119	C₂×D₄×C₆₁	—	2	1	no	—
120	C₂×D₄×C₆₁	—	2	1	no	—
121	C₂×D₄×C₆₁	—	2	1	no	—
122	C₂×D₄×C₆₁	—	2	1	no	—
123	C₂×D₄×C₆₁	C₂×Q₈	1	2	yes	—
124	C₂²×C₄×C₆₁	—	2	1	no	—
125	C₂²×C₄×C₆₁	—	2	2	no	—
126	C₂²×C₄×C₆₁	—	2	1	no	—
127	C₂²×C₄×C₆₁	—	2	1	no	—
128	C₂²×C₄×C₆₁	—	2	2	no	—
129	C₂²×C₄×C₆₁	—	4	2	no	—
130	C₂²×C₄×C₆₁	—	4	2	no	—
131	C₂²×C₄×C₆₁	—	4	2	no	—
132	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
137	SD₁₆×C₆₁	—	2	8	no	—
138	C₂×C₈×C₆₁	—	2	1	no	—
139	C₂×C₈×C₆₁	—	2	1	no	—
140	C₂×C₈×C₆₁	—	2	1	no	—
141	C₂×C₈×C₆₁	—	2	2	no	—
142	C₂×C₈×C₆₁	—	2	2	no	—
143	C₂×C₈×C₆₁	—	2	1	no	—
144	C₂×C₈×C₆₁	—	2	1	no	—
145	C₂×C₈×C₆₁	—	2	2	no	—
146	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
147	C₄⋊C₄×C₆₁	—	2	8	no	—
148	C₄⋊C₄×C₆₁	—	2	8	no	—
149	C₄⋊C₄×C₆₁	—	2	4	no	—
150	C₄⋊C₄×C₆₁	—	2	4	no	—
151	C₄⋊C₄×C₆₁	—	2	4	no	—
152	C₄⋊C₄×C₆₁	—	2	2	no	—
153	C₄⋊C₄×C₆₁	—	2	2	no	—
154	C₄⋊C₄×C₆₁	—	2	1	no	—
155	C₄⋊C₄×C₆₁	—	2	1	no	—
156	C₄⋊C₄×C₆₁	—	2	1	no	—
157	C₄⋊C₄×C₆₁	—	2	2	no	—
158	C₄⋊C₄×C₆₁	—	2	2	no	—
159	C₄⋊C₄×C₆₁	—	4	2	no	—
160	C₄⋊C₄×C₆₁	—	4	4	no	—
161	C₄⋊C₄×C₆₁	—	8	1	no	—
162	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
163	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
164	C₂²⋊C₄×C₆₁	—	4	2	no	—
165	C₂²⋊C₄×C₆₁	—	4	4	no	—
166	C₂²⋊C₄×C₆₁	—	8	2	no	—
167	C₄²×C₆₁	—	2	8	no	—
168	C₄²×C₆₁	—	2	4	no	—
169	C₄²×C₆₁	—	2	2	no	—
170	C₄²×C₆₁	—	2	2	no	—
171	C₄²×C₆₁	C₂×D₄	1	2	yes	—
172	C₄²×C₆₁	C₂×D₄	1	2	yes	—
173	C₂³×C₆₁	—	2	2	no	—
174	C₂³×C₆₁	—	4	1	no	—
175	C₂³×C₆₁	—	4	1	no	—
176	C₂³×C₆₁	—	4	1	no	—
177	C₂³×C₆₁	—	4	1	no	—
178	C₂³×C₆₁	—	4	1	no	—
179	C₂³×C₆₁	—	4	1	no	—
180	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
181	Q₈×C₆₁	—	2	2	no	—
182	Q₈×C₆₁	—	2	2	no	—
183	Q₈×C₆₁	—	2	2	no	—
184	Q₈×C₆₁	—	2	8	no	—
185	Q₈×C₆₁	—	8	4	no	—
186	Q₈×C₆₁	—	8	8	no	—
187	Q₈×C₆₁	C₂²⋊Q₈	1	4	yes	complemented
188	D₄×C₆₁	—	2	2	no	—
189	D₄×C₆₁	—	2	2	no	—
190	D₄×C₆₁	—	4	1	no	—
191	D₄×C₆₁	—	4	1	no	—
192	C₂×C₄×C₆₁	—	2	2	no	—
193	C₂×C₄×C₆₁	—	2	2	no	—
194	C₂×C₄×C₆₁	—	2	2	no	—
195	C₂×C₄×C₆₁	—	2	2	no	—
196	C₂×C₄×C₆₁	—	2	2	no	—
197	C₂×C₄×C₆₁	—	2	1	no	—
198	C₂×C₄×C₆₁	—	2	1	no	—
199	C₂×C₄×C₆₁	—	2	2	no	—
200	C₂×C₄×C₆₁	—	2	4	no	—
201	C₂×C₄×C₆₁	—	2	2	no	—
202	C₂×C₄×C₆₁	—	2	2	no	—
203	C₂×C₄×C₆₁	—	2	2	no	—
204	C₂×C₄×C₆₁	—	4	2	no	—
205	C₂×C₄×C₆₁	—	4	1	no	—
206	C₂×C₄×C₆₁	—	4	2	no	—
207	C₂×C₄×C₆₁	—	4	1	no	—
208	C₂×C₄×C₆₁	—	4	1	no	—
209	C₂×C₄×C₆₁	—	4	2	no	—
210	C₂×C₄×C₆₁	—	4	2	no	—
211	C₂×C₄×C₆₁	—	4	1	no	—
212	C₂×C₄×C₆₁	—	4	1	no	—
213	C₂×C₄×C₆₁	—	4	2	no	—
214	C₂×C₄×C₆₁	—	8	4	no	—
