Subgroups — C₂×C₆₁×#2439 #29538145

Order 15616.

2854 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#117×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2439	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₆₁×#396	C₂	1	2	yes	maximal
5	C₂×C₆₁×#393	C₂	1	2	yes	maximal
6	C₂×C₆₁×#391	C₂	1	2	yes	maximal, complemented
7	C₂×C₆₁×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₆₁×#389	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₆₁×#387	C₂	1	2	yes	maximal, complemented
10	C₂×C₆₁×#382	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₆₁×#380	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×C₂²⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
13	C₂³×Q₈×C₆₁	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₆₁	—	2	8	no	—
16	C₂×C₄⋊Q₈×C₆₁	C₂²	1	8	yes	complemented
17	C₂×#117×C₆₁	—	2	8	no	—
18	C₂×#117×C₆₁	C₂²	1	8	yes	—
19	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
30	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
31	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
32	C₂²×C₄²×C₆₁	C₂²	1	2	yes	—
33	C₆₁×#396	C₂²	1	16	yes	—
34	C₆₁×#393	C₂²	1	16	yes	—
35	C₆₁×#391	C₂²	1	16	yes	complemented
36	C₆₁×#389	C₂²	1	8	yes	complemented
37	C₆₁×#389	C₂²	1	8	yes	complemented
38	C₆₁×#387	C₂²	1	16	yes	complemented
39	C₆₁×#382	C₂²	1	8	yes	complemented
40	C₆₁×#380	C₂²	1	8	yes	—
41	C₄×C₂²⋊C₄×C₆₁	C₂²	1	8	yes	—
42	C₂×#95×C₆₁	C₂²	1	1	yes	characteristic
43	C₂×#95×C₆₁	C₂²	1	1	yes	characteristic
44	C₂×#95×C₆₁	C₂²	1	2	yes	—
45	C₂×#95×C₆₁	C₂²	1	2	yes	—
46	C₂×#95×C₆₁	C₂²	1	2	yes	—
47	C₂×#95×C₆₁	C₂²	1	2	yes	—
48	C₂×#95×C₆₁	C₂²	1	2	yes	—
49	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
50	C₂²×Q₈×C₆₁	—	2	8	no	—
51	C₂²×Q₈×C₆₁	—	2	8	no	—
52	C₂²×Q₈×C₆₁	—	4	4	no	—
53	C₂²×Q₈×C₆₁	C₂³	1	8	yes	—
54	C₂³×C₄×C₆₁	—	2	4	no	—
55	C₂³×C₄×C₆₁	—	2	4	no	—
56	C₂³×C₄×C₆₁	Q₈	1	8	yes	complemented
57	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
58	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
60	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
62	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
63	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
65	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₆₁	—	2	32	no	—
69	#117×C₆₁	—	2	32	no	—
70	C₂×C₄⋊C₄×C₆₁	—	2	8	no	—
71	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
72	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
73	C₂×C₄⋊C₄×C₆₁	—	2	8	no	—
74	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
75	C₂×C₄⋊C₄×C₆₁	—	4	4	no	—
76	C₂×C₄⋊C₄×C₆₁	D₄	1	8	yes	—
77	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
79	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
