Subgroups — C₄₁×#20929 #23797130

Order 10496.

1954 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,ℤ/8)×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#518	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#416	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#102×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1879	C₂	1	4	yes	maximal, complemented
7	C₄₁×#1877	C₂	1	4	yes	maximal, complemented
8	C₄₁×#1736	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1729	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
11	C₂³×D₄×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
13	C₂×AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
14	C₂×AGL(1,ℤ/8)×C₄₁	C₄	1	8	yes	complemented
15	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
16	C₂²×D₈×C₄₁	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
19	C₄₁×#518	—	2	4	no	—
20	C₄₁×#518	—	2	4	no	—
21	C₂×C₄×D₄×C₄₁	—	2	2	no	—
22	C₂×C₄×D₄×C₄₁	C₂²	1	4	yes	—
23	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₄₁	—	2	2	no	—
27	C₄₁×#416	—	2	4	no	—
28	C₂×#103×C₄₁	C₂²	1	2	yes	complemented
29	C₂×#102×C₄₁	—	2	4	no	—
30	C₂×#102×C₄₁	C₂²	1	4	yes	—
31	C₂×#102×C₄₁	C₂²	1	2	yes	complemented
32	C₄₁×#391	C₂²	1	2	yes	complemented
33	C₄₁×#390	C₂²	1	2	yes	complemented
34	C₄₁×#386	—	2	2	no	—
35	C₄₁×#380	—	2	2	no	—
36	C₄₁×#340	C₂²	1	4	yes	complemented
37	C₄₁×#336	C₂²	1	4	yes	complemented
38	C₂⁵×C₄₁	—	2	1	no	—
39	C₂×#41×C₄₁	—	2	2	no	—
40	C₂×#41×C₄₁	—	2	2	no	—
41	C₂×#41×C₄₁	—	2	2	no	—
42	C₂×#41×C₄₁	—	2	2	no	—
43	C₂×#41×C₄₁	—	2	2	no	—
44	C₂×#41×C₄₁	C₂×C₄	1	4	yes	complemented
45	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₄₁	—	2	1	no	—
47	C₂²×D₄×C₄₁	—	2	1	no	—
48	C₂²×D₄×C₄₁	—	2	2	no	—
49	C₂²×D₄×C₄₁	—	2	2	no	—
50	C₂²×D₄×C₄₁	—	2	2	no	—
51	C₂²×D₄×C₄₁	C₂×C₄	1	4	yes	—
52	C₂²×D₄×C₄₁	D₄	1	4	yes	—
53	C₂²×D₄×C₄₁	D₄	1	4	yes	—
54	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
55	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
56	C₂³×C₄×C₄₁	—	2	1	no	—
57	C₂³×C₄×C₄₁	—	2	4	no	—
58	C₂³×C₄×C₄₁	—	2	1	no	—
59	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
60	AGL(1,ℤ/8)×C₄₁	—	2	32	no	—
61	C₂×SD₁₆×C₄₁	—	2	4	no	—
62	C₂×SD₁₆×C₄₁	—	2	4	no	—
63	C₂×SD₁₆×C₄₁	C₂×C₄	1	8	yes	complemented
64	C₂×D₈×C₄₁	—	2	4	no	—
65	C₂×D₈×C₄₁	—	2	4	no	—
66	C₂×D₈×C₄₁	C₂×C₄	1	8	yes	complemented
67	C₂×M₁₆×C₄₁	—	2	2	no	—
68	C₂×M₁₆×C₄₁	—	2	2	no	—
69	C₂×M₁₆×C₄₁	C₂×C₄	1	4	yes	complemented
70	C₂²×C₈×C₄₁	—	2	2	no	—
71	C₂²×C₈×C₄₁	—	2	2	no	—
72	C₂²×C₈×C₄₁	—	2	2	no	—
73	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
74	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
75	C₄×D₄×C₄₁	—	2	16	no	—
76	C₄×D₄×C₄₁	—	2	8	no	—
77	C₄×D₄×C₄₁	—	2	8	no	—
