Subgroups — C₂×C₄₁×#3324 #23837345

Order 10496.

2566 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄²×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#3324	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₁×#482	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₁×#474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₁×#468	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₁×#460	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄₁×#450	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×SD₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄₁×#418	C₂	1	2	yes	maximal, complemented
13	C₂³×D₄×C₄₁	—	2	1	no	—
14	C₂³×D₄×C₄₁	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
16	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	complemented
17	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	complemented
18	C₄₁×#556	—	2	4	no	—
19	C₄₁×#556	C₂²	1	8	yes	complemented
20	D₄²×C₄₁	—	2	4	no	—
21	D₄²×C₄₁	C₂²	1	8	yes	—
22	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
23	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×#126×C₄₁	—	2	2	no	—
25	C₂×#124×C₄₁	—	2	1	no	—
26	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	complemented
27	C₂×#121×C₄₁	—	2	1	no	—
28	C₂×#121×C₄₁	C₂²	1	2	yes	—
29	C₂×#120×C₄₁	—	2	2	no	—
30	C₂×C₄×Q₈×C₄₁	—	2	1	no	—
31	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
34	C₂×#117×C₄₁	C₂²	1	2	yes	complemented
35	C₄₁×#486	C₂²	1	8	yes	complemented
36	C₄₁×#482	C₂²	1	16	yes	complemented
37	C₄₁×#474	C₂²	1	8	yes	complemented
38	C₄₁×#468	C₂²	1	16	yes	complemented
39	C₄₁×#460	C₂²	1	8	yes	complemented
40	C₄₁×#450	C₂²	1	16	yes	complemented
41	C₄×SD₁₆×C₄₁	C₂²	1	8	yes	complemented
42	C₄₁×#436	C₂²	1	8	yes	complemented
43	C₂×#107×C₄₁	C₂²	1	2	yes	complemented
44	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic, complemented
45	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
46	C₄₁×#418	C₂²	1	16	yes	complemented
47	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic, complemented
48	C₂×#102×C₄₁	C₂²	1	2	yes	—
49	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
50	C₂×#102×C₄₁	C₂²	1	2	yes	complemented
51	C₂×#102×C₄₁	C₂²	1	2	yes	complemented
52	C₂×#102×C₄₁	C₂²	1	2	yes	complemented
53	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	complemented
54	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
55	C₂⁵×C₄₁	—	2	2	no	—
56	C₂²×Q₈×C₄₁	—	2	1	no	—
57	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₄₁	—	2	8	no	—
59	C₂²×D₄×C₄₁	—	2	2	no	—
60	C₂²×D₄×C₄₁	—	2	8	no	—
61	C₂²×D₄×C₄₁	—	2	2	no	—
62	C₂²×D₄×C₄₁	—	2	1	no	—
63	C₂²×D₄×C₄₁	—	2	4	no	—
64	C₂²×D₄×C₄₁	—	2	1	no	—
65	C₂²×D₄×C₄₁	—	4	4	no	—
66	C₂²×D₄×C₄₁	—	4	8	no	—
67	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
68	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
69	C₂²×D₄×C₄₁	C₂³	1	8	yes	—
