Subgroups — C₆₁×#24322 #29527755

Order 15616.

988 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3503	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#3078	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#137×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2568	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2061	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2028	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1997	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1986	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1882	C₂	1	2	yes	maximal
10	C₆₁×#1860	C₂	1	2	yes	maximal
11	C₆₁×#1806	C₂	1	1	yes	maximal, characteristic
12	C₂×#137×C₆₁	C₂²	1	1	yes	characteristic
13	Q₈²×C₆₁	—	2	1	no	—
14	C₆₁×#557	—	2	1	no	—
15	C₆₁×#554	—	2	1	no	—
16	C₆₁×#554	—	2	2	no	—
17	D₄×Q₈×C₆₁	—	2	2	no	—
18	D₄×Q₈×C₆₁	—	2	1	no	—
19	C₆₁×#544	—	2	2	no	—
20	C₆₁×#544	C₂²	1	2	yes	—
21	C₆₁×#533	—	2	1	no	—
22	C₆₁×#533	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
24	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₆₁×#520	C₂²	1	2	yes	—
26	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#479	—	2	2	no	—
29	C₆₁×#477	—	2	1	no	—
30	C₆₁×#475	—	2	1	no	—
31	C₆₁×#441	—	2	1	no	—
32	C₆₁×#440	—	2	1	no	—
33	C₄×Q₁₆×C₆₁	—	2	1	no	—
34	C₄×SD₁₆×C₆₁	—	2	1	no	—
35	C₆₁×#428	C₂²	1	2	yes	—
36	C₆₁×#428	C₂²	1	1	yes	characteristic
37	C₆₁×#423	C₂²	1	1	yes	characteristic
38	C₂×#104×C₆₁	C₂²	1	2	yes	—
39	C₆₁×#419	C₂²	1	1	yes	characteristic
40	C₆₁×#419	C₂²	1	2	yes	—
41	C₆₁×#417	C₂²	1	1	yes	characteristic
42	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
43	C₆₁×#400	—	2	2	no	—
44	C₆₁×#391	—	2	2	no	—
45	C₆₁×#391	C₂²	1	2	yes	—
46	C₆₁×#382	C₂²	1	2	yes	—
47	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
48	C₆₁×#343	C₂²	1	2	yes	—
49	C₆₁×#343	C₂²	1	2	yes	—
50	C₆₁×#339	C₂²	1	2	yes	—
51	C₆₁×#339	C₂²	1	2	yes	—
52	C₆₁×#339	C₂²	1	2	yes	—
53	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
54	C₂²×Q₈×C₆₁	—	2	2	no	—
55	C₂²×Q₈×C₆₁	D₄	1	2	yes	—
56	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
57	#137×C₆₁	—	2	2	no	—
58	#137×C₆₁	Q₈	1	4	yes	complemented
59	C₂×Q₁₆×C₆₁	—	2	1	no	—
60	C₂×SD₁₆×C₆₁	—	2	1	no	—
61	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
62	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
63	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₆₁	—	2	2	no	—
65	C₄⋊Q₈×C₆₁	—	2	2	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	1	no	—
70	C₄⋊Q₈×C₆₁	—	2	1	no	—
71	C₄⋊Q₈×C₆₁	—	2	1	no	—
72	C₄⋊Q₈×C₆₁	—	2	1	no	—
73	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
74	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
75	#125×C₆₁	—	2	4	no	—
76	#125×C₆₁	—	2	2	no	—
