Subgroups — C₄₁×#24216 #23800417

Order 10496.

1482 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#464	C₂	1	2	yes	maximal, complemented
3	C₂×C₄₁×#452	C₂	1	2	yes	maximal, complemented
4	C₄×C₂²⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2043	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#2041	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#2016	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#2014	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1983	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1797	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1788	C₂	1	2	yes	maximal, complemented
12	C₂²×#41×C₄₁	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₄₁	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
15	C₄₁×#532	—	2	8	no	—
16	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×#124×C₄₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×#121×C₄₁	—	2	1	no	—
21	C₂×#121×C₄₁	—	2	1	no	—
22	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₄₁	—	2	1	no	—
24	C₂×C₄×D₄×C₄₁	—	2	1	no	—
25	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
26	C₄₁×#464	—	2	8	no	—
27	C₄₁×#452	—	2	8	no	—
28	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
29	C₂×#103×C₄₁	C₂²	1	2	yes	—
30	C₂×#103×C₄₁	C₂²	1	2	yes	—
31	C₂×#103×C₄₁	C₂²	1	2	yes	—
32	C₂×#102×C₄₁	C₂²	1	2	yes	—
33	C₂×#102×C₄₁	C₂²	1	2	yes	—
34	C₂×#102×C₄₁	C₂²	1	2	yes	—
35	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
36	C₄₁×#391	C₂²	1	1	yes	characteristic
37	C₄₁×#389	C₂²	1	1	yes	characteristic
38	C₄₁×#387	—	2	1	no	—
39	C₄₁×#385	—	2	1	no	—
40	C₄₁×#380	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₄₁	—	2	1	no	—
42	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
44	C₄₁×#336	C₂²	1	1	yes	characteristic
45	C₄₁×#336	C₂²	1	1	yes	characteristic
46	C₂×#41×C₄₁	—	2	4	no	—
47	C₂×#41×C₄₁	—	2	4	no	—
48	C₂×#41×C₄₁	—	2	4	no	—
49	C₂×#41×C₄₁	—	2	4	no	—
50	C₂×#41×C₄₁	—	4	2	no	—
51	C₂×#41×C₄₁	—	4	2	no	—
52	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
53	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
54	C₂²×D₄×C₄₁	—	2	2	no	—
55	C₂²×D₄×C₄₁	—	2	1	no	—
56	C₂²×D₄×C₄₁	—	2	1	no	—
57	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
58	C₂³×C₄×C₄₁	—	2	2	no	—
59	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
61	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
62	C₂×Q₁₆×C₄₁	—	2	4	no	—
63	C₂×Q₁₆×C₄₁	—	2	4	no	—
64	C₂×Q₁₆×C₄₁	—	4	2	no	—
65	C₂×SD₁₆×C₄₁	—	2	4	no	—
66	C₂×SD₁₆×C₄₁	—	2	4	no	—
67	C₂×SD₁₆×C₄₁	—	4	2	no	—