78	#117×C₄₁	—	2	2	no	—
79	#117×C₄₁	—	2	2	no	—
80	#117×C₄₁	—	2	2	no	—
81	#117×C₄₁	D₄	1	4	yes	—
82	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
83	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
84	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
86	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
88	C₂×C₂²⋊C₄×C₄₁	—	2	16	no	—
89	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
91	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
92	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
93	C₂×C₄²×C₄₁	—	2	1	no	—
94	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
95	#103×C₄₁	—	2	4	no	—
96	#102×C₄₁	—	2	8	no	—
97	#102×C₄₁	—	2	4	no	—
98	#95×C₄₁	—	2	4	no	—
99	C₂⁴×C₄₁	—	2	2	no	—
100	C₂⁴×C₄₁	—	2	4	no	—
101	C₂⁴×C₄₁	—	2	4	no	—
102	C₂⁴×C₄₁	—	2	4	no	—
103	C₂⁴×C₄₁	—	2	1	no	—
104	C₂⁴×C₄₁	—	8	2	no	—
105	C₂⁴×C₄₁	—	8	2	no	—
106	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
107	#41×C₄₁	—	2	8	no	—
108	#41×C₄₁	—	2	8	no	—
109	#41×C₄₁	—	4	4	no	—
110	#41×C₄₁	—	4	4	no	—
111	C₂×Q₈×C₄₁	—	2	1	no	—
112	C₂×Q₈×C₄₁	—	2	1	no	—
113	C₂×Q₈×C₄₁	—	2	1	no	—
114	C₂×Q₈×C₄₁	—	2	1	no	—
115	C₂×Q₈×C₄₁	—	2	1	no	—
116	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
117	C₂×D₄×C₄₁	—	2	4	no	—
118	C₂×D₄×C₄₁	—	2	8	no	—
119	C₂×D₄×C₄₁	—	2	4	no	—
120	C₂×D₄×C₄₁	—	2	4	no	—
121	C₂×D₄×C₄₁	—	2	8	no	—
122	C₂×D₄×C₄₁	—	2	4	no	—
123	C₂×D₄×C₄₁	—	2	8	no	—
124	C₂×D₄×C₄₁	—	2	2	no	—
125	C₂×D₄×C₄₁	—	2	1	no	—
126	C₂×D₄×C₄₁	—	2	1	no	—
127	C₂×D₄×C₄₁	—	2	1	no	—
128	C₂×D₄×C₄₁	—	2	2	no	—
129	C₂×D₄×C₄₁	—	2	2	no	—
130	C₂×D₄×C₄₁	—	2	1	no	—
131	C₂×D₄×C₄₁	—	2	1	no	—
132	C₂×D₄×C₄₁	—	4	2	no	—
133	C₂×D₄×C₄₁	—	4	2	no	—
134	C₂×D₄×C₄₁	C₂²⋊C₄	1	16	yes	complemented
135	C₂×D₄×C₄₁	C₂²×C₄	1	2	yes	—
136	C₂×D₄×C₄₁	C₂²×C₄	1	4	yes	—
137	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
138	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
139	C₂²×C₄×C₄₁	—	2	1	no	—
140	C₂²×C₄×C₄₁	—	2	2	no	—
141	C₂²×C₄×C₄₁	—	2	16	no	—
142	C₂²×C₄×C₄₁	—	2	4	no	—
143	C₂²×C₄×C₄₁	—	2	2	no	—
144	C₂²×C₄×C₄₁	—	4	2	no	—
145	C₂²×C₄×C₄₁	—	4	2	no	—
146	C₂²×C₄×C₄₁	—	4	2	no	—
147	C₂²×C₄×C₄₁	—	8	4	no	—
148	C₂²×C₄×C₄₁	—	8	4	no	—
149	C₂²×C₄×C₄₁	—	8	2	no	—
150	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
151	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
152	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₄₁	#41	1	2	yes	—
157	C₂²×C₄×C₄₁	#41	1	2	yes	—
158	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
159	SD₁₆×C₄₁	—	2	16	no	—
160	D₈×C₄₁	—	2	16	no	—
161	M₁₆×C₄₁	—	2	8	no	—
162	C₂×C₈×C₄₁	—	2	2	no	—
163	C₂×C₈×C₄₁	—	2	2	no	—
164	C₂×C₈×C₄₁	—	2	2	no	—
165	C₂×C₈×C₄₁	—	2	2	no	—
166	C₂×C₈×C₄₁	—	2	2	no	—
167	C₂×C₈×C₄₁	—	4	4	no	—