77	#125×C₆₁	—	2	2	no	—
78	#125×C₆₁	D₄	1	2	yes	—
79	C₂²⋊Q₈×C₆₁	—	2	2	no	—
80	C₂²⋊Q₈×C₆₁	—	2	2	no	—
81	C₂²⋊Q₈×C₆₁	—	2	4	no	—
82	C₂²⋊Q₈×C₆₁	—	2	4	no	—
83	C₂²⋊Q₈×C₆₁	—	2	1	no	—
84	C₂²⋊Q₈×C₆₁	—	2	1	no	—
85	C₂²⋊Q₈×C₆₁	—	2	4	no	—
86	C₄×Q₈×C₆₁	—	2	2	no	—
87	C₄×Q₈×C₆₁	—	2	2	no	—
88	C₄×Q₈×C₆₁	—	2	2	no	—
89	C₄×Q₈×C₆₁	—	2	1	no	—
90	C₄×Q₈×C₆₁	—	2	1	no	—
91	C₄×Q₈×C₆₁	—	2	1	no	—
92	C₄×Q₈×C₆₁	—	2	1	no	—
93	C₄×Q₈×C₆₁	—	2	1	no	—
94	C₄×Q₈×C₆₁	—	2	1	no	—
95	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
96	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
97	C₄×D₄×C₆₁	—	2	2	no	—
98	C₄×D₄×C₆₁	—	2	1	no	—
99	C₄×D₄×C₆₁	—	2	4	no	—
100	C₄×D₄×C₆₁	—	2	1	no	—
101	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
102	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
103	#117×C₆₁	—	2	2	no	—
104	#117×C₆₁	—	2	1	no	—
105	#117×C₆₁	C₂³	1	2	yes	—
106	#117×C₆₁	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
108	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
109	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
110	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
111	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
112	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
114	C₂×C₄²×C₆₁	—	2	1	no	—
115	C₂×C₄²×C₆₁	—	2	1	no	—
116	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
117	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
118	#107×C₆₁	—	2	1	no	—
119	#107×C₆₁	—	2	2	no	—
120	#106×C₆₁	—	2	1	no	—
121	#106×C₆₁	—	2	2	no	—
122	C₄⋊C₈×C₆₁	—	2	1	no	—
123	C₄⋊C₈×C₆₁	—	2	1	no	—
124	#104×C₆₁	—	2	4	no	—
125	#103×C₆₁	—	2	1	no	—
126	#103×C₆₁	—	2	1	no	—
127	#103×C₆₁	—	2	1	no	—
128	#103×C₆₁	—	2	2	no	—
129	#103×C₆₁	—	2	2	no	—
130	#102×C₆₁	—	2	1	no	—
131	#97×C₆₁	—	2	1	no	—
132	C₄×C₈×C₆₁	—	2	1	no	—
133	#95×C₆₁	—	2	1	no	—
134	#95×C₆₁	—	2	4	no	—
135	#95×C₆₁	—	2	4	no	—
136	#41×C₆₁	—	2	1	no	—
137	#41×C₆₁	—	2	1	no	—
138	#41×C₆₁	C₂×Q₈	1	2	yes	—
139	#41×C₆₁	#41	1	2	yes	—
140	C₂×Q₈×C₆₁	—	2	4	no	—
141	C₂×Q₈×C₆₁	—	2	1	no	—
142	C₂×Q₈×C₆₁	—	2	2	no	—
143	C₂×Q₈×C₆₁	—	2	1	no	—
144	C₂×Q₈×C₆₁	—	2	2	no	—
145	C₂×Q₈×C₆₁	—	2	1	no	—
146	C₂×Q₈×C₆₁	—	2	1	no	—
147	C₂×Q₈×C₆₁	—	2	1	no	—
148	C₂×Q₈×C₆₁	—	2	1	no	—
149	C₂×Q₈×C₆₁	—	2	1	no	—
150	C₂×Q₈×C₆₁	—	4	4	no	—
151	C₂×Q₈×C₆₁	—	8	4	no	—
152	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
153	C₂×Q₈×C₆₁	C₂×Q₈	1	2	yes	complemented
154	C₂×D₄×C₆₁	—	2	1	no	—
155	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₁	—	2	1	no	—
157	C₂²×C₄×C₆₁	—	2	1	no	—
158	C₂²×C₄×C₆₁	—	2	4	no	—
159	C₂²×C₄×C₆₁	—	2	1	no	—
160	C₂²×C₄×C₆₁	—	2	1	no	—
161	C₂²×C₄×C₆₁	—	2	2	no	—
162	C₂²×C₄×C₆₁	—	2	1	no	—
163	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
165	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
