Subgroups — #72231968

Order 1024.

631 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#512795	C₂	1	1	yes	maximal, characteristic
2	#512789	C₂	1	1	yes	maximal, characteristic
3	#479944	C₂	1	1	yes	maximal, characteristic
4	#149514	C₂	1	2	yes	maximal, complemented
5	#149513	C₂	1	1	yes	maximal, characteristic, complemented
6	#149509	C₂	1	1	yes	maximal, characteristic, complemented
7	#149509	C₂	1	1	yes	maximal, characteristic, complemented
8	#149509	C₂	1	2	yes	maximal, complemented
9	#149508	C₂	1	2	yes	maximal, complemented
10	#149502	C₂	1	1	yes	maximal, characteristic, complemented
11	#149501	C₂	1	2	yes	maximal, complemented
12	#33479	C₄	1	1	yes	characteristic
13	C₁₆×#41	C₂²	1	1	yes	characteristic
14	#17328	C₄	1	1	yes	characteristic
15	#13660	C₄	1	2	yes	—
16	Q₈×C₃₂	C₄	1	2	yes	—
17	#13646	C₄	1	4	yes	—
18	#13646	C₄	1	2	yes	—
19	#13645	C₄	1	4	yes	—
20	D₄×C₃₂	C₄	1	2	yes	—
21	#13640	C₂²	1	2	yes	—
22	#13640	C₂²	1	1	yes	characteristic
23	#13640	C₂²	1	1	yes	characteristic
24	#13640	C₂²	1	2	yes	—
25	C₄×M₆₄	C₂²	1	2	yes	—
26	C₄×M₆₄	C₂²	1	1	yes	characteristic
27	C₄×M₆₄	C₂²	1	1	yes	characteristic
28	C₂×#1397	C₂²	1	2	yes	—
29	#11412	C₄	1	2	yes	—
30	#11410	C₄	1	1	yes	characteristic
31	#11409	C₄	1	2	yes	—
32	#11408	C₂²	1	2	yes	—
33	#11408	C₂²	1	1	yes	characteristic
34	C₁₆×M₁₆	C₄	1	1	yes	characteristic
35	#11399	C₂²	1	2	yes	—
36	C₂×#1311	C₂²	1	1	yes	characteristic
37	#7332	C₂²	1	1	yes	characteristic
38	#7332	C₂²	1	2	yes	—
39	#7332	C₂²	1	2	yes	—
40	#7332	C₂²	1	1	yes	characteristic
41	#7332	C₂²	1	2	yes	—
42	#7332	C₂²	1	1	yes	characteristic
43	#7332	C₂²	1	1	yes	characteristic
44	#7330	C₂²	1	2	yes	—
45	#7330	C₂²	1	2	yes	—
46	#7330	C₂²	1	2	yes	—
47	#2965	C₈	1	1	yes	characteristic
48	C₈×#41	C₂×C₄	1	1	yes	characteristic
49	#2922	C₈	1	1	yes	characteristic
50	C₂×M₆₄	C₂×C₄	1	2	yes	—
51	C₂×M₆₄	C₂×C₄	1	4	yes	—
52	C₂×M₆₄	C₂×C₄	1	2	yes	—
53	C₂²×C₃₂	C₂×C₄	1	4	yes	—
54	#2183	C₈	1	2	yes	—
55	Q₈×C₁₆	C₂×C₄	1	1	yes	characteristic
56	Q₈×C₁₆	C₂×C₄	1	1	yes	characteristic
57	#2169	C₈	1	2	yes	—
58	#2168	C₈	1	4	yes	—
59	D₄×C₁₆	C₂×C₄	1	1	yes	characteristic
60	D₄×C₁₆	C₂×C₄	1	2	yes	—
61	D₄×C₁₆	C₂×C₄	1	1	yes	characteristic
62	D₄×C₁₆	C₂×C₄	1	2	yes	—
63	#2162	C₂×C₄	1	1	yes	characteristic
64	#2162	C₂×C₄	1	2	yes	—