168	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
169	C₄⋊C₄×C₄₁	—	2	4	no	—
170	C₄⋊C₄×C₄₁	—	2	2	no	—
171	C₄⋊C₄×C₄₁	—	2	2	no	—
172	C₄⋊C₄×C₄₁	—	2	2	no	—
173	C₄⋊C₄×C₄₁	—	4	2	no	—
174	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
175	C₂²⋊C₄×C₄₁	—	2	16	no	—
176	C₂²⋊C₄×C₄₁	—	2	4	no	—
177	C₂²⋊C₄×C₄₁	—	2	4	no	—
178	C₂²⋊C₄×C₄₁	—	8	4	no	—
179	C₄²×C₄₁	—	2	4	no	—
180	C₄²×C₄₁	—	2	2	no	—
181	C₄²×C₄₁	—	2	2	no	—
182	C₂³×C₄₁	—	2	2	no	—
183	C₂³×C₄₁	—	2	2	no	—
184	C₂³×C₄₁	—	2	4	no	—
185	C₂³×C₄₁	—	2	4	no	—
186	C₂³×C₄₁	—	2	4	no	—
187	C₂³×C₄₁	—	2	16	no	—
188	C₂³×C₄₁	—	2	2	no	—
189	C₂³×C₄₁	—	4	4	no	—
190	C₂³×C₄₁	—	4	4	no	—
191	C₂³×C₄₁	—	4	1	no	—
192	C₂³×C₄₁	—	4	1	no	—
193	C₂³×C₄₁	—	4	2	no	—
194	C₂³×C₄₁	—	4	2	no	—
195	C₂³×C₄₁	—	8	8	no	—
196	C₂³×C₄₁	—	8	8	no	—
197	C₂³×C₄₁	—	8	4	no	—
198	C₂³×C₄₁	—	8	4	no	—
199	C₂³×C₄₁	C₄×D₄	1	2	yes	—
200	C₂³×C₄₁	#120	1	2	yes	—
201	C₂³×C₄₁	#121	1	2	yes	—
202	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
203	Q₈×C₄₁	—	2	2	no	—
204	Q₈×C₄₁	—	2	2	no	—
205	Q₈×C₄₁	—	4	1	no	—
206	Q₈×C₄₁	—	4	1	no	—
207	D₄×C₄₁	—	2	2	no	—
208	D₄×C₄₁	—	2	4	no	—
209	D₄×C₄₁	—	2	2	no	—
210	D₄×C₄₁	—	2	4	no	—
211	D₄×C₄₁	—	2	4	no	—
212	D₄×C₄₁	—	2	16	no	—
213	D₄×C₄₁	—	4	8	no	—
214	D₄×C₄₁	—	4	1	no	—
215	D₄×C₄₁	—	4	1	no	—
216	D₄×C₄₁	C₂×C₂²⋊C₄	1	8	yes	—
217	C₂×C₄×C₄₁	—	2	2	no	—
218	C₂×C₄×C₄₁	—	2	4	no	—
219	C₂×C₄×C₄₁	—	2	4	no	—
220	C₂×C₄×C₄₁	—	2	4	no	—
221	C₂×C₄×C₄₁	—	2	4	no	—
222	C₂×C₄×C₄₁	—	4	1	no	—
223	C₂×C₄×C₄₁	—	4	1	no	—
224	C₂×C₄×C₄₁	—	4	2	no	—
225	C₂×C₄×C₄₁	—	4	2	no	—
226	C₂×C₄×C₄₁	—	4	8	no	—
227	C₂×C₄×C₄₁	—	4	2	no	—
228	C₂×C₄×C₄₁	—	4	4	no	—
229	C₂×C₄×C₄₁	—	8	16	no	—
230	C₂×C₄×C₄₁	—	8	4	no	—
231	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
232	C₂×C₄×C₄₁	C₄×D₄	1	4	yes	—
233	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
234	C₂×C₄×C₄₁	#120	1	2	yes	—
235	C₂×C₄×C₄₁	#121	1	4	yes	—
236	C₂×C₄×C₄₁	#121	1	2	yes	—
237	C₂×C₄×C₄₁	#123	1	4	yes	—
238	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
239	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
243	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
244	C₈×C₄₁	—	2	4	no	—
245	C₈×C₄₁	—	4	2	no	—
246	#20929	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
247	C₂²×C₄₁	—	2	4	no	—
248	C₂²×C₄₁	—	2	4	no	—
249	C₂²×C₄₁	—	2	2	no	—
250	C₂²×C₄₁	—	2	2	no	—
251	C₂²×C₄₁	—	2	8	no	—
252	C₂²×C₄₁	—	4	8	no	—
253	C₂²×C₄₁	—	4	8	no	—
254	C₂²×C₄₁	—	4	4	no	—
255	C₂²×C₄₁	—	4	4	no	—
256	C₂²×C₄₁	—	4	4	no	—
257	C₂²×C₄₁	—	4	2	no	—
258	C₂²×C₄₁	—	8	16	no	—
259	C₂²×C₄₁	—	8	2	no	—
260	C₂²×C₄₁	—	8	1	no	—
261	C₂²×C₄₁	—	8	1	no	—
262	C₂²×C₄₁	#386	1	4	yes	—
263	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