82	C₂×C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
83	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
84	C₂×C₂²⋊C₄×C₆₁	—	2	8	no	—
85	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
86	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
87	C₂×C₂²⋊C₄×C₆₁	—	4	4	no	—
88	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
91	C₂×C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
92	C₂×C₄²×C₆₁	—	2	4	no	—
93	C₂×C₄²×C₆₁	—	2	8	no	—
94	C₂×C₄²×C₆₁	—	2	4	no	—
95	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
96	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
97	C₂×C₄²×C₆₁	C₂³	1	8	yes	—
98	#95×C₆₁	C₂³	1	4	yes	—
99	#95×C₆₁	C₂³	1	4	yes	—
100	#95×C₆₁	C₂³	1	8	yes	—
101	#95×C₆₁	C₂³	1	8	yes	—
102	#95×C₆₁	C₂³	1	8	yes	—
103	#95×C₆₁	C₂³	1	8	yes	—
104	#95×C₆₁	C₂³	1	8	yes	—
105	C₂⁴×C₆₁	—	2	2	no	—
106	C₂⁴×C₆₁	—	2	2	no	—
107	C₂⁴×C₆₁	—	2	2	no	—
108	C₂⁴×C₆₁	—	2	2	no	—
109	C₂⁴×C₆₁	—	2	2	no	—
110	C₂⁴×C₆₁	—	2	2	no	—
111	C₂⁴×C₆₁	C₂×Q₈	1	4	yes	—
112	C₂⁴×C₆₁	C₂⁴	1	2	yes	—
113	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
114	C₂×Q₈×C₆₁	—	2	16	no	—
115	C₂×Q₈×C₆₁	—	2	16	no	—
116	C₂×Q₈×C₆₁	—	4	8	no	—
117	C₂×Q₈×C₆₁	—	4	32	no	—
118	C₂²×C₄×C₆₁	—	2	2	no	—
119	C₂²×C₄×C₆₁	—	2	2	no	—
120	C₂²×C₄×C₆₁	—	2	2	no	—
121	C₂²×C₄×C₆₁	—	2	2	no	—
122	C₂²×C₄×C₆₁	—	2	4	no	—
123	C₂²×C₄×C₆₁	—	2	4	no	—
124	C₂²×C₄×C₆₁	—	2	2	no	—
125	C₂²×C₄×C₆₁	—	2	2	no	—
126	C₂²×C₄×C₆₁	—	2	32	no	—
127	C₂²×C₄×C₆₁	—	2	8	no	—
128	C₂²×C₄×C₆₁	—	2	8	no	—
129	C₂²×C₄×C₆₁	—	2	4	no	—
130	C₂²×C₄×C₆₁	—	2	4	no	—
131	C₂²×C₄×C₆₁	—	2	2	no	—
132	C₂²×C₄×C₆₁	—	2	2	no	—
133	C₂²×C₄×C₆₁	—	2	2	no	—
134	C₂²×C₄×C₆₁	—	2	2	no	—
135	C₂²×C₄×C₆₁	—	2	4	no	—
136	C₂²×C₄×C₆₁	—	2	4	no	—
137	C₂²×C₄×C₆₁	—	2	4	no	—
138	C₂²×C₄×C₆₁	—	2	4	no	—
139	C₂²×C₄×C₆₁	—	2	4	no	—
140	C₂²×C₄×C₆₁	—	2	4	no	—
141	C₂²×C₄×C₆₁	—	2	2	no	—
142	C₂²×C₄×C₆₁	—	2	4	no	—
143	C₂²×C₄×C₆₁	—	2	2	no	—
144	C₂²×C₄×C₆₁	C₂×D₄	1	8	yes	—
145	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
146	C₂²×C₄×C₆₁	C₂×Q₈	1	4	yes	—
147	C₂²×C₄×C₆₁	C₂×Q₈	1	4	yes	—
148	C₂²×C₄×C₆₁	C₂×Q₈	1	16	yes	complemented
149	C₂²×C₄×C₆₁	#41	1	4	yes	—
150	C₂²×C₄×C₆₁	#41	1	4	yes	—
151	C₂²×C₄×C₆₁	#41	1	4	yes	—
152	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
162	C₄⋊C₄×C₆₁	—	2	8	no	—
163	C₄⋊C₄×C₆₁	—	2	16	no	—
164	C₄⋊C₄×C₆₁	—	2	8	no	—
165	C₄⋊C₄×C₆₁	—	2	16	no	—
166	C₄⋊C₄×C₆₁	—	2	8	no	—
167	C₄⋊C₄×C₆₁	—	4	8	no	—
168	C₄⋊C₄×C₆₁	C₂×D₄	1	16	yes	—
169	C₂²⋊C₄×C₆₁	—	2	16	no	—
170	C₂²⋊C₄×C₆₁	—	2	8	no	—
171	C₂²⋊C₄×C₆₁	—	2	8	no	—
172	C₂²⋊C₄×C₆₁	—	4	8	no	—
173	C₄²×C₆₁	—	2	16	no	—
174	C₄²×C₆₁	—	2	8	no	—
175	C₄²×C₆₁	—	2	8	no	—
176	C₂³×C₆₁	—	2	8	no	—
177	C₂³×C₆₁	—	2	8	no	—