68	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
69	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
70	C₄⋊Q₈×C₄₁	—	2	4	no	—
71	#127×C₄₁	—	2	4	no	—
72	#124×C₄₁	—	2	4	no	—
73	#124×C₄₁	D₄	1	8	yes	—
74	C₂²⋊Q₈×C₄₁	—	2	2	no	—
75	C₂²⋊Q₈×C₄₁	D₄	1	4	yes	complemented
76	#121×C₄₁	—	2	8	no	—
77	#121×C₄₁	—	2	8	no	—
78	C₄×Q₈×C₄₁	—	2	4	no	—
79	C₄×D₄×C₄₁	—	2	8	no	—
80	C₄×D₄×C₄₁	—	2	8	no	—
81	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
93	C₂×C₄²×C₄₁	—	2	4	no	—
94	C₂×C₄²×C₄₁	—	2	1	no	—
95	C₂×C₄²×C₄₁	—	2	1	no	—
96	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
100	C₄⋊C₈×C₄₁	—	2	4	no	—
101	#103×C₄₁	—	2	4	no	—
102	#103×C₄₁	—	2	4	no	—
103	#103×C₄₁	—	4	2	no	—
104	#102×C₄₁	—	2	4	no	—
105	#102×C₄₁	—	2	4	no	—
106	#102×C₄₁	—	4	2	no	—
107	C₂²⋊C₈×C₄₁	—	2	4	no	—
108	#95×C₄₁	—	2	1	no	—
109	#95×C₄₁	—	2	1	no	—
110	#95×C₄₁	—	2	1	no	—
111	#95×C₄₁	—	2	1	no	—
112	#95×C₄₁	—	2	1	no	—
113	C₂⁴×C₄₁	—	2	2	no	—
114	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
115	#41×C₄₁	—	4	8	no	—
116	#41×C₄₁	—	4	8	no	—
117	#41×C₄₁	—	4	8	no	—
118	#41×C₄₁	—	4	8	no	—
119	C₂×Q₈×C₄₁	—	2	2	no	—
120	C₂×Q₈×C₄₁	—	2	1	no	—
121	C₂×Q₈×C₄₁	—	2	2	no	—
122	C₂×Q₈×C₄₁	—	2	2	no	—
123	C₂×Q₈×C₄₁	—	2	2	no	—
124	C₂×Q₈×C₄₁	—	2	2	no	—
125	C₂×Q₈×C₄₁	—	2	2	no	—
126	C₂×Q₈×C₄₁	—	2	2	no	—
127	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
128	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
129	C₂×D₄×C₄₁	—	2	4	no	—
130	C₂×D₄×C₄₁	—	2	4	no	—
131	C₂×D₄×C₄₁	—	2	8	no	—
132	C₂×D₄×C₄₁	—	2	2	no	—
133	C₂×D₄×C₄₁	—	2	2	no	—
134	C₂×D₄×C₄₁	—	2	2	no	—
135	C₂×D₄×C₄₁	—	2	2	no	—
136	C₂×D₄×C₄₁	—	2	2	no	—
137	C₂×D₄×C₄₁	—	4	2	no	—
138	C₂×D₄×C₄₁	—	4	2	no	—
139	C₂×D₄×C₄₁	—	4	4	no	—
140	C₂×D₄×C₄₁	—	4	4	no	—
141	C₂×D₄×C₄₁	—	4	2	no	—
142	C₂×D₄×C₄₁	—	4	2	no	—
143	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
144	C₂²×C₄×C₄₁	—	2	2	no	—
145	C₂²×C₄×C₄₁	—	2	2	no	—
146	C₂²×C₄×C₄₁	—	2	8	no	—
147	C₂²×C₄×C₄₁	—	2	1	no	—
148	C₂²×C₄×C₄₁	—	2	1	no	—
149	C₂²×C₄×C₄₁	—	2	2	no	—
150	C₂²×C₄×C₄₁	—	2	2	no	—
151	C₂²×C₄×C₄₁	—	2	2	no	—
152	C₂²×C₄×C₄₁	—	2	2	no	—
153	C₂²×C₄×C₄₁	—	2	2	no	—
154	C₂²×C₄×C₄₁	—	2	1	no	—
155	C₂²×C₄×C₄₁	—	2	1	no	—
156	C₂²×C₄×C₄₁	—	4	1	no	—
157	C₂²×C₄×C₄₁	—	4	1	no	—
158	C₂²×C₄×C₄₁	—	4	4	no	—
159	C₂²×C₄×C₄₁	—	4	4	no	—
160	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₄₁	#41	1	2	yes	—
167	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
168	Q₁₆×C₄₁	—	4	8	no	—
169	SD₁₆×C₄₁	—	4	8	no	—
170	C₂×C₈×C₄₁	—	2	2	no	—
171	C₂×C₈×C₄₁	—	2	2	no	—
172	C₂×C₈×C₄₁	—	2	2	no	—
173	C₂×C₈×C₄₁	—	2	2	no	—
174	C₂×C₈×C₄₁	—	2	2	no	—
