Subgroups — C₄₁×#18028 #23794229

Order 10496.

616 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3101	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2986	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#2974	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1592	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1590	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1588	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1585	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1583	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1582	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₁×#1572	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄₁×#1497	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄₁×#1478	C₂	1	1	yes	maximal, characteristic
15	C₄₁×#1454	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄₁×#533	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
18	C₄₁×#483	C₄	1	1	yes	characteristic
19	C₄₁×#483	C₄	1	1	yes	characteristic
20	C₄₁×#476	C₄	1	1	yes	characteristic
21	C₄₁×#476	C₄	1	1	yes	characteristic
22	C₄₁×#461	C₄	1	1	yes	characteristic
23	C₄₁×#461	C₄	1	1	yes	characteristic
24	C₄₁×#450	C₄	1	1	yes	characteristic
25	C₄₁×#450	C₄	1	1	yes	characteristic
26	C₄₁×#446	—	2	1	no	—
27	C₄₁×#446	C₂²	1	1	yes	characteristic, complemented
28	C₄₁×#446	C₂²	1	1	yes	characteristic, complemented
29	C₈×Q₈×C₄₁	—	2	1	no	—
30	C₈×Q₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
31	C₈×Q₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
32	C₄×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
33	C₄×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
34	C₄×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
35	C₄×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
36	C₄₁×#436	C₂²	1	1	yes	characteristic
37	C₄₁×#436	C₂²	1	1	yes	characteristic
38	C₄₁×#435	—	2	1	no	—
39	C₄₁×#435	—	2	1	no	—
40	C₈×D₄×C₄₁	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₄₁	C₂²	1	1	yes	characteristic
42	C₄₁×#433	—	2	1	no	—
43	C₄₁×#433	—	2	1	no	—
44	C₄₁×#432	C₂²	1	1	yes	characteristic
45	C₄₁×#431	C₂²	1	1	yes	characteristic
46	C₄₁×#431	C₂²	1	1	yes	characteristic
47	C₄₁×#431	C₂²	1	1	yes	characteristic
48	C₄₁×#431	C₂²	1	1	yes	characteristic
49	C₄₁×#407	—	2	1	no	—
50	C₂×C₂²⋊C₈×C₄₁	—	2	1	no	—
51	C₄₁×#335	C₂²	1	1	yes	characteristic, complemented
52	C₄₁×#335	C₂²	1	1	yes	characteristic, complemented
53	C₄₁×#334	C₂²	1	1	yes	characteristic, complemented
54	C₄₁×#334	C₂²	1	1	yes	characteristic, complemented
55	C₄₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
56	C₄₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
57	C₄₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
58	C₄₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
59	C₄₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
60	C₄₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
61	C₄₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
62	C₄₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
63	C₄₁×#322	C₂²	1	1	yes	characteristic, complemented
64	C₄₁×#322	C₂²	1	1	yes	characteristic, complemented
65	C₄₁×#322	C₂²	1	1	yes	characteristic, complemented
66	C₈²×C₄₁	C₂²	1	1	yes	characteristic, complemented
67	C₂²×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
