Subgroups — #35375838

Order 21248.

716 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#13050574	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#13050577	C₂	1	1	yes	maximal, characteristic
3	#24916198	C₂	1	1	yes	maximal, characteristic
4	#24906350	C₂	1	2	yes	maximal, complemented
5	#24906303	C₂	1	2	yes	maximal, complemented
6	#24906302	C₂	1	2	yes	maximal, complemented
7	#24906260	C₂	1	2	yes	maximal, complemented
8	#24906218	C₂	1	1	yes	maximal, characteristic, complemented
9	#24905889	C₂	1	1	yes	maximal, characteristic
10	#24905744	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₃×#1487	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#13050577	C₄	1	1	yes	characteristic
13	C₂×M₁₆×D₈₃	—	2	1	no	—
14	C₂×C₄×D₃₃₂	C₂²	1	1	yes	characteristic
15	#19127832	—	2	2	no	—
16	#19127832	C₂²	1	2	yes	complemented
17	Q₈×C₈₃⋊C₈	C₂²	1	2	yes	complemented
18	C₄×#13050577	C₄	1	1	yes	characteristic
19	C₄×#13050577	C₄	1	1	yes	characteristic
20	C₄×#13050577	C₄	1	1	yes	characteristic
21	C₄×#13050577	C₄	1	1	yes	characteristic
22	C₄×#13050577	C₂²	1	1	yes	characteristic
23	C₄×#13050577	C₂²	1	1	yes	characteristic
24	C₄×#13050577	C₂²	1	1	yes	characteristic
25	C₄×#13050577	C₂²	1	1	yes	characteristic
26	#19127778	C₂²	1	2	yes	—
27	#19127771	—	2	2	no	—
28	#19127766	C₂²	1	2	yes	—
29	C₂×#14581723	C₄	1	1	yes	characteristic
30	#19127725	—	2	1	no	—
31	#19127725	C₂²	1	1	yes	characteristic
32	C₂×#14581716	C₄	1	1	yes	characteristic
33	C₂×#14581712	C₄	1	1	yes	characteristic
34	#19127664	—	2	1	no	—
35	C₄×#13050574	—	2	1	no	—
36	C₄×#13050574	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
38	#19127451	C₂²	1	2	yes	complemented
39	#19127451	C₂²	1	2	yes	complemented
40	#19127430	C₂²	1	2	yes	—
41	#19127430	C₂²	1	2	yes	—
42	#19127423	C₂²	1	2	yes	complemented
43	C₈×C₈₃⋊C₈	C₂²	1	2	yes	complemented
44	C₂×C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
45	#431×C₈₃	C₂²	1	1	yes	characteristic
46	#423×C₈₃	C₂²	1	1	yes	characteristic
47	Q₈⋊C₈×C₈₃	C₂²	1	2	yes	complemented
48	Q₈⋊C₈×C₈₃	C₂²	1	2	yes	complemented
49	C₂²×D₃₃₂	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×D₈₃	—	2	1	no	—
51	C₂×#13050577	—	2	4	no	—
52	C₂×#13050577	C₂×C₄	1	1	yes	characteristic
53	C₂×#13050577	C₂×C₄	1	1	yes	characteristic
54	C₂×#13050577	C₂×C₄	1	1	yes	characteristic
55	C₂×#13050577	C₂×C₄	1	1	yes	characteristic
56	C₂×#13050574	—	2	1	no	—
57	C₂×#13050574	D₄	1	1	yes	characteristic
58	C₂×#13050574	D₄	1	1	yes	characteristic