215	C₂×C₄×C₆₁	#120	1	2	yes	—
216	C₂×C₄×C₆₁	#120	1	2	yes	—
217	C₂×C₄×C₆₁	C₂×SD₁₆	1	4	yes	—
218	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
219	C₂×C₄×C₆₁	#137	1	2	yes	—
220	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
223	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
225	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
226	C₈×C₆₁	—	2	2	no	—
227	C₈×C₆₁	—	4	1	no	—
228	C₈×C₆₁	—	4	2	no	—
229	#24329	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
230	C₂²×C₆₁	—	4	2	no	—
231	C₂²×C₆₁	—	4	2	no	—
232	C₂²×C₆₁	—	4	2	no	—
233	C₂²×C₆₁	—	8	1	no	—
234	C₂²×C₆₁	—	8	1	no	—
235	C₂²×C₆₁	—	8	1	no	—
236	C₂²×C₆₁	—	8	1	no	—
237	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
238	C₂²×C₆₁	#548	1	1	yes	characteristic
239	C₂²×C₆₁	D₄×Q₈	1	1	yes	characteristic
240	C₂²×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
241	C₂²×C₆₁	C₂×#137	1	1	yes	characteristic
242	C₂²×C₆₁	#576	1	1	yes	characteristic
243	C₂²×C₆₁	#577	1	1	yes	characteristic
244	C₄×C₆₁	—	2	4	no	—
245	C₄×C₆₁	—	2	4	no	—
246	C₄×C₆₁	—	2	2	no	—
247	C₄×C₆₁	—	4	2	no	—
248	C₄×C₆₁	—	4	4	no	—
249	C₄×C₆₁	—	4	2	no	—
250	C₄×C₆₁	—	8	2	no	—
251	C₄×C₆₁	—	8	1	no	—
252	C₄×C₆₁	#448	1	4	yes	—
253	C₄×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
254	C₄×C₆₁	#536	1	1	yes	characteristic
255	C₄×C₆₁	#548	1	1	yes	characteristic
256	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
257	C₂×Q₈²	C₂×C₆₁	1	1	yes	characteristic, complemented
258	C₂×#476	C₂×C₆₁	1	1	yes	characteristic
259	C₂×C₄×SD₁₆	C₂×C₆₁	1	1	yes	characteristic
260	#2538	C₂×C₆₁	1	1	yes	characteristic
261	#2058	C₂×C₆₁	1	2	yes	complemented
262	#2023	C₂×C₆₁	1	2	yes	—
263	#1990	C₂×C₆₁	1	1	yes	characteristic, complemented
264	#1984	C₂×C₆₁	1	1	yes	characteristic, complemented
265	#1880	C₂×C₆₁	1	2	yes	—
266	#1865	C₂×C₆₁	1	2	yes	complemented
267	#1803	C₂×C₆₁	1	1	yes	characteristic, complemented
268	C₂×C₆₁	—	8	2	no	—
269	C₂×C₆₁	#2409	1	1	yes	characteristic
270	C₂×C₆₁	C₂×#448	1	1	yes	characteristic
271	C₂×C₆₁	#3012	1	1	yes	characteristic
272	C₂×C₆₁	#3362	1	1	yes	characteristic
273	C₂×C₆₁	#3369	1	1	yes	characteristic
274	C₂×C₆₁	Q₈×SD₁₆	1	1	yes	characteristic
275	C₂×C₆₁	#3385	1	1	yes	characteristic
276	C₂²×SD₁₆	C₂²×C₆₁	1	1	yes	characteristic
277	Q₈²	—	2	4	no	—
278	Q₈²	—	2	4	no	—
279	C₂×C₄⋊Q₈	—	2	1	no	—
280	C₂×C₄⋊Q₈	—	2	2	no	—
281	C₂×C₄⋊Q₈	C₂²×C₆₁	1	2	yes	—
282	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
283	C₂×C₂²⋊Q₈	—	2	1	no	—
284	C₂×C₄×Q₈	—	2	1	no	—
285	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