70	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₄₁	—	2	1	no	—
72	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
73	C₂×SD₁₆×C₄₁	—	2	4	no	—
74	C₂×SD₁₆×C₄₁	—	4	4	no	—
75	C₂×SD₁₆×C₄₁	C₂³	1	4	yes	—
76	C₂×SD₁₆×C₄₁	C₂³	1	8	yes	complemented
77	C₂×M₁₆×C₄₁	—	4	4	no	—
78	C₂×M₁₆×C₄₁	C₂³	1	8	yes	complemented
79	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
81	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
82	C₄⋊Q₈×C₄₁	—	2	2	no	—
83	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	complemented
84	#127×C₄₁	—	2	2	no	—
85	#127×C₄₁	C₂³	1	4	yes	—
86	#126×C₄₁	—	2	16	no	—
87	#124×C₄₁	—	2	8	no	—
88	C₂²⋊Q₈×C₄₁	—	2	4	no	—
89	C₂²⋊Q₈×C₄₁	C₂³	1	8	yes	complemented
90	#121×C₄₁	—	2	4	no	—
91	#121×C₄₁	—	2	8	no	—
92	#121×C₄₁	C₂³	1	8	yes	—
93	#120×C₄₁	—	2	16	no	—
94	C₄×Q₈×C₄₁	—	2	8	no	—
95	C₄×Q₈×C₄₁	—	2	2	no	—
96	C₄×Q₈×C₄₁	C₂³	1	4	yes	complemented
97	C₄×D₄×C₄₁	—	2	2	no	—
98	C₄×D₄×C₄₁	D₄	1	4	yes	—
99	C₄×D₄×C₄₁	C₂³	1	4	yes	—
100	C₄×D₄×C₄₁	C₂³	1	4	yes	—
101	#117×C₄₁	—	2	4	no	—
102	#117×C₄₁	C₂³	1	8	yes	complemented
103	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
104	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
105	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
106	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
110	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
114	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
115	C₂×C₄²×C₄₁	—	2	2	no	—
116	C₂×C₄²×C₄₁	—	2	1	no	—
117	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
118	#107×C₄₁	C₂³	1	8	yes	complemented
119	#106×C₄₁	C₂³	1	4	yes	complemented
120	C₄⋊C₈×C₄₁	C₂³	1	4	yes	complemented
121	#103×C₄₁	C₂³	1	4	yes	complemented
122	#102×C₄₁	C₂³	1	4	yes	—
123	#102×C₄₁	C₂³	1	8	yes	—
124	#102×C₄₁	C₂³	1	8	yes	complemented
125	#102×C₄₁	C₂³	1	8	yes	complemented
126	#102×C₄₁	C₂³	1	8	yes	complemented
127	C₂²⋊C₈×C₄₁	C₂³	1	8	yes	complemented
128	C₄×C₈×C₄₁	C₂³	1	4	yes	complemented
129	C₂⁴×C₄₁	—	2	1	no	—
130	C₂⁴×C₄₁	—	2	2	no	—
131	C₂⁴×C₄₁	—	2	8	no	—
132	C₂⁴×C₄₁	—	4	8	no	—
133	C₂⁴×C₄₁	—	4	8	no	—
134	C₂⁴×C₄₁	—	8	4	no	—
135	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
136	C₂×Q₈×C₄₁	—	2	1	no	—
137	C₂×Q₈×C₄₁	—	2	4	no	—
138	C₂×Q₈×C₄₁	—	2	2	no	—
139	C₂×Q₈×C₄₁	—	2	2	no	—
140	C₂×Q₈×C₄₁	—	4	4	no	—
141	C₂×Q₈×C₄₁	C₂⁴	1	2	yes	—
142	C₂×D₄×C₄₁	—	2	16	no	—
143	C₂×D₄×C₄₁	—	2	8	no	—
144	C₂×D₄×C₄₁	—	2	4	no	—
145	C₂×D₄×C₄₁	—	2	1	no	—
146	C₂×D₄×C₄₁	—	2	8	no	—
147	C₂×D₄×C₄₁	—	2	2	no	—
148	C₂×D₄×C₄₁	—	2	4	no	—
149	C₂×D₄×C₄₁	—	2	4	no	—
150	C₂×D₄×C₄₁	—	2	4	no	—
151	C₂×D₄×C₄₁	—	2	2	no	—
152	C₂×D₄×C₄₁	—	2	4	no	—
153	C₂×D₄×C₄₁	—	4	8	no	—
154	C₂×D₄×C₄₁	—	4	16	no	—
155	C₂×D₄×C₄₁	—	4	8	no	—
156	C₂×D₄×C₄₁	—	4	32	no	—
157	C₂×D₄×C₄₁	—	4	8	no	—
158	C₂×D₄×C₄₁	—	4	4	no	—
159	C₂×D₄×C₄₁	—	8	2	no	—
160	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
161	C₂×D₄×C₄₁	#41	1	4	yes	—
162	C₂×D₄×C₄₁	C₂⁴	1	4	yes	—
