Subgroups — C₂×C₄₁×#3221 #23836995

Order 10496.

2150 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#540	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#536	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3221	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄₁×#496	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₁×#491	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₁×#488	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₁×#463	C₂	1	2	yes	maximal, complemented
9	C₂×C₄₁×#461	C₂	1	2	yes	maximal, complemented
10	C₂×C₄₁×#449	C₂	1	2	yes	maximal, complemented
11	C₂×C₄₁×#448	C₂	1	2	yes	maximal, complemented
12	C₂×C₄₁×#433	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
14	C₂³×Q₈×C₄₁	C₂²	1	1	yes	characteristic
15	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
16	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
17	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
18	C₂²×D₈×C₄₁	C₂²	1	2	yes	—
19	C₄₁×#540	—	2	4	no	—
20	C₄₁×#540	C₂²	1	8	yes	complemented
21	C₄₁×#536	—	2	4	no	—
22	C₄₁×#536	C₂²	1	8	yes	complemented
23	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×#125×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×#124×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×#124×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×#123×C₄₁	—	2	2	no	—
30	C₂×#123×C₄₁	—	2	2	no	—
31	C₂×C₂²⋊Q₈×C₄₁	—	2	2	no	—
32	C₂×#121×C₄₁	—	2	2	no	—
33	C₂×#121×C₄₁	C₂²	1	2	yes	—
34	C₂×C₄×D₄×C₄₁	C₂²	1	2	yes	—
35	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
36	C₄₁×#496	C₂²	1	8	yes	complemented
37	C₄₁×#492	C₂²	1	8	yes	complemented
38	C₄₁×#491	C₂²	1	8	yes	complemented
39	C₄₁×#488	C₂²	1	8	yes	complemented
40	C₄₁×#463	C₂²	1	16	yes	complemented
41	C₄₁×#461	C₂²	1	16	yes	complemented
42	C₄₁×#449	C₂²	1	16	yes	complemented
43	C₄₁×#448	C₂²	1	16	yes	complemented
44	C₄₁×#433	C₂²	1	8	yes	complemented
45	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
46	C₂×#103×C₄₁	C₂²	1	2	yes	—
47	C₂×#103×C₄₁	C₂²	1	2	yes	—
48	C₂×#103×C₄₁	C₂²	1	2	yes	—
49	C₂×#102×C₄₁	C₂²	1	2	yes	—
50	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
51	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
52	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
53	C₂×#41×C₄₁	—	2	2	no	—
54	C₂×#41×C₄₁	—	4	4	no	—
55	C₂×#41×C₄₁	C₂³	1	4	yes	—
56	C₂²×Q₈×C₄₁	—	2	1	no	—
57	C₂²×Q₈×C₄₁	—	2	2	no	—
58	C₂²×Q₈×C₄₁	—	4	4	no	—
59	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
60	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
61	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₄₁	C₂³	1	4	yes	—
63	C₂²×D₄×C₄₁	—	2	1	no	—
64	C₂²×D₄×C₄₁	—	2	2	no	—
65	C₂²×D₄×C₄₁	—	2	2	no	—
