Subgroups — C₄₁×#22997 #23799198

Order 10496.

1066 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#460	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×SD₁₆×C₄₁	C₂	1	2	yes	maximal, complemented
3	C₄₁×#2366	C₂	1	1	yes	maximal, characteristic
4	Q₈×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1957	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1934	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1880	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1864	C₂	1	2	yes	maximal, complemented
10	C₈×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#103×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
13	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₄₁	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₄₁	—	2	1	no	—
17	C₂×C₄×D₄×C₄₁	C₂²	1	2	yes	—
18	C₄₁×#460	C₄	1	8	yes	complemented
19	C₄×SD₁₆×C₄₁	—	2	8	no	—
20	C₂×#106×C₄₁	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×#102×C₄₁	C₂²	1	2	yes	—
25	C₂×#102×C₄₁	C₂²	1	2	yes	—
26	C₂×#102×C₄₁	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
29	C₄₁×#390	—	2	1	no	—
30	C₄₁×#390	C₂²	1	2	yes	—
31	C₄₁×#389	—	2	2	no	—
32	C₄₁×#389	C₂²	1	2	yes	—
33	C₄₁×#386	—	2	2	no	—
34	C₄₁×#384	—	2	1	no	—
35	C₄₁×#384	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₄₁	—	2	1	no	—
37	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
39	C₄₁×#340	C₂²	1	1	yes	characteristic
40	C₄₁×#340	C₂²	1	1	yes	characteristic
41	C₄₁×#340	C₂²	1	2	yes	—
42	C₄₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₄₁	—	2	1	no	—
45	C₂²×D₄×C₄₁	—	2	1	no	—
46	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
47	C₂³×C₄×C₄₁	—	2	2	no	—
48	C₂×SD₁₆×C₄₁	—	2	4	no	—
49	C₂×SD₁₆×C₄₁	—	2	4	no	—
50	C₂×SD₁₆×C₄₁	C₂×C₄	1	8	yes	complemented
51	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
54	#127×C₄₁	—	2	2	no	—
55	#127×C₄₁	C₂×C₄	1	4	yes	—
56	C₄×Q₈×C₄₁	—	2	2	no	—
57	C₄×Q₈×C₄₁	—	2	4	no	—
58	C₄×Q₈×C₄₁	—	4	2	no	—
59	C₄×Q₈×C₄₁	C₂×C₄	1	4	yes	complemented
60	C₄×D₄×C₄₁	—	2	4	no	—
61	C₄×D₄×C₄₁	—	4	2	no	—
62	C₄×D₄×C₄₁	—	4	4	no	—
63	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
65	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
67	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
75	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
77	C₂×C₄²×C₄₁	—	2	1	no	—
78	C₂×C₄²×C₄₁	—	2	1	no	—
79	C₂×C₄²×C₄₁	—	2	2	no	—
80	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
81	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
82	#106×C₄₁	—	2	4	no	—
83	C₄⋊C₈×C₄₁	C₂×C₄	1	4	yes	complemented
84	#103×C₄₁	—	2	2	no	—
85	#103×C₄₁	D₄	1	4	yes	complemented
86	#102×C₄₁	—	2	4	no	—
87	#102×C₄₁	—	4	2	no	—
88	#102×C₄₁	C₂×C₄	1	8	yes	complemented