167	Q₁₆×C₆₁	—	2	4	no	—
168	SD₁₆×C₆₁	—	2	4	no	—
169	M₁₆×C₆₁	—	2	1	no	—
170	M₁₆×C₆₁	—	2	2	no	—
171	M₁₆×C₆₁	—	2	1	no	—
172	M₁₆×C₆₁	—	2	1	no	—
173	M₁₆×C₆₁	—	2	2	no	—
174	M₁₆×C₆₁	C₂×Q₈	1	2	yes	complemented
175	C₂×C₈×C₆₁	—	2	1	no	—
176	C₂×C₈×C₆₁	—	2	1	no	—
177	C₂×C₈×C₆₁	—	2	2	no	—
178	C₄⋊C₄×C₆₁	—	2	4	no	—
179	C₄⋊C₄×C₆₁	—	2	4	no	—
180	C₄⋊C₄×C₆₁	—	2	4	no	—
181	C₄⋊C₄×C₆₁	—	2	4	no	—
182	C₄⋊C₄×C₆₁	—	2	1	no	—
183	C₄⋊C₄×C₆₁	—	2	2	no	—
184	C₄⋊C₄×C₆₁	—	2	2	no	—
185	C₄⋊C₄×C₆₁	—	2	1	no	—
186	C₄⋊C₄×C₆₁	—	2	1	no	—
187	C₄⋊C₄×C₆₁	—	2	1	no	—
188	C₄⋊C₄×C₆₁	—	2	2	no	—
189	C₄⋊C₄×C₆₁	—	2	2	no	—
190	C₄⋊C₄×C₆₁	—	2	2	no	—
191	C₄⋊C₄×C₆₁	—	2	2	no	—
192	C₄⋊C₄×C₆₁	—	2	1	no	—
193	C₄⋊C₄×C₆₁	—	2	1	no	—
194	C₄⋊C₄×C₆₁	—	4	2	no	—
195	C₄⋊C₄×C₆₁	—	4	1	no	—
196	C₄⋊C₄×C₆₁	—	4	1	no	—
197	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
198	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
199	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
200	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
201	C₂²⋊C₄×C₆₁	—	2	4	no	—
202	C₂²⋊C₄×C₆₁	—	2	2	no	—
203	C₂²⋊C₄×C₆₁	—	2	1	no	—
204	C₂²⋊C₄×C₆₁	—	2	1	no	—
205	C₂²⋊C₄×C₆₁	—	2	1	no	—
206	C₄²×C₆₁	—	2	4	no	—
207	C₄²×C₆₁	—	2	1	no	—
208	C₄²×C₆₁	—	2	1	no	—
209	C₄²×C₆₁	—	2	1	no	—
210	C₄²×C₆₁	—	2	2	no	—
211	C₄²×C₆₁	—	2	2	no	—
212	C₄²×C₆₁	—	2	2	no	—
213	C₄²×C₆₁	—	4	1	no	—
214	C₄²×C₆₁	—	4	1	no	—
215	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
216	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
218	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
219	C₂³×C₆₁	—	2	1	no	—
220	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
221	Q₈×C₆₁	—	2	1	no	—
222	Q₈×C₆₁	—	2	2	no	—
223	Q₈×C₆₁	—	2	2	no	—
224	Q₈×C₆₁	—	4	2	no	—
225	Q₈×C₆₁	—	4	1	no	—
226	Q₈×C₆₁	—	4	1	no	—
227	Q₈×C₆₁	—	4	4	no	—
228	Q₈×C₆₁	—	4	2	no	—
229	Q₈×C₆₁	—	8	2	no	—
230	Q₈×C₆₁	—	8	4	no	—
231	Q₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
232	D₄×C₆₁	—	2	2	no	—
233	D₄×C₆₁	—	2	2	no	—
234	D₄×C₆₁	—	2	1	no	—
235	D₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
236	C₂×C₄×C₆₁	—	2	1	no	—
237	C₂×C₄×C₆₁	—	2	2	no	—
238	C₂×C₄×C₆₁	—	2	2	no	—
239	C₂×C₄×C₆₁	—	2	2	no	—
240	C₂×C₄×C₆₁	—	2	2	no	—
241	C₂×C₄×C₆₁	—	2	1	no	—
242	C₂×C₄×C₆₁	—	2	1	no	—
243	C₂×C₄×C₆₁	—	2	4	no	—
244	C₂×C₄×C₆₁	—	2	2	no	—
245	C₂×C₄×C₆₁	—	2	2	no	—
246	C₂×C₄×C₆₁	—	4	1	no	—
247	C₂×C₄×C₆₁	—	4	2	no	—
248	C₂×C₄×C₆₁	—	4	1	no	—
249	C₂×C₄×C₆₁	—	4	1	no	—
250	C₂×C₄×C₆₁	—	4	2	no	—
251	C₂×C₄×C₆₁	—	4	4	no	—
252	C₂×C₄×C₆₁	—	4	2	no	—
253	C₂×C₄×C₆₁	—	4	2	no	—
254	C₂×C₄×C₆₁	—	4	2	no	—
255	C₂×C₄×C₆₁	—	4	1	no	—
256	C₂×C₄×C₆₁	—	4	2	no	—
257	C₂×C₄×C₆₁	—	8	4	no	—
258	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
259	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