68	C₂³×C₄×C₄₁	—	2	1	no	—
69	C₂×SD₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
70	C₂×SD₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
71	C₂×SD₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
72	C₂×SD₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
73	C₂×M₁₆×C₄₁	—	2	1	no	—
74	C₂×M₁₆×C₄₁	—	2	1	no	—
75	C₂²×C₈×C₄₁	—	2	1	no	—
76	C₂²×C₈×C₄₁	—	2	1	no	—
77	C₄⋊Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
78	C₄⋊Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
79	#125×C₄₁	—	2	1	no	—
80	C₂²⋊Q₈×C₄₁	—	2	1	no	—
81	C₂²⋊Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
82	C₂²⋊Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₄₁	—	2	1	no	—
84	C₄×Q₈×C₄₁	C₂³	1	1	yes	characteristic
85	C₄×Q₈×C₄₁	C₂³	1	1	yes	characteristic
86	C₄×D₄×C₄₁	—	2	1	no	—
87	C₄×D₄×C₄₁	—	2	1	no	—
88	C₄×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
89	C₄×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
90	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
91	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
92	#117×C₄₁	—	2	1	no	—
93	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
95	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
96	#106×C₄₁	C₂×C₄	1	1	yes	characteristic
97	#106×C₄₁	C₂×C₄	1	1	yes	characteristic
98	#106×C₄₁	C₂×C₄	1	1	yes	characteristic
99	#106×C₄₁	C₂×C₄	1	1	yes	characteristic
100	C₄⋊C₈×C₄₁	—	2	1	no	—
101	C₄⋊C₈×C₄₁	—	2	1	no	—
102	C₄⋊C₈×C₄₁	—	2	1	no	—
103	C₄⋊C₈×C₄₁	—	2	1	no	—
104	C₄⋊C₈×C₄₁	—	2	1	no	—
105	C₄⋊C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
106	C₄⋊C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
107	C₄⋊C₈×C₄₁	D₄	1	1	yes	characteristic
108	C₄⋊C₈×C₄₁	D₄	1	1	yes	characteristic
109	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
110	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
111	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
112	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
113	#103×C₄₁	C₂×C₄	1	1	yes	characteristic
114	#103×C₄₁	C₂×C₄	1	1	yes	characteristic
115	#103×C₄₁	C₂×C₄	1	1	yes	characteristic
116	#103×C₄₁	C₂×C₄	1	1	yes	characteristic
117	#102×C₄₁	C₂×C₄	1	1	yes	characteristic
118	#102×C₄₁	C₂×C₄	1	1	yes	characteristic
119	#102×C₄₁	C₂×C₄	1	1	yes	characteristic
120	#102×C₄₁	C₂×C₄	1	1	yes	characteristic
121	C₂²⋊C₈×C₄₁	—	2	1	no	—
122	C₂²⋊C₈×C₄₁	—	2	1	no	—
123	C₂²⋊C₈×C₄₁	—	2	1	no	—
124	C₂²⋊C₈×C₄₁	—	2	1	no	—
125	C₂²⋊C₈×C₄₁	—	2	1	no	—
126	C₂²⋊C₈×C₄₁	—	2	1	no	—
127	C₂²⋊C₈×C₄₁	—	2	1	no	—
128	C₂²⋊C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
129	C₂²⋊C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
130	C₂²⋊C₈×C₄₁	D₄	1	1	yes	characteristic
131	C₂²⋊C₈×C₄₁	D₄	1	1	yes	characteristic
132	#97×C₄₁	—	2	1	no	—
133	#97×C₄₁	—	2	1	no	—
134	#97×C₄₁	—	2	1	no	—
135	C₄×C₈×C₄₁	—	2	1	no	—
136	C₄×C₈×C₄₁	—	2	1	no	—
137	C₄×C₈×C₄₁	—	2	1	no	—
138	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
144	C₂⁴×C₄₁	—	2	1	no	—
145	C₂×Q₈×C₄₁	—	2	1	no	—
146	C₂×Q₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
147	C₂×Q₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
148	C₂×D₄×C₄₁	—	2	2	no	—
149	C₂×D₄×C₄₁	—	2	1	no	—
150	C₂×D₄×C₄₁	—	4	1	no	—
151	C₂×D₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
152	C₂×D₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₄₁	—	2	1	no	—