286	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
287	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
288	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
289	#476	—	2	4	no	—
290	C₄×SD₁₆	—	2	4	no	—
291	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
292	C₂×#106	C₂²×C₆₁	1	2	yes	—
293	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
294	C₂×#103	C₂²×C₆₁	1	2	yes	—
295	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
296	C₂×#102	C₂²×C₆₁	1	2	yes	—
297	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
298	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
299	#396	—	2	2	no	—
300	#390	C₂²×C₆₁	1	2	yes	—
301	#389	C₂²×C₆₁	1	2	yes	—
302	#384	C₂²×C₆₁	1	1	yes	characteristic
303	#380	—	2	1	no	—
304	#340	C₂²×C₆₁	1	2	yes	—
305	#340	C₂²×C₆₁	1	2	yes	—
306	#340	C₂²×C₆₁	1	2	yes	—
307	#336	C₂²×C₆₁	1	2	yes	—
308	#336	C₂²×C₆₁	1	2	yes	—
309	C₆₁	#24329	1	1	yes	characteristic, direct product factor, Sylow
310	C₂²×Q₈	—	2	2	no	—
311	C₂²×Q₈	—	2	1	no	—
312	C₂²×Q₈	D₄×C₆₁	1	1	yes	characteristic
313	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
314	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
315	C₂³×C₄	—	2	1	no	—
316	C₂×SD₁₆	—	2	2	no	—
317	C₂×SD₁₆	—	2	2	no	—
318	C₂×SD₁₆	Q₈×C₆₁	1	4	yes	complemented
319	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
320	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
321	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
322	C₄⋊Q₈	—	2	16	no	—
323	C₄⋊Q₈	—	2	8	no	—
324	C₄⋊Q₈	—	2	4	no	—
325	C₄⋊Q₈	—	2	4	no	—
326	C₄⋊Q₈	—	2	2	no	—
327	C₄⋊Q₈	D₄×C₆₁	1	4	yes	—
328	C₂²⋊Q₈	—	4	4	no	—
329	C₄×Q₈	—	2	8	no	—
330	C₄×Q₈	—	2	4	no	—
331	C₄×Q₈	—	2	4	no	—
332	C₄×Q₈	—	2	2	no	—
333	C₄×Q₈	—	2	2	no	—
334	C₄×Q₈	—	2	2	no	—
335	C₄×Q₈	D₄×C₆₁	1	4	yes	—
336	C₄×D₄	—	2	2	no	—
337	C₄×D₄	—	2	2	no	—
338	C₂×C₄⋊C₄	—	2	2	no	—
339	C₂×C₄⋊C₄	—	2	1	no	—
340	C₂×C₄⋊C₄	—	2	2	no	—
341	C₂×C₄⋊C₄	—	2	1	no	—
342	C₂×C₄⋊C₄	—	2	1	no	—
343	C₂×C₄⋊C₄	—	2	2	no	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
346	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
347	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
348	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
349	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
350	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
351	C₂×C₂²⋊C₄	—	2	1	no	—
352	C₂×C₂²⋊C₄	—	2	2	no	—
353	C₂×C₂²⋊C₄	—	2	2	no	—
354	C₂×C₄²	—	2	2	no	—
355	C₂×C₄²	—	2	1	no	—
356	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
