Subgroups — C₄₁×#19809 #23796010

Order 10496.

632 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Q₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×SD₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3362	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#3278	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#1714	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1695	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1678	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1672	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1666	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1645	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1637	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₁×#1631	C₂	1	1	yes	maximal, characteristic
13	C₄₁×#1623	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄₁×#1583	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₁×#1579	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#137×C₄₁	—	2	1	no	—
17	Q₈²×C₄₁	C₂²	1	1	yes	characteristic
18	Q₈²×C₄₁	C₂²	1	1	yes	characteristic
19	C₄₁×#550	C₂²	1	1	yes	characteristic
20	D₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
21	C₄₁×#500	—	2	1	no	—
22	C₄₁×#500	C₂²	1	1	yes	characteristic
23	C₄₁×#498	—	2	1	no	—
24	C₄₁×#498	C₂²	1	1	yes	characteristic
25	C₄₁×#492	C₂²	1	1	yes	characteristic
26	C₄₁×#492	C₂²	1	1	yes	characteristic
27	C₄₁×#487	C₂²	1	1	yes	characteristic
28	C₄₁×#486	—	2	1	no	—
29	C₄₁×#486	C₂²	1	1	yes	characteristic
30	C₄₁×#477	—	2	1	no	—
31	C₄₁×#477	—	2	1	no	—
32	C₄₁×#477	C₂²	1	1	yes	characteristic
33	C₄₁×#477	C₂²	1	1	yes	characteristic
34	C₄₁×#476	—	2	1	no	—
35	C₄₁×#476	—	2	1	no	—
36	C₄₁×#476	C₂²	1	1	yes	characteristic
37	C₄₁×#476	C₂²	1	1	yes	characteristic
38	C₄₁×#475	—	2	1	no	—
39	C₄₁×#475	C₂²	1	1	yes	characteristic
40	C₄₁×#468	—	2	1	no	—
41	C₄₁×#462	C₂²	1	1	yes	characteristic
42	C₄₁×#460	—	2	1	no	—
43	C₄₁×#460	C₂²	1	1	yes	characteristic
44	C₄₁×#460	C₂²	1	1	yes	characteristic
45	C₄₁×#452	—	2	1	no	—
46	C₄₁×#448	—	2	1	no	—
47	C₈×Q₈×C₄₁	C₂²	1	1	yes	characteristic
48	C₈×Q₈×C₄₁	C₂²	1	1	yes	characteristic
49	C₈×Q₈×C₄₁	C₂²	1	1	yes	characteristic
50	C₄×Q₁₆×C₄₁	—	2	1	no	—
51	C₄×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
52	C₄×SD₁₆×C₄₁	—	2	1	no	—
53	C₄×SD₁₆×C₄₁	—	2	1	no	—
54	C₄×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
55	C₄₁×#436	C₂²	1	1	yes	characteristic
56	C₄₁×#334	C₂²	1	1	yes	characteristic
57	C₄₁×#334	C₂²	1	1	yes	characteristic
58	C₄₁×#333	C₂²	1	1	yes	characteristic
59	C₄₁×#333	C₂²	1	1	yes	characteristic
60	C₄₁×#332	C₂²	1	1	yes	characteristic
61	C₄₁×#331	C₂²	1	1	yes	characteristic