65	#2162	C₂×C₄	1	1	yes	characteristic
66	#2162	C₂×C₄	1	2	yes	—
67	C₄×M₃₂	C₂×C₄	1	2	yes	—
68	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₄×C₁₆	C₂³	1	2	yes	—
71	#1454	C₈	1	2	yes	—
72	#1453	C₈	1	2	yes	—
73	#1452	C₈	1	1	yes	characteristic
74	#1451	C₈	1	1	yes	characteristic
75	#1450	C₂×C₄	1	2	yes	—
76	C₈×M₁₆	C₂×C₄	1	1	yes	characteristic
77	C₈×M₁₆	C₂×C₄	1	2	yes	—
78	C₂×C₈²	C₂×C₄	1	1	yes	characteristic
79	C₄⋊C₃₂	C₂×C₄	1	2	yes	—
80	C₄⋊C₃₂	C₂×C₄	1	4	yes	—
81	C₄⋊C₃₂	C₂×C₄	1	4	yes	—
82	C₄⋊C₃₂	C₂×C₄	1	2	yes	—
83	C₂²⋊C₃₂	C₂×C₄	1	4	yes	—
84	C₂²⋊C₃₂	C₂×C₄	1	2	yes	—
85	C₂²⋊C₃₂	C₂×C₄	1	4	yes	—
86	C₂²⋊C₃₂	C₂×C₄	1	2	yes	—
87	#1397	C₂×C₄	1	4	yes	—
88	#1397	C₂³	1	2	yes	—
89	#1397	C₂³	1	1	yes	characteristic
90	#1397	C₂³	1	2	yes	—
91	#1397	C₂³	1	1	yes	characteristic
92	#1397	C₂³	1	1	yes	characteristic
93	#1397	C₂³	1	1	yes	characteristic
94	C₄×C₃₂	C₂×C₄	1	2	yes	—
95	C₄×C₃₂	C₂×C₄	1	4	yes	—
96	C₄×C₃₂	C₂×C₄	1	2	yes	—
97	#1312	C₂×C₄	1	2	yes	—
98	#1312	C₂×C₄	1	2	yes	—
99	#1312	C₂×C₄	1	2	yes	—
100	#1311	C₂×C₄	1	1	yes	characteristic
101	#1311	C₂×C₄	1	1	yes	characteristic
102	#1311	C₂³	1	1	yes	characteristic
103	#1311	C₂³	1	1	yes	characteristic
104	#1311	C₂³	1	2	yes	—
105	C₈×C₁₆	C₂×C₄	1	2	yes	—
106	C₈×C₁₆	C₂×C₄	1	2	yes	—
107	C₄×#41	C₂×C₈	1	1	yes	characteristic
108	C₂×M₃₂	C₂×C₈	1	4	yes	—
109	C₂²×C₁₆	C₂×C₈	1	2	yes	—
110	C₂²×C₁₆	C₂²×C₄	1	2	yes	—
111	C₂²×C₁₆	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₁₆	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₁₆	C₂²×C₄	1	2	yes	—
114	#446	—	2	1	no	—
115	C₈×Q₈	C₄²	1	1	yes	characteristic
116	C₈×Q₈	C₂×C₈	1	1	yes	characteristic
117	#436	—	2	1	no	—
118	#435	—	2	2	no	—
119	C₈×D₄	C₄²	1	1	yes	characteristic
120	C₈×D₄	C₄²	1	2	yes	—
121	C₈×D₄	C₂×C₈	1	2	yes	—
122	C₈×D₄	C₂×C₈	1	1	yes	characteristic
123	#433	—	2	2	no	—
124	#431	C₂×C₈	1	2	yes	—
125	#431	C₂×C₈	1	2	yes	—
126	#431	C₂×C₈	1	1	yes	characteristic
127	#431	C₂×C₈	1	2	yes	—
128	#431	C₂×C₈	1	1	yes	characteristic
129	#431	C₂×C₈	1	2	yes	—
130	#424	—	2	1	no	—
131	#408	—	2	1	no	—
132	C₄×M₁₆	C₂×C₈	1	1	yes	characteristic
133	C₄×M₁₆	C₂×C₈	1	2	yes	—
134	C₂×C₄×C₈	C₂×C₈	1	1	yes	characteristic
135	C₂×C₄×C₈	C₂×C₈	1	1	yes	characteristic
136	C₂×C₄×C₈	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂²×C₄	1	2	yes	—
138	M₆₄	—	4	2	no	—