89	C₄×C₈×C₄₁	—	2	4	no	—
90	C₄×C₈×C₄₁	—	2	2	no	—
91	#95×C₄₁	—	2	2	no	—
92	#95×C₄₁	—	2	1	no	—
93	#95×C₄₁	D₄	1	2	yes	—
94	C₂⁴×C₄₁	—	2	2	no	—
95	C₂×Q₈×C₄₁	—	2	1	no	—
96	C₂×Q₈×C₄₁	—	2	1	no	—
97	C₂×Q₈×C₄₁	—	2	1	no	—
98	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
99	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
100	C₂×Q₈×C₄₁	#41	1	2	yes	—
101	C₂×D₄×C₄₁	—	2	4	no	—
102	C₂×D₄×C₄₁	—	2	4	no	—
103	C₂×D₄×C₄₁	—	2	2	no	—
104	C₂×D₄×C₄₁	—	2	2	no	—
105	C₂×D₄×C₄₁	—	2	2	no	—
106	C₂×D₄×C₄₁	—	2	2	no	—
107	C₂×D₄×C₄₁	—	2	2	no	—
108	C₂×D₄×C₄₁	—	4	2	no	—
109	C₂×D₄×C₄₁	—	4	2	no	—
110	C₂×D₄×C₄₁	—	8	1	no	—
111	C₂×D₄×C₄₁	—	8	1	no	—
112	C₂×D₄×C₄₁	C₂²×C₄	1	4	yes	—
113	C₂²×C₄×C₄₁	—	2	1	no	—
114	C₂²×C₄×C₄₁	—	2	1	no	—
115	C₂²×C₄×C₄₁	—	2	1	no	—
116	C₂²×C₄×C₄₁	—	2	1	no	—
117	C₂²×C₄×C₄₁	—	2	2	no	—
118	C₂²×C₄×C₄₁	—	4	4	no	—
119	C₂²×C₄×C₄₁	—	4	4	no	—
120	C₂²×C₄×C₄₁	—	8	2	no	—
121	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
126	SD₁₆×C₄₁	—	2	16	no	—
127	C₂×C₈×C₄₁	—	2	2	no	—
128	C₂×C₈×C₄₁	—	2	1	no	—
129	C₂×C₈×C₄₁	—	2	2	no	—
130	C₂×C₈×C₄₁	—	2	1	no	—
131	C₂×C₈×C₄₁	—	2	1	no	—
132	C₂×C₈×C₄₁	—	2	1	no	—
133	C₂×C₈×C₄₁	—	2	1	no	—
134	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
135	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
136	C₄⋊C₄×C₄₁	—	2	4	no	—
137	C₄⋊C₄×C₄₁	—	2	2	no	—
138	C₄⋊C₄×C₄₁	—	2	1	no	—
139	C₄⋊C₄×C₄₁	—	2	1	no	—
140	C₄⋊C₄×C₄₁	—	2	2	no	—
141	C₄⋊C₄×C₄₁	—	2	1	no	—
142	C₄⋊C₄×C₄₁	—	2	2	no	—
143	C₄⋊C₄×C₄₁	—	2	1	no	—
144	C₄⋊C₄×C₄₁	—	4	4	no	—
145	C₄⋊C₄×C₄₁	—	4	2	no	—
146	C₄⋊C₄×C₄₁	—	8	1	no	—
147	C₄⋊C₄×C₄₁	—	8	2	no	—
148	C₄⋊C₄×C₄₁	SD₁₆	1	4	yes	complemented
149	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
150	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
151	C₂²⋊C₄×C₄₁	—	4	4	no	—
152	C₂²⋊C₄×C₄₁	—	4	4	no	—
153	C₂²⋊C₄×C₄₁	—	8	2	no	—
154	C₄²×C₄₁	—	2	4	no	—
155	C₄²×C₄₁	—	2	2	no	—
156	C₄²×C₄₁	—	2	2	no	—
157	C₄²×C₄₁	—	4	4	no	—
158	C₄²×C₄₁	—	4	2	no	—
159	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
160	C₄²×C₄₁	#41	1	2	yes	—
161	C₂³×C₄₁	—	2	4	no	—
162	C₂³×C₄₁	—	4	2	no	—
163	C₂³×C₄₁	—	4	2	no	—
164	C₂³×C₄₁	—	4	2	no	—
165	C₂³×C₄₁	—	4	2	no	—
166	C₂³×C₄₁	—	4	2	no	—
167	C₂³×C₄₁	—	4	2	no	—
168	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
169	Q₈×C₄₁	—	2	2	no	—
170	Q₈×C₄₁	—	2	2	no	—
171	Q₈×C₄₁	—	2	2	no	—
172	Q₈×C₄₁	C₄×D₄	1	4	yes	complemented
173	D₄×C₄₁	—	2	4	no	—
174	D₄×C₄₁	—	2	4	no	—
175	D₄×C₄₁	—	4	2	no	—
176	D₄×C₄₁	—	4	2	no	—
177	D₄×C₄₁	—	4	8	no	—
178	D₄×C₄₁	—	8	4	no	—
179	C₂×C₄×C₄₁	—	2	2	no	—
180	C₂×C₄×C₄₁	—	2	2	no	—
181	C₂×C₄×C₄₁	—	2	2	no	—
182	C₂×C₄×C₄₁	—	2	2	no	—
183	C₂×C₄×C₄₁	—	2	1	no	—
184	C₂×C₄×C₄₁	—	2	2	no	—
185	C₂×C₄×C₄₁	—	2	1	no	—
186	C₂×C₄×C₄₁	—	2	2	no	—
