Subgroups — C₄₁×#34203 #23810404

Order 10496.

3038 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#519	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#117×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#2362	C₂	1	4	yes	maximal, complemented
5	C₄₁×#2360	C₂	1	4	yes	maximal, complemented
6	C₄₁×#2344	C₂	1	4	yes	maximal, complemented
7	C₄₁×#2342	C₂	1	4	yes	maximal
8	C₄₁×#2323	C₂	1	4	yes	maximal, complemented
9	C₄₁×#2316	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#2310	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊Q₈×C₄₁	C₂	1	4	yes	maximal, complemented
12	C₄₁×#2280	C₂	1	1	yes	maximal, characteristic
13	C₄₁×#2274	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂³×Q₈×C₄₁	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Q₈×C₄₁	C₄	1	16	yes	complemented
17	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
18	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
21	C₄₁×#519	—	2	8	no	—
22	C₂×C₄×Q₈×C₄₁	C₂²	1	4	yes	complemented
23	C₂×C₄×Q₈×C₄₁	C₂²	1	4	yes	complemented
24	C₂×#117×C₄₁	—	2	8	no	—
25	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×#117×C₄₁	C₂²	1	2	yes	—
29	C₂×#117×C₄₁	C₂²	1	4	yes	—
30	C₂×#117×C₄₁	C₂²	1	2	yes	—
31	C₂×#117×C₄₁	C₂²	1	2	yes	—
32	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
33	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
35	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
36	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
37	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
38	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
39	C₄₁×#391	C₂²	1	4	yes	complemented
40	C₄₁×#391	C₂²	1	4	yes	complemented
41	C₄₁×#389	C₂²	1	4	yes	complemented
42	C₄₁×#389	C₂²	1	4	yes	complemented
43	C₄₁×#389	C₂²	1	8	yes	complemented
44	C₄₁×#388	C₂²	1	8	yes	—
45	C₄₁×#388	C₂²	1	8	yes	—
46	C₄₁×#387	C₂²	1	8	yes	complemented
47	C₄₁×#387	C₂²	1	8	yes	complemented
48	C₄₁×#385	C₂²	1	8	yes	—
49	C₄₁×#381	C₂²	1	2	yes	—
50	C₄₁×#381	C₂²	1	4	yes	—
51	C₄₁×#380	C₂²	1	2	yes	—
52	C₄₁×#380	C₂²	1	2	yes	—
53	C₄₁×#380	C₂²	1	4	yes	—
54	C₄₁×#379	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	complemented
56	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	complemented
57	C₄×C₄⋊C₄×C₄₁	C₂²	1	8	yes	complemented
58	C₄×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
59	C₄×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
60	C₄×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
61	C₄×C₂²⋊C₄×C₄₁	C₂²	1	8	yes	—
62	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₄₁	—	2	4	no	—
64	C₂²×Q₈×C₄₁	—	2	4	no	—
65	C₂²×Q₈×C₄₁	C₂×C₄	1	8	yes	complemented
66	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
67	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
68	C₂³×C₄×C₄₁	—	2	4	no	—
69	C₂³×C₄×C₄₁	—	2	4	no	—
70	C₂³×C₄×C₄₁	C₂×C₄	1	8	yes	—
71	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
72	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