59	C₂×#13050573	—	2	2	no	—
60	M₁₆×D₈₃	—	4	4	no	—
61	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₈×D₈₃	—	2	2	no	—
64	C₄×D₃₃₂	—	2	1	no	—
65	C₄×D₃₃₂	—	2	1	no	—
66	C₄×D₃₃₂	C₂×C₄	1	1	yes	characteristic
67	C₄×D₃₃₂	C₂×C₄	1	1	yes	characteristic
68	C₄×D₃₃₂	C₂³	1	1	yes	characteristic
69	C₄×D₃₃₂	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊Dic₈₃	C₂×C₄	1	1	yes	characteristic
71	C₂×(C₂×D₈₃)⋊C₄	—	2	1	no	—
72	#14581723	C₂×C₄	1	1	yes	characteristic
73	#14581723	C₂×C₄	1	1	yes	characteristic
74	#14581723	C₂×C₄	1	1	yes	characteristic
75	#14581723	C₂×C₄	1	1	yes	characteristic
76	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
77	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
78	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
79	C₄⋊(C₈₃⋊C₈)	C₂³	1	2	yes	—
80	#14581716	C₂×C₄	1	1	yes	characteristic
81	#14581716	C₂×C₄	1	1	yes	characteristic
82	#14581716	C₂×C₄	1	1	yes	characteristic
83	#14581716	C₂×C₄	1	1	yes	characteristic
84	#14581712	C₂×C₄	1	1	yes	characteristic
85	#14581712	C₂×C₄	1	1	yes	characteristic
86	#14581712	C₂×C₄	1	1	yes	characteristic
87	#14581712	C₂×C₄	1	1	yes	characteristic
88	(C₂×D₈₃)⋊C₈	—	2	2	no	—
89	(C₂×D₈₃)⋊C₈	—	2	2	no	—
90	#14581703	—	2	2	no	—
91	#14581703	D₄	1	2	yes	—
92	C₄×C₈₃⋊C₈	—	2	2	no	—
93	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
94	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
95	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
96	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
97	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
98	C₂²×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
99	C₂×M₁₆×C₈₃	—	2	1	no	—
100	C₄×Q₈×C₈₃	—	2	1	no	—
101	C₄×Q₈×C₈₃	—	2	1	no	—
102	C₄×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
103	C₄×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
104	C₄×Q₈×C₈₃	C₂³	1	1	yes	characteristic
105	C₄×Q₈×C₈₃	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
107	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
108	C₂×C₄²×C₈₃	—	2	1	no	—
109	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
110	C₄⋊C₈×C₈₃	—	2	1	no	—
111	C₄⋊C₈×C₈₃	—	2	1	no	—
112	C₄⋊C₈×C₈₃	C₂³	1	2	yes	—
113	C₂²⋊C₈×C₈₃	—	2	1	no	—
114	C₄×C₈×C₈₃	C₂³	1	2	yes	—
115	C₂³×D₈₃	—	2	1	no	—
116	C₂×D₃₃₂	—	2	2	no	—
117	C₂×D₃₃₂	—	2	1	no	—
118	C₂×D₃₃₂	—	4	1	no	—
119	C₂×D₃₃₂	C₂²×C₄	1	1	yes	characteristic
120	C₂×D₃₃₂	C₂²×C₄	1	1	yes	characteristic
121	C₂²×Dic₈₃	—	2	1	no	—
122	C₂×C₄×D₈₃	—	2	1	no	—
123	C₂×C₄×D₈₃	—	2	1	no	—