163	C₂×D₄×C₄₁	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₄₁	—	2	4	no	—
165	C₂²×C₄×C₄₁	—	2	4	no	—
166	C₂²×C₄×C₄₁	—	2	1	no	—
167	C₂²×C₄×C₄₁	—	2	1	no	—
168	C₂²×C₄×C₄₁	—	2	2	no	—
169	C₂²×C₄×C₄₁	—	2	1	no	—
170	C₂²×C₄×C₄₁	—	2	4	no	—
171	C₂²×C₄×C₄₁	—	4	4	no	—
172	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
173	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
178	SD₁₆×C₄₁	—	2	8	no	—
179	SD₁₆×C₄₁	—	4	8	no	—
180	M₁₆×C₄₁	—	4	8	no	—
181	C₂×C₈×C₄₁	—	2	2	no	—
182	C₂×C₈×C₄₁	—	2	2	no	—
183	C₂×C₈×C₄₁	—	2	4	no	—
184	C₂×C₈×C₄₁	C₂⁴	1	2	yes	—
185	C₂×C₈×C₄₁	C₂⁴	1	2	yes	—
186	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
187	C₄⋊C₄×C₄₁	—	2	8	no	—
188	C₄⋊C₄×C₄₁	—	2	4	no	—
189	C₄⋊C₄×C₄₁	—	2	4	no	—
190	C₄⋊C₄×C₄₁	—	2	1	no	—
191	C₄⋊C₄×C₄₁	—	2	2	no	—
192	C₄⋊C₄×C₄₁	—	2	1	no	—
193	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
194	C₄⋊C₄×C₄₁	C₂⁴	1	2	yes	—
195	C₄⋊C₄×C₄₁	C₂⁴	1	4	yes	—
196	C₄⋊C₄×C₄₁	C₂⁴	1	2	yes	—
197	C₂²⋊C₄×C₄₁	—	2	8	no	—
198	C₂²⋊C₄×C₄₁	—	2	8	no	—
199	C₂²⋊C₄×C₄₁	—	2	4	no	—
200	C₂²⋊C₄×C₄₁	—	2	8	no	—
201	C₂²⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
202	C₄²×C₄₁	—	2	8	no	—
203	C₄²×C₄₁	—	2	4	no	—
204	C₄²×C₄₁	C₂×D₄	1	2	yes	—
205	C₄²×C₄₁	C₂⁴	1	2	yes	—
206	C₂³×C₄₁	—	2	4	no	—
207	C₂³×C₄₁	—	2	4	no	—
208	C₂³×C₄₁	—	2	4	no	—
209	C₂³×C₄₁	—	2	4	no	—
210	C₂³×C₄₁	—	2	2	no	—
211	C₂³×C₄₁	—	2	4	no	—
212	C₂³×C₄₁	—	4	16	no	—
213	C₂³×C₄₁	—	4	16	no	—
214	C₂³×C₄₁	—	4	2	no	—
215	C₂³×C₄₁	—	4	2	no	—
216	C₂³×C₄₁	—	4	4	no	—
217	C₂³×C₄₁	—	4	4	no	—
218	C₂³×C₄₁	—	4	4	no	—
219	C₂³×C₄₁	—	8	8	no	—
220	C₂³×C₄₁	—	8	32	no	—
221	C₂³×C₄₁	C₂²×D₄	1	4	yes	—
222	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
223	Q₈×C₄₁	—	2	2	no	—
224	Q₈×C₄₁	—	2	2	no	—
225	Q₈×C₄₁	—	4	2	no	—
226	Q₈×C₄₁	—	4	8	no	—
227	D₄×C₄₁	—	2	4	no	—
228	D₄×C₄₁	—	2	8	no	—
229	D₄×C₄₁	—	2	4	no	—
230	D₄×C₄₁	—	2	2	no	—
231	D₄×C₄₁	—	2	4	no	—
232	D₄×C₄₁	—	4	16	no	—
233	D₄×C₄₁	—	4	16	no	—
234	D₄×C₄₁	—	4	4	no	—
235	D₄×C₄₁	—	4	8	no	—
236	D₄×C₄₁	—	8	4	no	—
237	D₄×C₄₁	C₂×#41	1	4	yes	—
238	C₂×C₄×C₄₁	—	2	4	no	—
239	C₂×C₄×C₄₁	—	2	2	no	—
240	C₂×C₄×C₄₁	—	2	8	no	—
241	C₂×C₄×C₄₁	—	2	4	no	—
242	C₂×C₄×C₄₁	—	2	4	no	—
243	C₂×C₄×C₄₁	—	2	2	no	—
244	C₂×C₄×C₄₁	—	2	2	no	—
245	C₂×C₄×C₄₁	—	2	4	no	—
246	C₂×C₄×C₄₁	—	2	2	no	—
247	C₂×C₄×C₄₁	—	4	2	no	—
248	C₂×C₄×C₄₁	—	4	2	no	—
249	C₂×C₄×C₄₁	—	4	4	no	—
250	C₂×C₄×C₄₁	—	4	2	no	—
251	C₂×C₄×C₄₁	—	4	8	no	—
252	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	4	yes	—
253	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
254	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
258	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
259	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
260	C₈×C₄₁	—	2	2	no	—
261	C₈×C₄₁	—	2	2	no	—
262	C₈×C₄₁	—	2	4	no	—
263	C₂×#3324	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
264	C₂²×C₄₁	—	2	4	no	—
