Subgroups — C₂×C₄₁×#3289 #23837300

Order 10496.

2620 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#578	C₂	1	2	yes	maximal, complemented
2	C₂×C₄₁×#577	C₂	1	2	yes	maximal, complemented
3	C₄₁×#3289	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄₁×#497	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₁×#473	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₁×#471	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₁×#456	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₁×#454	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄₁×#455	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄₁×#444	C₂	1	2	yes	maximal, complemented
11	C₂×2₋¹⁺⁴×C₄₁	C₂²	1	2	yes	—
12	C₂×2₋¹⁺⁴×C₄₁	C₂²	1	1	yes	characteristic, complemented
13	C₂×H(5,𝔽₂)×C₄₁	C₂²	1	1	yes	characteristic
14	C₄₁×#578	—	2	8	no	—
15	C₄₁×#578	C₂²	1	16	yes	complemented
16	C₄₁×#577	—	2	8	no	—
17	C₄₁×#577	C₂²	1	16	yes	complemented
18	C₂×#137×C₄₁	—	2	2	no	—
19	C₂×#137×C₄₁	—	2	2	no	—
20	C₂×#137×C₄₁	—	2	2	no	—
21	C₂×#137×C₄₁	C₂²	1	2	yes	—
22	C₂×#137×C₄₁	C₂²	1	2	yes	—
23	C₂×#137×C₄₁	C₂²	1	2	yes	complemented
24	C₂×AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
25	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
26	C₂×#135×C₄₁	—	2	2	no	—
27	C₂×#135×C₄₁	—	2	2	no	—
28	C₂×#135×C₄₁	C₂²	1	2	yes	—
29	C₂²×Q₁₆×C₄₁	—	2	2	no	—
30	C₂²×Q₁₆×C₄₁	C₂²	1	2	yes	complemented
31	C₂²×SD₁₆×C₄₁	—	2	2	no	—
32	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
33	C₂×#131×C₄₁	C₂²	1	2	yes	—
34	C₂×#131×C₄₁	C₂²	1	2	yes	complemented
35	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
36	C₂×#124×C₄₁	C₂²	1	1	yes	characteristic
37	C₄₁×#497	C₂²	1	8	yes	complemented
38	C₄₁×#473	C₂²	1	16	yes	complemented
39	C₄₁×#471	C₂²	1	16	yes	complemented
40	C₄₁×#456	C₂²	1	16	yes	complemented
41	C₄₁×#455	C₂²	1	8	yes	complemented
42	C₄₁×#454	C₂²	1	8	yes	complemented
43	C₄₁×#444	C₂²	1	16	yes	complemented
44	C₂×#108×C₄₁	C₂²	1	2	yes	complemented
45	C₂×#104×C₄₁	C₂²	1	2	yes	—
46	C₂×#104×C₄₁	C₂²	1	2	yes	complemented
47	C₂×#101×C₄₁	C₂²	1	2	yes	complemented
48	C₂×#101×C₄₁	C₂²	1	1	yes	characteristic, complemented
49	C₂×#100×C₄₁	C₂²	1	1	yes	characteristic
50	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic, complemented
51	2₋¹⁺⁴×C₄₁	—	2	4	no	—
52	2₋¹⁺⁴×C₄₁	—	2	4	no	—
53	2₋¹⁺⁴×C₄₁	—	2	4	no	—
54	2₋¹⁺⁴×C₄₁	—	4	2	no	—
55	2₋¹⁺⁴×C₄₁	—	4	1	no	—
56	2₋¹⁺⁴×C₄₁	C₂³	1	8	yes	—
57	2₋¹⁺⁴×C₄₁	C₂³	1	4	yes	complemented
58	H(5,𝔽₂)×C₄₁	—	2	4	no	—
59	H(5,𝔽₂)×C₄₁	—	4	1	no	—
60	H(5,𝔽₂)×C₄₁	C₂³	1	4	yes	—
61	C₂×#41×C₄₁	—	2	2	no	—
62	C₂×#41×C₄₁	—	2	2	no	—
63	C₂×#41×C₄₁	—	2	2	no	—
64	C₂×#41×C₄₁	—	2	2	no	—
65	C₂×#41×C₄₁	—	4	2	no	—
66	C₂×#41×C₄₁	—	4	1	no	—
67	C₂×#41×C₄₁	C₂³	1	2	yes	—
68	C₂×#41×C₄₁	C₂³	1	2	yes	—
69	C₂²×Q₈×C₄₁	—	2	2	no	—
70	C₂²×Q₈×C₄₁	—	2	2	no	—
71	C₂²×Q₈×C₄₁	—	2	2	no	—