178	C₂³×C₆₁	—	2	8	no	—
179	C₂³×C₆₁	—	2	2	no	—
180	C₂³×C₆₁	—	2	2	no	—
181	C₂³×C₆₁	—	2	2	no	—
182	C₂³×C₆₁	—	2	2	no	—
183	C₂³×C₆₁	—	2	2	no	—
184	C₂³×C₆₁	—	2	2	no	—
185	C₂³×C₆₁	—	4	4	no	—
186	C₂³×C₆₁	—	4	4	no	—
187	C₂³×C₆₁	—	4	4	no	—
188	C₂³×C₆₁	—	4	4	no	—
189	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
190	C₂³×C₆₁	C₂²×Q₈	1	4	yes	—
191	C₂³×C₆₁	C₂²×Q₈	1	2	yes	—
192	C₂³×C₆₁	C₂×#41	1	2	yes	—
193	C₂³×C₆₁	C₂×#41	1	2	yes	—
194	C₂³×C₆₁	C₂×#41	1	2	yes	—
195	C₂³×C₆₁	H(5,𝔽₂)	1	2	yes	—
196	C₂³×C₆₁	2₋¹⁺⁴	1	2	yes	—
197	C₂³×C₆₁	C₂⁵	1	1	yes	characteristic
198	Q₈×C₆₁	—	4	32	no	—
199	C₂×C₄×C₆₁	—	2	8	no	—
200	C₂×C₄×C₆₁	—	2	16	no	—
201	C₂×C₄×C₆₁	—	2	8	no	—
202	C₂×C₄×C₆₁	—	2	8	no	—
203	C₂×C₄×C₆₁	—	2	2	no	—
204	C₂×C₄×C₆₁	—	2	2	no	—
205	C₂×C₄×C₆₁	—	2	2	no	—
206	C₂×C₄×C₆₁	—	2	2	no	—
207	C₂×C₄×C₆₁	—	2	4	no	—
208	C₂×C₄×C₆₁	—	2	4	no	—
209	C₂×C₄×C₆₁	—	2	2	no	—
210	C₂×C₄×C₆₁	—	2	2	no	—
211	C₂×C₄×C₆₁	—	2	32	no	—
212	C₂×C₄×C₆₁	—	2	4	no	—
213	C₂×C₄×C₆₁	—	2	4	no	—
214	C₂×C₄×C₆₁	—	2	2	no	—
215	C₂×C₄×C₆₁	—	2	2	no	—
216	C₂×C₄×C₆₁	—	2	2	no	—
217	C₂×C₄×C₆₁	—	2	2	no	—
218	C₂×C₄×C₆₁	—	2	4	no	—
219	C₂×C₄×C₆₁	—	2	4	no	—
220	C₂×C₄×C₆₁	—	2	4	no	—
221	C₂×C₄×C₆₁	—	2	4	no	—
222	C₂×C₄×C₆₁	—	2	4	no	—
223	C₂×C₄×C₆₁	—	2	4	no	—
224	C₂×C₄×C₆₁	—	2	2	no	—
225	C₂×C₄×C₆₁	—	2	4	no	—
226	C₂×C₄×C₆₁	—	2	2	no	—
227	C₂×C₄×C₆₁	—	4	8	no	—
228	C₂×C₄×C₆₁	—	4	8	no	—
229	C₂×C₄×C₆₁	—	4	4	no	—
230	C₂×C₄×C₆₁	—	4	8	no	—
231	C₂×C₄×C₆₁	—	4	4	no	—
232	C₂×C₄×C₆₁	C₂²⋊Q₈	1	16	yes	—
233	C₂×C₄×C₆₁	C₂²×D₄	1	8	yes	—
234	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
235	C₂×C₄×C₆₁	C₂²×Q₈	1	4	yes	—
236	C₂×C₄×C₆₁	C₂²×Q₈	1	4	yes	—
237	C₂×C₄×C₆₁	C₂×#41	1	4	yes	—
238	C₂×C₄×C₆₁	C₂×#41	1	4	yes	—
239	C₂×C₄×C₆₁	C₂×#41	1	4	yes	—
240	C₂×#2439	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
241	C₂²×C₆₁	—	2	4	no	—
242	C₂²×C₆₁	—	2	4	no	—
243	C₂²×C₆₁	—	2	4	no	—
244	C₂²×C₆₁	—	4	16	no	—
245	C₂²×C₆₁	—	4	2	no	—
246	C₂²×C₆₁	—	4	2	no	—
247	C₂²×C₆₁	—	4	2	no	—
248	C₂²×C₆₁	—	4	2	no	—
249	C₂²×C₆₁	C₂×C₂²⋊Q₈	1	4	yes	—
250	C₂²×C₆₁	#533	1	4	yes	—
251	C₂²×C₆₁	#544	1	4	yes	—
252	C₂²×C₆₁	#546	1	4	yes	—
253	C₂²×C₆₁	#548	1	4	yes	—
254	C₂²×C₆₁	#555	1	4	yes	—
255	C₂²×C₆₁	#556	1	4	yes	—
256	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
257	C₂²×C₆₁	C₂³×Q₈	1	1	yes	characteristic
258	C₂²×C₆₁	C₂²×#41	1	1	yes	characteristic
259	C₂²×C₆₁	C₂²×#41	1	1	yes	characteristic
260	C₂²×C₆₁	C₂²×#41	1	1	yes	characteristic
261	C₂²×C₆₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
262	C₂²×C₆₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
263	C₄×C₆₁	—	2	4	no	—
264	C₄×C₆₁	—	2	8	no	—
265	C₄×C₆₁	—	2	4	no	—
266	C₄×C₆₁	—	2	4	no	—
267	C₄×C₆₁	—	4	4	no	—