74	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
79	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
80	C₂²⋊Q₈×C₄₁	C₂×C₄	1	16	yes	—
81	C₂²⋊Q₈×C₄₁	C₂×C₄	1	16	yes	—
82	C₂²⋊Q₈×C₄₁	C₂×C₄	1	16	yes	complemented
83	C₂²⋊Q₈×C₄₁	C₂×C₄	1	16	yes	complemented
84	C₄×Q₈×C₄₁	—	2	16	no	—
85	C₄×Q₈×C₄₁	—	2	16	no	—
86	#117×C₄₁	—	2	4	no	—
87	#117×C₄₁	—	2	4	no	—
88	#117×C₄₁	—	2	4	no	—
89	#117×C₄₁	—	2	8	no	—
90	#117×C₄₁	—	2	8	no	—
91	#117×C₄₁	—	2	16	no	—
92	#117×C₄₁	—	2	8	no	—
93	#117×C₄₁	D₄	1	8	yes	—
94	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
95	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	16	yes	complemented
96	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	8	yes	complemented
97	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
101	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
103	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
109	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	16	yes	—
110	C₂×C₂²⋊C₄×C₄₁	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₄₁	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₄₁	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
117	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
118	C₂×C₂²⋊C₄×C₄₁	C₂³	1	2	yes	—
119	C₂×C₄²×C₄₁	—	2	8	no	—
120	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
121	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
122	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
123	C₂×C₄²×C₄₁	C₂³	1	8	yes	—
124	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
125	C₂×C₄²×C₄₁	C₂³	1	8	yes	—
126	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
127	#95×C₄₁	C₂³	1	4	yes	—
128	#95×C₄₁	C₂³	1	8	yes	—
129	#95×C₄₁	C₂³	1	4	yes	—
130	#95×C₄₁	C₂³	1	8	yes	—
131	#95×C₄₁	C₂³	1	8	yes	—
132	#95×C₄₁	C₂³	1	8	yes	—
133	C₂⁴×C₄₁	—	2	2	no	—
134	C₂⁴×C₄₁	—	2	2	no	—
135	C₂⁴×C₄₁	—	2	4	no	—
136	C₂⁴×C₄₁	—	4	2	no	—
137	C₂⁴×C₄₁	C₂²×C₄	1	4	yes	—
138	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
139	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
140	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
141	C₂×Q₈×C₄₁	—	2	32	no	—
142	C₂×Q₈×C₄₁	—	2	4	no	—
143	C₂×Q₈×C₄₁	—	2	4	no	—
144	C₂×Q₈×C₄₁	—	2	4	no	—
145	C₂×Q₈×C₄₁	—	2	4	no	—
146	C₂×Q₈×C₄₁	C₂²×C₄	1	8	yes	—
147	C₂×Q₈×C₄₁	C₂²×C₄	1	8	yes	—
148	C₂²×C₄×C₄₁	—	2	2	no	—
149	C₂²×C₄×C₄₁	—	2	4	no	—
150	C₂²×C₄×C₄₁	—	2	2	no	—
151	C₂²×C₄×C₄₁	—	2	4	no	—
152	C₂²×C₄×C₄₁	—	2	32	no	—
153	C₂²×C₄×C₄₁	—	2	2	no	—
154	C₂²×C₄×C₄₁	—	2	2	no	—
155	C₂²×C₄×C₄₁	—	2	2	no	—
156	C₂²×C₄×C₄₁	—	2	2	no	—
157	C₂²×C₄×C₄₁	—	2	2	no	—
158	C₂²×C₄×C₄₁	—	2	2	no	—
159	C₂²×C₄×C₄₁	—	2	2	no	—
160	C₂²×C₄×C₄₁	—	2	2	no	—
161	C₂²×C₄×C₄₁	—	2	8	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	C₂²×C₄	1	8	yes	—
166	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
167	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
168	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