72	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
73	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
74	C₂²×D₄×C₄₁	—	2	2	no	—
75	C₂²×D₄×C₄₁	—	4	1	no	—
76	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
77	#137×C₄₁	—	2	4	no	—
78	#137×C₄₁	—	2	4	no	—
79	#137×C₄₁	—	2	4	no	—
80	#137×C₄₁	—	2	16	no	—
81	#137×C₄₁	—	2	16	no	—
82	#137×C₄₁	—	2	16	no	—
83	#137×C₄₁	C₂³	1	8	yes	—
84	#137×C₄₁	C₂³	1	8	yes	—
85	#137×C₄₁	C₂³	1	8	yes	complemented
86	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
87	AGL(1,ℤ/8)×C₄₁	—	2	16	no	—
88	AGL(1,ℤ/8)×C₄₁	C₂³	1	8	yes	—
89	#135×C₄₁	—	2	16	no	—
90	#135×C₄₁	—	2	16	no	—
91	#135×C₄₁	D₄	1	8	yes	—
92	#135×C₄₁	C₂³	1	8	yes	—
93	C₂×Q₁₆×C₄₁	—	2	2	no	—
94	C₂×Q₁₆×C₄₁	—	2	2	no	—
95	C₂×Q₁₆×C₄₁	—	2	2	no	—
96	C₂×Q₁₆×C₄₁	—	2	16	no	—
97	C₂×Q₁₆×C₄₁	—	2	2	no	—
98	C₂×Q₁₆×C₄₁	—	2	2	no	—
99	C₂×Q₁₆×C₄₁	—	2	4	no	—
100	C₂×Q₁₆×C₄₁	D₄	1	2	yes	—
101	C₂×Q₁₆×C₄₁	C₂³	1	8	yes	complemented
102	C₂×SD₁₆×C₄₁	—	2	2	no	—
103	C₂×SD₁₆×C₄₁	—	2	2	no	—
104	C₂×SD₁₆×C₄₁	—	2	2	no	—
105	C₂×SD₁₆×C₄₁	—	2	2	no	—
106	C₂×SD₁₆×C₄₁	—	2	16	no	—
107	C₂×SD₁₆×C₄₁	—	2	2	no	—
108	C₂×SD₁₆×C₄₁	—	2	2	no	—
109	C₂×SD₁₆×C₄₁	—	2	4	no	—
110	C₂×SD₁₆×C₄₁	D₄	1	2	yes	—
111	C₂×SD₁₆×C₄₁	D₄	1	2	yes	—
112	C₂×SD₁₆×C₄₁	C₂³	1	8	yes	—
113	C₂×D₈×C₄₁	—	2	2	no	—
114	C₂×D₈×C₄₁	D₄	1	2	yes	—
115	#131×C₄₁	—	2	4	no	—
116	#131×C₄₁	—	2	4	no	—
117	#131×C₄₁	C₂³	1	8	yes	—
118	#131×C₄₁	C₂³	1	8	yes	complemented
119	C₂×M₁₆×C₄₁	—	2	2	no	—
120	C₂×M₁₆×C₄₁	—	2	2	no	—
121	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
122	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
123	C₂²×C₈×C₄₁	—	2	2	no	—
124	C₂²×C₈×C₄₁	—	2	2	no	—
125	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
126	C₄⋊Q₈×C₄₁	—	4	1	no	—
127	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	complemented
128	#124×C₄₁	—	4	1	no	—
129	#124×C₄₁	C₂³	1	4	yes	—
130	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
131	C₂×C₂²⋊C₄×C₄₁	—	4	1	no	—
132	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
133	#108×C₄₁	C₂³	1	8	yes	complemented
134	#104×C₄₁	C₂³	1	8	yes	—
135	#104×C₄₁	C₂³	1	8	yes	complemented
136	#101×C₄₁	C₂³	1	4	yes	complemented
137	#101×C₄₁	C₂³	1	8	yes	complemented
138	#100×C₄₁	C₂³	1	4	yes	—
139	#97×C₄₁	C₂³	1	4	yes	complemented
140	C₂⁴×C₄₁	—	2	1	no	—
141	C₂⁴×C₄₁	—	4	1	no	—
142	#41×C₄₁	—	2	2	no	—
143	#41×C₄₁	—	2	2	no	—
144	#41×C₄₁	—	2	2	no	—
145	#41×C₄₁	—	2	2	no	—
146	#41×C₄₁	—	2	8	no	—
147	#41×C₄₁	—	2	8	no	—
148	#41×C₄₁	—	2	8	no	—
149	#41×C₄₁	—	2	8	no	—
150	#41×C₄₁	—	4	8	no	—
151	#41×C₄₁	—	4	4	no	—
152	#41×C₄₁	—	4	4	no	—
153	#41×C₄₁	—	4	4	no	—
154	#41×C₄₁	—	4	4	no	—
155	#41×C₄₁	—	4	4	no	—
156	#41×C₄₁	—	4	8	no	—
157	#41×C₄₁	—	4	4	no	—
158	#41×C₄₁	C₂×D₄	1	4	yes	—
159	#41×C₄₁	C₂×D₄	1	4	yes	—
160	#41×C₄₁	C₂⁴	1	4	yes	—
161	#41×C₄₁	C₂⁴	1	4	yes	—
162	C₂×Q₈×C₄₁	—	2	1	no	—
163	C₂×Q₈×C₄₁	—	2	2	no	—
164	C₂×Q₈×C₄₁	—	2	8	no	—