154	C₂²×C₄×C₄₁	—	2	1	no	—
155	C₂²×C₄×C₄₁	—	2	1	no	—
156	C₂²×C₄×C₄₁	—	2	1	no	—
157	C₂²×C₄×C₄₁	—	2	1	no	—
158	C₂²×C₄×C₄₁	—	8	1	no	—
159	C₂²×C₄×C₄₁	—	8	1	no	—
160	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
163	SD₁₆×C₄₁	—	2	2	no	—
164	SD₁₆×C₄₁	—	2	2	no	—
165	SD₁₆×C₄₁	—	4	1	no	—
166	SD₁₆×C₄₁	—	4	1	no	—
167	M₁₆×C₄₁	—	8	1	no	—
168	M₁₆×C₄₁	—	8	1	no	—
169	C₂×C₈×C₄₁	—	2	1	no	—
170	C₂×C₈×C₄₁	—	2	1	no	—
171	C₂×C₈×C₄₁	—	2	1	no	—
172	C₂×C₈×C₄₁	—	2	1	no	—
173	C₂×C₈×C₄₁	—	2	1	no	—
174	C₂×C₈×C₄₁	—	2	1	no	—
175	C₂×C₈×C₄₁	—	8	1	no	—
176	C₂×C₈×C₄₁	—	8	1	no	—
177	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
178	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
179	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
180	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
181	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
185	C₄⋊C₄×C₄₁	—	2	1	no	—
186	C₄⋊C₄×C₄₁	—	2	1	no	—
187	C₄⋊C₄×C₄₁	—	2	1	no	—
188	C₄⋊C₄×C₄₁	—	2	1	no	—
189	C₄⋊C₄×C₄₁	—	2	1	no	—
190	C₄⋊C₄×C₄₁	—	2	1	no	—
191	C₄⋊C₄×C₄₁	—	2	1	no	—
192	C₄⋊C₄×C₄₁	—	2	1	no	—
193	C₄⋊C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
194	C₄⋊C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
195	C₄⋊C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
196	C₄⋊C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
197	C₂²⋊C₄×C₄₁	—	2	1	no	—
198	C₂²⋊C₄×C₄₁	—	2	1	no	—
199	C₂²⋊C₄×C₄₁	—	2	1	no	—
200	C₂²⋊C₄×C₄₁	—	2	1	no	—
201	C₂²⋊C₄×C₄₁	—	2	1	no	—
202	C₂²⋊C₄×C₄₁	—	2	1	no	—
203	C₄²×C₄₁	—	2	1	no	—
204	C₄²×C₄₁	—	2	1	no	—
205	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
206	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₄₁	#41	1	1	yes	characteristic
208	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
209	C₂³×C₄₁	—	2	1	no	—
210	C₂³×C₄₁	—	2	1	no	—
211	C₂³×C₄₁	—	4	2	no	—
212	C₂³×C₄₁	—	4	2	no	—
213	C₂³×C₄₁	—	8	1	no	—
214	C₂³×C₄₁	C₄×D₄	1	1	yes	characteristic
215	Q₈×C₄₁	—	2	1	no	—
216	Q₈×C₄₁	—	2	1	no	—
217	Q₈×C₄₁	—	2	1	no	—
218	Q₈×C₄₁	—	2	1	no	—
219	Q₈×C₄₁	—	4	2	no	—
220	D₄×C₄₁	—	2	1	no	—
221	D₄×C₄₁	—	2	1	no	—
222	D₄×C₄₁	—	2	1	no	—
223	D₄×C₄₁	—	4	2	no	—
224	D₄×C₄₁	#131	1	2	yes	—
225	C₂×C₄×C₄₁	—	2	1	no	—
226	C₂×C₄×C₄₁	—	2	1	no	—
227	C₂×C₄×C₄₁	—	2	1	no	—
228	C₂×C₄×C₄₁	—	2	1	no	—
229	C₂×C₄×C₄₁	—	2	1	no	—
230	C₂×C₄×C₄₁	—	2	1	no	—
231	C₂×C₄×C₄₁	—	2	1	no	—
232	C₂×C₄×C₄₁	—	2	1	no	—
233	C₂×C₄×C₄₁	—	2	1	no	—
234	C₂×C₄×C₄₁	—	2	1	no	—
235	C₂×C₄×C₄₁	—	4	1	no	—
236	C₂×C₄×C₄₁	—	4	1	no	—
237	C₂×C₄×C₄₁	—	4	1	no	—
238	C₂×C₄×C₄₁	—	4	1	no	—
239	C₂×C₄×C₄₁	—	4	1	no	—
240	C₂×C₄×C₄₁	—	8	1	no	—
241	C₂×C₄×C₄₁	—	8	1	no	—
242	C₂×C₄×C₄₁	—	8	1	no	—
243	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
244	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
245	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
246	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
247	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
249	C₈×C₄₁	—	2	1	no	—
250	C₈×C₄₁	—	2	1	no	—
251	C₈×C₄₁	—	2	1	no	—
252	C₈×C₄₁	—	2	1	no	—
253	C₈×C₄₁	—	2	1	no	—
254	C₈×C₄₁	—	2	1	no	—
255	C₈×C₄₁	—	2	1	no	—
256	C₈×C₄₁	—	2	1	no	—
257	C₈×C₄₁	—	4	1	no	—
258	C₈×C₄₁	—	4	1	no	—
259	C₈×C₄₁	—	4	1	no	—
260	C₈×C₄₁	—	4	1	no	—
261	C₈×C₄₁	—	4	1	no	—
262	C₈×C₄₁	—	4	1	no	—
263	#18028	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
264	C₂²×C₄₁	—	2	1	no	—
265	C₂²×C₄₁	—	2	2	no	—