124	C₂×C₄×D₈₃	—	2	1	no	—
125	C₂×C₄×D₈₃	—	2	1	no	—
126	C₂×C₄×D₈₃	—	4	2	no	—
127	C₂×C₄×D₈₃	—	4	2	no	—
128	#13050577	—	2	2	no	—
129	#13050577	—	2	2	no	—
130	#13050577	—	2	2	no	—
131	#13050577	M₁₆	1	4	yes	complemented
132	#13050574	—	2	2	no	—
133	#13050574	—	4	2	no	—
134	#13050574	SD₁₆	1	4	yes	complemented
135	#13050573	—	4	4	no	—
136	C₂×C₈₃⋊C₈	—	2	2	no	—
137	C₂×C₈₃⋊C₈	—	2	2	no	—
138	C₂×C₈₃⋊C₈	—	4	1	no	—
139	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
144	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
145	C₈×D₈₃	—	4	4	no	—
146	C₄⋊Dic₈₃	—	2	1	no	—
147	C₄⋊Dic₈₃	C₂²×C₄	1	1	yes	characteristic
148	C₄⋊Dic₈₃	C₂²×C₄	1	1	yes	characteristic
149	(C₂×D₈₃)⋊C₄	—	2	1	no	—
150	(C₂×D₈₃)⋊C₄	—	2	1	no	—
151	(C₂×D₈₃)⋊C₄	—	2	1	no	—
152	(C₂×D₈₃)⋊C₄	—	2	1	no	—
153	C₂×Q₈×C₈₃	—	2	1	no	—
154	C₂×Q₈×C₈₃	—	2	2	no	—
155	C₂×Q₈×C₈₃	—	4	1	no	—
156	C₂×Q₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
157	C₂×Q₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₈₃	—	2	1	no	—
159	C₂²×C₄×C₈₃	—	2	1	no	—
160	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
163	M₁₆×C₈₃	—	4	2	no	—
164	C₂×C₈×C₈₃	—	2	2	no	—
165	C₂×C₈×C₈₃	—	2	2	no	—
166	C₄⋊C₄×C₈₃	—	2	1	no	—
167	C₄⋊C₄×C₈₃	—	2	1	no	—
168	C₄⋊C₄×C₈₃	—	2	1	no	—
169	C₄⋊C₄×C₈₃	—	2	1	no	—
170	C₄⋊C₄×C₈₃	—	2	1	no	—
171	C₄⋊C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
172	C₄⋊C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
173	C₄²×C₈₃	—	2	1	no	—
174	C₄²×C₈₃	—	2	1	no	—
175	C₄²×C₈₃	—	2	1	no	—
176	C₄²×C₈₃	—	2	1	no	—
177	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
178	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
179	C₄²×C₈₃	#41	1	1	yes	characteristic
180	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
181	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
182	C₂²×D₈₃	—	2	1	no	—
183	C₂²×D₈₃	—	2	1	no	—
184	C₂²×D₈₃	—	4	2	no	—
185	C₂²×D₈₃	—	4	2	no	—
186	C₂²×D₈₃	—	8	1	no	—
187	Q₈×C₈₃	—	2	1	no	—
188	Q₈×C₈₃	—	2	1	no	—
189	Q₈×C₈₃	—	2	1	no	—
190	Q₈×C₈₃	—	4	2	no	—
191	Q₈×C₈₃	C₂×M₁₆	1	2	yes	complemented
192	C₂×C₄×C₈₃	—	2	1	no	—
193	C₂×C₄×C₈₃	—	2	1	no	—
194	C₂×C₄×C₈₃	—	2	1	no	—
195	C₂×C₄×C₈₃	—	2	1	no	—
196	C₂×C₄×C₈₃	—	2	1	no	—
197	C₂×C₄×C₈₃	—	2	1	no	—
198	C₂×C₄×C₈₃	—	4	1	no	—
199	C₂×C₄×C₈₃	—	4	2	no	—
200	C₂×C₄×C₈₃	—	8	1	no	—
201	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
202	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
203	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