264	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
265	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
266	C₂²×C₄₁	#576	1	2	yes	—
267	C₂²×C₄₁	#577	1	2	yes	—
268	C₄×C₄₁	—	4	4	no	—
269	C₄×C₄₁	—	4	4	no	—
270	C₄×C₄₁	—	4	2	no	—
271	C₄×C₄₁	—	8	2	no	—
272	C₄×C₄₁	#386	1	4	yes	—
273	C₄×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
274	C₄×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
275	C₄×C₄₁	C₂×#121	1	1	yes	characteristic
276	C₄×C₄₁	C₂×#123	1	1	yes	characteristic
277	C₂²×AGL(1,ℤ/8)	C₂×C₄₁	1	1	yes	characteristic
278	C₂×#518	C₂×C₄₁	1	1	yes	characteristic, complemented
279	C₂×#416	C₂×C₄₁	1	1	yes	characteristic, complemented
280	C₂²×#102	C₂×C₄₁	1	1	yes	characteristic, complemented
281	#2311	C₂×C₄₁	1	1	yes	characteristic, complemented
282	#1879	C₂×C₄₁	1	4	yes	complemented
283	#1877	C₂×C₄₁	1	4	yes	complemented
284	#1736	C₂×C₄₁	1	1	yes	characteristic, complemented
285	#1729	C₂×C₄₁	1	1	yes	characteristic, complemented
286	C₂×C₄₁	—	2	4	no	—
287	C₂×C₄₁	—	4	8	no	—
288	C₂×C₄₁	—	8	2	no	—
289	C₂×C₄₁	C₂×#386	1	1	yes	characteristic
290	C₂×C₄₁	#2946	1	2	yes	—
291	C₂×C₄₁	#3012	1	2	yes	—
292	C₂×C₄₁	#3041	1	1	yes	characteristic
293	C₂×C₄₁	#3042	1	1	yes	characteristic
294	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
295	C₂³×D₄	C₂²×C₄₁	1	1	yes	characteristic
296	C₂×AGL(1,ℤ/8)	—	2	4	no	—
297	C₂×AGL(1,ℤ/8)	—	2	4	no	—
298	C₂×AGL(1,ℤ/8)	C₄×C₄₁	1	8	yes	complemented
299	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
300	C₂²×D₈	C₂²×C₄₁	1	2	yes	—
301	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
302	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
303	#518	—	2	4	no	—
304	#518	—	2	4	no	—
305	C₂×C₄×D₄	—	2	2	no	—
306	C₂×C₄×D₄	C₂²×C₄₁	1	4	yes	—
307	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
308	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
309	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
310	C₂²×C₂²⋊C₄	—	2	2	no	—
311	#416	—	2	4	no	—
312	C₂×#103	C₂²×C₄₁	1	2	yes	complemented
313	C₂×#102	—	2	4	no	—
314	C₂×#102	C₂²×C₄₁	1	4	yes	—
315	C₂×#102	C₂²×C₄₁	1	2	yes	complemented
316	#391	C₂²×C₄₁	1	2	yes	complemented
317	#390	C₂²×C₄₁	1	2	yes	complemented
318	#386	—	2	2	no	—
319	#380	—	2	2	no	—
320	#340	C₂²×C₄₁	1	4	yes	complemented
321	#336	C₂²×C₄₁	1	4	yes	complemented
322	C₄₁	#20929	1	1	yes	characteristic, direct product factor, Sylow
323	C₂⁵	—	2	1	no	—
324	C₂×#41	—	2	2	no	—
325	C₂×#41	—	2	2	no	—
326	C₂×#41	—	2	2	no	—
327	C₂×#41	—	2	2	no	—
328	C₂×#41	—	2	2	no	—
329	C₂×#41	C₂×C₄×C₄₁	1	4	yes	complemented
330	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
331	C₂²×D₄	—	2	1	no	—
332	C₂²×D₄	—	2	1	no	—
333	C₂²×D₄	—	2	2	no	—
334	C₂²×D₄	—	2	2	no	—
335	C₂²×D₄	—	2	2	no	—