169	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
170	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
171	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
172	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
173	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
174	C₂²×C₄×C₄₁	#41	1	4	yes	—
175	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
178	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
181	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
183	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
185	C₄⋊C₄×C₄₁	—	2	8	no	—
186	C₄⋊C₄×C₄₁	—	2	4	no	—
187	C₄⋊C₄×C₄₁	—	2	16	no	—
188	C₄⋊C₄×C₄₁	—	2	4	no	—
189	C₄⋊C₄×C₄₁	—	2	4	no	—
190	C₄⋊C₄×C₄₁	—	4	8	no	—
191	C₄⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
192	C₄⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
193	C₄⋊C₄×C₄₁	C₂²×C₄	1	16	yes	—
194	C₄⋊C₄×C₄₁	C₂²×C₄	1	16	yes	—
195	C₄⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
196	C₂²⋊C₄×C₄₁	—	2	4	no	—
197	C₂²⋊C₄×C₄₁	—	2	4	no	—
198	C₂²⋊C₄×C₄₁	—	2	4	no	—
199	C₂²⋊C₄×C₄₁	—	2	8	no	—
200	C₂²⋊C₄×C₄₁	—	2	8	no	—
201	C₂²⋊C₄×C₄₁	—	4	4	no	—
202	C₂²⋊C₄×C₄₁	C₂²×C₄	1	16	yes	—
203	C₂²⋊C₄×C₄₁	C₂²×C₄	1	16	yes	—
204	C₂²⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
205	C₄²×C₄₁	—	2	8	no	—
206	C₄²×C₄₁	—	2	4	no	—
207	C₄²×C₄₁	—	2	16	no	—
208	C₄²×C₄₁	—	2	4	no	—
209	C₄²×C₄₁	—	2	4	no	—
210	C₄²×C₄₁	—	4	8	no	—
211	C₄²×C₄₁	C₂×D₄	1	8	yes	—
212	C₂³×C₄₁	—	2	1	no	—
213	C₂³×C₄₁	—	2	1	no	—
214	C₂³×C₄₁	—	2	2	no	—
215	C₂³×C₄₁	—	2	2	no	—
216	C₂³×C₄₁	—	2	2	no	—
217	C₂³×C₄₁	—	2	8	no	—
218	C₂³×C₄₁	—	2	8	no	—
219	C₂³×C₄₁	—	2	1	no	—
220	C₂³×C₄₁	—	2	1	no	—
221	C₂³×C₄₁	—	2	2	no	—
222	C₂³×C₄₁	—	2	2	no	—
223	C₂³×C₄₁	—	4	4	no	—
224	C₂³×C₄₁	—	4	8	no	—
225	C₂³×C₄₁	—	4	8	no	—
226	C₂³×C₄₁	C₂³×C₄	1	2	yes	—
227	C₂³×C₄₁	C₂³×C₄	1	2	yes	—
228	C₂³×C₄₁	C₂³×C₄	1	2	yes	—
229	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
230	C₂³×C₄₁	C₂×#41	1	2	yes	—
231	C₂³×C₄₁	2₋¹⁺⁴	1	4	yes	—
232	C₂³×C₄₁	C₂⁵	1	1	yes	characteristic
233	Q₈×C₄₁	—	2	16	no	—
234	Q₈×C₄₁	—	2	16	no	—
235	C₂×C₄×C₄₁	—	2	8	no	—
236	C₂×C₄×C₄₁	—	2	16	no	—
237	C₂×C₄×C₄₁	—	2	1	no	—
238	C₂×C₄×C₄₁	—	2	4	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	4	no	—
243	C₂×C₄×C₄₁	—	2	4	no	—
244	C₂×C₄×C₄₁	—	2	4	no	—
245	C₂×C₄×C₄₁	—	2	8	no	—
246	C₂×C₄×C₄₁	—	2	8	no	—
247	C₂×C₄×C₄₁	—	2	8	no	—
248	C₂×C₄×C₄₁	—	2	8	no	—
249	C₂×C₄×C₄₁	—	2	8	no	—
250	C₂×C₄×C₄₁	—	2	4	no	—
251	C₂×C₄×C₄₁	—	2	4	no	—
252	C₂×C₄×C₄₁	—	2	4	no	—
253	C₂×C₄×C₄₁	—	2	4	no	—
254	C₂×C₄×C₄₁	—	2	4	no	—
255	C₂×C₄×C₄₁	—	2	4	no	—
256	C₂×C₄×C₄₁	—	2	4	no	—
257	C₂×C₄×C₄₁	—	2	4	no	—
258	C₂×C₄×C₄₁	—	2	4	no	—
259	C₂×C₄×C₄₁	—	4	8	no	—
260	C₂×C₄×C₄₁	—	4	16	no	—
261	C₂×C₄×C₄₁	C₄×D₄	1	16	yes	—
262	C₂×C₄×C₄₁	C₂³×C₄	1	8	yes	—
263	C₂×C₄×C₄₁	C₂³×C₄	1	8	yes	—
264	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
265	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