265	C₂²×C₄₁	—	2	4	no	—
266	C₂²×C₄₁	—	2	4	no	—
267	C₂²×C₄₁	—	2	4	no	—
268	C₂²×C₄₁	—	4	2	no	—
269	C₂²×C₄₁	—	4	8	no	—
270	C₂²×C₄₁	—	4	8	no	—
271	C₂²×C₄₁	—	4	2	no	—
272	C₂²×C₄₁	—	4	4	no	—
273	C₂²×C₄₁	—	4	4	no	—
274	C₂²×C₄₁	—	4	4	no	—
275	C₂²×C₄₁	—	8	32	no	—
276	C₂²×C₄₁	—	8	8	no	—
277	C₂²×C₄₁	#546	1	2	yes	—
278	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
279	C₂²×C₄₁	#577	1	2	yes	—
280	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
281	C₄×C₄₁	—	2	4	no	—
282	C₄×C₄₁	—	2	2	no	—
283	C₄×C₄₁	—	4	2	no	—
284	C₄×C₄₁	—	4	2	no	—
285	C₄×C₄₁	—	4	4	no	—
286	C₄×C₄₁	—	4	2	no	—
287	C₄×C₄₁	#546	1	2	yes	—
288	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
289	C₄×C₄₁	C₂²×#41	1	1	yes	characteristic
290	C₄×C₄₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
291	C₂×#556	C₂×C₄₁	1	1	yes	characteristic, complemented
292	C₂×D₄²	C₂×C₄₁	1	1	yes	characteristic
293	#3324	C₂×C₄₁	1	16	yes	direct product factor
294	C₂×#486	C₂×C₄₁	1	1	yes	characteristic, complemented
295	C₂×#482	C₂×C₄₁	1	2	yes	complemented
296	C₂×#474	C₂×C₄₁	1	1	yes	characteristic, complemented
297	C₂×#468	C₂×C₄₁	1	2	yes	complemented
298	C₂×#460	C₂×C₄₁	1	1	yes	characteristic, complemented
299	C₂×#450	C₂×C₄₁	1	2	yes	complemented
300	C₂×C₄×SD₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
301	C₂×#436	C₂×C₄₁	1	1	yes	characteristic, complemented
302	C₂×#418	C₂×C₄₁	1	2	yes	complemented
303	C₂×C₄₁	—	4	4	no	—
304	C₂×C₄₁	—	4	4	no	—
305	C₂×C₄₁	—	8	4	no	—
306	C₂×C₄₁	#3324	1	4	yes	direct product factor
307	C₂×C₄₁	C₂×#546	1	1	yes	characteristic
308	C₂×C₄₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
309	C₂×C₄₁	C₂×#577	1	1	yes	characteristic
310	C₂³×D₄	—	2	1	no	—
311	C₂³×D₄	C₂²×C₄₁	1	2	yes	—
312	C₂²×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
313	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	complemented
314	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	complemented
315	#556	—	2	4	no	—
316	#556	C₂²×C₄₁	1	8	yes	complemented
317	D₄²	—	2	4	no	—
318	D₄²	C₂²×C₄₁	1	8	yes	—
319	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
320	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
321	C₂×#126	—	2	2	no	—
322	C₂×#124	—	2	1	no	—
323	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	complemented
324	C₂×#121	—	2	1	no	—
325	C₂×#121	C₂²×C₄₁	1	2	yes	—
326	C₂×#120	—	2	2	no	—
327	C₂×C₄×Q₈	—	2	1	no	—
328	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
329	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
330	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
331	C₂×#117	C₂²×C₄₁	1	2	yes	complemented
332	#486	C₂²×C₄₁	1	8	yes	complemented
333	#482	C₂²×C₄₁	1	16	yes	complemented
334	#474	C₂²×C₄₁	1	8	yes	complemented
335	#468	C₂²×C₄₁	1	16	yes	complemented
336	#460	C₂²×C₄₁	1	8	yes	complemented
337	#450	C₂²×C₄₁	1	16	yes	complemented