266	C₂²×C₄₁	—	2	1	no	—
267	C₂²×C₄₁	—	2	1	no	—
268	C₂²×C₄₁	—	4	1	no	—
269	C₂²×C₄₁	—	4	1	no	—
270	C₂²×C₄₁	—	4	1	no	—
271	C₂²×C₄₁	—	4	1	no	—
272	C₂²×C₄₁	—	4	1	no	—
273	C₂²×C₄₁	—	8	4	no	—
274	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
275	C₄×C₄₁	—	2	1	no	—
276	C₄×C₄₁	—	2	1	no	—
277	C₄×C₄₁	—	2	1	no	—
278	C₄×C₄₁	—	2	1	no	—
279	C₄×C₄₁	—	4	1	no	—
280	C₄×C₄₁	—	4	1	no	—
281	C₄×C₄₁	—	4	1	no	—
282	C₄×C₄₁	—	4	1	no	—
283	C₄×C₄₁	—	4	1	no	—
284	C₄×C₄₁	—	4	1	no	—
285	C₄×C₄₁	C₂×#131	1	1	yes	characteristic
286	C₄×C₄₁	#568	1	1	yes	characteristic
287	C₄×C₄₁	#577	1	1	yes	characteristic
288	C₄×C₄₁	#577	1	1	yes	characteristic
289	#3101	C₂×C₄₁	1	1	yes	characteristic
290	#2986	C₂×C₄₁	1	1	yes	characteristic, complemented
291	#2974	C₂×C₄₁	1	1	yes	characteristic
292	#1592	C₂×C₄₁	1	1	yes	characteristic, complemented
293	#1590	C₂×C₄₁	1	1	yes	characteristic, complemented
294	#1588	C₂×C₄₁	1	1	yes	characteristic, complemented
295	#1585	C₂×C₄₁	1	1	yes	characteristic, complemented
296	#1583	C₂×C₄₁	1	1	yes	characteristic, complemented
297	#1582	C₂×C₄₁	1	1	yes	characteristic, complemented
298	SD₁₆⋊C₈	C₂×C₄₁	1	1	yes	characteristic, complemented
299	C₈×SD₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
300	#1572	C₂×C₄₁	1	1	yes	characteristic, complemented
301	#1497	C₂×C₄₁	1	1	yes	characteristic, complemented
302	#1478	C₂×C₄₁	1	1	yes	characteristic
303	#1454	C₂×C₄₁	1	1	yes	characteristic, complemented
304	C₂×C₄₁	—	4	1	no	—
305	C₂×C₄₁	—	4	2	no	—
306	C₂×C₄₁	—	8	1	no	—
307	C₂×C₄₁	#2933	1	1	yes	characteristic
308	C₂×C₄₁	#2949	1	1	yes	characteristic
309	C₂×C₄₁	#2955	1	1	yes	characteristic
310	#533	C₂²×C₄₁	1	1	yes	characteristic
311	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
312	#483	C₄×C₄₁	1	1	yes	characteristic
313	#483	C₄×C₄₁	1	1	yes	characteristic
314	#476	C₄×C₄₁	1	1	yes	characteristic
315	#476	C₄×C₄₁	1	1	yes	characteristic
316	#461	C₄×C₄₁	1	1	yes	characteristic
317	#461	C₄×C₄₁	1	1	yes	characteristic
318	#450	C₄×C₄₁	1	1	yes	characteristic
319	#450	C₄×C₄₁	1	1	yes	characteristic
320	#446	—	2	1	no	—
321	#446	C₂²×C₄₁	1	1	yes	characteristic, complemented
322	#446	C₂²×C₄₁	1	1	yes	characteristic, complemented
323	C₈×Q₈	—	2	1	no	—
324	C₈×Q₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
325	C₈×Q₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
326	C₄×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
327	C₄×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
328	C₄×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
329	C₄×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
330	#436	C₂²×C₄₁	1	1	yes	characteristic
331	#436	C₂²×C₄₁	1	1	yes	characteristic
332	#435	—	2	1	no	—
333	#435	—	2	1	no	—
334	C₈×D₄	C₂²×C₄₁	1	1	yes	characteristic
335	C₈×D₄	C₂²×C₄₁	1	1	yes	characteristic
336	#433	—	2	1	no	—
337	#433	—	2	1	no	—
338	#432	C₂²×C₄₁	1	1	yes	characteristic
339	#431	C₂²×C₄₁	1	1	yes	characteristic
340	#431	C₂²×C₄₁	1	1	yes	characteristic
341	#431	C₂²×C₄₁	1	1	yes	characteristic
342	#431	C₂²×C₄₁	1	1	yes	characteristic
343	#407	—	2	1	no	—
344	C₂×C₂²⋊C₈	—	2	1	no	—
345	#335	C₂²×C₄₁	1	1	yes	characteristic, complemented
346	#335	C₂²×C₄₁	1	1	yes	characteristic, complemented
347	#334	C₂²×C₄₁	1	1	yes	characteristic, complemented
348	#334	C₂²×C₄₁	1	1	yes	characteristic, complemented