357	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
358	#106	—	2	2	no	—
359	#106	—	2	4	no	—
360	C₄⋊C₈	—	2	2	no	—
361	#103	—	2	2	no	—
362	#103	—	2	4	no	—
363	#102	—	2	2	no	—
364	#102	—	2	4	no	—
365	C₄×C₈	—	2	2	no	—
366	#95	—	2	2	no	—
367	C₂⁴	—	2	1	no	—
368	C₂×Q₈	—	2	8	no	—
369	C₂×Q₈	—	2	2	no	—
370	C₂×Q₈	—	2	4	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	2	1	no	—
373	C₂×Q₈	—	2	1	no	—
374	C₂×Q₈	—	2	1	no	—
375	C₂×Q₈	—	8	1	no	—
376	C₂×Q₈	—	8	1	no	—
377	C₂×Q₈	—	8	2	no	—
378	C₂×Q₈	—	8	2	no	—
379	C₂×Q₈	SD₁₆×C₆₁	1	4	yes	complemented
380	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
381	C₂×Q₈	C₂×Q₈×C₆₁	1	2	yes	—
382	C₂×Q₈	#41×C₆₁	1	2	yes	—
383	C₂×D₄	—	2	1	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	C₂×Q₈×C₆₁	1	2	yes	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	4	2	no	—
395	C₂²×C₄	—	4	2	no	—
396	C₂²×C₄	—	4	2	no	—
397	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
398	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
399	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
400	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
401	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
402	SD₁₆	—	2	8	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	1	no	—
405	C₂×C₈	—	2	1	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
412	C₄⋊C₄	—	2	8	no	—
413	C₄⋊C₄	—	2	8	no	—
414	C₄⋊C₄	—	2	4	no	—
415	C₄⋊C₄	—	2	4	no	—
416	C₄⋊C₄	—	2	4	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	4	2	no	—
425	C₄⋊C₄	—	4	4	no	—
426	C₄⋊C₄	—	8	1	no	—
427	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
428	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
429	C₂²⋊C₄	—	4	2	no	—
430	C₂²⋊C₄	—	4	4	no	—
431	C₂²⋊C₄	—	8	2	no	—
432	C₄²	—	2	8	no	—
433	C₄²	—	2	4	no	—
434	C₄²	—	2	2	no	—
435	C₄²	—	2	2	no	—
436	C₄²	C₂×D₄×C₆₁	1	2	yes	—
437	C₄²	C₂×D₄×C₆₁	1	2	yes	—
438	C₂³	—	2	2	no	—
439	C₂³	—	4	1	no	—
440	C₂³	—	4	1	no	—
441	C₂³	—	4	1	no	—
442	C₂³	—	4	1	no	—
443	C₂³	—	4	1	no	—
444	C₂³	—	4	1	no	—
445	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
446	Q₈	—	2	2	no	—
447	Q₈	—	2	2	no	—
448	Q₈	—	2	2	no	—
449	Q₈	—	2	8	no	—
450	Q₈	—	8	4	no	—
451	Q₈	—	8	8	no	—
452	Q₈	C₂²⋊Q₈×C₆₁	1	4	yes	complemented
453	D₄	—	2	2	no	—
454	D₄	—	2	2	no	—
455	D₄	—	4	1	no	—
456	D₄	—	4	1	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	8	4	no	—
480	C₂×C₄	#120×C₆₁	1	2	yes	—
481	C₂×C₄	#120×C₆₁	1	2	yes	—
482	C₂×C₄	C₂×SD₁₆×C₆₁	1	4	yes	—
483	C₂×C₄	C₂×SD₁₆×C₆₁	1	2	yes	—
484	C₂×C₄	#137×C₆₁	1	2	yes	—
485	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