165	C₂×Q₈×C₄₁	—	2	8	no	—
166	C₂×Q₈×C₄₁	—	2	8	no	—
167	C₂×Q₈×C₄₁	—	2	2	no	—
168	C₂×Q₈×C₄₁	—	2	2	no	—
169	C₂×Q₈×C₄₁	—	2	2	no	—
170	C₂×Q₈×C₄₁	—	2	2	no	—
171	C₂×Q₈×C₄₁	—	2	2	no	—
172	C₂×Q₈×C₄₁	—	2	2	no	—
173	C₂×Q₈×C₄₁	—	4	2	no	—
174	C₂×Q₈×C₄₁	—	4	1	no	—
175	C₂×Q₈×C₄₁	—	4	4	no	—
176	C₂×Q₈×C₄₁	—	4	4	no	—
177	C₂×Q₈×C₄₁	—	4	4	no	—
178	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
179	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
180	C₂×Q₈×C₄₁	C₂⁴	1	2	yes	—
181	C₂×Q₈×C₄₁	C₂⁴	1	4	yes	—
182	C₂×D₄×C₄₁	—	2	2	no	—
183	C₂×D₄×C₄₁	—	2	2	no	—
184	C₂×D₄×C₄₁	—	2	8	no	—
185	C₂×D₄×C₄₁	—	2	2	no	—
186	C₂×D₄×C₄₁	—	2	2	no	—
187	C₂×D₄×C₄₁	—	2	1	no	—
188	C₂×D₄×C₄₁	—	4	4	no	—
189	C₂×D₄×C₄₁	—	4	2	no	—
190	C₂×D₄×C₄₁	—	4	4	no	—
191	C₂×D₄×C₄₁	—	4	4	no	—
192	C₂×D₄×C₄₁	—	4	1	no	—
193	C₂×D₄×C₄₁	—	4	2	no	—
194	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
195	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
196	C₂×D₄×C₄₁	C₂⁴	1	2	yes	—
197	C₂²×C₄×C₄₁	—	2	2	no	—
198	C₂²×C₄×C₄₁	—	2	2	no	—
199	C₂²×C₄×C₄₁	—	2	1	no	—
200	C₂²×C₄×C₄₁	—	2	1	no	—
201	C₂²×C₄×C₄₁	—	2	2	no	—
202	C₂²×C₄×C₄₁	—	4	1	no	—
203	C₂²×C₄×C₄₁	—	4	2	no	—
204	C₂²×C₄×C₄₁	—	4	2	no	—
205	C₂²×C₄×C₄₁	—	4	2	no	—
206	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
207	Q₁₆×C₄₁	—	2	8	no	—
208	Q₁₆×C₄₁	—	2	8	no	—
209	Q₁₆×C₄₁	—	2	8	no	—
210	Q₁₆×C₄₁	—	2	8	no	—
211	Q₁₆×C₄₁	—	2	8	no	—
212	Q₁₆×C₄₁	C₂×D₄	1	8	yes	—
213	SD₁₆×C₄₁	—	2	8	no	—
214	SD₁₆×C₄₁	—	2	8	no	—
215	SD₁₆×C₄₁	—	2	8	no	—
216	SD₁₆×C₄₁	—	2	8	no	—
217	SD₁₆×C₄₁	—	2	8	no	—
218	SD₁₆×C₄₁	—	2	8	no	—
219	SD₁₆×C₄₁	C₂×D₄	1	8	yes	—
220	SD₁₆×C₄₁	C₂×D₄	1	8	yes	—
221	D₈×C₄₁	—	2	8	no	—
222	D₈×C₄₁	C₂×D₄	1	8	yes	—
223	M₁₆×C₄₁	—	2	2	no	—
224	M₁₆×C₄₁	—	2	2	no	—
225	M₁₆×C₄₁	—	2	8	no	—
226	M₁₆×C₄₁	—	2	8	no	—
227	M₁₆×C₄₁	C₂⁴	1	4	yes	—
228	M₁₆×C₄₁	C₂⁴	1	4	yes	—
229	C₂×C₈×C₄₁	—	2	2	no	—
230	C₂×C₈×C₄₁	—	2	2	no	—
231	C₂×C₈×C₄₁	—	2	8	no	—
232	C₂×C₈×C₄₁	—	2	8	no	—
233	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
234	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
235	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
236	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
237	C₄⋊C₄×C₄₁	—	4	4	no	—
238	C₂²⋊C₄×C₄₁	—	4	4	no	—
239	C₄²×C₄₁	—	2	1	no	—
240	C₄²×C₄₁	C₂⁴	1	2	yes	—
241	C₂³×C₄₁	—	2	2	no	—
242	C₂³×C₄₁	—	2	2	no	—
243	C₂³×C₄₁	—	2	2	no	—
244	C₂³×C₄₁	—	4	4	no	—
245	C₂³×C₄₁	—	4	1	no	—
246	C₂³×C₄₁	—	4	1	no	—
247	C₂³×C₄₁	—	8	2	no	—
248	C₂³×C₄₁	—	16	2	no	—
249	C₂³×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
250	Q₈×C₄₁	—	2	2	no	—
251	Q₈×C₄₁	—	2	2	no	—
252	Q₈×C₄₁	—	2	4	no	—
253	Q₈×C₄₁	—	2	4	no	—
254	Q₈×C₄₁	—	2	4	no	—
255	Q₈×C₄₁	—	4	8	no	—
256	Q₈×C₄₁	—	4	4	no	—
257	Q₈×C₄₁	—	4	2	no	—
258	Q₈×C₄₁	—	4	2	no	—
259	Q₈×C₄₁	—	4	2	no	—
260	Q₈×C₄₁	C₂²×D₄	1	4	yes	—
261	Q₈×C₄₁	C₂²×D₄	1	4	yes	—
262	D₄×C₄₁	—	2	4	no	—