260	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
261	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
262	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
265	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
266	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
267	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
268	C₈×C₆₁	—	2	2	no	—
269	C₈×C₆₁	—	4	1	no	—
270	C₈×C₆₁	—	4	2	no	—
271	#24322	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
272	C₂²×C₆₁	—	2	1	no	—
273	C₂²×C₆₁	—	2	1	no	—
274	C₂²×C₆₁	—	2	2	no	—
275	C₂²×C₆₁	—	4	1	no	—
276	C₂²×C₆₁	—	4	1	no	—
277	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
278	C₂²×C₆₁	#548	1	1	yes	characteristic
279	C₂²×C₆₁	D₄×Q₈	1	1	yes	characteristic
280	C₄×C₆₁	—	2	2	no	—
281	C₄×C₆₁	—	2	1	no	—
282	C₄×C₆₁	—	2	1	no	—
283	C₄×C₆₁	—	2	1	no	—
284	C₄×C₆₁	—	4	2	no	—
285	C₄×C₆₁	—	4	1	no	—
286	C₄×C₆₁	—	4	4	no	—
287	C₄×C₆₁	—	4	2	no	—
288	C₄×C₆₁	—	4	1	no	—
289	C₄×C₆₁	—	4	2	no	—
290	C₄×C₆₁	—	8	2	no	—
291	C₄×C₆₁	—	8	1	no	—
292	C₄×C₆₁	#455	1	2	yes	—
293	C₄×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
294	C₄×C₆₁	#536	1	1	yes	characteristic
295	C₄×C₆₁	#548	1	1	yes	characteristic
296	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
297	#3503	C₂×C₆₁	1	1	yes	characteristic
298	#3078	C₂×C₆₁	1	1	yes	characteristic, complemented
299	C₄×#137	C₂×C₆₁	1	1	yes	characteristic
300	#2568	C₂×C₆₁	1	1	yes	characteristic, complemented
301	#2061	C₂×C₆₁	1	2	yes	complemented
302	#2028	C₂×C₆₁	1	2	yes	complemented
303	#1997	C₂×C₆₁	1	1	yes	characteristic, complemented
304	#1986	C₂×C₆₁	1	1	yes	characteristic, complemented
305	#1882	C₂×C₆₁	1	2	yes	—
306	#1860	C₂×C₆₁	1	2	yes	—
307	#1806	C₂×C₆₁	1	1	yes	characteristic
308	C₂×C₆₁	—	2	1	no	—
309	C₂×C₆₁	—	2	1	no	—
310	C₂×C₆₁	—	4	2	no	—
311	C₂×C₆₁	#2409	1	1	yes	characteristic
312	C₂×C₆₁	C₂×#455	1	1	yes	characteristic
313	C₂×C₆₁	#3020	1	1	yes	characteristic
314	C₂×#137	C₂²×C₆₁	1	1	yes	characteristic
315	Q₈²	—	2	1	no	—
316	#557	—	2	1	no	—
317	#554	—	2	1	no	—
318	#554	—	2	2	no	—
319	D₄×Q₈	—	2	2	no	—
320	D₄×Q₈	—	2	1	no	—
321	#544	—	2	2	no	—
322	#544	C₂²×C₆₁	1	2	yes	—
323	#533	—	2	1	no	—
324	#533	C₂²×C₆₁	1	1	yes	characteristic
325	C₂×C₄⋊Q₈	—	2	1	no	—
326	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
327	#520	C₂²×C₆₁	1	2	yes	—
328	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
329	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
330	#479	—	2	2	no	—
331	#477	—	2	1	no	—
332	#475	—	2	1	no	—
333	#441	—	2	1	no	—
334	#440	—	2	1	no	—
335	C₄×Q₁₆	—	2	1	no	—
336	C₄×SD₁₆	—	2	1	no	—
337	#428	C₂²×C₆₁	1	2	yes	—
338	#428	C₂²×C₆₁	1	1	yes	characteristic
339	#423	C₂²×C₆₁	1	1	yes	characteristic