266	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
267	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
268	C₂×C₄×C₄₁	C₂²×D₄	1	8	yes	—
269	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
270	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
271	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
272	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
273	#34203	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
274	C₂²×C₄₁	—	2	2	no	—
275	C₂²×C₄₁	—	2	2	no	—
276	C₂²×C₄₁	—	2	2	no	—
277	C₂²×C₄₁	—	2	2	no	—
278	C₂²×C₄₁	—	2	4	no	—
279	C₂²×C₄₁	—	2	4	no	—
280	C₂²×C₄₁	—	2	4	no	—
281	C₂²×C₄₁	—	2	4	no	—
282	C₂²×C₄₁	—	2	2	no	—
283	C₂²×C₄₁	—	2	2	no	—
284	C₂²×C₄₁	—	2	2	no	—
285	C₂²×C₄₁	—	4	16	no	—
286	C₂²×C₄₁	—	4	2	no	—
287	C₂²×C₄₁	—	4	2	no	—
288	C₂²×C₄₁	C₂×C₄×D₄	1	4	yes	—
289	C₂²×C₄₁	#519	1	4	yes	complemented
290	C₂²×C₄₁	C₂⁴×C₄	1	1	yes	characteristic
291	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
292	C₂²×C₄₁	C₂²×#41	1	1	yes	characteristic
293	C₂²×C₄₁	C₂×2₋¹⁺⁴	1	2	yes	—
294	C₂²×C₄₁	#585	1	2	yes	—
295	C₄×C₄₁	—	2	2	no	—
296	C₄×C₄₁	—	2	2	no	—
297	C₄×C₄₁	—	2	8	no	—
298	C₄×C₄₁	—	2	8	no	—
299	C₄×C₄₁	—	2	8	no	—
300	C₄×C₄₁	—	4	4	no	—
301	C₄×C₄₁	—	4	4	no	—
302	C₄×C₄₁	—	4	4	no	—
303	C₄×C₄₁	C₂×C₄×D₄	1	4	yes	—
304	C₄×C₄₁	C₂×C₄×D₄	1	4	yes	—
305	C₂²×C₂²⋊Q₈	C₂×C₄₁	1	1	yes	characteristic
306	C₂×#519	C₂×C₄₁	1	1	yes	characteristic, complemented
307	C₂²×#117	C₂×C₄₁	1	1	yes	characteristic
308	#2362	C₂×C₄₁	1	4	yes	complemented
309	#2360	C₂×C₄₁	1	4	yes	complemented
310	#2344	C₂×C₄₁	1	4	yes	complemented
311	#2342	C₂×C₄₁	1	4	yes	—
312	#2323	C₂×C₄₁	1	4	yes	complemented
313	#2316	C₂×C₄₁	1	1	yes	characteristic, complemented
314	#2310	C₂×C₄₁	1	1	yes	characteristic, complemented
315	C₄×C₂²⋊Q₈	C₂×C₄₁	1	4	yes	complemented
316	#2280	C₂×C₄₁	1	1	yes	characteristic
317	#2274	C₂×C₄₁	1	1	yes	characteristic, complemented
318	C₂×C₄₁	—	2	4	no	—
319	C₂×C₄₁	—	2	4	no	—
320	C₂×C₄₁	—	4	2	no	—
321	C₂×C₄₁	C₂²×C₄×D₄	1	1	yes	characteristic
322	C₂×C₄₁	C₂×#519	1	1	yes	characteristic
323	C₂×C₄₁	#3428	1	1	yes	characteristic
324	C₂×C₄₁	#3489	1	2	yes	—
325	C₂×C₄₁	#3495	1	2	yes	—
326	C₂³×Q₈	C₂²×C₄₁	1	1	yes	characteristic
327	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
328	C₂×C₂²⋊Q₈	C₄×C₄₁	1	16	yes	complemented
329	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
330	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
331	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
332	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
333	#519	—	2	8	no	—
334	C₂×C₄×Q₈	C₂²×C₄₁	1	4	yes	complemented
335	C₂×C₄×Q₈	C₂²×C₄₁	1	4	yes	complemented
336	C₂×#117	—	2	8	no	—
337	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
338	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
339	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
340	C₂×#117	C₂²×C₄₁	1	2	yes	—
341	C₂×#117	C₂²×C₄₁	1	4	yes	—
342	C₂×#117	C₂²×C₄₁	1	2	yes	—