263	D₄×C₄₁	—	2	4	no	—
264	D₄×C₄₁	—	2	4	no	—
265	D₄×C₄₁	—	2	4	no	—
266	D₄×C₄₁	—	2	2	no	—
267	D₄×C₄₁	—	2	2	no	—
268	D₄×C₄₁	—	2	4	no	—
269	D₄×C₄₁	—	4	8	no	—
270	D₄×C₄₁	—	4	4	no	—
271	D₄×C₄₁	—	4	8	no	—
272	D₄×C₄₁	—	4	2	no	—
273	D₄×C₄₁	C₂²×D₄	1	4	yes	—
274	D₄×C₄₁	C₂²×D₄	1	4	yes	—
275	C₂×C₄×C₄₁	—	2	2	no	—
276	C₂×C₄×C₄₁	—	2	2	no	—
277	C₂×C₄×C₄₁	—	2	4	no	—
278	C₂×C₄×C₄₁	—	2	4	no	—
279	C₂×C₄×C₄₁	—	2	4	no	—
280	C₂×C₄×C₄₁	—	2	4	no	—
281	C₂×C₄×C₄₁	—	2	2	no	—
282	C₂×C₄×C₄₁	—	2	2	no	—
283	C₂×C₄×C₄₁	—	2	4	no	—
284	C₂×C₄×C₄₁	—	4	4	no	—
285	C₂×C₄×C₄₁	—	4	8	no	—
286	C₂×C₄×C₄₁	—	4	8	no	—
287	C₂×C₄×C₄₁	—	4	8	no	—
288	C₂×C₄×C₄₁	—	4	2	no	—
289	C₂×C₄×C₄₁	—	4	1	no	—
290	C₂×C₄×C₄₁	—	4	2	no	—
291	C₂×C₄×C₄₁	—	4	1	no	—
292	C₂×C₄×C₄₁	—	4	2	no	—
293	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
294	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
295	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
296	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
297	C₈×C₄₁	—	2	4	no	—
298	C₈×C₄₁	—	2	4	no	—
299	C₈×C₄₁	C₂²×D₄	1	4	yes	—
300	C₈×C₄₁	C₂²×D₄	1	4	yes	—
301	C₂×#3289	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
302	C₂²×C₄₁	—	2	2	no	—
303	C₂²×C₄₁	—	2	4	no	—
304	C₂²×C₄₁	—	2	4	no	—
305	C₂²×C₄₁	—	4	4	no	—
306	C₂²×C₄₁	—	4	4	no	—
307	C₂²×C₄₁	—	4	2	no	—
308	C₂²×C₄₁	—	8	2	no	—
309	C₂²×C₄₁	—	8	2	no	—
310	C₂²×C₄₁	—	16	1	no	—
311	C₂²×C₄₁	—	16	4	no	—
312	C₂²×C₄₁	D₄²	1	1	yes	characteristic
313	C₂²×C₄₁	D₄²	1	1	yes	characteristic
314	C₂²×C₄₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
315	C₄×C₄₁	—	2	4	no	—
316	C₄×C₄₁	—	2	4	no	—
317	C₄×C₄₁	—	4	2	no	—
318	C₄×C₄₁	—	4	2	no	—
319	C₄×C₄₁	—	4	4	no	—
320	C₄×C₄₁	D₄²	1	2	yes	—
321	C₄×C₄₁	C₂³×D₄	1	2	yes	—
322	C₂×#578	C₂×C₄₁	1	2	yes	complemented
323	C₂×#577	C₂×C₄₁	1	2	yes	complemented
324	#3289	C₂×C₄₁	1	16	yes	direct product factor
325	C₂×#497	C₂×C₄₁	1	1	yes	characteristic, complemented
326	C₂×#473	C₂×C₄₁	1	2	yes	complemented
327	C₂×#471	C₂×C₄₁	1	2	yes	complemented
328	C₂×#456	C₂×C₄₁	1	2	yes	complemented
329	C₂×#454	C₂×C₄₁	1	1	yes	characteristic, complemented
330	C₂×#455	C₂×C₄₁	1	1	yes	characteristic, complemented
331	C₂×#444	C₂×C₄₁	1	2	yes	complemented
332	C₂×C₄₁	—	2	1	no	—
333	C₂×C₄₁	—	2	1	no	—
334	C₂×C₄₁	—	4	4	no	—
335	C₂×C₄₁	—	4	4	no	—
336	C₂×C₄₁	—	8	2	no	—
337	C₂×C₄₁	#3289	1	2	yes	direct product factor
338	C₂×C₄₁	C₂×D₄²	1	1	yes	characteristic
339	C₂×2₋¹⁺⁴	C₂²×C₄₁	1	2	yes	—
340	C₂×2₋¹⁺⁴	C₂²×C₄₁	1	1	yes	characteristic, complemented
341	C₂×H(5,𝔽₂)	C₂²×C₄₁	1	1	yes	characteristic
342	#578	—	2	8	no	—
343	#578	C₂²×C₄₁	1	16	yes	complemented
344	#577	—	2	8	no	—
345	#577	C₂²×C₄₁	1	16	yes	complemented
346	C₂×#137	—	2	2	no	—
347	C₂×#137	—	2	2	no	—
348	C₂×#137	—	2	2	no	—
349	C₂×#137	C₂²×C₄₁	1	2	yes	—
350	C₂×#137	C₂²×C₄₁	1	2	yes	—
351	C₂×#137	C₂²×C₄₁	1	2	yes	complemented
352	C₂×AGL(1,ℤ/8)	—	2	2	no	—