268	C₄×C₆₁	—	4	4	no	—
269	C₄×C₆₁	—	4	2	no	—
270	C₄×C₆₁	—	4	4	no	—
271	C₄×C₆₁	—	4	2	no	—
272	C₄×C₆₁	C₂×C₂²⋊Q₈	1	8	yes	—
273	C₂²×C₄⋊Q₈	C₂×C₆₁	1	1	yes	characteristic, complemented
274	C₂²×#117	C₂×C₆₁	1	1	yes	characteristic
275	#2439	C₂×C₆₁	1	16	yes	direct product factor
276	C₂×#396	C₂×C₆₁	1	2	yes	—
277	C₂×#393	C₂×C₆₁	1	2	yes	—
278	C₂×#391	C₂×C₆₁	1	2	yes	complemented
279	C₂×#389	C₂×C₆₁	1	1	yes	characteristic, complemented
280	C₂×#389	C₂×C₆₁	1	1	yes	characteristic, complemented
281	C₂×#387	C₂×C₆₁	1	2	yes	complemented
282	C₂×#382	C₂×C₆₁	1	1	yes	characteristic, complemented
283	C₂×#380	C₂×C₆₁	1	1	yes	characteristic
284	C₂×C₄×C₂²⋊C₄	C₂×C₆₁	1	1	yes	characteristic
285	C₂×C₆₁	—	4	4	no	—
286	C₂×C₆₁	#2439	1	8	yes	direct product factor
287	C₂×C₆₁	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
288	C₂×C₆₁	C₂×#533	1	1	yes	characteristic
289	C₂×C₆₁	C₂×#544	1	1	yes	characteristic
290	C₂×C₆₁	C₂×#546	1	1	yes	characteristic
291	C₂×C₆₁	C₂×#548	1	1	yes	characteristic
292	C₂×C₆₁	C₂×#555	1	1	yes	characteristic
293	C₂×C₆₁	C₂×#556	1	1	yes	characteristic
294	C₂³×Q₈	C₂²×C₆₁	1	2	yes	—
295	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
296	C₂×C₄⋊Q₈	—	2	8	no	—
297	C₂×C₄⋊Q₈	C₂²×C₆₁	1	8	yes	complemented
298	C₂×#117	—	2	8	no	—
299	C₂×#117	C₂²×C₆₁	1	8	yes	—
300	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
301	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
302	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
303	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
304	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
305	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
306	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
307	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
308	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
309	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
310	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
311	C₂²×C₄²	C₂²×C₆₁	1	2	yes	—
312	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
313	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
314	#396	C₂²×C₆₁	1	16	yes	—
315	#393	C₂²×C₆₁	1	16	yes	—
316	#391	C₂²×C₆₁	1	16	yes	complemented
317	#389	C₂²×C₆₁	1	8	yes	complemented
318	#389	C₂²×C₆₁	1	8	yes	complemented
319	#387	C₂²×C₆₁	1	16	yes	complemented
320	#382	C₂²×C₆₁	1	8	yes	complemented
321	#380	C₂²×C₆₁	1	8	yes	—
322	C₄×C₂²⋊C₄	C₂²×C₆₁	1	8	yes	—
323	C₂×#95	C₂²×C₆₁	1	2	yes	—
324	C₂×#95	C₂²×C₆₁	1	2	yes	—
325	C₂×#95	C₂²×C₆₁	1	2	yes	—
326	C₂×#95	C₂²×C₆₁	1	2	yes	—
327	C₂×#95	C₂²×C₆₁	1	1	yes	characteristic
328	C₂×#95	C₂²×C₆₁	1	1	yes	characteristic
329	C₂×#95	C₂²×C₆₁	1	2	yes	—