62	C₄₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₄₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₄₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₄₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₄₁×#322	C₂²	1	1	yes	characteristic
67	C₂×#41×C₄₁	—	2	1	no	—
68	C₂²×Q₈×C₄₁	—	2	1	no	—
69	#137×C₄₁	—	4	4	no	—
70	C₂×Q₁₆×C₄₁	—	2	1	no	—
71	C₂×Q₁₆×C₄₁	D₄	1	1	yes	characteristic
72	C₂×SD₁₆×C₄₁	—	2	1	no	—
73	C₂×SD₁₆×C₄₁	—	2	1	no	—
74	C₂×SD₁₆×C₄₁	—	2	1	no	—
75	C₂×SD₁₆×C₄₁	—	2	1	no	—
76	C₂×SD₁₆×C₄₁	—	4	1	no	—
77	C₂×SD₁₆×C₄₁	D₄	1	1	yes	characteristic
78	C₂×M₁₆×C₄₁	—	2	1	no	—
79	C₄⋊Q₈×C₄₁	—	2	1	no	—
80	C₄⋊Q₈×C₄₁	—	2	1	no	—
81	C₄⋊Q₈×C₄₁	—	2	1	no	—
82	C₄⋊Q₈×C₄₁	—	2	1	no	—
83	C₄⋊Q₈×C₄₁	—	2	1	no	—
84	C₄⋊Q₈×C₄₁	—	4	1	no	—
85	C₄⋊Q₈×C₄₁	—	4	1	no	—
86	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
87	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
88	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
89	#127×C₄₁	—	2	1	no	—
90	#127×C₄₁	C₂³	1	1	yes	characteristic
91	C₂²⋊Q₈×C₄₁	—	2	1	no	—
92	C₂²⋊Q₈×C₄₁	—	4	1	no	—
93	#121×C₄₁	—	2	1	no	—
94	#121×C₄₁	—	4	1	no	—
95	C₄×Q₈×C₄₁	—	2	1	no	—
96	C₄×Q₈×C₄₁	—	2	1	no	—
97	C₄×Q₈×C₄₁	—	2	1	no	—
98	C₄×Q₈×C₄₁	—	2	1	no	—
99	C₄×Q₈×C₄₁	C₂³	1	1	yes	characteristic
100	C₄×Q₈×C₄₁	C₂³	1	1	yes	characteristic
101	C₄×Q₈×C₄₁	C₂³	1	1	yes	characteristic
102	C₄×D₄×C₄₁	—	2	1	no	—
103	C₄×D₄×C₄₁	—	2	1	no	—
104	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
105	#107×C₄₁	—	2	1	no	—
106	#107×C₄₁	—	2	1	no	—
107	#107×C₄₁	—	2	1	no	—
108	#107×C₄₁	—	2	1	no	—
109	#107×C₄₁	D₄	1	1	yes	characteristic
110	#106×C₄₁	—	2	1	no	—
111	#106×C₄₁	—	2	1	no	—
112	#106×C₄₁	—	2	1	no	—
113	#106×C₄₁	—	2	1	no	—
114	#106×C₄₁	—	2	1	no	—
115	#106×C₄₁	D₄	1	1	yes	characteristic
116	C₄⋊C₈×C₄₁	—	2	1	no	—
117	C₄⋊C₈×C₄₁	—	2	1	no	—
118	C₄⋊C₈×C₄₁	—	2	1	no	—
119	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
120	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
121	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
122	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
123	#103×C₄₁	—	2	1	no	—
124	#103×C₄₁	—	2	1	no	—
125	#103×C₄₁	—	2	1	no	—
126	#103×C₄₁	—	2	1	no	—
127	#103×C₄₁	—	4	1	no	—
128	#103×C₄₁	D₄	1	1	yes	characteristic
129	#103×C₄₁	D₄	1	1	yes	characteristic
130	#103×C₄₁	D₄	1	1	yes	characteristic
131	#102×C₄₁	—	2	1	no	—
132	#102×C₄₁	—	2	1	no	—
133	#102×C₄₁	—	2	1	no	—
134	#102×C₄₁	—	2	1	no	—
135	#102×C₄₁	—	2	1	no	—
136	#102×C₄₁	—	2	1	no	—
137	#102×C₄₁	D₄	1	1	yes	characteristic
138	C₂²⋊C₈×C₄₁	—	2	1	no	—
139	C₄×C₈×C₄₁	—	2	1	no	—
140	C₄×C₈×C₄₁	—	2	1	no	—
141	C₄×C₈×C₄₁	—	2	1	no	—
142	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
145	#41×C₄₁	—	4	2	no	—