204	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
205	C₂×C₄×C₈₃	#135	1	2	yes	—
206	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
207	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
209	C₂×Dic₈₃	—	2	1	no	—
210	C₂×Dic₈₃	—	2	1	no	—
211	C₂×Dic₈₃	—	4	2	no	—
212	D₃₃₂	—	2	1	no	—
213	D₃₃₂	—	2	1	no	—
214	D₃₃₂	—	2	1	no	—
215	D₃₃₂	—	4	2	no	—
216	D₃₃₂	C₂×M₁₆	1	2	yes	—
217	C₄×D₈₃	—	4	1	no	—
218	C₄×D₈₃	—	4	1	no	—
219	C₄×D₈₃	—	4	1	no	—
220	C₄×D₈₃	—	4	1	no	—
221	C₄×D₈₃	—	4	2	no	—
222	C₄×D₈₃	—	4	2	no	—
223	C₈×C₈₃	—	4	2	no	—
224	C₈×C₈₃	—	4	2	no	—
225	C₈₃⋊C₈	—	2	1	no	—
226	C₈₃⋊C₈	—	2	1	no	—
227	C₈₃⋊C₈	—	2	1	no	—
228	C₈₃⋊C₈	—	2	2	no	—
229	C₈₃⋊C₈	—	4	2	no	—
230	C₈₃⋊C₈	C₂×M₁₆	1	2	yes	complemented
231	C₈₃⋊C₈	C₂×SD₁₆	1	4	yes	complemented
232	C₂²×C₈₃	—	2	1	no	—
233	C₂²×C₈₃	—	2	1	no	—
234	C₂²×C₈₃	C₄×SD₁₆	1	2	yes	—
235	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
236	C₂×D₈₃	—	2	2	no	—
237	C₂×D₈₃	—	4	1	no	—
238	C₂×D₈₃	—	4	1	no	—
239	C₂×D₈₃	—	4	1	no	—
240	C₂×D₈₃	—	4	1	no	—
241	C₂×D₈₃	—	4	1	no	—
242	C₂×D₈₃	—	8	4	no	—
243	C₄×C₈₃	—	2	1	no	—
244	C₄×C₈₃	—	2	2	no	—
245	C₄×C₈₃	—	2	1	no	—
246	C₄×C₈₃	—	2	1	no	—
247	C₄×C₈₃	—	4	1	no	—
248	C₄×C₈₃	—	4	1	no	—
249	C₄×C₈₃	—	4	1	no	—
250	C₄×C₈₃	C₄×SD₁₆	1	2	yes	—
251	C₄×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
252	C₄×C₈₃	#568	1	1	yes	characteristic
253	C₄×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
254	C₄×C₈₃	C₂×#135	1	1	yes	characteristic
255	Dic₈₃	—	4	1	no	—
256	Dic₈₃	—	4	1	no	—
257	M₁₆×SD₁₆	—	83	1	no	maximal, Sylow
258	C₂×C₈₃	—	2	2	no	—
259	C₂×C₈₃	D₄×M₁₆	1	1	yes	characteristic
260	C₂×C₈₃	C₂×C₄×SD₁₆	1	1	yes	characteristic
261	C₂×C₈₃	#2957	1	1	yes	characteristic
262	D₈₃	—	4	2	no	—
263	D₈₃	—	8	1	no	—
264	Q₈×M₁₆	—	83	1	no	—
265	C₂×C₄×SD₁₆	—	83	1	no	—
266	D₄×M₁₆	—	83	1	no	—
267	#1590	—	83	2	no	—
268	#1583	—	83	2	no	—
269	#1582	—	83	2	no	—
270	C₈×SD₁₆	—	83	2	no	—
271	#1566	—	83	1	no	—
272	#1487	D₈₃	1	1	yes	characteristic, complemented
273	#1486	—	83	1	no	—
274	C₈×M₁₆	—	83	1	no	—
275	C₈₃	M₁₆×SD₁₆	1	1	yes	characteristic, complemented, Sylow
276	C₂²×SD₁₆	—	83	1	no	—
277	C₂²×M₁₆	—	166	1	no	—
278	C₂×C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
279	C₂×C₄×D₄	—	83	1	no	—
280	#446	—	83	2	no	—
281	#446	—	166	2	no	—
282	C₈×Q₈	—	83	2	no	—
283	C₄×SD₁₆	—	83	1	no	—
284	C₄×SD₁₆	—	83	1	no	—
285	C₄×SD₁₆	—	83	1	no	—
286	C₄×SD₁₆	—	83	1	no	—
287	C₄×SD₁₆	—	83	1	no	—
288	C₄×SD₁₆	—	83	1	no	—
289	C₄×SD₁₆	—	83	1	no	—
290	C₄×SD₁₆	—	83	1	no	—