343	C₂×#117	C₂²×C₄₁	1	2	yes	—
344	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
345	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
346	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
347	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
348	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
349	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
350	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
351	#391	C₂²×C₄₁	1	4	yes	complemented
352	#391	C₂²×C₄₁	1	4	yes	complemented
353	#389	C₂²×C₄₁	1	4	yes	complemented
354	#389	C₂²×C₄₁	1	4	yes	complemented
355	#389	C₂²×C₄₁	1	8	yes	complemented
356	#388	C₂²×C₄₁	1	8	yes	—
357	#388	C₂²×C₄₁	1	8	yes	—
358	#387	C₂²×C₄₁	1	8	yes	complemented
359	#387	C₂²×C₄₁	1	8	yes	complemented
360	#385	C₂²×C₄₁	1	8	yes	—
361	#381	C₂²×C₄₁	1	4	yes	—
362	#381	C₂²×C₄₁	1	2	yes	—
363	#380	C₂²×C₄₁	1	4	yes	—
364	#380	C₂²×C₄₁	1	2	yes	—
365	#380	C₂²×C₄₁	1	2	yes	—
366	#379	C₂²×C₄₁	1	2	yes	—
367	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	complemented
368	C₄×C₄⋊C₄	C₂²×C₄₁	1	8	yes	complemented
369	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	complemented
370	C₄×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
371	C₄×C₂²⋊C₄	C₂²×C₄₁	1	8	yes	—
372	C₄×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
373	C₄×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
374	C₄₁	#34203	1	1	yes	characteristic, direct product factor, Sylow
375	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
376	C₂²×Q₈	—	2	4	no	—
377	C₂²×Q₈	—	2	4	no	—
378	C₂²×Q₈	C₂×C₄×C₄₁	1	8	yes	complemented
379	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
380	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
381	C₂³×C₄	—	2	4	no	—
382	C₂³×C₄	—	2	4	no	—
383	C₂³×C₄	C₂×C₄×C₄₁	1	8	yes	—
384	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
385	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
386	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
387	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
388	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
389	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
390	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
391	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
392	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
393	C₂²⋊Q₈	C₂×C₄×C₄₁	1	16	yes	—
394	C₂²⋊Q₈	C₂×C₄×C₄₁	1	16	yes	—
395	C₂²⋊Q₈	C₂×C₄×C₄₁	1	16	yes	complemented
396	C₂²⋊Q₈	C₂×C₄×C₄₁	1	16	yes	complemented
397	C₄×Q₈	—	2	16	no	—
398	C₄×Q₈	—	2	16	no	—
399	#117	—	2	4	no	—
400	#117	—	2	4	no	—
401	#117	—	2	4	no	—
402	#117	—	2	8	no	—
403	#117	—	2	8	no	—
404	#117	—	2	16	no	—
405	#117	—	2	8	no	—
406	#117	D₄×C₄₁	1	8	yes	—
407	C₂×C₄⋊C₄	—	2	8	no	—
408	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	16	yes	complemented
409	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	8	yes	complemented
410	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