330	C₆₁	C₂×#2439	1	1	yes	characteristic, direct product factor, Sylow
331	C₂⁵	C₂³×C₆₁	1	1	yes	characteristic
332	C₂²×Q₈	—	2	8	no	—
333	C₂²×Q₈	—	2	8	no	—
334	C₂²×Q₈	—	4	4	no	—
335	C₂²×Q₈	C₂³×C₆₁	1	8	yes	—
336	C₂³×C₄	—	2	4	no	—
337	C₂³×C₄	—	2	4	no	—
338	C₂³×C₄	Q₈×C₆₁	1	8	yes	complemented
339	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
340	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
341	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
342	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
343	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
344	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
345	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
346	C₂³×C₄	C₂³×C₆₁	1	4	yes	—
347	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
348	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
349	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
350	C₄⋊Q₈	—	2	32	no	—
351	#117	—	2	32	no	—
352	C₂×C₄⋊C₄	—	2	4	no	—
353	C₂×C₄⋊C₄	—	2	8	no	—
354	C₂×C₄⋊C₄	—	2	4	no	—
355	C₂×C₄⋊C₄	—	2	8	no	—
356	C₂×C₄⋊C₄	—	2	4	no	—
357	C₂×C₄⋊C₄	—	4	4	no	—
358	C₂×C₄⋊C₄	D₄×C₆₁	1	8	yes	—
359	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
360	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
361	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
362	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
363	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
364	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
365	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
366	C₂×C₂²⋊C₄	—	2	8	no	—
367	C₂×C₂²⋊C₄	—	2	4	no	—
368	C₂×C₂²⋊C₄	—	2	4	no	—
369	C₂×C₂²⋊C₄	—	4	4	no	—
370	C₂×C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
371	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
372	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
373	C₂×C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
374	C₂×C₄²	—	2	8	no	—
375	C₂×C₄²	—	2	4	no	—
376	C₂×C₄²	—	2	4	no	—
377	C₂×C₄²	C₂³×C₆₁	1	8	yes	—
378	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
379	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
380	#95	C₂³×C₆₁	1	4	yes	—
381	#95	C₂³×C₆₁	1	4	yes	—
382	#95	C₂³×C₆₁	1	8	yes	—
383	#95	C₂³×C₆₁	1	8	yes	—
384	#95	C₂³×C₆₁	1	8	yes	—
385	#95	C₂³×C₆₁	1	8	yes	—
386	#95	C₂³×C₆₁	1	8	yes	—
387	C₂⁴	—	2	2	no	—
388	C₂⁴	—	2	2	no	—
389	C₂⁴	—	2	2	no	—
390	C₂⁴	—	2	2	no	—
391	C₂⁴	—	2	2	no	—
392	C₂⁴	—	2	2	no	—
393	C₂⁴	C₂×Q₈×C₆₁	1	4	yes	—
394	C₂⁴	C₂⁴×C₆₁	1	2	yes	—
395	C₂⁴	C₂⁴×C₆₁	1	1	yes	characteristic
396	C₂×Q₈	—	2	16	no	—
397	C₂×Q₈	—	2	16	no	—
398	C₂×Q₈	—	4	8	no	—
399	C₂×Q₈	—	4	32	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	4	no	—