187	C₂×C₄×C₄₁	—	2	2	no	—
188	C₂×C₄×C₄₁	—	2	2	no	—
189	C₂×C₄×C₄₁	—	2	2	no	—
190	C₂×C₄×C₄₁	—	2	2	no	—
191	C₂×C₄×C₄₁	—	4	1	no	—
192	C₂×C₄×C₄₁	—	4	1	no	—
193	C₂×C₄×C₄₁	—	4	1	no	—
194	C₂×C₄×C₄₁	—	4	1	no	—
195	C₂×C₄×C₄₁	—	4	2	no	—
196	C₂×C₄×C₄₁	—	4	1	no	—
197	C₂×C₄×C₄₁	—	4	1	no	—
198	C₂×C₄×C₄₁	—	4	2	no	—
199	C₂×C₄×C₄₁	—	4	1	no	—
200	C₂×C₄×C₄₁	—	4	2	no	—
201	C₂×C₄×C₄₁	—	8	8	no	—
202	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
203	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
204	C₂×C₄×C₄₁	C₂×SD₁₆	1	4	yes	—
205	C₂×C₄×C₄₁	C₂×SD₁₆	1	2	yes	—
206	C₂×C₄×C₄₁	#135	1	2	yes	—
207	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
208	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
213	C₈×C₄₁	—	2	4	no	—
214	C₈×C₄₁	—	2	2	no	—
215	C₈×C₄₁	—	4	1	no	—
216	#22997	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂²×C₄₁	—	4	4	no	—
218	C₂²×C₄₁	—	4	4	no	—
219	C₂²×C₄₁	—	4	4	no	—
220	C₂²×C₄₁	—	8	2	no	—
221	C₂²×C₄₁	—	8	2	no	—
222	C₂²×C₄₁	—	8	2	no	—
223	C₂²×C₄₁	—	8	2	no	—
224	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
225	C₂²×C₄₁	D₄²	1	1	yes	characteristic
226	C₂²×C₄₁	#548	1	1	yes	characteristic
227	C₂²×C₄₁	C₂²×SD₁₆	1	1	yes	characteristic
228	C₂²×C₄₁	C₂×#135	1	1	yes	characteristic
229	C₂²×C₄₁	#576	1	1	yes	characteristic
230	C₂²×C₄₁	#577	1	1	yes	characteristic
231	C₄×C₄₁	—	2	4	no	—
232	C₄×C₄₁	—	2	2	no	—
233	C₄×C₄₁	—	2	4	no	—
234	C₄×C₄₁	—	4	2	no	—
235	C₄×C₄₁	—	4	2	no	—
236	C₄×C₄₁	—	4	2	no	—
237	C₄×C₄₁	—	8	1	no	—
238	C₄×C₄₁	—	8	2	no	—
239	C₄×C₄₁	C₄×SD₁₆	1	4	yes	complemented
240	C₄×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
241	C₄×C₄₁	#520	1	1	yes	characteristic
242	C₄×C₄₁	#547	1	1	yes	characteristic
243	C₄×C₄₁	#553	1	1	yes	characteristic
244	C₂×#460	C₂×C₄₁	1	1	yes	characteristic
245	C₂×C₄×SD₁₆	C₂×C₄₁	1	2	yes	complemented
246	#2366	C₂×C₄₁	1	1	yes	characteristic
247	Q₈×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic, complemented
248	#1957	C₂×C₄₁	1	1	yes	characteristic
249	#1948	C₂×C₄₁	1	1	yes	characteristic, complemented
250	#1934	C₂×C₄₁	1	2	yes	complemented
251	#1880	C₂×C₄₁	1	2	yes	complemented
252	#1864	C₂×C₄₁	1	2	yes	complemented
253	C₈×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic, complemented
254	C₄×#103	C₂×C₄₁	1	1	yes	characteristic, complemented
255	C₂×C₄₁	—	8	4	no	—
256	C₂×C₄₁	#2361	1	1	yes	characteristic
257	C₂×C₄₁	C₂×C₄×SD₁₆	1	1	yes	characteristic
258	C₂×C₄₁	#2946	1	1	yes	characteristic
259	C₂×C₄₁	D₄×SD₁₆	1	1	yes	characteristic
260	C₂×C₄₁	#3287	1	1	yes	characteristic
261	C₂×C₄₁	#3362	1	1	yes	characteristic
262	C₂×C₄₁	#3365	1	1	yes	characteristic
263	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
264	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