146	#41×C₄₁	—	8	1	no	—
147	C₂×Q₈×C₄₁	—	2	1	no	—
148	C₂×Q₈×C₄₁	—	2	1	no	—
149	C₂×Q₈×C₄₁	—	2	1	no	—
150	C₂×Q₈×C₄₁	—	4	1	no	—
151	C₂×Q₈×C₄₁	—	4	1	no	—
152	C₂×Q₈×C₄₁	—	4	1	no	—
153	C₂×Q₈×C₄₁	—	4	2	no	—
154	C₂×Q₈×C₄₁	—	8	1	no	—
155	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
156	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
157	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
158	C₂×D₄×C₄₁	—	2	1	no	—
159	C₂×D₄×C₄₁	—	2	1	no	—
160	C₂×D₄×C₄₁	—	2	1	no	—
161	C₂×D₄×C₄₁	—	4	1	no	—
162	C₂×D₄×C₄₁	—	4	1	no	—
163	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₄₁	—	2	1	no	—
165	C₂²×C₄×C₄₁	—	4	1	no	—
166	C₂²×C₄×C₄₁	—	4	1	no	—
167	Q₁₆×C₄₁	—	4	2	no	—
168	Q₁₆×C₄₁	SD₁₆	1	4	yes	complemented
169	SD₁₆×C₄₁	—	2	2	no	—
170	SD₁₆×C₄₁	—	4	2	no	—
171	SD₁₆×C₄₁	—	4	2	no	—
172	SD₁₆×C₄₁	—	4	2	no	—
173	SD₁₆×C₄₁	—	8	1	no	—
174	SD₁₆×C₄₁	—	8	2	no	—
175	M₁₆×C₄₁	—	4	2	no	—
176	C₂×C₈×C₄₁	—	2	1	no	—
177	C₂×C₈×C₄₁	—	2	1	no	—
178	C₂×C₈×C₄₁	—	2	1	no	—
179	C₂×C₈×C₄₁	—	2	1	no	—
180	C₂×C₈×C₄₁	—	2	1	no	—
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	—	2	1	no	—
194	C₄⋊C₄×C₄₁	—	4	1	no	—
195	C₄⋊C₄×C₄₁	—	4	1	no	—
196	C₄⋊C₄×C₄₁	—	4	1	no	—
197	C₄⋊C₄×C₄₁	—	4	1	no	—
198	C₄⋊C₄×C₄₁	—	4	1	no	—
199	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
200	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
201	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
202	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
203	C₂²⋊C₄×C₄₁	—	2	1	no	—
204	C₂²⋊C₄×C₄₁	—	4	1	no	—
205	C₂²⋊C₄×C₄₁	—	4	1	no	—
206	C₄²×C₄₁	—	2	1	no	—
207	C₄²×C₄₁	—	2	1	no	—
208	C₄²×C₄₁	—	2	1	no	—
209	C₄²×C₄₁	—	4	1	no	—
210	C₄²×C₄₁	—	4	1	no	—
211	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
212	C₂³×C₄₁	—	2	1	no	—
213	C₂³×C₄₁	—	4	1	no	—
214	Q₈×C₄₁	—	2	1	no	—
215	Q₈×C₄₁	—	2	1	no	—
216	Q₈×C₄₁	—	2	1	no	—
217	Q₈×C₄₁	—	4	1	no	—
218	Q₈×C₄₁	—	4	1	no	—
219	Q₈×C₄₁	—	8	2	no	—
220	Q₈×C₄₁	—	8	1	no	—
221	Q₈×C₄₁	—	8	1	no	—
222	Q₈×C₄₁	—	16	1	no	—
223	Q₈×C₄₁	—	16	1	no	—
224	Q₈×C₄₁	C₂×SD₁₆	1	2	yes	—
225	Q₈×C₄₁	C₂×SD₁₆	1	2	yes	—
226	Q₈×C₄₁	#137	1	2	yes	complemented
227	D₄×C₄₁	—	2	1	no	—
228	D₄×C₄₁	—	2	1	no	—
229	D₄×C₄₁	—	4	1	no	—
230	D₄×C₄₁	—	4	1	no	—
231	D₄×C₄₁	—	4	1	no	—
232	D₄×C₄₁	—	4	1	no	—
233	D₄×C₄₁	—	8	2	no	—
234	D₄×C₄₁	—	8	1	no	—
235	D₄×C₄₁	—	16	1	no	—
236	C₂×C₄×C₄₁	—	2	1	no	—
237	C₂×C₄×C₄₁	—	2	1	no	—
238	C₂×C₄×C₄₁	—	2	1	no	—
239	C₂×C₄×C₄₁	—	2	1	no	—
240	C₂×C₄×C₄₁	—	2	1	no	—
241	C₂×C₄×C₄₁	—	2	1	no	—
242	C₂×C₄×C₄₁	—	2	1	no	—
243	C₂×C₄×C₄₁	—	4	1	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	—	8	2	no	—
249	C₂×C₄×C₄₁	—	8	2	no	—
250	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
253	C₈×C₄₁	—	2	1	no	—