349	Q₈⋊C₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
350	Q₈⋊C₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
351	Q₈⋊C₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
352	Q₈⋊C₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
353	D₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
354	D₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
355	D₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
356	D₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
357	#322	C₂²×C₄₁	1	1	yes	characteristic, complemented
358	#322	C₂²×C₄₁	1	1	yes	characteristic, complemented
359	#322	C₂²×C₄₁	1	1	yes	characteristic, complemented
360	C₈²	C₂²×C₄₁	1	1	yes	characteristic, complemented
361	C₄₁	#18028	1	1	yes	characteristic, direct product factor, Sylow
362	C₂²×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
363	C₂³×C₄	—	2	1	no	—
364	C₂×SD₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
365	C₂×SD₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
366	C₂×SD₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
367	C₂×SD₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
368	C₂×M₁₆	—	2	1	no	—
369	C₂×M₁₆	—	2	1	no	—
370	C₂²×C₈	—	2	1	no	—
371	C₂²×C₈	—	2	1	no	—
372	C₄⋊Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
373	C₄⋊Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
374	#125	—	2	1	no	—
375	C₂²⋊Q₈	—	2	1	no	—
376	C₂²⋊Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
377	C₂²⋊Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
378	C₄×Q₈	—	2	1	no	—
379	C₄×Q₈	C₂³×C₄₁	1	1	yes	characteristic
380	C₄×Q₈	C₂³×C₄₁	1	1	yes	characteristic
381	C₄×D₄	—	2	1	no	—
382	C₄×D₄	—	2	1	no	—
383	C₄×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
384	C₄×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
385	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
386	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
387	#117	—	2	1	no	—
388	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
389	C₂×C₂²⋊C₄	—	2	1	no	—
390	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
391	#106	C₂×C₄×C₄₁	1	1	yes	characteristic
392	#106	C₂×C₄×C₄₁	1	1	yes	characteristic
393	#106	C₂×C₄×C₄₁	1	1	yes	characteristic
394	#106	C₂×C₄×C₄₁	1	1	yes	characteristic
395	C₄⋊C₈	—	2	1	no	—
396	C₄⋊C₈	—	2	1	no	—
397	C₄⋊C₈	—	2	1	no	—
398	C₄⋊C₈	—	2	1	no	—
399	C₄⋊C₈	—	2	1	no	—
400	C₄⋊C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
401	C₄⋊C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
402	C₄⋊C₈	D₄×C₄₁	1	1	yes	characteristic
403	C₄⋊C₈	D₄×C₄₁	1	1	yes	characteristic
404	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
405	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
406	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
407	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
408	#103	C₂×C₄×C₄₁	1	1	yes	characteristic
409	#103	C₂×C₄×C₄₁	1	1	yes	characteristic
410	#103	C₂×C₄×C₄₁	1	1	yes	characteristic
411	#103	C₂×C₄×C₄₁	1	1	yes	characteristic
412	#102	C₂×C₄×C₄₁	1	1	yes	characteristic
413	#102	C₂×C₄×C₄₁	1	1	yes	characteristic
414	#102	C₂×C₄×C₄₁	1	1	yes	characteristic
415	#102	C₂×C₄×C₄₁	1	1	yes	characteristic
416	C₂²⋊C₈	—	2	1	no	—
417	C₂²⋊C₈	—	2	1	no	—
418	C₂²⋊C₈	—	2	1	no	—
419	C₂²⋊C₈	—	2	1	no	—
420	C₂²⋊C₈	—	2	1	no	—
421	C₂²⋊C₈	—	2	1	no	—