175	C₂×C₈×C₄₁	—	2	2	no	—
176	C₄⋊C₄×C₄₁	—	2	4	no	—
177	C₄⋊C₄×C₄₁	—	2	4	no	—
178	C₄⋊C₄×C₄₁	—	2	4	no	—
179	C₄⋊C₄×C₄₁	—	2	1	no	—
180	C₄⋊C₄×C₄₁	—	2	2	no	—
181	C₄⋊C₄×C₄₁	—	2	2	no	—
182	C₄⋊C₄×C₄₁	—	4	2	no	—
183	C₄⋊C₄×C₄₁	—	4	2	no	—
184	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
185	C₂²⋊C₄×C₄₁	—	2	4	no	—
186	C₂²⋊C₄×C₄₁	—	2	8	no	—
187	C₂²⋊C₄×C₄₁	—	2	8	no	—
188	C₂²⋊C₄×C₄₁	—	4	2	no	—
189	C₄²×C₄₁	—	2	2	no	—
190	C₄²×C₄₁	—	2	2	no	—
191	C₄²×C₄₁	—	2	2	no	—
192	C₄²×C₄₁	—	4	2	no	—
193	C₄²×C₄₁	—	4	2	no	—
194	C₄²×C₄₁	C₂×D₄	1	4	yes	—
195	C₂³×C₄₁	—	2	1	no	—
196	C₂³×C₄₁	—	2	1	no	—
197	C₂³×C₄₁	—	2	1	no	—
198	C₂³×C₄₁	—	2	1	no	—
199	C₂³×C₄₁	—	2	1	no	—
200	C₂³×C₄₁	—	2	1	no	—
201	C₂³×C₄₁	—	2	4	no	—
202	C₂³×C₄₁	—	4	2	no	—
203	C₂³×C₄₁	—	4	2	no	—
204	C₂³×C₄₁	—	4	2	no	—
205	C₂³×C₄₁	—	4	2	no	—
206	C₂³×C₄₁	—	4	2	no	—
207	C₂³×C₄₁	—	4	2	no	—
208	C₂³×C₄₁	#121	1	2	yes	—
209	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
210	Q₈×C₄₁	—	2	4	no	—
211	Q₈×C₄₁	—	2	4	no	—
212	Q₈×C₄₁	—	2	4	no	—
213	Q₈×C₄₁	—	4	2	no	—
214	Q₈×C₄₁	—	4	2	no	—
215	Q₈×C₄₁	—	4	2	no	—
216	D₄×C₄₁	—	2	4	no	—
217	D₄×C₄₁	—	2	4	no	—
218	D₄×C₄₁	—	4	8	no	—
219	D₄×C₄₁	—	4	8	no	—
220	D₄×C₄₁	—	4	16	no	—
221	D₄×C₄₁	—	4	2	no	—
222	D₄×C₄₁	—	4	2	no	—
223	C₂×C₄×C₄₁	—	2	1	no	—
224	C₂×C₄×C₄₁	—	2	1	no	—
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	4	no	—
234	C₂×C₄×C₄₁	—	2	4	no	—
235	C₂×C₄×C₄₁	—	2	4	no	—
236	C₂×C₄×C₄₁	—	2	2	no	—
237	C₂×C₄×C₄₁	—	2	2	no	—
238	C₂×C₄×C₄₁	—	4	4	no	—
239	C₂×C₄×C₄₁	—	4	4	no	—
240	C₂×C₄×C₄₁	—	4	16	no	—
241	C₂×C₄×C₄₁	—	4	1	no	—
242	C₂×C₄×C₄₁	—	4	2	no	—
243	C₂×C₄×C₄₁	—	4	2	no	—
244	C₂×C₄×C₄₁	—	4	1	no	—
245	C₂×C₄×C₄₁	—	4	1	no	—
246	C₂×C₄×C₄₁	—	4	1	no	—
247	C₂×C₄×C₄₁	—	4	1	no	—
248	C₂×C₄×C₄₁	—	4	1	no	—
249	C₂×C₄×C₄₁	—	4	1	no	—
250	C₂×C₄×C₄₁	—	4	1	no	—
251	C₂×C₄×C₄₁	—	4	1	no	—
252	C₂×C₄×C₄₁	—	4	1	no	—
253	C₂×C₄×C₄₁	#120	1	2	yes	—
254	C₂×C₄×C₄₁	#120	1	2	yes	—
255	C₂×C₄×C₄₁	#121	1	2	yes	—
256	C₂×C₄×C₄₁	#135	1	2	yes	—
257	C₂×C₄×C₄₁	#137	1	2	yes	—
258	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
261	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
262	C₈×C₄₁	—	4	2	no	—
263	C₈×C₄₁	—	4	2	no	—
264	#24216	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
265	C₂²×C₄₁	—	2	2	no	—
266	C₂²×C₄₁	—	2	2	no	—
267	C₂²×C₄₁	—	2	2	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	4	no	—
273	C₂²×C₄₁	—	4	4	no	—
274	C₂²×C₄₁	—	4	4	no	—
275	C₂²×C₄₁	—	8	2	no	—
276	C₂²×C₄₁	—	8	2	no	—
277	C₂²×C₄₁	—	8	2	no	—
278	C₂²×C₄₁	—	8	2	no	—
279	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