338	C₄×SD₁₆	C₂²×C₄₁	1	8	yes	complemented
339	#436	C₂²×C₄₁	1	8	yes	complemented
340	C₂×#107	C₂²×C₄₁	1	2	yes	complemented
341	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic, complemented
342	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
343	#418	C₂²×C₄₁	1	16	yes	complemented
344	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic, complemented
345	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
346	C₂×#102	C₂²×C₄₁	1	2	yes	—
347	C₂×#102	C₂²×C₄₁	1	2	yes	complemented
348	C₂×#102	C₂²×C₄₁	1	2	yes	complemented
349	C₂×#102	C₂²×C₄₁	1	2	yes	complemented
350	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	complemented
351	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
352	C₄₁	C₂×#3324	1	1	yes	characteristic, direct product factor, Sylow
353	C₂⁵	—	2	2	no	—
354	C₂²×Q₈	—	2	1	no	—
355	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
356	C₂²×D₄	—	2	8	no	—
357	C₂²×D₄	—	2	2	no	—
358	C₂²×D₄	—	2	8	no	—
359	C₂²×D₄	—	2	2	no	—
360	C₂²×D₄	—	2	1	no	—
361	C₂²×D₄	—	2	4	no	—
362	C₂²×D₄	—	2	1	no	—
363	C₂²×D₄	—	4	4	no	—
364	C₂²×D₄	—	4	8	no	—
365	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
366	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
367	C₂²×D₄	C₂³×C₄₁	1	8	yes	—
368	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
369	C₂³×C₄	—	2	1	no	—
370	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
371	C₂×SD₁₆	—	2	4	no	—
372	C₂×SD₁₆	—	4	4	no	—
373	C₂×SD₁₆	C₂³×C₄₁	1	4	yes	—
374	C₂×SD₁₆	C₂³×C₄₁	1	8	yes	complemented
375	C₂×M₁₆	—	4	4	no	—
376	C₂×M₁₆	C₂³×C₄₁	1	8	yes	complemented
377	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
378	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
379	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
380	C₄⋊Q₈	—	2	2	no	—
381	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	complemented
382	#127	—	2	2	no	—
383	#127	C₂³×C₄₁	1	4	yes	—
384	#126	—	2	16	no	—
385	#124	—	2	8	no	—
386	C₂²⋊Q₈	—	2	4	no	—
387	C₂²⋊Q₈	C₂³×C₄₁	1	8	yes	complemented
388	#121	—	2	4	no	—
389	#121	—	2	8	no	—
390	#121	C₂³×C₄₁	1	8	yes	—
391	#120	—	2	16	no	—
392	C₄×Q₈	—	2	8	no	—
393	C₄×Q₈	—	2	2	no	—
394	C₄×Q₈	C₂³×C₄₁	1	4	yes	complemented
395	C₄×D₄	—	2	2	no	—
396	C₄×D₄	D₄×C₄₁	1	4	yes	—
397	C₄×D₄	C₂³×C₄₁	1	4	yes	—
398	C₄×D₄	C₂³×C₄₁	1	4	yes	—
399	#117	—	2	4	no	—
400	#117	C₂³×C₄₁	1	8	yes	complemented
401	C₂×C₄⋊C₄	—	2	2	no	—
402	C₂×C₄⋊C₄	—	2	1	no	—
403	C₂×C₄⋊C₄	—	2	1	no	—
404	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
405	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
406	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
407	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
408	C₂×C₂²⋊C₄	—	2	2	no	—
409	C₂×C₂²⋊C₄	—	2	2	no	—
410	C₂×C₂²⋊C₄	—	2	1	no	—
411	C₂×C₂²⋊C₄	—	2	2	no	—
412	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
413	C₂×C₄²	—	2	2	no	—
414	C₂×C₄²	—	2	1	no	—