66	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
67	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₄₁	—	2	1	no	—
69	C₂³×C₄×C₄₁	—	2	1	no	—
70	C₂³×C₄×C₄₁	—	2	2	no	—
71	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
72	C₂×SD₁₆×C₄₁	—	4	4	no	—
73	C₂×SD₁₆×C₄₁	—	4	4	no	—
74	C₂×SD₁₆×C₄₁	—	4	4	no	—
75	C₂×SD₁₆×C₄₁	C₂³	1	8	yes	—
76	C₂×SD₁₆×C₄₁	C₂³	1	8	yes	—
77	C₂×SD₁₆×C₄₁	C₂³	1	8	yes	—
78	C₂×D₈×C₄₁	—	4	4	no	—
79	C₂×D₈×C₄₁	C₂³	1	8	yes	—
80	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
81	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
82	C₄⋊Q₈×C₄₁	—	2	2	no	—
83	C₄⋊Q₈×C₄₁	—	2	2	no	—
84	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	—
85	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	—
86	#127×C₄₁	—	2	2	no	—
87	#127×C₄₁	C₂³	1	4	yes	—
88	#125×C₄₁	—	2	2	no	—
89	#125×C₄₁	C₂³	1	4	yes	—
90	#124×C₄₁	—	2	2	no	—
91	#124×C₄₁	—	2	2	no	—
92	#124×C₄₁	C₂³	1	4	yes	—
93	#124×C₄₁	C₂³	1	4	yes	—
94	#123×C₄₁	—	2	16	no	—
95	#123×C₄₁	—	2	16	no	—
96	C₂²⋊Q₈×C₄₁	—	2	16	no	—
97	#121×C₄₁	—	2	4	no	—
98	#121×C₄₁	—	2	16	no	—
99	#121×C₄₁	C₂³	1	8	yes	—
100	C₄×D₄×C₄₁	—	2	4	no	—
101	C₄×D₄×C₄₁	C₂³	1	8	yes	—
102	#117×C₄₁	D₄	1	4	yes	—
103	#117×C₄₁	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
106	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
107	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
108	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
114	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
116	C₄⋊C₈×C₄₁	C₂³	1	8	yes	—
117	#103×C₄₁	C₂³	1	8	yes	—
118	#103×C₄₁	C₂³	1	8	yes	—
119	#103×C₄₁	C₂³	1	8	yes	—
120	#102×C₄₁	C₂³	1	8	yes	—
121	C₂²⋊C₈×C₄₁	C₂³	1	8	yes	—
122	#97×C₄₁	C₂³	1	4	yes	—
123	C₄×C₈×C₄₁	C₂³	1	4	yes	—
124	C₂⁴×C₄₁	—	2	1	no	—
125	C₂⁴×C₄₁	—	2	2	no	—
126	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
127	#41×C₄₁	—	4	8	no	—
128	#41×C₄₁	—	4	8	no	—
129	C₂×Q₈×C₄₁	—	2	4	no	—
130	C₂×Q₈×C₄₁	—	2	1	no	—
131	C₂×Q₈×C₄₁	—	2	1	no	—
132	C₂×Q₈×C₄₁	—	2	1	no	—
133	C₂×Q₈×C₄₁	—	2	4	no	—
134	C₂×Q₈×C₄₁	—	2	4	no	—
135	C₂×Q₈×C₄₁	—	2	4	no	—
136	C₂×Q₈×C₄₁	—	4	8	no	—
137	C₂×Q₈×C₄₁	—	4	4	no	—
138	C₂×Q₈×C₄₁	—	4	8	no	—
139	C₂×Q₈×C₄₁	C₂⁴	1	2	yes	—
140	C₂×Q₈×C₄₁	C₂⁴	1	2	yes	—
141	C₂×Q₈×C₄₁	C₂⁴	1	2	yes	—
142	C₂×D₄×C₄₁	—	2	4	no	—
143	C₂×D₄×C₄₁	—	2	8	no	—
144	C₂×D₄×C₄₁	—	2	2	no	—
145	C₂×D₄×C₄₁	—	2	1	no	—
146	C₂×D₄×C₄₁	—	2	8	no	—
147	C₂×D₄×C₄₁	—	2	4	no	—
148	C₂×D₄×C₄₁	—	2	4	no	—
149	C₂×D₄×C₄₁	—	2	4	no	—
150	C₂×D₄×C₄₁	—	4	4	no	—
151	C₂×D₄×C₄₁	—	8	4	no	—
152	C₂×D₄×C₄₁	—	8	4	no	—
153	C₂×D₄×C₄₁	C₂⁴	1	2	yes	—
154	C₂×D₄×C₄₁	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₄₁	—	2	4	no	—
156	C₂²×C₄×C₄₁	—	2	4	no	—
157	C₂²×C₄×C₄₁	—	2	2	no	—
158	C₂²×C₄×C₄₁	—	2	1	no	—
159	C₂²×C₄×C₄₁	—	2	1	no	—
160	C₂²×C₄×C₄₁	—	2	1	no	—