353	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
354	C₂×#135	—	2	2	no	—
355	C₂×#135	—	2	2	no	—
356	C₂×#135	C₂²×C₄₁	1	2	yes	—
357	C₂²×Q₁₆	—	2	2	no	—
358	C₂²×Q₁₆	C₂²×C₄₁	1	2	yes	complemented
359	C₂²×SD₁₆	—	2	2	no	—
360	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
361	C₂×#131	C₂²×C₄₁	1	2	yes	—
362	C₂×#131	C₂²×C₄₁	1	2	yes	complemented
363	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
364	C₂×#124	C₂²×C₄₁	1	1	yes	characteristic
365	#497	C₂²×C₄₁	1	8	yes	complemented
366	#473	C₂²×C₄₁	1	16	yes	complemented
367	#471	C₂²×C₄₁	1	16	yes	complemented
368	#456	C₂²×C₄₁	1	16	yes	complemented
369	#455	C₂²×C₄₁	1	8	yes	complemented
370	#454	C₂²×C₄₁	1	8	yes	complemented
371	#444	C₂²×C₄₁	1	16	yes	complemented
372	C₂×#108	C₂²×C₄₁	1	2	yes	complemented
373	C₂×#104	C₂²×C₄₁	1	2	yes	—
374	C₂×#104	C₂²×C₄₁	1	2	yes	complemented
375	C₂×#101	C₂²×C₄₁	1	1	yes	characteristic, complemented
376	C₂×#101	C₂²×C₄₁	1	2	yes	complemented
377	C₂×#100	C₂²×C₄₁	1	1	yes	characteristic
378	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic, complemented
379	C₄₁	C₂×#3289	1	1	yes	characteristic, direct product factor, Sylow
380	2₋¹⁺⁴	—	2	4	no	—
381	2₋¹⁺⁴	—	2	4	no	—
382	2₋¹⁺⁴	—	2	4	no	—
383	2₋¹⁺⁴	—	4	2	no	—
384	2₋¹⁺⁴	—	4	1	no	—
385	2₋¹⁺⁴	C₂³×C₄₁	1	8	yes	—
386	2₋¹⁺⁴	C₂³×C₄₁	1	4	yes	complemented
387	H(5,𝔽₂)	—	2	4	no	—
388	H(5,𝔽₂)	—	4	1	no	—
389	H(5,𝔽₂)	C₂³×C₄₁	1	4	yes	—
390	C₂×#41	—	2	2	no	—
391	C₂×#41	—	2	2	no	—
392	C₂×#41	—	2	2	no	—
393	C₂×#41	—	2	2	no	—
394	C₂×#41	—	4	2	no	—
395	C₂×#41	—	4	1	no	—
396	C₂×#41	C₂³×C₄₁	1	2	yes	—
397	C₂×#41	C₂³×C₄₁	1	2	yes	—
398	C₂²×Q₈	—	2	2	no	—
399	C₂²×Q₈	—	2	2	no	—
400	C₂²×Q₈	—	2	2	no	—
401	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
402	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
403	C₂²×D₄	—	2	2	no	—
404	C₂²×D₄	—	4	1	no	—
405	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
406	#137	—	2	4	no	—
407	#137	—	2	4	no	—
408	#137	—	2	4	no	—
409	#137	—	2	16	no	—
410	#137	—	2	16	no	—
411	#137	—	2	16	no	—
412	#137	C₂³×C₄₁	1	8	yes	—
413	#137	C₂³×C₄₁	1	8	yes	—
414	#137	C₂³×C₄₁	1	8	yes	complemented
415	AGL(1,ℤ/8)	—	2	4	no	—
416	AGL(1,ℤ/8)	—	2	16	no	—
417	AGL(1,ℤ/8)	C₂³×C₄₁	1	8	yes	—
418	#135	—	2	16	no	—
419	#135	—	2	16	no	—
420	#135	D₄×C₄₁	1	8	yes	—
421	#135	C₂³×C₄₁	1	8	yes	—
422	C₂×Q₁₆	—	2	2	no	—
423	C₂×Q₁₆	—	2	2	no	—
424	C₂×Q₁₆	—	2	2	no	—
425	C₂×Q₁₆	—	2	16	no	—
426	C₂×Q₁₆	—	2	2	no	—
427	C₂×Q₁₆	—	2	2	no	—
428	C₂×Q₁₆	—	2	4	no	—
429	C₂×Q₁₆	D₄×C₄₁	1	2	yes	—
430	C₂×Q₁₆	C₂³×C₄₁	1	8	yes	complemented
431	C₂×SD₁₆	—	2	2	no	—
432	C₂×SD₁₆	—	2	2	no	—
433	C₂×SD₁₆	—	2	2	no	—
434	C₂×SD₁₆	—	2	2	no	—
435	C₂×SD₁₆	—	2	16	no	—
436	C₂×SD₁₆	—	2	2	no	—
437	C₂×SD₁₆	—	2	2	no	—
438	C₂×SD₁₆	—	2	4	no	—
439	C₂×SD₁₆	D₄×C₄₁	1	2	yes	—
440	C₂×SD₁₆	D₄×C₄₁	1	2	yes	—
441	C₂×SD₁₆	C₂³×C₄₁	1	8	yes	—
442	C₂×D₈	—	2	2	no	—
