Subgroups — #147257

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2159	C₂	1	1	yes	maximal, characteristic
2	#20451	C₂	1	1	yes	maximal, characteristic
3	C₂×#1386	C₂	1	1	yes	maximal, characteristic
4	C₂×#1386	C₂	1	1	yes	maximal, characteristic
5	#12557	C₂	1	1	yes	maximal, characteristic
6	C₂×#1377	C₂	1	1	yes	maximal, characteristic
7	#10863	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×#427	C₂²	1	1	yes	characteristic
11	C₂×#427	C₂²	1	1	yes	characteristic
12	C₂²×#107	C₂²	1	1	yes	characteristic
13	C₂²×#106	C₂²	1	1	yes	characteristic
14	C₂²×#97	C₂²	1	1	yes	characteristic
15	C₄×#117	—	2	1	no	—
16	#2159	—	2	2	no	—
17	#2159	C₄	1	4	yes	complemented
18	C₂×#367	—	2	1	no	—
19	C₂×#366	—	2	1	no	—
20	#1776	—	2	1	no	—
21	#1776	—	2	1	no	—
22	#1776	—	2	2	no	—
23	#1735	—	2	1	no	—
24	C₂×#340	—	2	1	no	—
25	C₄⋊#95	—	2	1	no	—
26	#1386	—	2	2	no	—
27	#1386	C₄	1	4	yes	complemented
28	#1380	—	2	2	no	—
29	#1377	C₄	1	4	yes	complemented
30	C₂³×C₈	D₄	1	1	yes	characteristic
31	C₂³×C₈	C₂³	1	1	yes	characteristic
32	C₂×#117	—	2	1	no	—
33	C₂×#117	—	2	1	no	—
34	C₂²×C₄⋊C₄	—	2	1	no	—
35	C₂²×C₄⋊C₄	—	2	1	no	—
36	C₂²×C₄²	D₄	1	1	yes	characteristic
37	C₂×M₃₂	—	2	1	no	—
38	C₂×M₃₂	—	2	1	no	—
39	C₂×M₃₂	—	2	1	no	—
40	C₂×M₃₂	—	2	1	no	—
41	C₂×M₃₂	—	2	1	no	—
42	C₂×M₃₂	—	2	1	no	—
43	C₂×M₃₂	—	2	1	no	—
44	C₂×M₃₂	—	2	1	no	—
45	C₂×M₃₂	C₂×C₄	1	2	yes	—
46	C₂×M₃₂	C₂×C₄	1	2	yes	—
47	C₂×M₃₂	C₂×C₄	1	2	yes	—
48	C₂×M₃₂	C₂×C₄	1	2	yes	—
49	C₂²×C₁₆	—	2	1	no	—
50	C₂²×C₁₆	—	2	1	no	—
51	#427	—	2	1	no	—
52	#427	—	2	1	no	—
53	#427	—	2	1	no	—
54	#427	—	2	1	no	—
55	#427	C₂×C₄	1	2	yes	—
56	#427	C₂×C₄	1	2	yes	—
57	#427	D₄	1	2	yes	—
58	#427	Q₈	1	2	yes	—
59	C₂×#107	—	2	1	no	—
60	C₂×#107	—	2	1	no	—
61	C₂×#107	—	2	1	no	—
62	C₂×#107	—	2	1	no	—
63	C₂×#107	—	2	1	no	—
64	C₂×#107	C₂×C₄	1	2	yes	—
65	C₂×#107	D₄	1	1	yes	characteristic
66	C₂×#107	D₄	1	1	yes	characteristic
67	C₂×#107	D₄	1	2	yes	—
68	C₂×#106	—	2	1	no	—
69	C₂×#106	—	2	1	no	—
70	C₂×#106	—	2	1	no	—
71	C₂×#106	—	2	1	no	—
72	C₂×#106	—	2	1	no	—
73	C₂×#106	—	2	1	no	—
74	C₂×#106	—	2	1	no	—
75	C₂×#106	C₂×C₄	1	2	yes	—
76	C₂×#97	—	2	1	no	—
77	C₂×#97	—	2	1	no	—
78	C₂×#97	—	2	1	no	—
79	C₂×#97	—	2	1	no	—
80	C₂×#97	C₂×C₄	1	2	yes	—
81	C₂×#97	Q₈	1	2	yes	—
82	C₄×C₄⋊C₄	—	2	2	no	—
83	C₄×C₄⋊C₄	—	2	2	no	—
84	C₄×C₂²⋊C₄	—	4	2	no	—
85	#376	—	4	1	no	—
86	C₂×#95	—	4	2	no	—
87	C₄³	—	4	1	no	—
88	#367	—	2	4	no	—
89	#366	—	2	4	no	—
90	#340	—	2	2	no	—
91	#340	—	2	2	no	—
92	#340	—	4	2	no	—
93	C₂³×C₄	—	4	1	no	—
94	C₂³×C₄	—	4	1	no	—
95	C₂³×C₄	D₈	1	2	yes	—
96	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₈	—	2	1	no	—
98	C₂²×C₈	—	2	1	no	—