411	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
412	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
413	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
414	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
415	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
416	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
417	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
418	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
419	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
420	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
421	C₂×C₂²⋊C₄	—	2	8	no	—
422	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	16	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₄₁	1	2	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
425	C₂×C₂²⋊C₄	C₂³×C₄₁	1	2	yes	—
426	C₂×C₂²⋊C₄	C₂³×C₄₁	1	2	yes	—
427	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₄₁	1	2	yes	—
432	C₂×C₄²	—	2	8	no	—
433	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
434	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
435	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
436	C₂×C₄²	C₂³×C₄₁	1	8	yes	—
437	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
438	C₂×C₄²	C₂³×C₄₁	1	8	yes	—
439	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
440	#95	C₂³×C₄₁	1	4	yes	—
441	#95	C₂³×C₄₁	1	8	yes	—
442	#95	C₂³×C₄₁	1	4	yes	—
443	#95	C₂³×C₄₁	1	8	yes	—
444	#95	C₂³×C₄₁	1	8	yes	—
445	#95	C₂³×C₄₁	1	8	yes	—
446	C₂⁴	—	2	2	no	—
447	C₂⁴	—	2	2	no	—
448	C₂⁴	—	2	4	no	—
449	C₂⁴	—	4	2	no	—
450	C₂⁴	C₂²×C₄×C₄₁	1	4	yes	—
451	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
452	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
453	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
454	C₂×Q₈	—	2	32	no	—
455	C₂×Q₈	—	2	4	no	—
456	C₂×Q₈	—	2	4	no	—
457	C₂×Q₈	—	2	4	no	—
458	C₂×Q₈	—	2	4	no	—
459	C₂×Q₈	C₂²×C₄×C₄₁	1	8	yes	—
460	C₂×Q₈	C₂²×C₄×C₄₁	1	8	yes	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	32	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	8	no	—
475	C₂²×C₄	—	2	8	no	—
476	C₂²×C₄	—	4	4	no	—
477	C₂²×C₄	—	4	4	no	—
478	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
479	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
480	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
481	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
482	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
483	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
484	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
485	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
486	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
487	C₂²×C₄	#41×C₄₁	1	4	yes	—
488	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
489	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
490	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