336	C₂²×D₄	C₂×C₄×C₄₁	1	4	yes	—
337	C₂²×D₄	D₄×C₄₁	1	4	yes	—
338	C₂²×D₄	D₄×C₄₁	1	4	yes	—
339	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
340	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
341	C₂³×C₄	—	2	1	no	—
342	C₂³×C₄	—	2	4	no	—
343	C₂³×C₄	—	2	1	no	—
344	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
345	AGL(1,ℤ/8)	—	2	32	no	—
346	C₂×SD₁₆	—	2	4	no	—
347	C₂×SD₁₆	—	2	4	no	—
348	C₂×SD₁₆	C₂×C₄×C₄₁	1	8	yes	complemented
349	C₂×D₈	—	2	4	no	—
350	C₂×D₈	—	2	4	no	—
351	C₂×D₈	C₂×C₄×C₄₁	1	8	yes	complemented
352	C₂×M₁₆	—	2	2	no	—
353	C₂×M₁₆	—	2	2	no	—
354	C₂×M₁₆	C₂×C₄×C₄₁	1	4	yes	complemented
355	C₂²×C₈	—	2	2	no	—
356	C₂²×C₈	—	2	2	no	—
357	C₂²×C₈	—	2	2	no	—
358	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
359	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
360	C₄×D₄	—	2	16	no	—
361	C₄×D₄	—	2	8	no	—
362	C₄×D₄	—	2	8	no	—
363	#117	—	2	2	no	—
364	#117	—	2	2	no	—
365	#117	—	2	2	no	—
366	#117	D₄×C₄₁	1	4	yes	—
367	C₂×C₄⋊C₄	—	2	2	no	—
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
371	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
373	C₂×C₂²⋊C₄	—	2	16	no	—
374	C₂×C₂²⋊C₄	—	2	1	no	—
375	C₂×C₂²⋊C₄	—	2	4	no	—
376	C₂×C₂²⋊C₄	—	2	4	no	—
377	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
378	C₂×C₄²	—	2	1	no	—
379	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
380	#103	—	2	4	no	—
381	#102	—	2	8	no	—
382	#102	—	2	4	no	—
383	#95	—	2	4	no	—
384	C₂⁴	—	2	2	no	—
385	C₂⁴	—	2	4	no	—
386	C₂⁴	—	2	4	no	—
387	C₂⁴	—	2	4	no	—
388	C₂⁴	—	2	1	no	—
389	C₂⁴	—	8	2	no	—
390	C₂⁴	—	8	2	no	—
391	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
392	#41	—	2	8	no	—
393	#41	—	2	8	no	—
394	#41	—	4	4	no	—
395	#41	—	4	4	no	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	—	2	1	no	—
398	C₂×Q₈	—	2	1	no	—
399	C₂×Q₈	—	2	1	no	—
400	C₂×Q₈	—	2	1	no	—
401	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	2	8	no	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	4	no	—
406	C₂×D₄	—	2	8	no	—
407	C₂×D₄	—	2	4	no	—
408	C₂×D₄	—	2	8	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	1	no	—
411	C₂×D₄	—	2	1	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	2	2	no	—
414	C₂×D₄	—	2	2	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	—	4	2	no	—
418	C₂×D₄	—	4	2	no	—
419	C₂×D₄	C₂²⋊C₄×C₄₁	1	16	yes	complemented
420	C₂×D₄	C₂²×C₄×C₄₁	1	2	yes	—
421	C₂×D₄	C₂²×C₄×C₄₁	1	4	yes	—
422	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
423	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	2	16	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	4	2	no	—
430	C₂²×C₄	—	4	2	no	—
431	C₂²×C₄	—	4	2	no	—
432	C₂²×C₄	—	8	4	no	—
433	C₂²×C₄	—	8	4	no	—
434	C₂²×C₄	—	8	2	no	—