486	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
487	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
488	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
489	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
490	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
491	C₈	—	2	2	no	—
492	C₈	—	4	1	no	—
493	C₈	—	4	2	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	8	1	no	—
498	C₂²	—	8	1	no	—
499	C₂²	—	8	1	no	—
500	C₂²	—	8	1	no	—
501	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
502	C₂²	C₆₁×#548	1	1	yes	characteristic
503	C₂²	D₄×Q₈×C₆₁	1	1	yes	characteristic
504	C₂²	C₂²×SD₁₆×C₆₁	1	1	yes	characteristic
505	C₂²	C₂×#137×C₆₁	1	1	yes	characteristic
506	C₂²	C₆₁×#576	1	1	yes	characteristic
507	C₂²	C₆₁×#577	1	1	yes	characteristic
508	C₄	—	2	4	no	—
509	C₄	—	2	4	no	—
510	C₄	—	2	2	no	—
511	C₄	—	4	2	no	—
512	C₄	—	4	4	no	—
513	C₄	—	4	2	no	—
514	C₄	—	8	2	no	—
515	C₄	—	8	1	no	—
516	C₄	C₆₁×#448	1	4	yes	—
517	C₄	C₂×C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
518	C₄	C₆₁×#536	1	1	yes	characteristic
519	C₄	C₆₁×#548	1	1	yes	characteristic
520	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
521	C₂	—	8	2	no	—
522	C₂	C₆₁×#2409	1	1	yes	characteristic
523	C₂	C₂×C₆₁×#448	1	1	yes	characteristic
524	C₂	C₆₁×#3012	1	1	yes	characteristic
525	C₂	C₆₁×#3362	1	1	yes	characteristic
526	C₂	C₆₁×#3369	1	1	yes	characteristic
527	C₂	Q₈×SD₁₆×C₆₁	1	1	yes	characteristic
528	C₂	C₆₁×#3385	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂²×D₄×C₆₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂×C₄×Q₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈²×C₆₁	C₂	semidirect
5. C₆₁×#2058	C₂	semidirect
7. C₆₁×#1990	C₂	semidirect
8. C₆₁×#1984	C₂	semidirect
10. C₆₁×#1865	C₂	semidirect
11. C₆₁×#1803	C₂	semidirect
53. C₂×SD₁₆×C₆₁	Q₈	semidirect
114. C₂×Q₈×C₆₁	SD₁₆	semidirect
187. Q₈×C₆₁	C₂²⋊Q₈	semidirect
229. #24329	C₆₁	direct (N × H)
257. C₂×Q₈²	C₂×C₆₁	semidirect
261. #2058	C₂×C₆₁	semidirect
263. #1990	C₂×C₆₁	semidirect
264. #1984	C₂×C₆₁	semidirect
266. #1865	C₂×C₆₁	semidirect
267. #1803	C₂×C₆₁	semidirect
309. C₆₁	#24329	direct (N × H)
318. C₂×SD₁₆	Q₈×C₆₁	semidirect
379. C₂×Q₈	SD₁₆×C₆₁	semidirect
452. Q₈	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
C₂×Q₈²×C₆₁	2
C₂×C₆₁×#476	2
C₂×C₄×SD₁₆×C₆₁	2
C₆₁×#2538	2
C₆₁×#2058	2
C₆₁×#2023	2
C₆₁×#1990	2
C₆₁×#1984	2
C₆₁×#1880	2
C₆₁×#1865	2
C₆₁×#1803	2
#24329	61

Subgroups — C61×#24329 #29527762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#24329 #29527762