99	C₂²×C₈	—	2	1	no	—
100	C₂²×C₈	—	2	1	no	—
101	C₂²×C₈	—	2	1	no	—
102	C₂²×C₈	—	2	1	no	—
103	C₂²×C₈	C₄⋊C₄	1	2	yes	—
104	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
111	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
112	C₂²×C₈	#41	1	1	yes	characteristic
113	C₂²×C₈	#41	1	1	yes	characteristic
114	C₂²×C₈	#41	1	1	yes	characteristic
115	C₂²×C₈	#41	1	1	yes	characteristic
116	C₂²×C₈	#41	1	1	yes	characteristic
117	C₂²×C₈	#41	1	1	yes	characteristic
118	#117	—	2	2	no	—
119	#117	—	2	2	no	—
120	#117	—	2	2	no	—
121	#117	—	2	2	no	—
122	#117	—	4	1	no	—
123	#117	—	4	1	no	—
124	#117	—	4	1	no	—
125	#117	—	4	1	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄⋊C₄	—	4	1	no	—
133	C₂×C₄⋊C₄	—	4	1	no	—
134	C₂×C₄⋊C₄	—	4	1	no	—
135	C₂×C₄⋊C₄	—	4	1	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄⋊C₄	—	4	1	no	—
138	C₂×C₄⋊C₄	—	4	1	no	—
139	C₂×C₄⋊C₄	—	4	1	no	—
140	C₂×C₄⋊C₄	—	4	1	no	—
141	C₂×C₂²⋊C₄	—	4	1	no	—
142	C₂×C₂²⋊C₄	—	4	1	no	—
143	C₂×C₄²	—	2	1	no	—
144	C₂×C₄²	—	2	1	no	—
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	—	4	1	no	—
148	C₂×C₄²	—	4	1	no	—
149	C₂×C₄²	—	4	2	no	—
150	C₂×C₄²	—	4	2	no	—
151	C₂×C₄²	C₄⋊C₄	1	2	yes	—
152	C₂×C₄²	Q₁₆	1	2	yes	—
153	M₃₂	—	2	2	no	—
154	M₃₂	—	2	2	no	—
155	M₃₂	—	2	2	no	—
156	M₃₂	—	2	2	no	—
157	M₃₂	—	2	2	no	—
158	M₃₂	—	4	1	no	—
159	M₃₂	—	4	1	no	—
160	M₃₂	C₄²	1	4	yes	complemented
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	—	2	2	no	—
163	C₂×C₁₆	—	2	2	no	—
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	—	4	1	no	—
166	C₂×C₁₆	—	4	1	no	—
167	#107	—	2	2	no	—
168	#107	—	2	1	no	—
169	#107	—	2	1	no	—
170	#107	—	2	2	no	—
171	#107	—	2	2	no	—
172	#107	—	4	1	no	—
173	#107	—	4	1	no	—
174	#107	C₂²⋊C₄	1	2	yes	—
175	#107	C₂²⋊C₄	1	2	yes	—
176	#107	C₄⋊C₄	1	4	yes	—
177	#106	—	2	2	no	—
178	#106	—	2	2	no	—
179	#106	—	2	2	no	—
180	#106	—	2	2	no	—
181	#106	—	2	2	no	—
182	#106	—	2	2	no	—
183	#106	—	4	1	no	—
184	#106	—	4	1	no	—
185	#97	—	2	2	no	—
186	#97	—	2	2	no	—
187	#97	—	2	2	no	—
188	#97	—	2	2	no	—
189	#95	—	4	2	no	—
190	#95	—	4	2	no	—
191	#95	—	8	4	no	—
192	C₂⁴	C₂×D₈	1	1	yes	characteristic
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	8	2	no	—
203	C₂²×C₄	—	8	2	no	—
204	C₂²×C₄	—	8	2	no	—
205	C₂²×C₄	—	8	2	no	—
206	C₂²×C₄	—	8	1	no	—
207	C₂²×C₄	—	8	1	no	—
208	C₂²×C₄	—	8	1	no	—
209	C₂²×C₄	—	8	1	no	—
210	C₂²×C₄	#107	1	4	yes	—
211	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
213	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
214	C₂²×C₄	#120	1	1	yes	characteristic
215	C₂²×C₄	#121	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
217	C₂²×C₄	#123	1	1	yes	characteristic