435	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
436	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
437	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
438	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
439	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
441	C₂²×C₄	#41×C₄₁	1	2	yes	—
442	C₂²×C₄	#41×C₄₁	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
444	SD₁₆	—	2	16	no	—
445	D₈	—	2	16	no	—
446	M₁₆	—	2	8	no	—
447	C₂×C₈	—	2	2	no	—
448	C₂×C₈	—	2	2	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	—	2	2	no	—
452	C₂×C₈	—	4	4	no	—
453	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	4	2	no	—
459	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
460	C₂²⋊C₄	—	2	16	no	—
461	C₂²⋊C₄	—	2	4	no	—
462	C₂²⋊C₄	—	2	4	no	—
463	C₂²⋊C₄	—	8	4	no	—
464	C₄²	—	2	4	no	—
465	C₄²	—	2	2	no	—
466	C₄²	—	2	2	no	—
467	C₂³	—	2	2	no	—
468	C₂³	—	2	2	no	—
469	C₂³	—	2	4	no	—
470	C₂³	—	2	4	no	—
471	C₂³	—	2	4	no	—
472	C₂³	—	2	16	no	—
473	C₂³	—	2	2	no	—
474	C₂³	—	4	4	no	—
475	C₂³	—	4	4	no	—
476	C₂³	—	4	1	no	—
477	C₂³	—	4	1	no	—
478	C₂³	—	4	2	no	—
479	C₂³	—	4	2	no	—
480	C₂³	—	8	4	no	—
481	C₂³	—	8	4	no	—
482	C₂³	—	8	8	no	—
483	C₂³	—	8	8	no	—
484	C₂³	C₄×D₄×C₄₁	1	2	yes	—
485	C₂³	#120×C₄₁	1	2	yes	—
486	C₂³	#121×C₄₁	1	2	yes	—
487	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
488	Q₈	—	2	2	no	—
489	Q₈	—	2	2	no	—
490	Q₈	—	4	1	no	—
491	Q₈	—	4	1	no	—
492	D₄	—	2	2	no	—
493	D₄	—	2	4	no	—
494	D₄	—	2	2	no	—
495	D₄	—	2	4	no	—
496	D₄	—	2	4	no	—
497	D₄	—	2	16	no	—
498	D₄	—	4	8	no	—
499	D₄	—	4	1	no	—
500	D₄	—	4	1	no	—
501	D₄	C₂×C₂²⋊C₄×C₄₁	1	8	yes	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	4	1	no	—
508	C₂×C₄	—	4	1	no	—
509	C₂×C₄	—	4	2	no	—
510	C₂×C₄	—	4	2	no	—
511	C₂×C₄	—	4	8	no	—
512	C₂×C₄	—	4	2	no	—
513	C₂×C₄	—	4	4	no	—
514	C₂×C₄	—	8	16	no	—
515	C₂×C₄	—	8	4	no	—
516	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
517	C₂×C₄	C₄×D₄×C₄₁	1	4	yes	—
518	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
519	C₂×C₄	#120×C₄₁	1	2	yes	—
520	C₂×C₄	#121×C₄₁	1	4	yes	—
521	C₂×C₄	#121×C₄₁	1	2	yes	—
522	C₂×C₄	#123×C₄₁	1	4	yes	—
523	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
524	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
525	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
526	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
527	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
528	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
529	C₈	—	2	4	no	—
530	C₈	—	4	2	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	2	2	no	—
533	C₂²	—	2	4	no	—
534	C₂²	—	2	4	no	—
535	C₂²	—	2	8	no	—
536	C₂²	—	4	4	no	—
537	C₂²	—	4	4	no	—
538	C₂²	—	4	8	no	—