491	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
492	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
493	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
494	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
495	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
496	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
497	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
498	C₄⋊C₄	—	2	8	no	—
499	C₄⋊C₄	—	2	4	no	—
500	C₄⋊C₄	—	2	16	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	4	8	no	—
504	C₄⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
505	C₄⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
506	C₄⋊C₄	C₂²×C₄×C₄₁	1	16	yes	—
507	C₄⋊C₄	C₂²×C₄×C₄₁	1	16	yes	—
508	C₄⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₂²⋊C₄	—	2	4	no	—
511	C₂²⋊C₄	—	2	4	no	—
512	C₂²⋊C₄	—	2	8	no	—
513	C₂²⋊C₄	—	2	8	no	—
514	C₂²⋊C₄	—	4	4	no	—
515	C₂²⋊C₄	C₂²×C₄×C₄₁	1	16	yes	—
516	C₂²⋊C₄	C₂²×C₄×C₄₁	1	16	yes	—
517	C₂²⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
518	C₄²	—	2	8	no	—
519	C₄²	—	2	4	no	—
520	C₄²	—	2	16	no	—
521	C₄²	—	2	4	no	—
522	C₄²	—	2	4	no	—
523	C₄²	—	4	8	no	—
524	C₄²	C₂×D₄×C₄₁	1	8	yes	—
525	C₂³	—	2	1	no	—
526	C₂³	—	2	1	no	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	2	no	—
529	C₂³	—	2	2	no	—
530	C₂³	—	2	8	no	—
531	C₂³	—	2	8	no	—
532	C₂³	—	2	1	no	—
533	C₂³	—	2	1	no	—
534	C₂³	—	2	2	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	4	4	no	—
537	C₂³	—	4	8	no	—
538	C₂³	—	4	8	no	—
539	C₂³	C₂³×C₄×C₄₁	1	2	yes	—
540	C₂³	C₂³×C₄×C₄₁	1	2	yes	—
541	C₂³	C₂³×C₄×C₄₁	1	2	yes	—
542	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
543	C₂³	C₂×#41×C₄₁	1	2	yes	—
544	C₂³	2₋¹⁺⁴×C₄₁	1	4	yes	—
545	C₂³	C₂⁵×C₄₁	1	1	yes	characteristic
546	Q₈	—	2	16	no	—
547	Q₈	—	2	16	no	—
548	C₂×C₄	—	2	8	no	—
549	C₂×C₄	—	2	16	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	8	no	—
559	C₂×C₄	—	2	8	no	—
560	C₂×C₄	—	2	8	no	—
561	C₂×C₄	—	2	8	no	—
562	C₂×C₄	—	2	8	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	4	8	no	—
573	C₂×C₄	—	4	16	no	—
574	C₂×C₄	C₄×D₄×C₄₁	1	16	yes	—
575	C₂×C₄	C₂³×C₄×C₄₁	1	8	yes	—
576	C₂×C₄	C₂³×C₄×C₄₁	1	8	yes	—
577	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
578	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
579	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
580	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
581	C₂×C₄	C₂²×D₄×C₄₁	1	8	yes	—
582	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
583	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
584	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
585	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	2	2	no	—
590	C₂²	—	2	4	no	—
591	C₂²	—	2	4	no	—
592	C₂²	—	2	4	no	—
593	C₂²	—	2	4	no	—
594	C₂²	—	2	2	no	—
595	C₂²	—	2	2	no	—
596	C₂²	—	2	2	no	—
597	C₂²	—	4	16	no	—
598	C₂²	—	4	2	no	—
599	C₂²	—	4	2	no	—
600	C₂²	C₂×C₄×D₄×C₄₁	1	4	yes	—
601	C₂²	C₄₁×#519	1	4	yes	complemented