340	C₂×#104	C₂²×C₆₁	1	2	yes	—
341	#419	C₂²×C₆₁	1	2	yes	—
342	#419	C₂²×C₆₁	1	1	yes	characteristic
343	#417	C₂²×C₆₁	1	1	yes	characteristic
344	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
345	#400	—	2	2	no	—
346	#391	—	2	2	no	—
347	#391	C₂²×C₆₁	1	2	yes	—
348	#382	C₂²×C₆₁	1	2	yes	—
349	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
350	#343	C₂²×C₆₁	1	2	yes	—
351	#343	C₂²×C₆₁	1	2	yes	—
352	#339	C₂²×C₆₁	1	2	yes	—
353	#339	C₂²×C₆₁	1	2	yes	—
354	#339	C₂²×C₆₁	1	2	yes	—
355	C₆₁	#24322	1	1	yes	characteristic, direct product factor, Sylow
356	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
357	C₂²×Q₈	—	2	2	no	—
358	C₂²×Q₈	D₄×C₆₁	1	2	yes	—
359	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
360	#137	—	2	2	no	—
361	#137	Q₈×C₆₁	1	4	yes	complemented
362	C₂×Q₁₆	—	2	1	no	—
363	C₂×SD₁₆	—	2	1	no	—
364	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
365	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
366	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
367	C₄⋊Q₈	—	2	2	no	—
368	C₄⋊Q₈	—	2	2	no	—
369	C₄⋊Q₈	—	2	4	no	—
370	C₄⋊Q₈	—	2	2	no	—
371	C₄⋊Q₈	—	2	2	no	—
372	C₄⋊Q₈	—	2	1	no	—
373	C₄⋊Q₈	—	2	1	no	—
374	C₄⋊Q₈	—	2	1	no	—
375	C₄⋊Q₈	—	2	1	no	—
376	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
377	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
378	#125	—	2	4	no	—
379	#125	—	2	2	no	—
380	#125	—	2	2	no	—
381	#125	D₄×C₆₁	1	2	yes	—
382	C₂²⋊Q₈	—	2	2	no	—
383	C₂²⋊Q₈	—	2	2	no	—
384	C₂²⋊Q₈	—	2	4	no	—
385	C₂²⋊Q₈	—	2	4	no	—
386	C₂²⋊Q₈	—	2	1	no	—
387	C₂²⋊Q₈	—	2	1	no	—
388	C₂²⋊Q₈	—	2	4	no	—
389	C₄×Q₈	—	2	2	no	—
390	C₄×Q₈	—	2	2	no	—
391	C₄×Q₈	—	2	2	no	—
392	C₄×Q₈	—	2	1	no	—
393	C₄×Q₈	—	2	1	no	—
394	C₄×Q₈	—	2	1	no	—
395	C₄×Q₈	—	2	1	no	—
396	C₄×Q₈	—	2	1	no	—
397	C₄×Q₈	—	2	1	no	—
398	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
399	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
400	C₄×D₄	—	2	2	no	—
401	C₄×D₄	—	2	1	no	—
402	C₄×D₄	—	2	1	no	—
403	C₄×D₄	—	2	4	no	—
404	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
405	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
406	#117	—	2	2	no	—
407	#117	—	2	1	no	—
408	#117	C₂³×C₆₁	1	1	yes	characteristic
409	#117	C₂³×C₆₁	1	2	yes	—
410	C₂×C₄⋊C₄	—	2	2	no	—
411	C₂×C₄⋊C₄	—	2	2	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	—	2	1	no	—
414	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
415	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
416	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
417	C₂×C₄²	—	2	1	no	—
418	C₂×C₄²	—	2	1	no	—
419	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
420	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
421	#107	—	2	1	no	—
422	#107	—	2	2	no	—
423	#106	—	2	1	no	—