280	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
281	C₂²×C₄₁	#535	1	1	yes	characteristic
282	C₂²×C₄₁	C₂×#135	1	1	yes	characteristic
283	C₂²×C₄₁	C₂×#137	1	1	yes	characteristic
284	C₂²×C₄₁	#575	1	1	yes	characteristic
285	C₂²×C₄₁	#575	1	1	yes	characteristic
286	C₄×C₄₁	—	2	2	no	—
287	C₄×C₄₁	—	2	2	no	—
288	C₄×C₄₁	—	2	2	no	—
289	C₄×C₄₁	—	2	2	no	—
290	C₄×C₄₁	—	2	2	no	—
291	C₄×C₄₁	—	4	4	no	—
292	C₄×C₄₁	—	4	2	no	—
293	C₄×C₄₁	—	4	2	no	—
294	C₄×C₄₁	—	8	1	no	—
295	C₄×C₄₁	—	8	1	no	—
296	C₄×C₄₁	D₄²	1	2	yes	—
297	C₄×C₄₁	#546	1	2	yes	—
298	C₂×#532	C₂×C₄₁	1	1	yes	characteristic
299	C₂×#464	C₂×C₄₁	1	2	yes	complemented
300	C₂×#452	C₂×C₄₁	1	2	yes	complemented
301	C₄×C₂²⋊Q₈	C₂×C₄₁	1	1	yes	characteristic, complemented
302	#2043	C₂×C₄₁	1	1	yes	characteristic, complemented
303	#2041	C₂×C₄₁	1	1	yes	characteristic, complemented
304	#2016	C₂×C₄₁	1	1	yes	characteristic, complemented
305	#2014	C₂×C₄₁	1	1	yes	characteristic, complemented
306	#1983	C₂×C₄₁	1	2	yes	complemented
307	#1797	C₂×C₄₁	1	1	yes	characteristic, complemented
308	#1788	C₂×C₄₁	1	2	yes	complemented
309	C₂×C₄₁	—	4	2	no	—
310	C₂×C₄₁	—	8	4	no	—
311	C₂×C₄₁	#2408	1	1	yes	characteristic
312	C₂×C₄₁	#3002	1	1	yes	characteristic
313	C₂×C₄₁	#3007	1	1	yes	characteristic
314	C₂×C₄₁	#3035	1	1	yes	characteristic
315	C₂×C₄₁	#3037	1	1	yes	characteristic
316	C₂×C₄₁	#3210	1	1	yes	characteristic
317	C₂×C₄₁	#3219	1	1	yes	characteristic
318	C₂²×#41	C₂²×C₄₁	1	2	yes	—
319	C₂²×Q₁₆	C₂²×C₄₁	1	2	yes	—
320	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
321	#532	—	2	8	no	—
322	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
323	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
324	C₂×#124	C₂²×C₄₁	1	2	yes	—
325	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
326	C₂×#121	—	2	1	no	—
327	C₂×#121	—	2	1	no	—
328	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
329	C₂×C₄×D₄	—	2	1	no	—
330	C₂×C₄×D₄	—	2	1	no	—
331	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
332	#464	—	2	8	no	—
333	#452	—	2	8	no	—
334	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
335	C₂×#103	C₂²×C₄₁	1	2	yes	—
336	C₂×#103	C₂²×C₄₁	1	2	yes	—
337	C₂×#103	C₂²×C₄₁	1	2	yes	—
338	C₂×#102	C₂²×C₄₁	1	2	yes	—
339	C₂×#102	C₂²×C₄₁	1	2	yes	—
340	C₂×#102	C₂²×C₄₁	1	2	yes	—
341	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
342	#391	C₂²×C₄₁	1	1	yes	characteristic
343	#389	C₂²×C₄₁	1	1	yes	characteristic
344	#387	—	2	1	no	—
345	#385	—	2	1	no	—
346	#380	C₂²×C₄₁	1	1	yes	characteristic
347	C₄×C₄⋊C₄	—	2	1	no	—
348	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