139	M₆₄	—	4	4	no	—
140	M₆₄	—	4	2	no	—
141	C₂×C₃₂	—	4	4	no	—
142	C₂×C₃₂	C₂²×C₄	1	2	yes	—
143	C₂×C₃₂	C₂²×C₄	1	2	yes	—
144	C₂×C₃₂	C₂²×C₄	1	2	yes	—
145	C₂×C₃₂	C₂²×C₄	1	4	yes	—
146	C₂×C₃₂	C₂²×C₄	1	2	yes	—
147	C₂×C₃₂	C₂²×C₄	1	4	yes	—
148	C₄⋊C₁₆	C₄²	1	1	yes	characteristic
149	C₄⋊C₁₆	C₄²	1	1	yes	characteristic
150	C₄⋊C₁₆	C₄²	1	2	yes	—
151	C₄⋊C₁₆	C₄²	1	2	yes	—
152	C₄⋊C₁₆	C₂×C₈	1	4	yes	—
153	C₄⋊C₁₆	C₂×C₈	1	2	yes	—
154	C₂²⋊C₁₆	C₄²	1	2	yes	—
155	C₂²⋊C₁₆	C₄²	1	2	yes	—
156	C₂²⋊C₁₆	C₄²	1	2	yes	—
157	C₂²⋊C₁₆	C₂×C₈	1	2	yes	—
158	C₂²⋊C₁₆	C₂×C₈	1	4	yes	—
159	#346	C₂×C₈	1	4	yes	—
160	C₄×C₁₆	C₂×C₈	1	2	yes	—
161	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
162	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
163	C₄×C₁₆	C₂²×C₄	1	2	yes	—
164	C₄×C₁₆	C₂²×C₄	1	2	yes	—
165	C₄×C₁₆	C₂²×C₄	1	2	yes	—
166	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
167	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
168	C₄×C₁₆	C₂²×C₄	1	2	yes	—
169	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
170	C₄×C₁₆	C₂⁴	1	1	yes	characteristic
171	#322	C₂×C₈	1	2	yes	—
172	#322	C₂×C₈	1	2	yes	—
173	#322	C₂×C₈	1	2	yes	—
174	#322	C₂×C₈	1	2	yes	—
175	#322	C₂×C₈	1	2	yes	—
176	C₈²	C₂×C₈	1	1	yes	characteristic
177	C₈²	C₂×C₈	1	1	yes	characteristic
178	C₈²	C₂²×C₄	1	1	yes	characteristic
179	C₈²	C₂²×C₄	1	2	yes	—
180	C₈²	C₂²×C₄	1	1	yes	characteristic
181	C₂×#41	M₃₂	1	1	yes	characteristic
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂²×C₈	—	2	1	no	—
185	C₂²×C₈	—	2	1	no	—
186	C₂²×C₈	C₂×C₄²	1	1	yes	characteristic
187	C₂²×C₈	C₂×C₄²	1	2	yes	—
188	C₂²×C₈	C₂²×C₈	1	1	yes	characteristic
189	C₂²×C₈	C₂²×C₈	1	2	yes	—
190	C₄×Q₈	C₄×C₈	1	1	yes	characteristic
191	C₄×Q₈	M₃₂	1	1	yes	characteristic
192	C₄×D₄	C₄×C₈	1	1	yes	characteristic
193	C₄×D₄	C₄×C₈	1	2	yes	—
194	C₄×D₄	M₃₂	1	2	yes	—
195	C₄×D₄	M₃₂	1	1	yes	characteristic
196	#117	M₃₂	1	1	yes	characteristic
197	#117	M₃₂	1	2	yes	—
198	C₂×C₄²	C₂²×C₈	1	1	yes	characteristic
199	C₂×C₄²	C₂²×C₈	1	2	yes	—
200	M₃₂	—	2	8	no	—
201	C₂×C₁₆	—	2	4	no	—
202	C₂×C₁₆	—	2	2	no	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	2	4	no	—
205	C₂×C₁₆	—	2	4	no	—
206	C₂×C₁₆	C₂×C₄²	1	2	yes	—
207	C₂×C₁₆	C₂×C₄²	1	1	yes	characteristic
208	C₂×C₁₆	C₂×C₄²	1	1	yes	characteristic
209	C₂×C₁₆	C₂×C₄²	1	2	yes	—
210	C₂×C₁₆	C₂²×C₈	1	2	yes	—