161	C₂²×C₄×C₄₁	—	2	4	no	—
162	C₂²×C₄×C₄₁	—	2	8	no	—
163	C₂²×C₄×C₄₁	—	4	4	no	—
164	C₂²×C₄×C₄₁	—	4	4	no	—
165	C₂²×C₄×C₄₁	—	8	4	no	—
166	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
172	SD₁₆×C₄₁	—	4	8	no	—
173	SD₁₆×C₄₁	—	4	8	no	—
174	SD₁₆×C₄₁	—	4	8	no	—
175	D₈×C₄₁	—	4	8	no	—
176	C₂×C₈×C₄₁	—	2	4	no	—
177	C₂×C₈×C₄₁	—	2	4	no	—
178	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
179	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
180	C₄⋊C₄×C₄₁	—	2	8	no	—
181	C₄⋊C₄×C₄₁	—	2	8	no	—
182	C₄⋊C₄×C₄₁	—	2	8	no	—
183	C₄⋊C₄×C₄₁	—	2	2	no	—
184	C₄⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
185	C₄⋊C₄×C₄₁	C₂⁴	1	4	yes	—
186	C₂²⋊C₄×C₄₁	—	2	8	no	—
187	C₂²⋊C₄×C₄₁	—	2	8	no	—
188	C₂²⋊C₄×C₄₁	—	2	8	no	—
189	C₂²⋊C₄×C₄₁	—	2	8	no	—
190	C₂²⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
191	C₄²×C₄₁	C₂×D₄	1	2	yes	—
192	C₄²×C₄₁	C₂×D₄	1	2	yes	—
193	C₄²×C₄₁	C₂⁴	1	2	yes	—
194	C₄²×C₄₁	C₂⁴	1	2	yes	—
195	C₂³×C₄₁	—	2	2	no	—
196	C₂³×C₄₁	—	2	4	no	—
197	C₂³×C₄₁	—	2	2	no	—
198	C₂³×C₄₁	—	2	4	no	—
199	C₂³×C₄₁	—	4	2	no	—
200	C₂³×C₄₁	—	4	4	no	—
201	C₂³×C₄₁	—	4	4	no	—
202	C₂³×C₄₁	—	4	4	no	—
203	C₂³×C₄₁	—	4	4	no	—
204	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
205	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
206	Q₈×C₄₁	—	2	4	no	—
207	Q₈×C₄₁	—	2	4	no	—
208	Q₈×C₄₁	—	2	4	no	—
209	Q₈×C₄₁	—	4	8	no	—
210	Q₈×C₄₁	—	4	2	no	—
211	Q₈×C₄₁	—	4	2	no	—
212	Q₈×C₄₁	—	4	2	no	—
213	D₄×C₄₁	—	2	4	no	—
214	D₄×C₄₁	—	2	4	no	—
215	D₄×C₄₁	—	2	4	no	—
216	D₄×C₄₁	—	4	8	no	—
217	D₄×C₄₁	—	4	4	no	—
218	D₄×C₄₁	—	4	2	no	—
219	D₄×C₄₁	—	8	8	no	—
220	D₄×C₄₁	—	8	8	no	—
221	C₂×C₄×C₄₁	—	2	2	no	—
222	C₂×C₄×C₄₁	—	2	4	no	—
223	C₂×C₄×C₄₁	—	2	4	no	—
224	C₂×C₄×C₄₁	—	2	4	no	—
225	C₂×C₄×C₄₁	—	2	4	no	—
226	C₂×C₄×C₄₁	—	2	4	no	—
227	C₂×C₄×C₄₁	—	2	2	no	—
228	C₂×C₄×C₄₁	—	2	2	no	—
229	C₂×C₄×C₄₁	—	2	2	no	—
230	C₂×C₄×C₄₁	—	4	8	no	—
231	C₂×C₄×C₄₁	—	4	8	no	—
232	C₂×C₄×C₄₁	—	4	2	no	—
233	C₂×C₄×C₄₁	—	4	2	no	—
234	C₂×C₄×C₄₁	—	4	2	no	—
235	C₂×C₄×C₄₁	—	4	4	no	—
236	C₂×C₄×C₄₁	—	8	8	no	—
237	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
238	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
239	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
240	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₄₁	H(5,𝔽₂)	1	4	yes	—
242	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
243	C₈×C₄₁	—	2	4	no	—
244	C₈×C₄₁	—	2	4	no	—
245	C₂×#3221	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
246	C₂²×C₄₁	—	2	2	no	—
247	C₂²×C₄₁	—	2	4	no	—
248	C₂²×C₄₁	—	4	4	no	—
249	C₂²×C₄₁	—	4	2	no	—
250	C₂²×C₄₁	—	4	4	no	—
251	C₂²×C₄₁	—	4	4	no	—
252	C₂²×C₄₁	—	4	4	no	—
253	C₂²×C₄₁	—	4	4	no	—
254	C₂²×C₄₁	—	8	4	no	—
255	C₂²×C₄₁	—	8	8	no	—