349	C₄×C₂²⋊C₄	—	2	1	no	—
350	#336	C₂²×C₄₁	1	1	yes	characteristic
351	#336	C₂²×C₄₁	1	1	yes	characteristic
352	C₄₁	#24216	1	1	yes	characteristic, direct product factor, Sylow
353	C₂×#41	—	2	4	no	—
354	C₂×#41	—	2	4	no	—
355	C₂×#41	—	2	4	no	—
356	C₂×#41	—	2	4	no	—
357	C₂×#41	—	4	2	no	—
358	C₂×#41	—	4	2	no	—
359	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
360	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
361	C₂²×D₄	—	2	2	no	—
362	C₂²×D₄	—	2	1	no	—
363	C₂²×D₄	—	2	1	no	—
364	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
365	C₂³×C₄	—	2	2	no	—
366	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
367	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
368	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
369	C₂×Q₁₆	—	2	4	no	—
370	C₂×Q₁₆	—	2	4	no	—
371	C₂×Q₁₆	—	4	2	no	—
372	C₂×SD₁₆	—	2	4	no	—
373	C₂×SD₁₆	—	2	4	no	—
374	C₂×SD₁₆	—	4	2	no	—
375	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
376	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
377	C₄⋊Q₈	—	2	4	no	—
378	#127	—	2	4	no	—
379	#124	—	2	4	no	—
380	#124	D₄×C₄₁	1	8	yes	—
381	C₂²⋊Q₈	—	2	2	no	—
382	C₂²⋊Q₈	D₄×C₄₁	1	4	yes	complemented
383	#121	—	2	8	no	—
384	#121	—	2	8	no	—
385	C₄×Q₈	—	2	4	no	—
386	C₄×D₄	—	2	8	no	—
387	C₄×D₄	—	2	8	no	—
388	C₂×C₄⋊C₄	—	2	1	no	—
389	C₂×C₄⋊C₄	—	2	1	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	C₂×C₄⋊C₄	—	2	1	no	—
392	C₂×C₄⋊C₄	—	2	1	no	—
393	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
394	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
395	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
396	C₂×C₂²⋊C₄	—	2	1	no	—
397	C₂×C₂²⋊C₄	—	2	2	no	—
398	C₂×C₂²⋊C₄	—	2	2	no	—
399	C₂×C₂²⋊C₄	—	2	1	no	—
400	C₂×C₄²	—	2	4	no	—
401	C₂×C₄²	—	2	1	no	—
402	C₂×C₄²	—	2	1	no	—
403	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
404	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
405	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
406	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
407	C₄⋊C₈	—	2	4	no	—
408	#103	—	2	4	no	—
409	#103	—	2	4	no	—
410	#103	—	4	2	no	—
411	#102	—	2	4	no	—
412	#102	—	2	4	no	—
413	#102	—	4	2	no	—
414	C₂²⋊C₈	—	2	4	no	—
415	#95	—	2	1	no	—
416	#95	—	2	1	no	—
417	#95	—	2	1	no	—
418	#95	—	2	1	no	—
419	#95	—	2	1	no	—
420	C₂⁴	—	2	2	no	—
421	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
422	#41	—	4	8	no	—
423	#41	—	4	8	no	—
424	#41	—	4	8	no	—
425	#41	—	4	8	no	—
426	C₂×Q₈	—	2	2	no	—
427	C₂×Q₈	—	2	1	no	—
428	C₂×Q₈	—	2	2	no	—
429	C₂×Q₈	—	2	2	no	—