422	C₂²⋊C₈	—	2	1	no	—
423	C₂²⋊C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
424	C₂²⋊C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
425	C₂²⋊C₈	D₄×C₄₁	1	1	yes	characteristic
426	C₂²⋊C₈	D₄×C₄₁	1	1	yes	characteristic
427	#97	—	2	1	no	—
428	#97	—	2	1	no	—
429	#97	—	2	1	no	—
430	C₄×C₈	—	2	1	no	—
431	C₄×C₈	—	2	1	no	—
432	C₄×C₈	—	2	1	no	—
433	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
434	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
435	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
436	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
437	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
438	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
439	C₂⁴	—	2	1	no	—
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
442	C₂×Q₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
443	C₂×D₄	—	2	2	no	—
444	C₂×D₄	—	2	1	no	—
445	C₂×D₄	—	4	1	no	—
446	C₂×D₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
447	C₂×D₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	8	1	no	—
454	C₂²×C₄	—	8	1	no	—
455	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
456	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
457	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
458	SD₁₆	—	2	2	no	—
459	SD₁₆	—	2	2	no	—
460	SD₁₆	—	4	1	no	—
461	SD₁₆	—	4	1	no	—
462	M₁₆	—	8	1	no	—
463	M₁₆	—	8	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	8	1	no	—
471	C₂×C₈	—	8	1	no	—
472	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
473	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
474	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
475	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
476	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
477	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
478	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
479	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
489	C₄⋊C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
490	C₄⋊C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
491	C₄⋊C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
492	C₂²⋊C₄	—	2	1	no	—
493	C₂²⋊C₄	—	2	1	no	—
494	C₂²⋊C₄	—	2	1	no	—
495	C₂²⋊C₄	—	2	1	no	—
496	C₂²⋊C₄	—	2	1	no	—
497	C₂²⋊C₄	—	2	1	no	—
498	C₄²	—	2	1	no	—
499	C₄²	—	2	1	no	—
500	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
501	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
502	C₄²	#41×C₄₁	1	1	yes	characteristic
503	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
504	C₂³	—	2	1	no	—
505	C₂³	—	2	1	no	—
506	C₂³	—	4	2	no	—
507	C₂³	—	4	2	no	—
508	C₂³	—	8	1	no	—
509	C₂³	C₄×D₄×C₄₁	1	1	yes	characteristic
510	Q₈	—	2	1	no	—
511	Q₈	—	2	1	no	—
512	Q₈	—	2	1	no	—
513	Q₈	—	2	1	no	—
514	Q₈	—	4	2	no	—
515	D₄	—	2	1	no	—
516	D₄	—	2	1	no	—
517	D₄	—	2	1	no	—
518	D₄	—	4	2	no	—
519	D₄	#131×C₄₁	1	2	yes	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	8	1	no	—
536	C₂×C₄	—	8	1	no	—
537	C₂×C₄	—	8	1	no	—
538	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
539	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
540	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
541	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
542	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
543	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