254	C₈×C₄₁	—	2	1	no	—
255	C₈×C₄₁	—	2	1	no	—
256	C₈×C₄₁	—	4	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	C₂×SD₁₆	1	2	yes	—
262	#19809	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
263	C₂²×C₄₁	—	4	1	no	—
264	C₂²×C₄₁	—	4	1	no	—
265	C₂²×C₄₁	—	4	1	no	—
266	C₂²×C₄₁	—	8	1	no	—
267	C₂²×C₄₁	—	8	1	no	—
268	C₂²×C₄₁	—	8	1	no	—
269	C₂²×C₄₁	D₄²	1	1	yes	characteristic
270	C₄×C₄₁	—	2	2	no	—
271	C₄×C₄₁	—	2	2	no	—
272	C₄×C₄₁	—	2	2	no	—
273	C₄×C₄₁	—	2	1	no	—
274	C₄×C₄₁	—	4	1	no	—
275	C₄×C₄₁	—	4	1	no	—
276	C₄×C₄₁	—	4	1	no	—
277	C₄×C₄₁	—	4	1	no	—
278	C₄×C₄₁	—	8	1	no	—
279	C₄×C₄₁	—	8	1	no	—
280	C₄×C₄₁	—	8	1	no	—
281	C₄×C₄₁	C₂²×SD₁₆	1	1	yes	characteristic
282	C₄×C₄₁	C₂×#137	1	1	yes	characteristic
283	C₄×C₄₁	#577	1	1	yes	characteristic
284	C₄×C₄₁	#578	1	1	yes	characteristic
285	Q₈×Q₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
286	Q₈×SD₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
287	#3362	C₂×C₄₁	1	1	yes	characteristic
288	#3278	C₂×C₄₁	1	1	yes	characteristic
289	#1714	C₂×C₄₁	1	1	yes	characteristic, complemented
290	#1695	C₂×C₄₁	1	1	yes	characteristic, complemented
291	#1678	C₂×C₄₁	1	1	yes	characteristic, complemented
292	#1672	C₂×C₄₁	1	1	yes	characteristic, complemented
293	#1666	C₂×C₄₁	1	1	yes	characteristic, complemented
294	#1645	C₂×C₄₁	1	1	yes	characteristic, complemented
295	#1637	C₂×C₄₁	1	1	yes	characteristic, complemented
296	#1631	C₂×C₄₁	1	1	yes	characteristic
297	#1623	C₂×C₄₁	1	1	yes	characteristic, complemented
298	#1583	C₂×C₄₁	1	1	yes	characteristic, complemented
299	#1579	C₂×C₄₁	1	1	yes	characteristic, complemented
300	C₂×C₄₁	—	8	1	no	—
301	C₂×C₄₁	—	16	1	no	—
302	C₂×C₄₁	#3270	1	1	yes	characteristic
303	C₂×C₄₁	D₄×SD₁₆	1	1	yes	characteristic
304	C₂×C₄₁	#3288	1	1	yes	characteristic
305	C₂×#137	—	2	1	no	—
306	Q₈²	C₂²×C₄₁	1	1	yes	characteristic
307	Q₈²	C₂²×C₄₁	1	1	yes	characteristic
308	#550	C₂²×C₄₁	1	1	yes	characteristic
309	D₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
310	#500	—	2	1	no	—
311	#500	C₂²×C₄₁	1	1	yes	characteristic
312	#498	—	2	1	no	—
313	#498	C₂²×C₄₁	1	1	yes	characteristic
314	#492	C₂²×C₄₁	1	1	yes	characteristic
315	#492	C₂²×C₄₁	1	1	yes	characteristic
316	#487	C₂²×C₄₁	1	1	yes	characteristic
317	#486	—	2	1	no	—
318	#486	C₂²×C₄₁	1	1	yes	characteristic
319	#477	—	2	1	no	—
320	#477	—	2	1	no	—
321	#477	C₂²×C₄₁	1	1	yes	characteristic
322	#477	C₂²×C₄₁	1	1	yes	characteristic
323	#476	—	2	1	no	—
324	#476	—	2	1	no	—
325	#476	C₂²×C₄₁	1	1	yes	characteristic
326	#476	C₂²×C₄₁	1	1	yes	characteristic
327	#475	—	2	1	no	—
328	#475	C₂²×C₄₁	1	1	yes	characteristic
329	#468	—	2	1	no	—
330	#462	C₂²×C₄₁	1	1	yes	characteristic
331	#460	—	2	1	no	—