430	C₂×Q₈	—	2	2	no	—
431	C₂×Q₈	—	2	2	no	—
432	C₂×Q₈	—	2	2	no	—
433	C₂×Q₈	—	2	2	no	—
434	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
435	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
436	C₂×D₄	—	2	4	no	—
437	C₂×D₄	—	2	4	no	—
438	C₂×D₄	—	2	8	no	—
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	2	2	no	—
444	C₂×D₄	—	4	2	no	—
445	C₂×D₄	—	4	2	no	—
446	C₂×D₄	—	4	4	no	—
447	C₂×D₄	—	4	4	no	—
448	C₂×D₄	—	4	2	no	—
449	C₂×D₄	—	4	2	no	—
450	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
451	C₂²×C₄	—	2	2	no	—
452	C₂²×C₄	—	2	2	no	—
453	C₂²×C₄	—	2	8	no	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	4	1	no	—
464	C₂²×C₄	—	4	1	no	—
465	C₂²×C₄	—	4	4	no	—
466	C₂²×C₄	—	4	4	no	—
467	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
468	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
469	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
470	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
471	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
473	C₂²×C₄	#41×C₄₁	1	2	yes	—
474	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
475	Q₁₆	—	4	8	no	—
476	SD₁₆	—	4	8	no	—
477	C₂×C₈	—	2	2	no	—
478	C₂×C₈	—	2	2	no	—
479	C₂×C₈	—	2	2	no	—
480	C₂×C₈	—	2	2	no	—
481	C₂×C₈	—	2	2	no	—
482	C₂×C₈	—	2	2	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	2	4	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	4	2	no	—
490	C₄⋊C₄	—	4	2	no	—
491	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₂²⋊C₄	—	2	8	no	—
494	C₂²⋊C₄	—	2	8	no	—
495	C₂²⋊C₄	—	4	2	no	—
496	C₄²	—	2	2	no	—
497	C₄²	—	2	2	no	—
498	C₄²	—	2	2	no	—
499	C₄²	—	4	2	no	—
500	C₄²	—	4	2	no	—
501	C₄²	C₂×D₄×C₄₁	1	4	yes	—
502	C₂³	—	2	1	no	—
503	C₂³	—	2	1	no	—
504	C₂³	—	2	1	no	—
505	C₂³	—	2	1	no	—
506	C₂³	—	2	1	no	—
507	C₂³	—	2	1	no	—
508	C₂³	—	2	4	no	—
509	C₂³	—	4	2	no	—
510	C₂³	—	4	2	no	—
511	C₂³	—	4	2	no	—
512	C₂³	—	4	2	no	—
513	C₂³	—	4	2	no	—
514	C₂³	—	4	2	no	—
515	C₂³	#121×C₄₁	1	2	yes	—
516	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
517	Q₈	—	2	4	no	—
518	Q₈	—	2	4	no	—
519	Q₈	—	2	4	no	—
520	Q₈	—	4	2	no	—
521	Q₈	—	4	2	no	—
522	Q₈	—	4	2	no	—
523	D₄	—	2	4	no	—
524	D₄	—	2	4	no	—
525	D₄	—	4	8	no	—
526	D₄	—	4	8	no	—
527	D₄	—	4	16	no	—
528	D₄	—	4	2	no	—
529	D₄	—	4	2	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	4	4	no	—
546	C₂×C₄	—	4	4	no	—
547	C₂×C₄	—	4	16	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	—	4	2	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	#120×C₄₁	1	2	yes	—
561	C₂×C₄	#120×C₄₁	1	2	yes	—
562	C₂×C₄	#121×C₄₁	1	2	yes	—