415	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
416	#107	C₂³×C₄₁	1	8	yes	complemented
417	#106	C₂³×C₄₁	1	4	yes	complemented
418	C₄⋊C₈	C₂³×C₄₁	1	4	yes	complemented
419	#103	C₂³×C₄₁	1	4	yes	complemented
420	#102	C₂³×C₄₁	1	4	yes	—
421	#102	C₂³×C₄₁	1	8	yes	—
422	#102	C₂³×C₄₁	1	8	yes	complemented
423	#102	C₂³×C₄₁	1	8	yes	complemented
424	#102	C₂³×C₄₁	1	8	yes	complemented
425	C₂²⋊C₈	C₂³×C₄₁	1	8	yes	complemented
426	C₄×C₈	C₂³×C₄₁	1	4	yes	complemented
427	C₂⁴	—	2	1	no	—
428	C₂⁴	—	2	2	no	—
429	C₂⁴	—	2	8	no	—
430	C₂⁴	—	4	8	no	—
431	C₂⁴	—	4	8	no	—
432	C₂⁴	—	8	4	no	—
433	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	—	2	4	no	—
436	C₂×Q₈	—	2	2	no	—
437	C₂×Q₈	—	2	2	no	—
438	C₂×Q₈	—	4	4	no	—
439	C₂×Q₈	C₂⁴×C₄₁	1	2	yes	—
440	C₂×D₄	—	2	16	no	—
441	C₂×D₄	—	2	8	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	1	no	—
444	C₂×D₄	—	2	8	no	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	—	2	4	no	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	2	4	no	—
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	2	4	no	—
451	C₂×D₄	—	4	8	no	—
452	C₂×D₄	—	4	16	no	—
453	C₂×D₄	—	4	8	no	—
454	C₂×D₄	—	4	32	no	—
455	C₂×D₄	—	4	8	no	—
456	C₂×D₄	—	4	4	no	—
457	C₂×D₄	—	8	2	no	—
458	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
459	C₂×D₄	#41×C₄₁	1	4	yes	—
460	C₂×D₄	C₂⁴×C₄₁	1	4	yes	—
461	C₂×D₄	C₂⁴×C₄₁	1	2	yes	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	4	4	no	—
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	4	yes	—
473	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
474	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
475	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
476	SD₁₆	—	2	8	no	—
477	SD₁₆	—	4	8	no	—
478	M₁₆	—	4	8	no	—
479	C₂×C₈	—	2	2	no	—
480	C₂×C₈	—	2	2	no	—
481	C₂×C₈	—	2	4	no	—
482	C₂×C₈	C₂⁴×C₄₁	1	2	yes	—
483	C₂×C₈	C₂⁴×C₄₁	1	2	yes	—
484	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
485	C₄⋊C₄	—	2	8	no	—
486	C₄⋊C₄	—	2	4	no	—
487	C₄⋊C₄	—	2	4	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
492	C₄⋊C₄	C₂⁴×C₄₁	1	2	yes	—
493	C₄⋊C₄	C₂⁴×C₄₁	1	4	yes	—
494	C₄⋊C₄	C₂⁴×C₄₁	1	2	yes	—
495	C₂²⋊C₄	—	2	8	no	—
496	C₂²⋊C₄	—	2	8	no	—
497	C₂²⋊C₄	—	2	4	no	—
498	C₂²⋊C₄	—	2	8	no	—
499	C₂²⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
500	C₄²	—	2	8	no	—
501	C₄²	—	2	4	no	—
502	C₄²	C₂×D₄×C₄₁	1	2	yes	—
503	C₄²	C₂⁴×C₄₁	1	2	yes	—
504	C₂³	—	2	4	no	—
505	C₂³	—	2	4	no	—
506	C₂³	—	2	4	no	—
507	C₂³	—	2	4	no	—
508	C₂³	—	2	2	no	—
509	C₂³	—	2	4	no	—
510	C₂³	—	4	16	no	—
511	C₂³	—	4	16	no	—
512	C₂³	—	4	2	no	—
513	C₂³	—	4	2	no	—
514	C₂³	—	4	4	no	—
515	C₂³	—	4	4	no	—
516	C₂³	—	4	4	no	—
517	C₂³	—	8	8	no	—
518	C₂³	—	8	32	no	—
519	C₂³	C₂²×D₄×C₄₁	1	4	yes	—
520	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
521	Q₈	—	2	2	no	—
522	Q₈	—	2	2	no	—
523	Q₈	—	4	2	no	—
524	Q₈	—	4	8	no	—
525	D₄	—	2	4	no	—
526	D₄	—	2	8	no	—