256	C₂²×C₄₁	#535	1	2	yes	—
257	C₂²×C₄₁	#577	1	4	yes	—
258	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
259	C₄×C₄₁	—	2	2	no	—
260	C₄×C₄₁	—	2	4	no	—
261	C₄×C₄₁	—	2	4	no	—
262	C₄×C₄₁	—	4	2	no	—
263	C₄×C₄₁	—	4	2	no	—
264	C₄×C₄₁	—	4	2	no	—
265	C₄×C₄₁	—	4	4	no	—
266	C₄×C₄₁	#535	1	2	yes	—
267	C₄×C₄₁	C₂×H(5,𝔽₂)	1	2	yes	—
268	C₂×#540	C₂×C₄₁	1	1	yes	characteristic, complemented
269	C₂×#536	C₂×C₄₁	1	1	yes	characteristic, complemented
270	#3221	C₂×C₄₁	1	16	yes	direct product factor
271	C₂×#496	C₂×C₄₁	1	1	yes	characteristic, complemented
272	C₂×#492	C₂×C₄₁	1	1	yes	characteristic, complemented
273	C₂×#491	C₂×C₄₁	1	1	yes	characteristic, complemented
274	C₂×#488	C₂×C₄₁	1	1	yes	characteristic, complemented
275	C₂×#463	C₂×C₄₁	1	2	yes	complemented
276	C₂×#461	C₂×C₄₁	1	2	yes	complemented
277	C₂×#449	C₂×C₄₁	1	2	yes	complemented
278	C₂×#448	C₂×C₄₁	1	2	yes	complemented
279	C₂×#433	C₂×C₄₁	1	1	yes	characteristic, complemented
280	C₂×C₄₁	—	4	2	no	—
281	C₂×C₄₁	—	8	2	no	—
282	C₂×C₄₁	—	8	4	no	—
283	C₂×C₄₁	#3221	1	4	yes	direct product factor
284	C₂×C₄₁	C₂×#535	1	1	yes	characteristic
285	C₂×C₄₁	C₂×#577	1	2	yes	—
286	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
287	C₂³×Q₈	C₂²×C₄₁	1	1	yes	characteristic
288	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
289	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
290	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
291	C₂²×D₈	C₂²×C₄₁	1	2	yes	—
292	#540	—	2	4	no	—
293	#540	C₂²×C₄₁	1	8	yes	complemented
294	#536	—	2	4	no	—
295	#536	C₂²×C₄₁	1	8	yes	complemented
296	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
297	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
298	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
299	C₂×#125	C₂²×C₄₁	1	1	yes	characteristic
300	C₂×#124	C₂²×C₄₁	1	1	yes	characteristic
301	C₂×#124	C₂²×C₄₁	1	1	yes	characteristic
302	C₂×#123	—	2	2	no	—
303	C₂×#123	—	2	2	no	—
304	C₂×C₂²⋊Q₈	—	2	2	no	—
305	C₂×#121	—	2	2	no	—
306	C₂×#121	C₂²×C₄₁	1	2	yes	—
307	C₂×C₄×D₄	C₂²×C₄₁	1	2	yes	—
308	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
309	#496	C₂²×C₄₁	1	8	yes	complemented
310	#492	C₂²×C₄₁	1	8	yes	complemented
311	#491	C₂²×C₄₁	1	8	yes	complemented
312	#488	C₂²×C₄₁	1	8	yes	complemented
313	#463	C₂²×C₄₁	1	16	yes	complemented
314	#461	C₂²×C₄₁	1	16	yes	complemented
315	#449	C₂²×C₄₁	1	16	yes	complemented
316	#448	C₂²×C₄₁	1	16	yes	complemented
317	#433	C₂²×C₄₁	1	8	yes	complemented
318	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
319	C₂×#103	C₂²×C₄₁	1	2	yes	—
320	C₂×#103	C₂²×C₄₁	1	2	yes	—
321	C₂×#103	C₂²×C₄₁	1	2	yes	—
322	C₂×#102	C₂²×C₄₁	1	2	yes	—
323	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