265	C₂×C₄×Q₈	C₂²×C₄₁	1	2	yes	—
266	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
267	C₂×C₄×D₄	—	2	1	no	—
268	C₂×C₄×D₄	C₂²×C₄₁	1	2	yes	—
269	#460	C₄×C₄₁	1	8	yes	complemented
270	C₄×SD₁₆	—	2	8	no	—
271	C₂×#106	C₂²×C₄₁	1	2	yes	—
272	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
273	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
274	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
275	C₂×#102	C₂²×C₄₁	1	2	yes	—
276	C₂×#102	C₂²×C₄₁	1	2	yes	—
277	C₂×#102	C₂²×C₄₁	1	2	yes	—
278	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
279	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
280	#390	—	2	1	no	—
281	#390	C₂²×C₄₁	1	2	yes	—
282	#389	—	2	2	no	—
283	#389	C₂²×C₄₁	1	2	yes	—
284	#386	—	2	2	no	—
285	#384	—	2	1	no	—
286	#384	C₂²×C₄₁	1	1	yes	characteristic
287	C₄×C₄⋊C₄	—	2	1	no	—
288	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
289	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
290	#340	C₂²×C₄₁	1	1	yes	characteristic
291	#340	C₂²×C₄₁	1	2	yes	—
292	#340	C₂²×C₄₁	1	1	yes	characteristic
293	#336	C₂²×C₄₁	1	2	yes	—
294	C₄₁	#22997	1	1	yes	characteristic, direct product factor, Sylow
295	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
296	C₂²×D₄	—	2	1	no	—
297	C₂²×D₄	—	2	1	no	—
298	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
299	C₂³×C₄	—	2	2	no	—
300	C₂×SD₁₆	—	2	4	no	—
301	C₂×SD₁₆	—	2	4	no	—
302	C₂×SD₁₆	C₂×C₄×C₄₁	1	8	yes	complemented
303	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
304	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
305	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
306	#127	—	2	2	no	—
307	#127	C₂×C₄×C₄₁	1	4	yes	—
308	C₄×Q₈	—	2	2	no	—
309	C₄×Q₈	—	2	4	no	—
310	C₄×Q₈	—	4	2	no	—
311	C₄×Q₈	C₂×C₄×C₄₁	1	4	yes	complemented
312	C₄×D₄	—	2	4	no	—
313	C₄×D₄	—	4	2	no	—
314	C₄×D₄	—	4	4	no	—
315	C₂×C₄⋊C₄	—	2	1	no	—
316	C₂×C₄⋊C₄	—	2	2	no	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄⋊C₄	—	2	2	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
321	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
322	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
323	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
325	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
326	C₂×C₂²⋊C₄	—	2	2	no	—
327	C₂×C₂²⋊C₄	—	2	2	no	—
328	C₂×C₂²⋊C₄	—	2	2	no	—
329	C₂×C₄²	—	2	1	no	—
330	C₂×C₄²	—	2	1	no	—
331	C₂×C₄²	—	2	2	no	—
332	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
333	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
334	#106	—	2	4	no	—
335	C₄⋊C₈	C₂×C₄×C₄₁	1	4	yes	complemented
336	#103	—	2	2	no	—
337	#103	D₄×C₄₁	1	4	yes	complemented
338	#102	—	2	4	no	—
339	#102	—	4	2	no	—