424	#106	—	2	2	no	—
425	C₄⋊C₈	—	2	1	no	—
426	C₄⋊C₈	—	2	1	no	—
427	#104	—	2	4	no	—
428	#103	—	2	1	no	—
429	#103	—	2	1	no	—
430	#103	—	2	1	no	—
431	#103	—	2	2	no	—
432	#103	—	2	2	no	—
433	#102	—	2	1	no	—
434	#97	—	2	1	no	—
435	C₄×C₈	—	2	1	no	—
436	#95	—	2	1	no	—
437	#95	—	2	4	no	—
438	#95	—	2	4	no	—
439	#41	—	2	1	no	—
440	#41	—	2	1	no	—
441	#41	C₂×Q₈×C₆₁	1	2	yes	—
442	#41	#41×C₆₁	1	2	yes	—
443	C₂×Q₈	—	2	4	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	2	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	2	no	—
448	C₂×Q₈	—	2	1	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	—	2	1	no	—
451	C₂×Q₈	—	2	1	no	—
452	C₂×Q₈	—	2	1	no	—
453	C₂×Q₈	—	4	4	no	—
454	C₂×Q₈	—	8	4	no	—
455	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
456	C₂×Q₈	C₂×Q₈×C₆₁	1	2	yes	complemented
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
467	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
468	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
469	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
470	Q₁₆	—	2	4	no	—
471	SD₁₆	—	2	4	no	—
472	M₁₆	—	2	1	no	—
473	M₁₆	—	2	2	no	—
474	M₁₆	—	2	1	no	—
475	M₁₆	—	2	1	no	—
476	M₁₆	—	2	2	no	—
477	M₁₆	C₂×Q₈×C₆₁	1	2	yes	complemented
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	2	no	—
481	C₄⋊C₄	—	2	4	no	—
482	C₄⋊C₄	—	2	4	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	2	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	4	2	no	—
498	C₄⋊C₄	—	4	1	no	—
499	C₄⋊C₄	—	4	1	no	—
500	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
501	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
502	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
503	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
504	C₂²⋊C₄	—	2	4	no	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	1	no	—
507	C₂²⋊C₄	—	2	1	no	—
508	C₂²⋊C₄	—	2	1	no	—
509	C₄²	—	2	4	no	—
510	C₄²	—	2	1	no	—
511	C₄²	—	2	1	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	2	2	no	—
514	C₄²	—	2	2	no	—
515	C₄²	—	2	2	no	—
516	C₄²	—	4	1	no	—
517	C₄²	—	4	1	no	—
518	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
519	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
520	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
521	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
522	C₂³	—	2	1	no	—
523	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
524	Q₈	—	2	1	no	—
525	Q₈	—	2	2	no	—
526	Q₈	—	2	2	no	—
527	Q₈	—	4	2	no	—
528	Q₈	—	4	1	no	—
529	Q₈	—	4	1	no	—
530	Q₈	—	4	4	no	—
531	Q₈	—	4	2	no	—
532	Q₈	—	8	2	no	—
533	Q₈	—	8	4	no	—
534	Q₈	C₂²⋊Q₈×C₆₁	1	2	yes	—
535	D₄	—	2	2	no	—
536	D₄	—	2	2	no	—
537	D₄	—	2	1	no	—
538	D₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	2	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	2	no	—