324	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
325	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
326	C₄₁	C₂×#3221	1	1	yes	characteristic, direct product factor, Sylow
327	C₂×#41	—	2	2	no	—
328	C₂×#41	—	4	4	no	—
329	C₂×#41	C₂³×C₄₁	1	4	yes	—
330	C₂²×Q₈	—	2	1	no	—
331	C₂²×Q₈	—	2	2	no	—
332	C₂²×Q₈	—	4	4	no	—
333	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
334	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
335	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
336	C₂²×Q₈	C₂³×C₄₁	1	4	yes	—
337	C₂²×D₄	—	2	1	no	—
338	C₂²×D₄	—	2	2	no	—
339	C₂²×D₄	—	2	2	no	—
340	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
341	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
342	C₂³×C₄	—	2	1	no	—
343	C₂³×C₄	—	2	1	no	—
344	C₂³×C₄	—	2	2	no	—
345	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
346	C₂×SD₁₆	—	4	4	no	—
347	C₂×SD₁₆	—	4	4	no	—
348	C₂×SD₁₆	—	4	4	no	—
349	C₂×SD₁₆	C₂³×C₄₁	1	8	yes	—
350	C₂×SD₁₆	C₂³×C₄₁	1	8	yes	—
351	C₂×SD₁₆	C₂³×C₄₁	1	8	yes	—
352	C₂×D₈	—	4	4	no	—
353	C₂×D₈	C₂³×C₄₁	1	8	yes	—
354	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
355	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
356	C₄⋊Q₈	—	2	2	no	—
357	C₄⋊Q₈	—	2	2	no	—
358	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	—
359	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	—
360	#127	—	2	2	no	—
361	#127	C₂³×C₄₁	1	4	yes	—
362	#125	—	2	2	no	—
363	#125	C₂³×C₄₁	1	4	yes	—
364	#124	—	2	2	no	—
365	#124	—	2	2	no	—
366	#124	C₂³×C₄₁	1	4	yes	—
367	#124	C₂³×C₄₁	1	4	yes	—
368	#123	—	2	16	no	—
369	#123	—	2	16	no	—
370	C₂²⋊Q₈	—	2	16	no	—
371	#121	—	2	4	no	—
372	#121	—	2	16	no	—
373	#121	C₂³×C₄₁	1	8	yes	—
374	C₄×D₄	—	2	4	no	—
375	C₄×D₄	C₂³×C₄₁	1	8	yes	—
376	#117	D₄×C₄₁	1	4	yes	—
377	#117	C₂³×C₄₁	1	4	yes	—
378	C₂×C₄⋊C₄	—	2	2	no	—
379	C₂×C₄⋊C₄	—	2	2	no	—
380	C₂×C₄⋊C₄	—	2	2	no	—
381	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
382	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
383	C₂×C₂²⋊C₄	—	2	2	no	—
384	C₂×C₂²⋊C₄	—	2	2	no	—
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₂²⋊C₄	—	2	2	no	—
387	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
388	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
389	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
390	C₄⋊C₈	C₂³×C₄₁	1	8	yes	—
391	#103	C₂³×C₄₁	1	8	yes	—
392	#103	C₂³×C₄₁	1	8	yes	—
393	#103	C₂³×C₄₁	1	8	yes	—
394	#102	C₂³×C₄₁	1	8	yes	—
395	C₂²⋊C₈	C₂³×C₄₁	1	8	yes	—
396	#97	C₂³×C₄₁	1	4	yes	—
397	C₄×C₈	C₂³×C₄₁	1	4	yes	—
398	C₂⁴	—	2	1	no	—
399	C₂⁴	—	2	2	no	—
400	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
401	#41	—	4	8	no	—
402	#41	—	4	8	no	—
403	C₂×Q₈	—	2	4	no	—
404	C₂×Q₈	—	2	1	no	—
405	C₂×Q₈	—	2	1	no	—
406	C₂×Q₈	—	2	1	no	—