332	#460	C₂²×C₄₁	1	1	yes	characteristic
333	#460	C₂²×C₄₁	1	1	yes	characteristic
334	#452	—	2	1	no	—
335	#448	—	2	1	no	—
336	C₈×Q₈	C₂²×C₄₁	1	1	yes	characteristic
337	C₈×Q₈	C₂²×C₄₁	1	1	yes	characteristic
338	C₈×Q₈	C₂²×C₄₁	1	1	yes	characteristic
339	C₄×Q₁₆	—	2	1	no	—
340	C₄×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
341	C₄×SD₁₆	—	2	1	no	—
342	C₄×SD₁₆	—	2	1	no	—
343	C₄×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
344	#436	C₂²×C₄₁	1	1	yes	characteristic
345	#334	C₂²×C₄₁	1	1	yes	characteristic
346	#334	C₂²×C₄₁	1	1	yes	characteristic
347	#333	C₂²×C₄₁	1	1	yes	characteristic
348	#333	C₂²×C₄₁	1	1	yes	characteristic
349	#332	C₂²×C₄₁	1	1	yes	characteristic
350	#331	C₂²×C₄₁	1	1	yes	characteristic
351	Q₈⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
352	Q₈⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
353	Q₈⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
354	D₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
355	#322	C₂²×C₄₁	1	1	yes	characteristic
356	C₄₁	#19809	1	1	yes	characteristic, direct product factor, Sylow
357	C₂×#41	—	2	1	no	—
358	C₂²×Q₈	—	2	1	no	—
359	#137	—	4	4	no	—
360	C₂×Q₁₆	—	2	1	no	—
361	C₂×Q₁₆	D₄×C₄₁	1	1	yes	characteristic
362	C₂×SD₁₆	—	2	1	no	—
363	C₂×SD₁₆	—	2	1	no	—
364	C₂×SD₁₆	—	2	1	no	—
365	C₂×SD₁₆	—	2	1	no	—
366	C₂×SD₁₆	—	4	1	no	—
367	C₂×SD₁₆	D₄×C₄₁	1	1	yes	characteristic
368	C₂×M₁₆	—	2	1	no	—
369	C₄⋊Q₈	—	2	1	no	—
370	C₄⋊Q₈	—	2	1	no	—
371	C₄⋊Q₈	—	2	1	no	—
372	C₄⋊Q₈	—	2	1	no	—
373	C₄⋊Q₈	—	2	1	no	—
374	C₄⋊Q₈	—	4	1	no	—
375	C₄⋊Q₈	—	4	1	no	—
376	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
377	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
378	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
379	#127	—	2	1	no	—
380	#127	C₂³×C₄₁	1	1	yes	characteristic
381	C₂²⋊Q₈	—	2	1	no	—
382	C₂²⋊Q₈	—	4	1	no	—
383	#121	—	2	1	no	—
384	#121	—	4	1	no	—
385	C₄×Q₈	—	2	1	no	—
386	C₄×Q₈	—	2	1	no	—
387	C₄×Q₈	—	2	1	no	—
388	C₄×Q₈	—	2	1	no	—
389	C₄×Q₈	C₂³×C₄₁	1	1	yes	characteristic
390	C₄×Q₈	C₂³×C₄₁	1	1	yes	characteristic
391	C₄×Q₈	C₂³×C₄₁	1	1	yes	characteristic
392	C₄×D₄	—	2	1	no	—
393	C₄×D₄	—	2	1	no	—
394	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
395	#107	—	2	1	no	—
396	#107	—	2	1	no	—
397	#107	—	2	1	no	—
398	#107	—	2	1	no	—
399	#107	D₄×C₄₁	1	1	yes	characteristic
400	#106	—	2	1	no	—
401	#106	—	2	1	no	—
402	#106	—	2	1	no	—
403	#106	—	2	1	no	—
404	#106	—	2	1	no	—
405	#106	D₄×C₄₁	1	1	yes	characteristic
406	C₄⋊C₈	—	2	1	no	—
407	C₄⋊C₈	—	2	1	no	—
408	C₄⋊C₈	—	2	1	no	—
409	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
410	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