544	C₈	—	2	1	no	—
545	C₈	—	2	1	no	—
546	C₈	—	2	1	no	—
547	C₈	—	2	1	no	—
548	C₈	—	2	1	no	—
549	C₈	—	2	1	no	—
550	C₈	—	2	1	no	—
551	C₈	—	2	1	no	—
552	C₈	—	4	1	no	—
553	C₈	—	4	1	no	—
554	C₈	—	4	1	no	—
555	C₈	—	4	1	no	—
556	C₈	—	4	1	no	—
557	C₈	—	4	1	no	—
558	C₂²	—	2	1	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	2	1	no	—
561	C₂²	—	2	1	no	—
562	C₂²	—	4	1	no	—
563	C₂²	—	4	1	no	—
564	C₂²	—	4	1	no	—
565	C₂²	—	4	1	no	—
566	C₂²	—	4	1	no	—
567	C₂²	—	8	4	no	—
568	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
569	C₄	—	2	1	no	—
570	C₄	—	2	1	no	—
571	C₄	—	2	1	no	—
572	C₄	—	2	1	no	—
573	C₄	—	4	1	no	—
574	C₄	—	4	1	no	—
575	C₄	—	4	1	no	—
576	C₄	—	4	1	no	—
577	C₄	—	4	1	no	—
578	C₄	—	4	1	no	—
579	C₄	C₂×#131×C₄₁	1	1	yes	characteristic
580	C₄	C₄₁×#568	1	1	yes	characteristic
581	C₄	C₄₁×#577	1	1	yes	characteristic
582	C₄	C₄₁×#577	1	1	yes	characteristic
583	C₂	—	4	1	no	—
584	C₂	—	4	2	no	—
585	C₂	—	8	1	no	—
586	C₂	C₄₁×#2933	1	1	yes	characteristic
587	C₂	C₄₁×#2949	1	1	yes	characteristic
588	C₂	C₄₁×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₄²	C₂⁴×C₄₁
socle	C₂²×C₄₁	C₂×C₄×D₄
prime radical	C₂²×D₄×C₄₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₄₁×#431 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2986	C₂	semidirect
4. C₄₁×#1592	C₂	semidirect
5. C₄₁×#1590	C₂	semidirect
6. C₄₁×#1588	C₂	semidirect
7. C₄₁×#1585	C₂	semidirect
8. C₄₁×#1583	C₂	semidirect
9. C₄₁×#1582	C₂	semidirect
10. C₄₁×SD₁₆⋊C₈	C₂	semidirect
11. C₈×SD₁₆×C₄₁	C₂	semidirect
12. C₄₁×#1572	C₂	semidirect
13. C₄₁×#1497	C₂	semidirect
15. C₄₁×#1454	C₂	semidirect
27. C₄₁×#446	C₂²	semidirect
28. C₄₁×#446	C₂²	semidirect
30. C₈×Q₈×C₄₁	C₂²	semidirect
31. C₈×Q₈×C₄₁	C₂²	semidirect
51. C₄₁×#335	C₂²	semidirect
52. C₄₁×#335	C₂²	semidirect
53. C₄₁×#334	C₂²	semidirect
54. C₄₁×#334	C₂²	semidirect
55. C₄₁×Q₈⋊C₈	C₂²	semidirect
56. C₄₁×Q₈⋊C₈	C₂²	semidirect
57. C₄₁×Q₈⋊C₈	C₂²	semidirect
58. C₄₁×Q₈⋊C₈	C₂²	semidirect
63. C₄₁×#322	C₂²	semidirect
64. C₄₁×#322	C₂²	semidirect
65. C₄₁×#322	C₂²	semidirect
66. C₈²×C₄₁	C₂²	semidirect
263. #18028	C₄₁	direct (N × H)
290. #2986	C₂×C₄₁	semidirect
292. #1592	C₂×C₄₁	semidirect
293. #1590	C₂×C₄₁	semidirect
294. #1588	C₂×C₄₁	semidirect
295. #1585	C₂×C₄₁	semidirect
296. #1583	C₂×C₄₁	semidirect
297. #1582	C₂×C₄₁	semidirect
298. SD₁₆⋊C₈	C₂×C₄₁	semidirect
299. C₈×SD₁₆	C₂×C₄₁	semidirect
300. #1572	C₂×C₄₁	semidirect
301. #1497	C₂×C₄₁	semidirect
303. #1454	C₂×C₄₁	semidirect
321. #446	C₂²×C₄₁	semidirect
322. #446	C₂²×C₄₁	semidirect
324. C₈×Q₈	C₂²×C₄₁	semidirect
325. C₈×Q₈	C₂²×C₄₁	semidirect
345. #335	C₂²×C₄₁	semidirect
346. #335	C₂²×C₄₁	semidirect
347. #334	C₂²×C₄₁	semidirect
348. #334	C₂²×C₄₁	semidirect
349. Q₈⋊C₈	C₂²×C₄₁	semidirect
350. Q₈⋊C₈	C₂²×C₄₁	semidirect
351. Q₈⋊C₈	C₂²×C₄₁	semidirect
352. Q₈⋊C₈	C₂²×C₄₁	semidirect
357. #322	C₂²×C₄₁	semidirect
358. #322	C₂²×C₄₁	semidirect
359. #322	C₂²×C₄₁	semidirect
360. C₈²	C₂²×C₄₁	semidirect
361. C₄₁	#18028	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#3101	2
C₄₁×#2986	2
C₄₁×#2974	2
C₄₁×#1592	2
C₄₁×#1590	2
C₄₁×#1588	2
C₄₁×#1585	2
C₄₁×#1583	2
C₄₁×#1582	2
C₄₁×SD₁₆⋊C₈	2
C₈×SD₁₆×C₄₁	2
C₄₁×#1572	2
C₄₁×#1497	2
C₄₁×#1478	2
C₄₁×#1454	2
#18028	41

Subgroups — C41×#18028 #23794229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#18028 #23794229