443	C₂×D₈	D₄×C₄₁	1	2	yes	—
444	#131	—	2	4	no	—
445	#131	—	2	4	no	—
446	#131	C₂³×C₄₁	1	8	yes	—
447	#131	C₂³×C₄₁	1	8	yes	complemented
448	C₂×M₁₆	—	2	2	no	—
449	C₂×M₁₆	—	2	2	no	—
450	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
451	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
452	C₂²×C₈	—	2	2	no	—
453	C₂²×C₈	—	2	2	no	—
454	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
455	C₄⋊Q₈	—	4	1	no	—
456	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	complemented
457	#124	—	4	1	no	—
458	#124	C₂³×C₄₁	1	4	yes	—
459	C₂×C₄⋊C₄	—	4	1	no	—
460	C₂×C₂²⋊C₄	—	4	1	no	—
461	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
462	#108	C₂³×C₄₁	1	8	yes	complemented
463	#104	C₂³×C₄₁	1	8	yes	—
464	#104	C₂³×C₄₁	1	8	yes	complemented
465	#101	C₂³×C₄₁	1	4	yes	complemented
466	#101	C₂³×C₄₁	1	8	yes	complemented
467	#100	C₂³×C₄₁	1	4	yes	—
468	#97	C₂³×C₄₁	1	4	yes	complemented
469	C₂⁴	—	2	1	no	—
470	C₂⁴	—	4	1	no	—
471	#41	—	2	2	no	—
472	#41	—	2	2	no	—
473	#41	—	2	2	no	—
474	#41	—	2	2	no	—
475	#41	—	2	8	no	—
476	#41	—	2	8	no	—
477	#41	—	2	8	no	—
478	#41	—	2	8	no	—
479	#41	—	4	8	no	—
480	#41	—	4	4	no	—
481	#41	—	4	4	no	—
482	#41	—	4	4	no	—
483	#41	—	4	4	no	—
484	#41	—	4	4	no	—
485	#41	—	4	8	no	—
486	#41	—	4	4	no	—
487	#41	C₂×D₄×C₄₁	1	4	yes	—
488	#41	C₂×D₄×C₄₁	1	4	yes	—
489	#41	C₂⁴×C₄₁	1	4	yes	—
490	#41	C₂⁴×C₄₁	1	4	yes	—
491	C₂×Q₈	—	2	1	no	—
492	C₂×Q₈	—	2	2	no	—
493	C₂×Q₈	—	2	8	no	—
494	C₂×Q₈	—	2	8	no	—
495	C₂×Q₈	—	2	8	no	—
496	C₂×Q₈	—	2	2	no	—
497	C₂×Q₈	—	2	2	no	—
498	C₂×Q₈	—	2	2	no	—
499	C₂×Q₈	—	2	2	no	—
500	C₂×Q₈	—	2	2	no	—
501	C₂×Q₈	—	2	2	no	—
502	C₂×Q₈	—	4	2	no	—
503	C₂×Q₈	—	4	1	no	—
504	C₂×Q₈	—	4	4	no	—
505	C₂×Q₈	—	4	4	no	—
506	C₂×Q₈	—	4	4	no	—
507	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
508	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
509	C₂×Q₈	C₂⁴×C₄₁	1	2	yes	—
510	C₂×Q₈	C₂⁴×C₄₁	1	4	yes	—
511	C₂×D₄	—	2	2	no	—
512	C₂×D₄	—	2	2	no	—
513	C₂×D₄	—	2	8	no	—
514	C₂×D₄	—	2	2	no	—
515	C₂×D₄	—	2	2	no	—
516	C₂×D₄	—	2	1	no	—
517	C₂×D₄	—	4	4	no	—
518	C₂×D₄	—	4	2	no	—
519	C₂×D₄	—	4	4	no	—
520	C₂×D₄	—	4	4	no	—
521	C₂×D₄	—	4	1	no	—
522	C₂×D₄	—	4	2	no	—
523	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
524	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
525	C₂×D₄	C₂⁴×C₄₁	1	2	yes	—
526	C₂²×C₄	—	2	2	no	—
527	C₂²×C₄	—	2	2	no	—
528	C₂²×C₄	—	2	1	no	—
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	—	2	2	no	—
531	C₂²×C₄	—	4	1	no	—
532	C₂²×C₄	—	4	2	no	—
533	C₂²×C₄	—	4	2	no	—
534	C₂²×C₄	—	4	2	no	—
535	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
536	Q₁₆	—	2	8	no	—
537	Q₁₆	—	2	8	no	—
538	Q₁₆	—	2	8	no	—
539	Q₁₆	—	2	8	no	—
540	Q₁₆	—	2	8	no	—
541	Q₁₆	C₂×D₄×C₄₁	1	8	yes	—
542	SD₁₆	—	2	8	no	—
543	SD₁₆	—	2	8	no	—
544	SD₁₆	—	2	8	no	—
545	SD₁₆	—	2	8	no	—
546	SD₁₆	—	2	8	no	—
547	SD₁₆	—	2	8	no	—