340	#102	C₂×C₄×C₄₁	1	8	yes	complemented
341	C₄×C₈	—	2	4	no	—
342	C₄×C₈	—	2	2	no	—
343	#95	—	2	2	no	—
344	#95	—	2	1	no	—
345	#95	D₄×C₄₁	1	2	yes	—
346	C₂⁴	—	2	2	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	1	no	—
350	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
351	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
352	C₂×Q₈	#41×C₄₁	1	2	yes	—
353	C₂×D₄	—	2	4	no	—
354	C₂×D₄	—	2	4	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	4	2	no	—
361	C₂×D₄	—	4	2	no	—
362	C₂×D₄	—	8	1	no	—
363	C₂×D₄	—	8	1	no	—
364	C₂×D₄	C₂²×C₄×C₄₁	1	4	yes	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	4	4	no	—
371	C₂²×C₄	—	4	4	no	—
372	C₂²×C₄	—	8	2	no	—
373	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
374	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
375	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
376	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
377	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
378	SD₁₆	—	2	16	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
387	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
388	C₄⋊C₄	—	2	4	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	4	4	no	—
397	C₄⋊C₄	—	4	2	no	—
398	C₄⋊C₄	—	8	1	no	—
399	C₄⋊C₄	—	8	2	no	—
400	C₄⋊C₄	SD₁₆×C₄₁	1	4	yes	complemented
401	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
402	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
403	C₂²⋊C₄	—	4	4	no	—
404	C₂²⋊C₄	—	4	4	no	—
405	C₂²⋊C₄	—	8	2	no	—
406	C₄²	—	2	4	no	—
407	C₄²	—	2	2	no	—
408	C₄²	—	2	2	no	—
409	C₄²	—	4	4	no	—
410	C₄²	—	4	2	no	—
411	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
412	C₄²	#41×C₄₁	1	2	yes	—
413	C₂³	—	2	4	no	—
414	C₂³	—	4	2	no	—
415	C₂³	—	4	2	no	—
416	C₂³	—	4	2	no	—
417	C₂³	—	4	2	no	—
418	C₂³	—	4	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
421	Q₈	—	2	2	no	—
422	Q₈	—	2	2	no	—
423	Q₈	—	2	2	no	—
424	Q₈	C₄×D₄×C₄₁	1	4	yes	complemented
425	D₄	—	2	4	no	—
426	D₄	—	2	4	no	—
427	D₄	—	4	2	no	—
428	D₄	—	4	2	no	—
429	D₄	—	4	8	no	—
430	D₄	—	8	4	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	8	8	no	—
454	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
455	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
456	C₂×C₄	C₂×SD₁₆×C₄₁	1	4	yes	—
457	C₂×C₄	C₂×SD₁₆×C₄₁	1	2	yes	—
458	C₂×C₄	#135×C₄₁	1	2	yes	—
459	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
460	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
461	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
462	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
463	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
464	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
465	C₈	—	2	4	no	—
466	C₈	—	2	2	no	—
467	C₈	—	4	1	no	—