563	C₂×C₄	#135×C₄₁	1	2	yes	—
564	C₂×C₄	#137×C₄₁	1	2	yes	—
565	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
566	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
567	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
568	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
569	C₈	—	4	2	no	—
570	C₈	—	4	2	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	4	1	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	1	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	8	2	no	—
582	C₂²	—	8	2	no	—
583	C₂²	—	8	2	no	—
584	C₂²	—	8	2	no	—
585	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
586	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
587	C₂²	C₄₁×#535	1	1	yes	characteristic
588	C₂²	C₂×#135×C₄₁	1	1	yes	characteristic
589	C₂²	C₂×#137×C₄₁	1	1	yes	characteristic
590	C₂²	C₄₁×#575	1	1	yes	characteristic
591	C₂²	C₄₁×#575	1	1	yes	characteristic
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	2	2	no	—
595	C₄	—	2	2	no	—
596	C₄	—	2	2	no	—
597	C₄	—	4	4	no	—
598	C₄	—	4	2	no	—
599	C₄	—	4	2	no	—
600	C₄	—	8	1	no	—
601	C₄	—	8	1	no	—
602	C₄	D₄²×C₄₁	1	2	yes	—
603	C₄	C₄₁×#546	1	2	yes	—
604	C₂	—	4	2	no	—
605	C₂	—	8	4	no	—
606	C₂	C₄₁×#2408	1	1	yes	characteristic
607	C₂	C₄₁×#3002	1	1	yes	characteristic
608	C₂	C₄₁×#3007	1	1	yes	characteristic
609	C₂	C₄₁×#3035	1	1	yes	characteristic
610	C₂	C₄₁×#3037	1	1	yes	characteristic
611	C₂	C₄₁×#3210	1	1	yes	characteristic
612	C₂	C₄₁×#3219	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂×C₄₁×#532	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#464	C₂	semidirect
3. C₂×C₄₁×#452	C₂	semidirect
4. C₄×C₂²⋊Q₈×C₄₁	C₂	semidirect
5. C₄₁×#2043	C₂	semidirect
6. C₄₁×#2041	C₂	semidirect
7. C₄₁×#2016	C₂	semidirect
8. C₄₁×#2014	C₂	semidirect
9. C₄₁×#1983	C₂	semidirect
10. C₄₁×#1797	C₂	semidirect
11. C₄₁×#1788	C₂	semidirect
75. C₂²⋊Q₈×C₄₁	D₄	semidirect
264. #24216	C₄₁	direct (N × H)
299. C₂×#464	C₂×C₄₁	semidirect
300. C₂×#452	C₂×C₄₁	semidirect
301. C₄×C₂²⋊Q₈	C₂×C₄₁	semidirect
302. #2043	C₂×C₄₁	semidirect
303. #2041	C₂×C₄₁	semidirect
304. #2016	C₂×C₄₁	semidirect
305. #2014	C₂×C₄₁	semidirect
306. #1983	C₂×C₄₁	semidirect
307. #1797	C₂×C₄₁	semidirect
308. #1788	C₂×C₄₁	semidirect
352. C₄₁	#24216	direct (N × H)
382. C₂²⋊Q₈	D₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#532	2
C₂×C₄₁×#464	2
C₂×C₄₁×#452	2
C₄×C₂²⋊Q₈×C₄₁	2
C₄₁×#2043	2
C₄₁×#2041	2
C₄₁×#2016	2
C₄₁×#2014	2
C₄₁×#1983	2
C₄₁×#1797	2
C₄₁×#1788	2
#24216	41

Subgroups — C41×#24216 #23800417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24216 #23800417