539	C₂²	—	4	8	no	—
540	C₂²	—	4	2	no	—
541	C₂²	—	4	4	no	—
542	C₂²	—	8	16	no	—
543	C₂²	—	8	1	no	—
544	C₂²	—	8	1	no	—
545	C₂²	—	8	2	no	—
546	C₂²	C₄₁×#386	1	4	yes	—
547	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
548	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
549	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
550	C₂²	C₄₁×#576	1	2	yes	—
551	C₂²	C₄₁×#577	1	2	yes	—
552	C₄	—	4	2	no	—
553	C₄	—	4	4	no	—
554	C₄	—	4	4	no	—
555	C₄	—	8	2	no	—
556	C₄	C₄₁×#386	1	4	yes	—
557	C₄	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
558	C₄	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
559	C₄	C₂×#121×C₄₁	1	1	yes	characteristic
560	C₄	C₂×#123×C₄₁	1	1	yes	characteristic
561	C₂	—	2	4	no	—
562	C₂	—	4	8	no	—
563	C₂	—	8	2	no	—
564	C₂	C₂×C₄₁×#386	1	1	yes	characteristic
565	C₂	C₄₁×#2946	1	2	yes	—
566	C₂	C₄₁×#3012	1	2	yes	—
567	C₂	C₄₁×#3041	1	1	yes	characteristic
568	C₂	C₄₁×#3042	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂²×AGL(1,ℤ/8)×C₄₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×#117×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#518	C₂	semidirect
3. C₂×C₄₁×#416	C₂	semidirect
4. C₂²×#102×C₄₁	C₂	semidirect
5. C₄₁×#2311	C₂	semidirect
6. C₄₁×#1879	C₂	semidirect
7. C₄₁×#1877	C₂	semidirect
8. C₄₁×#1736	C₂	semidirect
9. C₄₁×#1729	C₂	semidirect
14. C₂×AGL(1,ℤ/8)×C₄₁	C₄	semidirect
28. C₂×#103×C₄₁	C₂²	semidirect
31. C₂×#102×C₄₁	C₂²	semidirect
32. C₄₁×#391	C₂²	semidirect
33. C₄₁×#390	C₂²	semidirect
36. C₄₁×#340	C₂²	semidirect
37. C₄₁×#336	C₂²	semidirect
44. C₂×#41×C₄₁	C₂×C₄	semidirect
63. C₂×SD₁₆×C₄₁	C₂×C₄	semidirect
66. C₂×D₈×C₄₁	C₂×C₄	semidirect
69. C₂×M₁₆×C₄₁	C₂×C₄	semidirect
134. C₂×D₄×C₄₁	C₂²⋊C₄	semidirect
246. #20929	C₄₁	direct (N × H)
278. C₂×#518	C₂×C₄₁	semidirect
279. C₂×#416	C₂×C₄₁	semidirect
280. C₂²×#102	C₂×C₄₁	semidirect
281. #2311	C₂×C₄₁	semidirect
282. #1879	C₂×C₄₁	semidirect
283. #1877	C₂×C₄₁	semidirect
284. #1736	C₂×C₄₁	semidirect
285. #1729	C₂×C₄₁	semidirect
298. C₂×AGL(1,ℤ/8)	C₄×C₄₁	semidirect
312. C₂×#103	C₂²×C₄₁	semidirect
315. C₂×#102	C₂²×C₄₁	semidirect
316. #391	C₂²×C₄₁	semidirect
317. #390	C₂²×C₄₁	semidirect
320. #340	C₂²×C₄₁	semidirect
321. #336	C₂²×C₄₁	semidirect
322. C₄₁	#20929	direct (N × H)
329. C₂×#41	C₂×C₄×C₄₁	semidirect
348. C₂×SD₁₆	C₂×C₄×C₄₁	semidirect
351. C₂×D₈	C₂×C₄×C₄₁	semidirect
354. C₂×M₁₆	C₂×C₄×C₄₁	semidirect
419. C₂×D₄	C₂²⋊C₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×AGL(1,ℤ/8)×C₄₁	2
C₂×C₄₁×#518	2
C₂×C₄₁×#416	2
C₂²×#102×C₄₁	2
C₄₁×#2311	2
C₄₁×#1879	2
C₄₁×#1877	2
C₄₁×#1736	2
C₄₁×#1729	2
#20929	41

Subgroups — C41×#20929 #23797130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20929 #23797130