527	D₄	—	2	4	no	—
528	D₄	—	2	2	no	—
529	D₄	—	2	4	no	—
530	D₄	—	4	16	no	—
531	D₄	—	4	16	no	—
532	D₄	—	4	4	no	—
533	D₄	—	4	8	no	—
534	D₄	—	8	4	no	—
535	D₄	C₂×#41×C₄₁	1	4	yes	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	8	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	4	4	no	—
548	C₂×C₄	—	4	2	no	—
549	C₂×C₄	—	4	8	no	—
550	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	4	yes	—
551	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
552	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
553	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
554	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
555	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
556	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
557	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
558	C₈	—	2	2	no	—
559	C₈	—	2	2	no	—
560	C₈	—	2	4	no	—
561	C₂²	—	2	4	no	—
562	C₂²	—	2	4	no	—
563	C₂²	—	2	4	no	—
564	C₂²	—	2	4	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	4	8	no	—
567	C₂²	—	4	8	no	—
568	C₂²	—	4	2	no	—
569	C₂²	—	4	4	no	—
570	C₂²	—	4	4	no	—
571	C₂²	—	4	4	no	—
572	C₂²	—	8	32	no	—
573	C₂²	—	8	8	no	—
574	C₂²	C₄₁×#546	1	2	yes	—
575	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	2	yes	—
576	C₂²	C₄₁×#577	1	2	yes	—
577	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
578	C₄	—	2	2	no	—
579	C₄	—	2	4	no	—
580	C₄	—	4	2	no	—
581	C₄	—	4	4	no	—
582	C₄	—	4	2	no	—
583	C₄	—	4	2	no	—
584	C₄	C₄₁×#546	1	2	yes	—
585	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	4	yes	—
586	C₄	C₂²×#41×C₄₁	1	1	yes	characteristic
587	C₄	C₂×H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
588	C₂	—	4	4	no	—
589	C₂	—	4	4	no	—
590	C₂	—	8	4	no	—
591	C₂	C₄₁×#3324	1	4	yes	direct product factor
592	C₂	C₂×C₄₁×#546	1	1	yes	characteristic
593	C₂	C₂²×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
594	C₂	C₂×C₄₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
prime radical	C₂×D₄²×C₄₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄×D₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#556	C₂	semidirect
3. C₄₁×#3324	C₂	direct (N × H)
4. C₂×C₄₁×#486	C₂	semidirect
5. C₂×C₄₁×#482	C₂	semidirect
6. C₂×C₄₁×#474	C₂	semidirect
7. C₂×C₄₁×#468	C₂	semidirect
8. C₂×C₄₁×#460	C₂	semidirect
9. C₂×C₄₁×#450	C₂	semidirect
10. C₂×C₄×SD₁₆×C₄₁	C₂	semidirect
11. C₂×C₄₁×#436	C₂	semidirect
12. C₂×C₄₁×#418	C₂	semidirect
16. C₂²×SD₁₆×C₄₁	C₂²	semidirect
17. C₂²×M₁₆×C₄₁	C₂²	semidirect
19. C₄₁×#556	C₂²	semidirect
22. C₂×C₄⋊Q₈×C₄₁	C₂²	semidirect
26. C₂×C₂²⋊Q₈×C₄₁	C₂²	semidirect
31. C₂×C₄×Q₈×C₄₁	C₂²	semidirect
34. C₂×#117×C₄₁	C₂²	semidirect
35. C₄₁×#486	C₂²	semidirect
36. C₄₁×#482	C₂²	semidirect
37. C₄₁×#474	C₂²	semidirect
38. C₄₁×#468	C₂²	semidirect
39. C₄₁×#460	C₂²	semidirect
40. C₄₁×#450	C₂²	semidirect
41. C₄×SD₁₆×C₄₁	C₂²	semidirect
42. C₄₁×#436	C₂²	semidirect