291	#436	—	83	2	no	—
292	#435	—	166	2	no	—
293	C₈×D₄	—	83	2	no	—
294	#431	C₂×D₈₃	1	1	yes	characteristic
295	C₂×#106	—	83	1	no	—
296	#423	—	83	1	no	—
297	#423	—	166	1	no	—
298	#423	C₂×D₈₃	1	1	yes	characteristic
299	C₂×#103	—	83	1	no	—
300	C₂×#102	—	83	1	no	—
301	#407	—	166	1	no	—
302	C₄×M₁₆	—	83	1	no	—
303	C₄×M₁₆	—	166	1	no	—
304	C₂×C₄×C₈	—	83	1	no	—
305	#334	—	83	2	no	—
306	#334	—	83	2	no	—
307	Q₈⋊C₈	C₂×D₈₃	1	2	yes	complemented
308	Q₈⋊C₈	C₂×D₈₃	1	2	yes	complemented
309	D₄⋊C₈	—	83	2	no	—
310	D₄⋊C₈	—	83	2	no	—
311	#322	—	83	2	no	—
312	C₈²	—	83	2	no	—
313	C₂²×Q₈	C₄×D₈₃	1	1	yes	characteristic
314	C₂²×D₄	—	83	1	no	—
315	C₂³×C₄	—	166	1	no	—
316	C₂×SD₁₆	—	83	1	no	—
317	C₂×SD₁₆	—	83	1	no	—
318	C₂×SD₁₆	—	83	1	no	—
319	C₂×SD₁₆	—	83	1	no	—
320	C₂×SD₁₆	—	166	4	no	—
321	C₂×M₁₆	—	2	1	no	—
322	C₂×M₁₆	—	332	4	no	—
323	C₂×M₁₆	—	83	1	no	—
324	C₂×M₁₆	—	83	1	no	—
325	C₂×M₁₆	—	166	2	no	—
326	C₂×M₁₆	—	166	1	no	—
327	C₂²×C₈	—	83	1	no	—
328	C₂²×C₈	—	83	1	no	—
329	C₂²×C₈	—	166	2	no	—
330	C₄×Q₈	—	2	1	no	—
331	C₄×Q₈	—	2	1	no	—
332	C₄×Q₈	C₄×D₈₃	1	1	yes	characteristic
333	C₄×Q₈	C₄×D₈₃	1	1	yes	characteristic
334	C₄×Q₈	C₂²×D₈₃	1	1	yes	characteristic
335	C₄×Q₈	C₂²×D₈₃	1	1	yes	characteristic
336	C₄×D₄	—	83	1	no	—
337	C₄×D₄	—	83	1	no	—
338	C₄×D₄	—	83	1	no	—
339	C₄×D₄	—	83	1	no	—
340	C₄×D₄	—	166	1	no	—
341	C₄×D₄	—	166	1	no	—
342	C₂×C₄⋊C₄	—	2	1	no	—
343	C₂×C₄⋊C₄	—	83	1	no	—
344	C₂×C₄⋊C₄	C₄×D₈₃	1	1	yes	characteristic
345	C₂×C₂²⋊C₄	—	166	1	no	—
346	C₂×C₄²	—	2	1	no	—
347	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
348	#106	—	83	1	no	—
349	#106	—	83	1	no	—
350	#106	—	83	1	no	—
351	#106	—	83	1	no	—
352	C₄⋊C₈	—	2	1	no	—
353	C₄⋊C₈	—	2	1	no	—
354	C₄⋊C₈	—	83	2	no	—
355	C₄⋊C₈	—	166	2	no	—
356	C₄⋊C₈	—	166	2	no	—
357	C₄⋊C₈	—	166	1	no	—
358	C₄⋊C₈	C₂²×D₈₃	1	2	yes	—
359	#103	—	83	1	no	—
360	#103	—	83	1	no	—
361	#103	—	83	1	no	—
362	#103	—	83	1	no	—
363	#102	—	83	1	no	—
364	#102	—	83	1	no	—
365	#102	—	83	1	no	—
366	#102	—	83	1	no	—
367	C₂²⋊C₈	—	2	1	no	—
368	C₂²⋊C₈	—	166	2	no	—
369	C₂²⋊C₈	—	166	2	no	—
370	#97	—	83	2	no	—
371	#97	—	166	2	no	—
372	C₄×C₈	—	83	1	no	—
373	C₄×C₈	—	83	1	no	—
374	C₄×C₈	—	83	1	no	—
375	C₄×C₈	—	83	2	no	—
376	C₄×C₈	—	83	1	no	—
377	C₄×C₈	—	166	2	no	—
378	C₄×C₈	C₂²×D₈₃	1	2	yes	—
379	C₂⁴	—	166	1	no	—
380	C₂×Q₈	—	2	1	no	—
381	C₂×Q₈	—	2	2	no	—
382	C₂×Q₈	—	4	1	no	—
383	C₂×Q₈	C₂×C₄×D₈₃	1	1	yes	characteristic
384	C₂×Q₈	C₂×C₄×D₈₃	1	1	yes	characteristic
385	C₂×D₄	—	332	1	no	—
386	C₂×D₄	—	83	1	no	—