406	C₂²×C₄	—	2	2	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	32	no	—
409	C₂²×C₄	—	2	8	no	—
410	C₂²×C₄	—	2	8	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	C₂×D₄×C₆₁	1	8	yes	—
427	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
428	C₂²×C₄	C₂×Q₈×C₆₁	1	4	yes	—
429	C₂²×C₄	C₂×Q₈×C₆₁	1	4	yes	—
430	C₂²×C₄	C₂×Q₈×C₆₁	1	16	yes	complemented
431	C₂²×C₄	#41×C₆₁	1	4	yes	—
432	C₂²×C₄	#41×C₆₁	1	4	yes	—
433	C₂²×C₄	#41×C₆₁	1	4	yes	—
434	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
435	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
436	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
437	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
438	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
439	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
440	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
441	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
442	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
444	C₄⋊C₄	—	2	8	no	—
445	C₄⋊C₄	—	2	16	no	—
446	C₄⋊C₄	—	2	8	no	—
447	C₄⋊C₄	—	2	16	no	—
448	C₄⋊C₄	—	2	8	no	—
449	C₄⋊C₄	—	4	8	no	—
450	C₄⋊C₄	C₂×D₄×C₆₁	1	16	yes	—
451	C₂²⋊C₄	—	2	16	no	—
452	C₂²⋊C₄	—	2	8	no	—
453	C₂²⋊C₄	—	2	8	no	—
454	C₂²⋊C₄	—	4	8	no	—
455	C₄²	—	2	16	no	—
456	C₄²	—	2	8	no	—
457	C₄²	—	2	8	no	—
458	C₂³	—	2	8	no	—
459	C₂³	—	2	8	no	—
460	C₂³	—	2	2	no	—
461	C₂³	—	2	2	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	2	2	no	—
464	C₂³	—	2	2	no	—
465	C₂³	—	2	2	no	—
466	C₂³	—	2	8	no	—
467	C₂³	—	4	4	no	—
468	C₂³	—	4	4	no	—
469	C₂³	—	4	4	no	—
470	C₂³	—	4	4	no	—
471	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
472	C₂³	C₂²×Q₈×C₆₁	1	4	yes	—
473	C₂³	C₂²×Q₈×C₆₁	1	2	yes	—
474	C₂³	C₂×#41×C₆₁	1	2	yes	—
475	C₂³	C₂×#41×C₆₁	1	2	yes	—
476	C₂³	C₂×#41×C₆₁	1	2	yes	—
477	C₂³	H(5,𝔽₂)×C₆₁	1	2	yes	—
478	C₂³	2₋¹⁺⁴×C₆₁	1	2	yes	—
479	C₂³	C₂⁵×C₆₁	1	1	yes	characteristic
480	Q₈	—	4	32	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	2	no	—
487	C₂×C₄	—	2	2	no	—
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	8	no	—
493	C₂×C₄	—	2	32	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	4	8	no	—
510	C₂×C₄	—	4	8	no	—
511	C₂×C₄	—	4	4	no	—
512	C₂×C₄	—	4	8	no	—
513	C₂×C₄	—	4	4	no	—
514	C₂×C₄	C₂²⋊Q₈×C₆₁	1	16	yes	—
515	C₂×C₄	C₂²×D₄×C₆₁	1	8	yes	—
516	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
517	C₂×C₄	C₂²×Q₈×C₆₁	1	4	yes	—
518	C₂×C₄	C₂²×Q₈×C₆₁	1	4	yes	—
519	C₂×C₄	C₂×#41×C₆₁	1	4	yes	—
520	C₂×C₄	C₂×#41×C₆₁	1	4	yes	—
521	C₂×C₄	C₂×#41×C₆₁	1	4	yes	—
522	C₂²	—	2	4	no	—
523	C₂²	—	2	4	no	—
524	C₂²	—	2	4	no	—
525	C₂²	—	4	16	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	4	2	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	2	no	—