218	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
219	C₂²×C₄	#135	1	1	yes	characteristic
220	C₂²×C₄	#135	1	1	yes	characteristic
221	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₂²×C₄	#137	1	1	yes	characteristic
224	C₂²×C₄	#137	1	1	yes	characteristic
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
233	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
235	C₂×C₈	#117	1	1	yes	characteristic
236	C₂×C₈	#117	1	1	yes	characteristic
237	C₂×C₈	#117	1	1	yes	characteristic
238	C₂×C₈	C₄×D₄	1	1	yes	characteristic
239	C₂×C₈	C₄×D₄	1	1	yes	characteristic
240	C₂×C₈	C₄×D₄	1	1	yes	characteristic
241	C₂×C₈	C₄×D₄	1	1	yes	characteristic
242	C₂×C₈	C₄×D₄	1	1	yes	characteristic
243	C₂×C₈	C₄×D₄	1	1	yes	characteristic
244	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
245	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
246	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
247	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
248	C₂×C₈	#121	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
251	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
252	C₂×C₈	#123	1	1	yes	characteristic
253	C₂×C₈	#123	1	1	yes	characteristic
254	C₂×C₈	#124	1	1	yes	characteristic
255	C₂×C₈	#125	1	1	yes	characteristic
256	C₂×C₈	#125	1	1	yes	characteristic
257	C₂×C₈	#126	1	1	yes	characteristic
258	C₂×C₈	#126	1	1	yes	characteristic
259	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	8	1	no	—
271	C₄⋊C₄	—	8	1	no	—
272	C₄⋊C₄	—	8	1	no	—
273	C₄⋊C₄	—	8	1	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	4	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	—	4	2	no	—
285	C₄²	—	4	2	no	—
286	C₄²	—	8	4	no	—
287	C₄²	—	8	4	no	—
288	C₁₆	—	2	4	no	—
289	C₁₆	—	4	2	no	—
290	C₂³	—	2	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	C₂×#107	1	1	yes	characteristic
293	C₂³	C₄×D₈	1	1	yes	characteristic
294	C₂³	C₄×D₈	1	1	yes	characteristic
295	C₂³	#447	1	1	yes	characteristic
296	C₂³	#459	1	1	yes	characteristic
297	C₂³	#474	1	1	yes	characteristic
298	C₂³	#480	1	1	yes	characteristic
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	8	2	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	—	8	2	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	—	8	4	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	—	16	2	no	—
316	C₂×C₄	—	16	1	no	—
317	C₂×C₄	—	16	1	no	—
318	C₂×C₄	—	16	1	no	—
319	C₂×C₄	—	16	1	no	—
320	C₂×C₄	—	16	2	no	—
321	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
322	C₂×C₄	#385	1	1	yes	characteristic
323	C₂×C₄	#385	1	1	yes	characteristic
324	C₂×C₄	#386	1	1	yes	characteristic