548	SD₁₆	C₂×D₄×C₄₁	1	8	yes	—
549	SD₁₆	C₂×D₄×C₄₁	1	8	yes	—
550	D₈	—	2	8	no	—
551	D₈	C₂×D₄×C₄₁	1	8	yes	—
552	M₁₆	—	2	2	no	—
553	M₁₆	—	2	2	no	—
554	M₁₆	—	2	8	no	—
555	M₁₆	—	2	8	no	—
556	M₁₆	C₂⁴×C₄₁	1	4	yes	—
557	M₁₆	C₂⁴×C₄₁	1	4	yes	—
558	C₂×C₈	—	2	2	no	—
559	C₂×C₈	—	2	2	no	—
560	C₂×C₈	—	2	8	no	—
561	C₂×C₈	—	2	8	no	—
562	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
563	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
564	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
565	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
566	C₄⋊C₄	—	4	4	no	—
567	C₂²⋊C₄	—	4	4	no	—
568	C₄²	—	2	1	no	—
569	C₄²	C₂⁴×C₄₁	1	2	yes	—
570	C₂³	—	2	2	no	—
571	C₂³	—	2	2	no	—
572	C₂³	—	2	2	no	—
573	C₂³	—	4	4	no	—
574	C₂³	—	4	1	no	—
575	C₂³	—	4	1	no	—
576	C₂³	—	8	2	no	—
577	C₂³	—	16	2	no	—
578	C₂³	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
579	Q₈	—	2	2	no	—
580	Q₈	—	2	2	no	—
581	Q₈	—	2	4	no	—
582	Q₈	—	2	4	no	—
583	Q₈	—	2	4	no	—
584	Q₈	—	4	8	no	—
585	Q₈	—	4	4	no	—
586	Q₈	—	4	2	no	—
587	Q₈	—	4	2	no	—
588	Q₈	—	4	2	no	—
589	Q₈	C₂²×D₄×C₄₁	1	4	yes	—
590	Q₈	C₂²×D₄×C₄₁	1	4	yes	—
591	D₄	—	2	4	no	—
592	D₄	—	2	4	no	—
593	D₄	—	2	4	no	—
594	D₄	—	2	4	no	—
595	D₄	—	2	2	no	—
596	D₄	—	2	2	no	—
597	D₄	—	2	4	no	—
598	D₄	—	4	8	no	—
599	D₄	—	4	4	no	—
600	D₄	—	4	8	no	—
601	D₄	—	4	2	no	—
602	D₄	C₂²×D₄×C₄₁	1	4	yes	—
603	D₄	C₂²×D₄×C₄₁	1	4	yes	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	4	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	4	no	—
609	C₂×C₄	—	2	4	no	—
610	C₂×C₄	—	2	2	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	4	no	—
613	C₂×C₄	—	4	4	no	—
614	C₂×C₄	—	4	8	no	—
615	C₂×C₄	—	4	8	no	—
616	C₂×C₄	—	4	8	no	—
617	C₂×C₄	—	4	2	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	2	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	4	2	no	—
622	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
623	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
624	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
625	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
626	C₈	—	2	4	no	—
627	C₈	—	2	4	no	—
628	C₈	C₂²×D₄×C₄₁	1	4	yes	—
629	C₈	C₂²×D₄×C₄₁	1	4	yes	—
630	C₂²	—	2	2	no	—
631	C₂²	—	2	4	no	—
632	C₂²	—	2	4	no	—
633	C₂²	—	4	4	no	—
634	C₂²	—	4	4	no	—
635	C₂²	—	4	2	no	—
636	C₂²	—	8	2	no	—
637	C₂²	—	8	2	no	—
638	C₂²	—	16	1	no	—
639	C₂²	—	16	4	no	—
640	C₂²	D₄²×C₄₁	1	1	yes	characteristic
641	C₂²	D₄²×C₄₁	1	1	yes	characteristic
642	C₂²	C₂×H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
643	C₄	—	2	4	no	—
644	C₄	—	2	4	no	—
645	C₄	—	4	2	no	—
646	C₄	—	4	2	no	—
647	C₄	—	4	4	no	—
648	C₄	D₄²×C₄₁	1	2	yes	—
649	C₄	C₂³×D₄×C₄₁	1	2	yes	—
650	C₂	—	2	1	no	—
651	C₂	—	2	1	no	—
652	C₂	—	4	4	no	—
653	C₂	—	4	4	no	—
654	C₂	—	8	2	no	—
655	C₂	C₄₁×#3289	1	2	yes	direct product factor
656	C₂	C₂×D₄²×C₄₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#578	C₂	semidirect