411	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
412	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
413	#103	—	2	1	no	—
414	#103	—	2	1	no	—
415	#103	—	2	1	no	—
416	#103	—	2	1	no	—
417	#103	—	4	1	no	—
418	#103	D₄×C₄₁	1	1	yes	characteristic
419	#103	D₄×C₄₁	1	1	yes	characteristic
420	#103	D₄×C₄₁	1	1	yes	characteristic
421	#102	—	2	1	no	—
422	#102	—	2	1	no	—
423	#102	—	2	1	no	—
424	#102	—	2	1	no	—
425	#102	—	2	1	no	—
426	#102	—	2	1	no	—
427	#102	D₄×C₄₁	1	1	yes	characteristic
428	C₂²⋊C₈	—	2	1	no	—
429	C₄×C₈	—	2	1	no	—
430	C₄×C₈	—	2	1	no	—
431	C₄×C₈	—	2	1	no	—
432	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
433	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
434	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
435	#41	—	4	2	no	—
436	#41	—	8	1	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	4	1	no	—
441	C₂×Q₈	—	4	1	no	—
442	C₂×Q₈	—	4	1	no	—
443	C₂×Q₈	—	4	2	no	—
444	C₂×Q₈	—	8	1	no	—
445	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
446	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
447	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	4	1	no	—
452	C₂×D₄	—	4	1	no	—
453	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	4	1	no	—
456	C₂²×C₄	—	4	1	no	—
457	Q₁₆	—	4	2	no	—
458	Q₁₆	SD₁₆×C₄₁	1	4	yes	complemented
459	SD₁₆	—	2	2	no	—
460	SD₁₆	—	4	2	no	—
461	SD₁₆	—	4	2	no	—
462	SD₁₆	—	4	2	no	—
463	SD₁₆	—	8	1	no	—
464	SD₁₆	—	8	2	no	—
465	M₁₆	—	4	2	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	—	2	1	no	—
471	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
472	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
473	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
474	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
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	—	4	1	no	—
485	C₄⋊C₄	—	4	1	no	—
486	C₄⋊C₄	—	4	1	no	—
487	C₄⋊C₄	—	4	1	no	—
488	C₄⋊C₄	—	4	1	no	—
489	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
490	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
491	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
492	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
493	C₂²⋊C₄	—	2	1	no	—
494	C₂²⋊C₄	—	4	1	no	—
495	C₂²⋊C₄	—	4	1	no	—
496	C₄²	—	2	1	no	—
497	C₄²	—	2	1	no	—
498	C₄²	—	2	1	no	—
499	C₄²	—	4	1	no	—
500	C₄²	—	4	1	no	—
501	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
502	C₂³	—	2	1	no	—
503	C₂³	—	4	1	no	—
504	Q₈	—	2	1	no	—
505	Q₈	—	2	1	no	—
506	Q₈	—	2	1	no	—
507	Q₈	—	4	1	no	—
508	Q₈	—	4	1	no	—
509	Q₈	—	8	2	no	—
510	Q₈	—	8	1	no	—
511	Q₈	—	8	1	no	—
512	Q₈	—	16	1	no	—
513	Q₈	—	16	1	no	—
514	Q₈	C₂×SD₁₆×C₄₁	1	2	yes	—
515	Q₈	C₂×SD₁₆×C₄₁	1	2	yes	—
516	Q₈	#137×C₄₁	1	2	yes	complemented