468	C₂²	—	4	4	no	—
469	C₂²	—	4	4	no	—
470	C₂²	—	4	4	no	—
471	C₂²	—	8	2	no	—
472	C₂²	—	8	2	no	—
473	C₂²	—	8	2	no	—
474	C₂²	—	8	2	no	—
475	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
476	C₂²	D₄²×C₄₁	1	1	yes	characteristic
477	C₂²	C₄₁×#548	1	1	yes	characteristic
478	C₂²	C₂²×SD₁₆×C₄₁	1	1	yes	characteristic
479	C₂²	C₂×#135×C₄₁	1	1	yes	characteristic
480	C₂²	C₄₁×#576	1	1	yes	characteristic
481	C₂²	C₄₁×#577	1	1	yes	characteristic
482	C₄	—	2	4	no	—
483	C₄	—	2	2	no	—
484	C₄	—	2	4	no	—
485	C₄	—	4	2	no	—
486	C₄	—	4	2	no	—
487	C₄	—	4	2	no	—
488	C₄	—	8	1	no	—
489	C₄	—	8	2	no	—
490	C₄	C₄×SD₁₆×C₄₁	1	4	yes	complemented
491	C₄	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
492	C₄	C₄₁×#520	1	1	yes	characteristic
493	C₄	C₄₁×#547	1	1	yes	characteristic
494	C₄	C₄₁×#553	1	1	yes	characteristic
495	C₂	—	8	4	no	—
496	C₂	C₄₁×#2361	1	1	yes	characteristic
497	C₂	C₂×C₄×SD₁₆×C₄₁	1	1	yes	characteristic
498	C₂	C₄₁×#2946	1	1	yes	characteristic
499	C₂	D₄×SD₁₆×C₄₁	1	1	yes	characteristic
500	C₂	C₄₁×#3287	1	1	yes	characteristic
501	C₂	C₄₁×#3362	1	1	yes	characteristic
502	C₂	C₄₁×#3365	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	C₂×#127×C₄₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₄×C₄⋊C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×SD₁₆×C₄₁	C₂	semidirect
4. Q₈×C₄⋊C₄×C₄₁	C₂	semidirect
6. C₄₁×#1948	C₂	semidirect
7. C₄₁×#1934	C₂	semidirect
8. C₄₁×#1880	C₂	semidirect
9. C₄₁×#1864	C₂	semidirect
10. C₈×C₄⋊C₄×C₄₁	C₂	semidirect
11. C₄×#103×C₄₁	C₂	semidirect
18. C₄₁×#460	C₄	semidirect
50. C₂×SD₁₆×C₄₁	C₂×C₄	semidirect
59. C₄×Q₈×C₄₁	C₂×C₄	semidirect
83. C₄⋊C₈×C₄₁	C₂×C₄	semidirect
85. #103×C₄₁	D₄	semidirect
88. #102×C₄₁	C₂×C₄	semidirect
148. C₄⋊C₄×C₄₁	SD₁₆	semidirect
172. Q₈×C₄₁	C₄×D₄	semidirect
216. #22997	C₄₁	direct (N × H)
239. C₄×C₄₁	C₄×SD₁₆	semidirect
245. C₂×C₄×SD₁₆	C₂×C₄₁	semidirect
247. Q₈×C₄⋊C₄	C₂×C₄₁	semidirect
249. #1948	C₂×C₄₁	semidirect
250. #1934	C₂×C₄₁	semidirect
251. #1880	C₂×C₄₁	semidirect
252. #1864	C₂×C₄₁	semidirect
253. C₈×C₄⋊C₄	C₂×C₄₁	semidirect
254. C₄×#103	C₂×C₄₁	semidirect
269. #460	C₄×C₄₁	semidirect
294. C₄₁	#22997	direct (N × H)
302. C₂×SD₁₆	C₂×C₄×C₄₁	semidirect
311. C₄×Q₈	C₂×C₄×C₄₁	semidirect
335. C₄⋊C₈	C₂×C₄×C₄₁	semidirect
337. #103	D₄×C₄₁	semidirect
340. #102	C₂×C₄×C₄₁	semidirect
400. C₄⋊C₄	SD₁₆×C₄₁	semidirect
424. Q₈	C₄×D₄×C₄₁	semidirect
490. C₄	C₄×SD₁₆×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×#460	2
C₂×C₄×SD₁₆×C₄₁	2
C₄₁×#2366	2
Q₈×C₄⋊C₄×C₄₁	2
C₄₁×#1957	2
C₄₁×#1948	2
C₄₁×#1934	2
C₄₁×#1880	2
C₄₁×#1864	2
C₈×C₄⋊C₄×C₄₁	2
C₄×#103×C₄₁	2
#22997	41

Subgroups — C41×#22997 #23799198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22997 #23799198