407	C₂×Q₈	—	2	4	no	—
408	C₂×Q₈	—	2	4	no	—
409	C₂×Q₈	—	2	4	no	—
410	C₂×Q₈	—	4	8	no	—
411	C₂×Q₈	—	4	4	no	—
412	C₂×Q₈	—	4	8	no	—
413	C₂×Q₈	C₂⁴×C₄₁	1	2	yes	—
414	C₂×Q₈	C₂⁴×C₄₁	1	2	yes	—
415	C₂×Q₈	C₂⁴×C₄₁	1	2	yes	—
416	C₂×D₄	—	2	4	no	—
417	C₂×D₄	—	2	8	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	8	no	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	—	2	4	no	—
423	C₂×D₄	—	2	4	no	—
424	C₂×D₄	—	4	4	no	—
425	C₂×D₄	—	8	4	no	—
426	C₂×D₄	—	8	4	no	—
427	C₂×D₄	C₂⁴×C₄₁	1	2	yes	—
428	C₂×D₄	C₂⁴×C₄₁	1	4	yes	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	8	no	—
437	C₂²×C₄	—	4	4	no	—
438	C₂²×C₄	—	4	4	no	—
439	C₂²×C₄	—	8	4	no	—
440	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
441	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
442	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
443	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
444	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
445	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
446	SD₁₆	—	4	8	no	—
447	SD₁₆	—	4	8	no	—
448	SD₁₆	—	4	8	no	—
449	D₈	—	4	8	no	—
450	C₂×C₈	—	2	4	no	—
451	C₂×C₈	—	2	4	no	—
452	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
453	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
454	C₄⋊C₄	—	2	8	no	—
455	C₄⋊C₄	—	2	8	no	—
456	C₄⋊C₄	—	2	8	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
459	C₄⋊C₄	C₂⁴×C₄₁	1	4	yes	—
460	C₂²⋊C₄	—	2	8	no	—
461	C₂²⋊C₄	—	2	8	no	—
462	C₂²⋊C₄	—	2	8	no	—
463	C₂²⋊C₄	—	2	8	no	—
464	C₂²⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
465	C₄²	C₂×D₄×C₄₁	1	2	yes	—
466	C₄²	C₂×D₄×C₄₁	1	2	yes	—
467	C₄²	C₂⁴×C₄₁	1	2	yes	—
468	C₄²	C₂⁴×C₄₁	1	2	yes	—
469	C₂³	—	2	2	no	—
470	C₂³	—	2	4	no	—
471	C₂³	—	2	2	no	—
472	C₂³	—	2	4	no	—
473	C₂³	—	4	2	no	—
474	C₂³	—	4	4	no	—
475	C₂³	—	4	4	no	—
476	C₂³	—	4	4	no	—
477	C₂³	—	4	4	no	—
478	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
479	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
480	Q₈	—	2	4	no	—
481	Q₈	—	2	4	no	—
482	Q₈	—	2	4	no	—
483	Q₈	—	4	8	no	—
484	Q₈	—	4	2	no	—
485	Q₈	—	4	2	no	—
486	Q₈	—	4	2	no	—
487	D₄	—	2	4	no	—
488	D₄	—	2	4	no	—
489	D₄	—	2	4	no	—
490	D₄	—	4	8	no	—
491	D₄	—	4	2	no	—
492	D₄	—	4	4	no	—
493	D₄	—	8	8	no	—
494	D₄	—	8	8	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	4	8	no	—
505	C₂×C₄	—	4	8	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	4	2	no	—
509	C₂×C₄	—	4	4	no	—
510	C₂×C₄	—	8	8	no	—
511	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
512	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
513	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
514	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
515	C₂×C₄	H(5,𝔽₂)×C₄₁	1	4	yes	—