517	D₄	—	2	1	no	—
518	D₄	—	2	1	no	—
519	D₄	—	4	1	no	—
520	D₄	—	4	1	no	—
521	D₄	—	4	1	no	—
522	D₄	—	4	1	no	—
523	D₄	—	8	2	no	—
524	D₄	—	8	1	no	—
525	D₄	—	16	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	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	8	2	no	—
539	C₂×C₄	—	8	2	no	—
540	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
541	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
542	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
543	C₈	—	2	1	no	—
544	C₈	—	2	1	no	—
545	C₈	—	2	1	no	—
546	C₈	—	4	1	no	—
547	C₈	—	4	1	no	—
548	C₈	—	4	1	no	—
549	C₈	—	4	1	no	—
550	C₈	—	4	1	no	—
551	C₈	C₂×SD₁₆×C₄₁	1	2	yes	—
552	C₂²	—	4	1	no	—
553	C₂²	—	4	1	no	—
554	C₂²	—	4	1	no	—
555	C₂²	—	8	1	no	—
556	C₂²	—	8	1	no	—
557	C₂²	—	8	1	no	—
558	C₂²	D₄²×C₄₁	1	1	yes	characteristic
559	C₄	—	2	2	no	—
560	C₄	—	2	2	no	—
561	C₄	—	2	2	no	—
562	C₄	—	2	1	no	—
563	C₄	—	4	1	no	—
564	C₄	—	4	1	no	—
565	C₄	—	4	1	no	—
566	C₄	—	4	1	no	—
567	C₄	—	8	1	no	—
568	C₄	—	8	1	no	—
569	C₄	—	8	1	no	—
570	C₄	C₂²×SD₁₆×C₄₁	1	1	yes	characteristic
571	C₄	C₂×#137×C₄₁	1	1	yes	characteristic
572	C₄	C₄₁×#577	1	1	yes	characteristic
573	C₄	C₄₁×#578	1	1	yes	characteristic
574	C₂	—	8	1	no	—
575	C₂	—	16	1	no	—
576	C₂	C₄₁×#3270	1	1	yes	characteristic
577	C₂	D₄×SD₁₆×C₄₁	1	1	yes	characteristic
578	C₂	C₄₁×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₄₁×#550	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×Q₁₆×C₄₁	C₂	semidirect
2. Q₈×SD₁₆×C₄₁	C₂	semidirect
5. C₄₁×#1714	C₂	semidirect
6. C₄₁×#1695	C₂	semidirect
7. C₄₁×#1678	C₂	semidirect
8. C₄₁×#1672	C₂	semidirect
9. C₄₁×#1666	C₂	semidirect
10. C₄₁×#1645	C₂	semidirect
11. C₄₁×#1637	C₂	semidirect
13. C₄₁×#1623	C₂	semidirect
14. C₄₁×#1583	C₂	semidirect
15. C₄₁×#1579	C₂	semidirect
168. Q₁₆×C₄₁	SD₁₆	semidirect
226. Q₈×C₄₁	#137	semidirect
262. #19809	C₄₁	direct (N × H)
285. Q₈×Q₁₆	C₂×C₄₁	semidirect
286. Q₈×SD₁₆	C₂×C₄₁	semidirect
289. #1714	C₂×C₄₁	semidirect
290. #1695	C₂×C₄₁	semidirect
291. #1678	C₂×C₄₁	semidirect
292. #1672	C₂×C₄₁	semidirect
293. #1666	C₂×C₄₁	semidirect
294. #1645	C₂×C₄₁	semidirect
295. #1637	C₂×C₄₁	semidirect
297. #1623	C₂×C₄₁	semidirect
298. #1583	C₂×C₄₁	semidirect
299. #1579	C₂×C₄₁	semidirect
356. C₄₁	#19809	direct (N × H)
458. Q₁₆	SD₁₆×C₄₁	semidirect
516. Q₈	#137×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Q₁₆×C₄₁	2
Q₈×SD₁₆×C₄₁	2
C₄₁×#3362	2
C₄₁×#3278	2
C₄₁×#1714	2
C₄₁×#1695	2
C₄₁×#1678	2
C₄₁×#1672	2
C₄₁×#1666	2
C₄₁×#1645	2
C₄₁×#1637	2
C₄₁×#1631	2
C₄₁×#1623	2
C₄₁×#1583	2
C₄₁×#1579	2
#19809	41

Subgroups — C41×#19809 #23796010

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#19809 #23796010