211	C₂×C₁₆	C₂²×C₈	1	4	yes	—
212	C₂×C₁₆	C₂²×C₈	1	2	yes	—
213	C₂×C₁₆	C₂³×C₄	1	1	yes	characteristic
214	C₂×C₁₆	C₂³×C₄	1	1	yes	characteristic
215	C₄⋊C₈	—	2	2	no	—
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	2	2	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	C₄×C₈	1	1	yes	characteristic
220	C₄⋊C₈	C₄×C₈	1	2	yes	—
221	C₄⋊C₈	C₄×C₈	1	1	yes	characteristic
222	C₄⋊C₈	C₄×C₈	1	2	yes	—
223	C₂²⋊C₈	—	2	2	no	—
224	C₂²⋊C₈	—	2	2	no	—
225	C₂²⋊C₈	—	2	1	no	—
226	C₂²⋊C₈	—	2	2	no	—
227	C₂²⋊C₈	C₄×C₈	1	2	yes	—
228	C₂²⋊C₈	C₄×C₈	1	2	yes	—
229	C₂²⋊C₈	C₄×C₈	1	2	yes	—
230	#97	—	2	1	no	—
231	#97	—	2	2	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	C₂×C₄²	1	1	yes	characteristic
235	C₄×C₈	C₂×C₄²	1	2	yes	—
236	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
237	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
238	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
239	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
240	C₄×C₈	C₂²×C₈	1	2	yes	—
241	C₄×C₈	C₂²×C₈	1	2	yes	—
242	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
243	C₄×C₈	C₂²×C₈	1	2	yes	—
244	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
245	C₄×C₈	C₂²×C₈	1	2	yes	—
246	C₄×C₈	C₂³×C₄	1	1	yes	characteristic
247	C₃₂	—	2	2	no	—
248	C₃₂	—	2	4	no	—
249	C₃₂	—	2	2	no	—
250	C₃₂	—	2	4	no	—
251	C₃₂	—	2	2	no	—
252	C₃₂	—	2	2	no	—
253	#41	—	4	2	no	—
254	C₂×Q₈	#346	1	1	yes	characteristic
255	C₂×D₄	#346	1	1	yes	characteristic
256	C₂×D₄	#346	1	2	yes	—
257	C₂²×C₄	C₂×C₄×C₈	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₄×C₈	1	2	yes	—
259	C₂²×C₄	C₂×M₃₂	1	1	yes	characteristic
260	C₂²×C₄	C₂×M₃₂	1	2	yes	—
261	C₂²×C₄	C₂×M₃₂	1	1	yes	characteristic
262	C₂²×C₄	C₂×M₃₂	1	2	yes	—
263	M₁₆	—	4	2	no	—
264	M₁₆	—	8	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	4	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	4	2	no	—
278	C₂×C₈	—	8	1	no	—
279	C₂×C₈	C₂×C₄×C₈	1	1	yes	characteristic
280	C₂×C₈	C₂×C₄×C₈	1	2	yes	—
281	C₂×C₈	C₂×C₄×C₈	1	1	yes	characteristic
282	C₂×C₈	C₂×C₄×C₈	1	2	yes	—
283	C₂×C₈	C₂²×C₄²	1	1	yes	characteristic
284	C₂×C₈	C₂³×C₈	1	1	yes	characteristic
285	C₄⋊C₄	#346	1	2	yes	—
286	C₄⋊C₄	#346	1	1	yes	characteristic
287	C₄⋊C₄	#346	1	2	yes	—
288	C₄⋊C₄	#346	1	1	yes	characteristic
289	C₂²⋊C₄	#346	1	2	yes	—