554	C₂×C₄	—	4	4	no	—
555	C₂×C₄	—	4	2	no	—
556	C₂×C₄	—	4	2	no	—
557	C₂×C₄	—	4	2	no	—
558	C₂×C₄	—	4	2	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	8	4	no	—
561	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
562	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
563	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
564	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
565	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
566	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
567	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
568	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
569	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
570	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
571	C₈	—	2	2	no	—
572	C₈	—	4	1	no	—
573	C₈	—	4	2	no	—
574	C₂²	—	2	1	no	—
575	C₂²	—	2	1	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	1	no	—
579	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
580	C₂²	C₆₁×#548	1	1	yes	characteristic
581	C₂²	D₄×Q₈×C₆₁	1	1	yes	characteristic
582	C₄	—	2	2	no	—
583	C₄	—	2	1	no	—
584	C₄	—	2	1	no	—
585	C₄	—	2	1	no	—
586	C₄	—	4	2	no	—
587	C₄	—	4	1	no	—
588	C₄	—	4	4	no	—
589	C₄	—	4	2	no	—
590	C₄	—	4	1	no	—
591	C₄	—	4	2	no	—
592	C₄	—	8	2	no	—
593	C₄	—	8	1	no	—
594	C₄	C₆₁×#455	1	2	yes	—
595	C₄	C₂×C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
596	C₄	C₆₁×#536	1	1	yes	characteristic
597	C₄	C₆₁×#548	1	1	yes	characteristic
598	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
599	C₂	—	2	1	no	—
600	C₂	—	2	1	no	—
601	C₂	—	4	2	no	—
602	C₂	C₆₁×#2409	1	1	yes	characteristic
603	C₂	C₂×C₆₁×#455	1	1	yes	characteristic
604	C₂	C₆₁×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂×#41×C₆₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#3078	C₂	semidirect
4. C₆₁×#2568	C₂	semidirect
5. C₆₁×#2061	C₂	semidirect
6. C₆₁×#2028	C₂	semidirect
7. C₆₁×#1997	C₂	semidirect
8. C₆₁×#1986	C₂	semidirect
58. #137×C₆₁	Q₈	semidirect
153. C₂×Q₈×C₆₁	C₂×Q₈	semidirect
174. M₁₆×C₆₁	C₂×Q₈	semidirect
271. #24322	C₆₁	direct (N × H)
298. #3078	C₂×C₆₁	semidirect
300. #2568	C₂×C₆₁	semidirect
301. #2061	C₂×C₆₁	semidirect
302. #2028	C₂×C₆₁	semidirect
303. #1997	C₂×C₆₁	semidirect
304. #1986	C₂×C₆₁	semidirect
355. C₆₁	#24322	direct (N × H)
361. #137	Q₈×C₆₁	semidirect
456. C₂×Q₈	C₂×Q₈×C₆₁	semidirect
477. M₁₆	C₂×Q₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3503	2
C₆₁×#3078	2
C₄×#137×C₆₁	2
C₆₁×#2568	2
C₆₁×#2061	2
C₆₁×#2028	2
C₆₁×#1997	2
C₆₁×#1986	2
C₆₁×#1882	2
C₆₁×#1860	2
C₆₁×#1806	2
#24322	61

Subgroups — C61×#24322 #29527755

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#24322 #29527755