602	C₂²	C₂⁴×C₄×C₄₁	1	1	yes	characteristic
603	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
604	C₂²	C₂²×#41×C₄₁	1	1	yes	characteristic
605	C₂²	C₂×2₋¹⁺⁴×C₄₁	1	2	yes	—
606	C₂²	C₄₁×#585	1	2	yes	—
607	C₄	—	2	2	no	—
608	C₄	—	2	2	no	—
609	C₄	—	2	8	no	—
610	C₄	—	2	8	no	—
611	C₄	—	2	8	no	—
612	C₄	—	4	4	no	—
613	C₄	—	4	4	no	—
614	C₄	—	4	4	no	—
615	C₄	C₂×C₄×D₄×C₄₁	1	4	yes	—
616	C₄	C₂×C₄×D₄×C₄₁	1	4	yes	—
617	C₂	—	2	4	no	—
618	C₂	—	2	4	no	—
619	C₂	—	4	2	no	—
620	C₂	C₂²×C₄×D₄×C₄₁	1	1	yes	characteristic
621	C₂	C₂×C₄₁×#519	1	1	yes	characteristic
622	C₂	C₄₁×#3428	1	1	yes	characteristic
623	C₂	C₄₁×#3489	1	2	yes	—
624	C₂	C₄₁×#3495	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₄₁
Frattini, cosocle	C₂³	C₂⁵×C₄₁
prime radical	C₂⁵×C₄₁	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#519	C₂	semidirect
4. C₄₁×#2362	C₂	semidirect
5. C₄₁×#2360	C₂	semidirect
6. C₄₁×#2344	C₂	semidirect
8. C₄₁×#2323	C₂	semidirect
9. C₄₁×#2316	C₂	semidirect
10. C₄₁×#2310	C₂	semidirect
11. C₄×C₂²⋊Q₈×C₄₁	C₂	semidirect
13. C₄₁×#2274	C₂	semidirect
16. C₂×C₂²⋊Q₈×C₄₁	C₄	semidirect
22. C₂×C₄×Q₈×C₄₁	C₂²	semidirect
23. C₂×C₄×Q₈×C₄₁	C₂²	semidirect
39. C₄₁×#391	C₂²	semidirect
40. C₄₁×#391	C₂²	semidirect
41. C₄₁×#389	C₂²	semidirect
42. C₄₁×#389	C₂²	semidirect
43. C₄₁×#389	C₂²	semidirect
46. C₄₁×#387	C₂²	semidirect
47. C₄₁×#387	C₂²	semidirect
55. C₄×C₄⋊C₄×C₄₁	C₂²	semidirect
56. C₄×C₄⋊C₄×C₄₁	C₂²	semidirect
57. C₄×C₄⋊C₄×C₄₁	C₂²	semidirect
65. C₂²×Q₈×C₄₁	C₂×C₄	semidirect
82. C₂²⋊Q₈×C₄₁	C₂×C₄	semidirect
83. C₂²⋊Q₈×C₄₁	C₂×C₄	semidirect
95. C₂×C₄⋊C₄×C₄₁	C₂×C₄	semidirect
96. C₂×C₄⋊C₄×C₄₁	C₂×C₄	semidirect
273. #34203	C₄₁	direct (N × H)
289. C₂²×C₄₁	#519	semidirect
306. C₂×#519	C₂×C₄₁	semidirect
308. #2362	C₂×C₄₁	semidirect
309. #2360	C₂×C₄₁	semidirect
310. #2344	C₂×C₄₁	semidirect
312. #2323	C₂×C₄₁	semidirect
313. #2316	C₂×C₄₁	semidirect
314. #2310	C₂×C₄₁	semidirect
315. C₄×C₂²⋊Q₈	C₂×C₄₁	semidirect
317. #2274	C₂×C₄₁	semidirect
328. C₂×C₂²⋊Q₈	C₄×C₄₁	semidirect
334. C₂×C₄×Q₈	C₂²×C₄₁	semidirect
335. C₂×C₄×Q₈	C₂²×C₄₁	semidirect
351. #391	C₂²×C₄₁	semidirect
352. #391	C₂²×C₄₁	semidirect
353. #389	C₂²×C₄₁	semidirect
354. #389	C₂²×C₄₁	semidirect
355. #389	C₂²×C₄₁	semidirect
358. #387	C₂²×C₄₁	semidirect
359. #387	C₂²×C₄₁	semidirect
367. C₄×C₄⋊C₄	C₂²×C₄₁	semidirect
368. C₄×C₄⋊C₄	C₂²×C₄₁	semidirect
369. C₄×C₄⋊C₄	C₂²×C₄₁	semidirect
374. C₄₁	#34203	direct (N × H)
378. C₂²×Q₈	C₂×C₄×C₄₁	semidirect
395. C₂²⋊Q₈	C₂×C₄×C₄₁	semidirect
396. C₂²⋊Q₈	C₂×C₄×C₄₁	semidirect
408. C₂×C₄⋊C₄	C₂×C₄×C₄₁	semidirect
409. C₂×C₄⋊C₄	C₂×C₄×C₄₁	semidirect
601. C₂²	C₄₁×#519	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊Q₈×C₄₁	2
C₂×C₄₁×#519	2
C₂²×#117×C₄₁	2
C₄₁×#2362	2
C₄₁×#2360	2
C₄₁×#2344	2
C₄₁×#2342	2
C₄₁×#2323	2
C₄₁×#2316	2
C₄₁×#2310	2
C₄×C₂²⋊Q₈×C₄₁	2
C₄₁×#2280	2
C₄₁×#2274	2
#34203	41

Subgroups — C41×#34203 #23810404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#34203 #23810404