290	C₂²⋊C₄	#346	1	2	yes	—
291	C₂²⋊C₄	#346	1	2	yes	—
292	C₄²	C₂×C₄×C₈	1	2	yes	—
293	C₄²	C₂×C₄×C₈	1	1	yes	characteristic
294	C₄²	C₂×M₃₂	1	1	yes	characteristic
295	C₄²	C₂×M₃₂	1	1	yes	characteristic
296	C₄²	C₂×M₃₂	1	2	yes	—
297	C₄²	C₂×M₃₂	1	2	yes	—
298	C₄²	C₂³×C₈	1	1	yes	characteristic
299	C₁₆	—	2	2	no	—
300	C₁₆	—	2	1	no	—
301	C₁₆	—	2	1	no	—
302	C₁₆	—	2	2	no	—
303	C₁₆	—	2	2	no	—
304	C₁₆	—	2	4	no	—
305	C₁₆	#504	1	4	yes	—
306	C₁₆	#568	1	2	yes	—
307	C₁₆	#568	1	2	yes	—
308	C₂³	C₂×#346	1	1	yes	characteristic
309	C₂³	C₂×#346	1	2	yes	—
310	Q₈	—	2	2	no	—
311	D₄	—	2	2	no	—
312	D₄	—	4	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	C₂×#346	1	1	yes	characteristic
322	C₂×C₄	C₂×#346	1	2	yes	—
323	C₂×C₄	C₂×#346	1	2	yes	—
324	C₂×C₄	C₂×#346	1	1	yes	characteristic
325	C₂×C₄	C₂×#346	1	2	yes	—
326	C₂×C₄	C₂×#346	1	1	yes	characteristic
327	C₂×C₄	C₂²×C₄×C₈	1	1	yes	characteristic
328	C₂×C₄	C₂²×M₃₂	1	1	yes	characteristic
329	C₂×C₄	C₂²×M₃₂	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	2	2	no	—
332	C₈	—	2	2	no	—
333	C₈	—	2	1	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	1	no	—
337	C₈	—	4	1	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	2	no	—
340	C₈	#2875	1	1	yes	characteristic
341	C₈	#2875	1	1	yes	characteristic
342	C₈	C₂×#504	1	2	yes	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	2	no	—
349	C₂²	C₂²×#346	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	2	no	—
357	C₄	#13551	1	2	yes	—
358	C₄	#17316	1	1	yes	characteristic
359	C₄	C₄×#504	1	1	yes	characteristic
360	C₂	—	4	1	no	—
361	C₂	—	4	2	no	—
362	C₂	#479949	1	1	yes	characteristic
363	C₂	C₂×#13551	1	1	yes	characteristic
364	C₂	#512548	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×C₄²
commutator	C₂×C₄	C₂²×C₄×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁴
socle	C₂²	C₂²×#346
prime radical	C₂×#41	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #149514	C₂	semidirect
5. #149513	C₂	semidirect
6. #149509	C₂	semidirect
7. #149509	C₂	semidirect
8. #149509	C₂	semidirect
9. #149508	C₂	semidirect
10. #149502	C₂	semidirect
11. #149501	C₂	semidirect

Maximal subgroups

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

subgroup	index
#512795	2
#512789	2
#479944	2
#149514	2
#149513	2
#149509	2
#149508	2
#149502	2
#149501	2