530	C₂²	C₂×C₂²⋊Q₈×C₆₁	1	4	yes	—
531	C₂²	C₆₁×#533	1	4	yes	—
532	C₂²	C₆₁×#544	1	4	yes	—
533	C₂²	C₆₁×#546	1	4	yes	—
534	C₂²	C₆₁×#548	1	4	yes	—
535	C₂²	C₆₁×#555	1	4	yes	—
536	C₂²	C₆₁×#556	1	4	yes	—
537	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
538	C₂²	C₂³×Q₈×C₆₁	1	1	yes	characteristic
539	C₂²	C₂²×#41×C₆₁	1	1	yes	characteristic
540	C₂²	C₂²×#41×C₆₁	1	1	yes	characteristic
541	C₂²	C₂²×#41×C₆₁	1	1	yes	characteristic
542	C₂²	C₂×H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
543	C₂²	C₂×2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
544	C₄	—	2	4	no	—
545	C₄	—	2	8	no	—
546	C₄	—	2	4	no	—
547	C₄	—	2	4	no	—
548	C₄	—	4	4	no	—
549	C₄	—	4	4	no	—
550	C₄	—	4	2	no	—
551	C₄	—	4	4	no	—
552	C₄	—	4	2	no	—
553	C₄	C₂×C₂²⋊Q₈×C₆₁	1	8	yes	—
554	C₂	—	4	4	no	—
555	C₂	C₆₁×#2439	1	8	yes	direct product factor
556	C₂	C₂²×C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
557	C₂	C₂×C₆₁×#533	1	1	yes	characteristic
558	C₂	C₂×C₆₁×#544	1	1	yes	characteristic
559	C₂	C₂×C₆₁×#546	1	1	yes	characteristic
560	C₂	C₂×C₆₁×#548	1	1	yes	characteristic
561	C₂	C₂×C₆₁×#555	1	1	yes	characteristic
562	C₂	C₂×C₆₁×#556	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₁
prime radical	C₂⁵×C₆₁	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄⋊Q₈×C₆₁	C₂	semidirect
3. C₆₁×#2439	C₂	direct (N × H)
6. C₂×C₆₁×#391	C₂	semidirect
7. C₂×C₆₁×#389	C₂	semidirect
8. C₂×C₆₁×#389	C₂	semidirect
9. C₂×C₆₁×#387	C₂	semidirect
10. C₂×C₆₁×#382	C₂	semidirect
16. C₂×C₄⋊Q₈×C₆₁	C₂²	semidirect
35. C₆₁×#391	C₂²	semidirect
36. C₆₁×#389	C₂²	semidirect
37. C₆₁×#389	C₂²	semidirect
38. C₆₁×#387	C₂²	semidirect
39. C₆₁×#382	C₂²	semidirect
56. C₂³×C₄×C₆₁	Q₈	semidirect
148. C₂²×C₄×C₆₁	C₂×Q₈	semidirect
240. C₂×#2439	C₆₁	direct (N × H)
273. C₂²×C₄⋊Q₈	C₂×C₆₁	semidirect
275. #2439	C₂×C₆₁	direct (N × H)
278. C₂×#391	C₂×C₆₁	semidirect
279. C₂×#389	C₂×C₆₁	semidirect
280. C₂×#389	C₂×C₆₁	semidirect
281. C₂×#387	C₂×C₆₁	semidirect
282. C₂×#382	C₂×C₆₁	semidirect
286. C₂×C₆₁	#2439	direct (N × H)
297. C₂×C₄⋊Q₈	C₂²×C₆₁	semidirect
316. #391	C₂²×C₆₁	semidirect
317. #389	C₂²×C₆₁	semidirect
318. #389	C₂²×C₆₁	semidirect
319. #387	C₂²×C₆₁	semidirect
320. #382	C₂²×C₆₁	semidirect
330. C₆₁	C₂×#2439	direct (N × H)
338. C₂³×C₄	Q₈×C₆₁	semidirect
430. C₂²×C₄	C₂×Q₈×C₆₁	semidirect
555. C₂	C₆₁×#2439	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×C₄⋊Q₈×C₆₁	2
C₂²×#117×C₆₁	2
C₆₁×#2439	2
C₂×C₆₁×#396	2
C₂×C₆₁×#393	2
C₂×C₆₁×#391	2
C₂×C₆₁×#389	2
C₂×C₆₁×#387	2
C₂×C₆₁×#382	2
C₂×C₆₁×#380	2
C₂×C₄×C₂²⋊C₄×C₆₁	2
C₂×#2439	61

Subgroups — C2×C61×#2439 #29538145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#2439 #29538145