387	C₂×D₄	—	83	1	no	—
388	C₂×D₄	—	166	2	no	—
389	C₂×D₄	—	166	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	332	2	no	—
393	C₂²×C₄	—	332	2	no	—
394	C₂²×C₄	—	166	1	no	—
395	C₂²×C₄	—	166	1	no	—
396	C₂²×C₄	—	166	1	no	—
397	C₂²×C₄	—	166	1	no	—
398	C₂²×C₄	—	166	1	no	—
399	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
400	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
401	C₂²×C₄	#13050589	1	1	yes	characteristic
402	SD₁₆	—	83	4	no	—
403	SD₁₆	—	166	2	no	—
404	SD₁₆	—	166	2	no	—
405	SD₁₆	—	166	2	no	—
406	M₁₆	—	4	2	no	—
407	M₁₆	—	332	4	no	—
408	M₁₆	—	332	2	no	—
409	M₁₆	—	83	4	no	—
410	M₁₆	—	166	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	332	4	no	—
414	C₂×C₈	—	332	1	no	—
415	C₂×C₈	—	83	1	no	—
416	C₂×C₈	—	83	2	no	—
417	C₂×C₈	—	83	1	no	—
418	C₂×C₈	—	83	1	no	—
419	C₂×C₈	—	83	1	no	—
420	C₂×C₈	—	83	2	no	—
421	C₂×C₈	—	166	2	no	—
422	C₂×C₈	—	166	2	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	83	1	no	—
429	C₄⋊C₄	—	83	1	no	—
430	C₄⋊C₄	—	166	1	no	—
431	C₄⋊C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
432	C₄⋊C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
433	C₂²⋊C₄	—	166	1	no	—
434	C₂²⋊C₄	—	166	1	no	—
435	C₂²⋊C₄	—	166	1	no	—
436	C₂²⋊C₄	—	166	1	no	—
437	C₄²	—	2	1	no	—
438	C₄²	—	2	1	no	—
439	C₄²	—	2	1	no	—
440	C₄²	—	2	1	no	—
441	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
442	C₄²	D₄×D₈₃	1	1	yes	characteristic
443	C₄²	#13050589	1	1	yes	characteristic
444	C₄²	C₂³×D₈₃	1	1	yes	characteristic
445	C₂³	—	332	2	no	—
446	C₂³	—	332	2	no	—
447	C₂³	—	664	1	no	—
448	C₂³	—	166	1	no	—
449	C₂³	—	166	1	no	—
450	C₂³	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
451	Q₈	—	2	1	no	—
452	Q₈	—	2	1	no	—
453	Q₈	—	2	1	no	—
454	Q₈	—	4	2	no	—
455	Q₈	M₁₆×D₈₃	1	2	yes	complemented
456	D₄	—	332	2	no	—
457	D₄	—	83	2	no	—
458	D₄	—	166	1	no	—
459	D₄	—	166	1	no	—
460	D₄	—	166	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	4	1	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	8	1	no	—
470	C₂×C₄	—	332	2	no	—
471	C₂×C₄	—	332	1	no	—
472	C₂×C₄	—	332	1	no	—
473	C₂×C₄	—	332	1	no	—
474	C₂×C₄	—	332	1	no	—
475	C₂×C₄	—	332	2	no	—
476	C₂×C₄	—	332	2	no	—
477	C₂×C₄	—	166	1	no	—
478	C₂×C₄	—	166	1	no	—
479	C₂×C₄	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
480	C₂×C₄	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
481	C₂×C₄	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
482	C₂×C₄	SD₁₆×D₈₃	1	2	yes	—
483	C₂×C₄	#14581815	1	2	yes	—