516	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
517	C₈	—	2	4	no	—
518	C₈	—	2	4	no	—
519	C₂²	—	2	4	no	—
520	C₂²	—	2	2	no	—
521	C₂²	—	4	4	no	—
522	C₂²	—	4	4	no	—
523	C₂²	—	4	4	no	—
524	C₂²	—	4	4	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	4	no	—
527	C₂²	—	8	4	no	—
528	C₂²	—	8	8	no	—
529	C₂²	C₄₁×#535	1	2	yes	—
530	C₂²	C₄₁×#577	1	4	yes	—
531	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
532	C₄	—	2	4	no	—
533	C₄	—	2	4	no	—
534	C₄	—	2	2	no	—
535	C₄	—	4	4	no	—
536	C₄	—	4	2	no	—
537	C₄	—	4	2	no	—
538	C₄	—	4	2	no	—
539	C₄	C₄₁×#535	1	2	yes	—
540	C₄	C₂×H(5,𝔽₂)×C₄₁	1	2	yes	—
541	C₂	—	4	2	no	—
542	C₂	—	8	4	no	—
543	C₂	—	8	2	no	—
544	C₂	C₄₁×#3221	1	4	yes	direct product factor
545	C₂	C₂×C₄₁×#535	1	1	yes	characteristic
546	C₂	C₂×C₄₁×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
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
1. C₂×C₄₁×#540	C₂	semidirect
2. C₂×C₄₁×#536	C₂	semidirect
3. C₄₁×#3221	C₂	direct (N × H)
4. C₂×C₄₁×#496	C₂	semidirect
5. C₂×C₄₁×#492	C₂	semidirect
6. C₂×C₄₁×#491	C₂	semidirect
7. C₂×C₄₁×#488	C₂	semidirect
8. C₂×C₄₁×#463	C₂	semidirect
9. C₂×C₄₁×#461	C₂	semidirect
10. C₂×C₄₁×#449	C₂	semidirect
11. C₂×C₄₁×#448	C₂	semidirect
12. C₂×C₄₁×#433	C₂	semidirect
20. C₄₁×#540	C₂²	semidirect
22. C₄₁×#536	C₂²	semidirect
36. C₄₁×#496	C₂²	semidirect
37. C₄₁×#492	C₂²	semidirect
38. C₄₁×#491	C₂²	semidirect
39. C₄₁×#488	C₂²	semidirect
40. C₄₁×#463	C₂²	semidirect
41. C₄₁×#461	C₂²	semidirect
42. C₄₁×#449	C₂²	semidirect
43. C₄₁×#448	C₂²	semidirect
44. C₄₁×#433	C₂²	semidirect
245. C₂×#3221	C₄₁	direct (N × H)
268. C₂×#540	C₂×C₄₁	semidirect
269. C₂×#536	C₂×C₄₁	semidirect
270. #3221	C₂×C₄₁	direct (N × H)
271. C₂×#496	C₂×C₄₁	semidirect
272. C₂×#492	C₂×C₄₁	semidirect
273. C₂×#491	C₂×C₄₁	semidirect
274. C₂×#488	C₂×C₄₁	semidirect
275. C₂×#463	C₂×C₄₁	semidirect
276. C₂×#461	C₂×C₄₁	semidirect
277. C₂×#449	C₂×C₄₁	semidirect
278. C₂×#448	C₂×C₄₁	semidirect
279. C₂×#433	C₂×C₄₁	semidirect
283. C₂×C₄₁	#3221	direct (N × H)
293. #540	C₂²×C₄₁	semidirect
295. #536	C₂²×C₄₁	semidirect
309. #496	C₂²×C₄₁	semidirect
310. #492	C₂²×C₄₁	semidirect
311. #491	C₂²×C₄₁	semidirect
312. #488	C₂²×C₄₁	semidirect
313. #463	C₂²×C₄₁	semidirect
314. #461	C₂²×C₄₁	semidirect
315. #449	C₂²×C₄₁	semidirect
316. #448	C₂²×C₄₁	semidirect
317. #433	C₂²×C₄₁	semidirect
326. C₄₁	C₂×#3221	direct (N × H)
544. C₂	C₄₁×#3221	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#540	2
C₂×C₄₁×#536	2
C₄₁×#3221	2
C₂×C₄₁×#496	2
C₂×C₄₁×#492	2
C₂×C₄₁×#491	2
C₂×C₄₁×#488	2
C₂×C₄₁×#463	2
C₂×C₄₁×#461	2
C₂×C₄₁×#449	2
C₂×C₄₁×#448	2
C₂×C₄₁×#433	2
C₂×#3221	41

Subgroups — C2×C41×#3221 #23836995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#3221 #23836995