43. C₂×#107×C₄₁	C₂²	semidirect
44. C₂×#106×C₄₁	C₂²	semidirect
45. C₂×C₄⋊C₈×C₄₁	C₂²	semidirect
46. C₄₁×#418	C₂²	semidirect
47. C₂×#103×C₄₁	C₂²	semidirect
50. C₂×#102×C₄₁	C₂²	semidirect
51. C₂×#102×C₄₁	C₂²	semidirect
52. C₂×#102×C₄₁	C₂²	semidirect
53. C₂×C₂²⋊C₈×C₄₁	C₂²	semidirect
54. C₂×C₄×C₈×C₄₁	C₂²	semidirect
76. C₂×SD₁₆×C₄₁	C₂³	semidirect
78. C₂×M₁₆×C₄₁	C₂³	semidirect
83. C₄⋊Q₈×C₄₁	C₂³	semidirect
89. C₂²⋊Q₈×C₄₁	C₂³	semidirect
96. C₄×Q₈×C₄₁	C₂³	semidirect
102. #117×C₄₁	C₂³	semidirect
118. #107×C₄₁	C₂³	semidirect
119. #106×C₄₁	C₂³	semidirect
120. C₄⋊C₈×C₄₁	C₂³	semidirect
121. #103×C₄₁	C₂³	semidirect
124. #102×C₄₁	C₂³	semidirect
125. #102×C₄₁	C₂³	semidirect
126. #102×C₄₁	C₂³	semidirect
127. C₂²⋊C₈×C₄₁	C₂³	semidirect
128. C₄×C₈×C₄₁	C₂³	semidirect
263. C₂×#3324	C₄₁	direct (N × H)
291. C₂×#556	C₂×C₄₁	semidirect
293. #3324	C₂×C₄₁	direct (N × H)
294. C₂×#486	C₂×C₄₁	semidirect
295. C₂×#482	C₂×C₄₁	semidirect
296. C₂×#474	C₂×C₄₁	semidirect
297. C₂×#468	C₂×C₄₁	semidirect
298. C₂×#460	C₂×C₄₁	semidirect
299. C₂×#450	C₂×C₄₁	semidirect
300. C₂×C₄×SD₁₆	C₂×C₄₁	semidirect
301. C₂×#436	C₂×C₄₁	semidirect
302. C₂×#418	C₂×C₄₁	semidirect
306. C₂×C₄₁	#3324	direct (N × H)
313. C₂²×SD₁₆	C₂²×C₄₁	semidirect
314. C₂²×M₁₆	C₂²×C₄₁	semidirect
316. #556	C₂²×C₄₁	semidirect
319. C₂×C₄⋊Q₈	C₂²×C₄₁	semidirect
323. C₂×C₂²⋊Q₈	C₂²×C₄₁	semidirect
328. C₂×C₄×Q₈	C₂²×C₄₁	semidirect
331. C₂×#117	C₂²×C₄₁	semidirect
332. #486	C₂²×C₄₁	semidirect
333. #482	C₂²×C₄₁	semidirect
334. #474	C₂²×C₄₁	semidirect
335. #468	C₂²×C₄₁	semidirect
336. #460	C₂²×C₄₁	semidirect
337. #450	C₂²×C₄₁	semidirect
338. C₄×SD₁₆	C₂²×C₄₁	semidirect
339. #436	C₂²×C₄₁	semidirect
340. C₂×#107	C₂²×C₄₁	semidirect
341. C₂×#106	C₂²×C₄₁	semidirect
342. C₂×C₄⋊C₈	C₂²×C₄₁	semidirect
343. #418	C₂²×C₄₁	semidirect
344. C₂×#103	C₂²×C₄₁	semidirect
347. C₂×#102	C₂²×C₄₁	semidirect
348. C₂×#102	C₂²×C₄₁	semidirect
349. C₂×#102	C₂²×C₄₁	semidirect
350. C₂×C₂²⋊C₈	C₂²×C₄₁	semidirect
351. C₂×C₄×C₈	C₂²×C₄₁	semidirect
352. C₄₁	C₂×#3324	direct (N × H)
374. C₂×SD₁₆	C₂³×C₄₁	semidirect
376. C₂×M₁₆	C₂³×C₄₁	semidirect
381. C₄⋊Q₈	C₂³×C₄₁	semidirect
387. C₂²⋊Q₈	C₂³×C₄₁	semidirect
394. C₄×Q₈	C₂³×C₄₁	semidirect
400. #117	C₂³×C₄₁	semidirect
416. #107	C₂³×C₄₁	semidirect
417. #106	C₂³×C₄₁	semidirect
418. C₄⋊C₈	C₂³×C₄₁	semidirect
419. #103	C₂³×C₄₁	semidirect
422. #102	C₂³×C₄₁	semidirect
423. #102	C₂³×C₄₁	semidirect
424. #102	C₂³×C₄₁	semidirect
425. C₂²⋊C₈	C₂³×C₄₁	semidirect
426. C₄×C₈	C₂³×C₄₁	semidirect
591. C₂	C₄₁×#3324	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×#556	2
C₂×D₄²×C₄₁	2
C₄₁×#3324	2
C₂×C₄₁×#486	2
C₂×C₄₁×#482	2
C₂×C₄₁×#474	2
C₂×C₄₁×#468	2
C₂×C₄₁×#460	2
C₂×C₄₁×#450	2
C₂×C₄×SD₁₆×C₄₁	2
C₂×C₄₁×#436	2
C₂×C₄₁×#418	2
C₂×#3324	41

Subgroups — C2×C41×#3324 #23837345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#3324 #23837345