484	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
485	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
486	C₂×C₄	C₂×#13050589	1	1	yes	characteristic
487	C₈	—	4	2	no	—
488	C₈	—	4	2	no	—
489	C₈	—	332	2	no	—
490	C₈	—	83	2	no	—
491	C₈	—	83	4	no	—
492	C₈	—	166	1	no	—
493	C₈	—	166	1	no	—
494	C₈	—	166	1	no	—
495	C₈	—	166	2	no	—
496	C₂²	—	2	1	no	—
497	C₂²	—	2	1	no	—
498	C₂²	—	332	1	no	—
499	C₂²	—	332	1	no	—
500	C₂²	—	332	1	no	—
501	C₂²	—	332	1	no	—
502	C₂²	—	332	1	no	—
503	C₂²	—	664	4	no	—
504	C₂²	—	166	2	no	—
505	C₂²	#19127786	1	2	yes	—
506	C₂²	C₂×(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
507	C₄	—	2	1	no	—
508	C₄	—	2	1	no	—
509	C₄	—	2	1	no	—
510	C₄	—	2	2	no	—
511	C₄	—	4	1	no	—
512	C₄	—	4	1	no	—
513	C₄	—	4	1	no	—
514	C₄	—	332	1	no	—
515	C₄	—	332	1	no	—
516	C₄	#19127786	1	2	yes	—
517	C₄	C₂×M₁₆×D₈₃	1	1	yes	characteristic
518	C₄	#19128467	1	1	yes	characteristic
519	C₄	C₂×SD₁₆×D₈₃	1	1	yes	characteristic
520	C₄	C₂×#14581815	1	1	yes	characteristic
521	C₂	—	2	2	no	—
522	C₂	—	332	2	no	—
523	C₂	—	664	1	no	—
524	C₂	#24916195	1	1	yes	characteristic
525	C₂	C₂×#19127786	1	1	yes	characteristic
526	C₂	#24916376	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₄²	C₂³×D₈₃
Fitting	C₈₃×#1487	C₂
socle	C₂²×C₈₃	C₂×C₄×D₄
prime radical	C₂²×D₃₃₂	C₂×C₄
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	#1487	D₈₃
hypocenter	C₈₃	M₁₆×SD₁₆
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1487

Complements

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

N	H ≅ G/N	type
1. Q₈×#13050574	C₂	semidirect
4. #24906350	C₂	semidirect
5. #24906303	C₂	semidirect
6. #24906302	C₂	semidirect
7. #24906260	C₂	semidirect
8. #24906218	C₂	semidirect
10. #24905744	C₂	semidirect
11. C₈₃×#1487	C₂	semidirect
16. #19127832	C₂²	semidirect
17. Q₈×C₈₃⋊C₈	C₂²	semidirect
38. #19127451	C₂²	semidirect
39. #19127451	C₂²	semidirect
42. #19127423	C₂²	semidirect
43. C₈×C₈₃⋊C₈	C₂²	semidirect
47. Q₈⋊C₈×C₈₃	C₂²	semidirect
48. Q₈⋊C₈×C₈₃	C₂²	semidirect
131. #13050577	M₁₆	semidirect
134. #13050574	SD₁₆	semidirect
191. Q₈×C₈₃	C₂×M₁₆	semidirect
230. C₈₃⋊C₈	C₂×M₁₆	semidirect
231. C₈₃⋊C₈	C₂×SD₁₆	semidirect
272. #1487	D₈₃	semidirect
275. C₈₃	M₁₆×SD₁₆	semidirect
307. Q₈⋊C₈	C₂×D₈₃	semidirect
308. Q₈⋊C₈	C₂×D₈₃	semidirect
455. Q₈	M₁₆×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#13050574	2
C₂×C₄×#13050577	2
#24916198	2
#24906350	2
#24906303	2
#24906302	2
#24906260	2
#24906218	2
#24905889	2
#24905744	2
C₈₃×#1487	2
M₁₆×SD₁₆	83