325	C₂×C₄	#387	1	1	yes	characteristic
326	C₂×C₄	#388	1	1	yes	characteristic
327	C₂×C₄	#389	1	1	yes	characteristic
328	C₂×C₄	#427	1	1	yes	characteristic
329	C₂×C₄	#428	1	1	yes	characteristic
330	C₂×C₄	#428	1	1	yes	characteristic
331	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
332	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
333	C₂×C₄	#440	1	1	yes	characteristic
334	C₂×C₄	#440	1	1	yes	characteristic
335	C₂×C₄	#441	1	1	yes	characteristic
336	C₂×C₄	#442	1	1	yes	characteristic
337	C₂×C₄	#449	1	1	yes	characteristic
338	C₂×C₄	#451	1	1	yes	characteristic
339	C₂×C₄	#452	1	1	yes	characteristic
340	C₂×C₄	#462	1	1	yes	characteristic
341	C₂×C₄	#463	1	1	yes	characteristic
342	C₂×C₄	#464	1	1	yes	characteristic
343	C₂×C₄	#477	1	1	yes	characteristic
344	C₂×C₄	#478	1	1	yes	characteristic
345	C₂×C₄	#479	1	1	yes	characteristic
346	C₂×C₄	#482	1	1	yes	characteristic
347	C₂×C₄	#484	1	1	yes	characteristic
348	C₂×C₄	#485	1	1	yes	characteristic
349	C₈	—	4	2	no	—
350	C₈	C₄×C₂²⋊C₄	1	1	yes	characteristic
351	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
352	C₈	#387	1	1	yes	characteristic
353	C₈	#387	1	1	yes	characteristic
354	C₈	#387	1	1	yes	characteristic
355	C₈	#388	1	1	yes	characteristic
356	C₈	#389	1	1	yes	characteristic
357	C₈	#391	1	1	yes	characteristic
358	C₂²	—	2	2	no	—
359	C₂²	—	2	2	no	—
360	C₂²	—	2	2	no	—
361	C₂²	—	2	2	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	C₄×#107	1	1	yes	characteristic
367	C₂²	#1852	1	1	yes	characteristic
368	C₂²	#1862	1	1	yes	characteristic
369	C₂²	#1879	1	1	yes	characteristic
370	C₂²	#1918	1	1	yes	characteristic
371	C₂²	#1925	1	1	yes	characteristic
372	C₂²	#1947	1	1	yes	characteristic
373	C₂²	#2190	1	2	yes	—
374	C₂²	#2191	1	2	yes	—
375	C₄	—	4	4	no	—
376	C₄	—	8	4	no	—
377	C₄	—	8	2	no	—
378	C₄	—	8	2	no	—
379	C₄	—	16	2	no	—
380	C₄	—	16	2	no	—
381	C₄	#1436	1	1	yes	characteristic
382	C₄	#1776	1	1	yes	characteristic
383	C₄	#1854	1	1	yes	characteristic
384	C₄	#1863	1	1	yes	characteristic
385	C₄	#1878	1	1	yes	characteristic
386	C₄	#1921	1	1	yes	characteristic
387	C₄	#1928	1	1	yes	characteristic
388	C₄	#1949	1	1	yes	characteristic
389	C₂	—	2	1	no	—
390	C₂	—	2	1	no	—
391	C₂	—	2	1	no	—
392	C₂	—	2	1	no	—
393	C₂	#10907	1	1	yes	characteristic
394	C₂	#12923	1	2	yes	—
395	C₂	#12943	1	2	yes	—
396	C₂	#13025	1	1	yes	characteristic
397	C₂	#13026	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
17. #2159	C₄	semidirect
27. #1386	C₄	semidirect
29. #1377	C₄	semidirect
160. M₃₂	C₄²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2159	2
#20451	2
C₂×#1386	2
#12557	2
C₂×#1377	2
#10863	2