2. C₂×C₄₁×#577	C₂	semidirect
3. C₄₁×#3289	C₂	direct (N × H)
4. C₂×C₄₁×#497	C₂	semidirect
5. C₂×C₄₁×#473	C₂	semidirect
6. C₂×C₄₁×#471	C₂	semidirect
7. C₂×C₄₁×#456	C₂	semidirect
8. C₂×C₄₁×#454	C₂	semidirect
9. C₂×C₄₁×#455	C₂	semidirect
10. C₂×C₄₁×#444	C₂	semidirect
12. C₂×2₋¹⁺⁴×C₄₁	C₂²	semidirect
15. C₄₁×#578	C₂²	semidirect
17. C₄₁×#577	C₂²	semidirect
23. C₂×#137×C₄₁	C₂²	semidirect
30. C₂²×Q₁₆×C₄₁	C₂²	semidirect
34. C₂×#131×C₄₁	C₂²	semidirect
35. C₂×C₄⋊Q₈×C₄₁	C₂²	semidirect
37. C₄₁×#497	C₂²	semidirect
38. C₄₁×#473	C₂²	semidirect
39. C₄₁×#471	C₂²	semidirect
40. C₄₁×#456	C₂²	semidirect
41. C₄₁×#455	C₂²	semidirect
42. C₄₁×#454	C₂²	semidirect
43. C₄₁×#444	C₂²	semidirect
44. C₂×#108×C₄₁	C₂²	semidirect
46. C₂×#104×C₄₁	C₂²	semidirect
47. C₂×#101×C₄₁	C₂²	semidirect
48. C₂×#101×C₄₁	C₂²	semidirect
50. C₂×#97×C₄₁	C₂²	semidirect
57. 2₋¹⁺⁴×C₄₁	C₂³	semidirect
85. #137×C₄₁	C₂³	semidirect
101. C₂×Q₁₆×C₄₁	C₂³	semidirect
118. #131×C₄₁	C₂³	semidirect
127. C₄⋊Q₈×C₄₁	C₂³	semidirect
133. #108×C₄₁	C₂³	semidirect
135. #104×C₄₁	C₂³	semidirect
136. #101×C₄₁	C₂³	semidirect
137. #101×C₄₁	C₂³	semidirect
139. #97×C₄₁	C₂³	semidirect
301. C₂×#3289	C₄₁	direct (N × H)
322. C₂×#578	C₂×C₄₁	semidirect
323. C₂×#577	C₂×C₄₁	semidirect
324. #3289	C₂×C₄₁	direct (N × H)
325. C₂×#497	C₂×C₄₁	semidirect
326. C₂×#473	C₂×C₄₁	semidirect
327. C₂×#471	C₂×C₄₁	semidirect
328. C₂×#456	C₂×C₄₁	semidirect
329. C₂×#454	C₂×C₄₁	semidirect
330. C₂×#455	C₂×C₄₁	semidirect
331. C₂×#444	C₂×C₄₁	semidirect
337. C₂×C₄₁	#3289	direct (N × H)
340. C₂×2₋¹⁺⁴	C₂²×C₄₁	semidirect
343. #578	C₂²×C₄₁	semidirect
345. #577	C₂²×C₄₁	semidirect
351. C₂×#137	C₂²×C₄₁	semidirect
358. C₂²×Q₁₆	C₂²×C₄₁	semidirect
362. C₂×#131	C₂²×C₄₁	semidirect
363. C₂×C₄⋊Q₈	C₂²×C₄₁	semidirect
365. #497	C₂²×C₄₁	semidirect
366. #473	C₂²×C₄₁	semidirect
367. #471	C₂²×C₄₁	semidirect
368. #456	C₂²×C₄₁	semidirect
369. #455	C₂²×C₄₁	semidirect
370. #454	C₂²×C₄₁	semidirect
371. #444	C₂²×C₄₁	semidirect
372. C₂×#108	C₂²×C₄₁	semidirect
374. C₂×#104	C₂²×C₄₁	semidirect
375. C₂×#101	C₂²×C₄₁	semidirect
376. C₂×#101	C₂²×C₄₁	semidirect
378. C₂×#97	C₂²×C₄₁	semidirect
379. C₄₁	C₂×#3289	direct (N × H)
386. 2₋¹⁺⁴	C₂³×C₄₁	semidirect
414. #137	C₂³×C₄₁	semidirect
430. C₂×Q₁₆	C₂³×C₄₁	semidirect
447. #131	C₂³×C₄₁	semidirect
456. C₄⋊Q₈	C₂³×C₄₁	semidirect
462. #108	C₂³×C₄₁	semidirect
464. #104	C₂³×C₄₁	semidirect
465. #101	C₂³×C₄₁	semidirect
466. #101	C₂³×C₄₁	semidirect
468. #97	C₂³×C₄₁	semidirect
655. C₂	C₄₁×#3289	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×#578	2
C₂×C₄₁×#577	2
C₄₁×#3289	2
C₂×C₄₁×#497	2
C₂×C₄₁×#473	2
C₂×C₄₁×#471	2
C₂×C₄₁×#456	2
C₂×C₄₁×#454	2
C₂×C₄₁×#455	2
C₂×C₄₁×#444	2
C₂×#3289	41

Subgroups — C2×C41×#3289 #23837300

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#3289 #23837300