Subgroups — #498082

Order 512.

508 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#13264	C₂	1	2	yes	maximal
3	#13105	C₂	1	2	yes	maximal
4	#13104	C₂	1	2	yes	maximal
5	#12894	C₂	1	1	yes	maximal, characteristic
6	#12894	C₂	1	1	yes	maximal, characteristic
7	#12879	C₂	1	1	yes	maximal, characteristic
8	#12663	C₂	1	2	yes	maximal
9	#12653	C₂	1	1	yes	maximal, characteristic
10	#12591	C₂	1	1	yes	maximal, characteristic
11	#12590	C₂	1	1	yes	maximal, characteristic
12	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
13	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
14	C₄²×Q₈	C₄	1	1	yes	characteristic
15	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₂×#346	C₂²	1	2	yes	—
18	C₂×#346	C₂²	1	2	yes	—
19	C₂×C₄×C₁₆	C₂²	1	2	yes	—
20	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
21	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
22	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
23	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
28	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
29	C₄²×C₈	C₂²	1	1	yes	characteristic
30	C₄²×C₈	C₂²	1	1	yes	characteristic
31	C₄²×C₈	C₂²	1	1	yes	characteristic
32	#1374	C₂²	1	1	yes	characteristic
33	#1374	C₂²	1	2	yes	—
34	#1374	C₂²	1	1	yes	characteristic
35	#1374	C₂²	1	2	yes	—
36	#1374	C₂²	1	1	yes	characteristic
37	#1374	C₂²	1	2	yes	—
38	#1374	C₂²	1	2	yes	—
39	#1374	C₂²	1	2	yes	—
40	#1374	C₂²	1	2	yes	—
41	#1374	C₂²	1	2	yes	—
42	#1374	C₂²	1	1	yes	characteristic
43	C₂×C₄×Q₈	C₈	1	1	yes	characteristic
44	C₂×C₄×Q₈	C₈	1	1	yes	characteristic
45	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₁₆	C₂³	1	2	yes	—
47	C₂²×C₁₆	C₂³	1	2	yes	—
48	C₂²×C₁₆	C₂³	1	2	yes	—
49	C₂²×C₁₆	C₂³	1	2	yes	—
50	C₈×Q₈	—	2	4	no	—
51	C₈×Q₈	—	2	2	no	—
52	C₈×Q₈	—	2	2	no	—
53	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
75	C₄×C₄⋊C₄	C₈	1	1	yes	characteristic
76	C₄×C₄⋊C₄	C₈	1	1	yes	characteristic
77	C₄×C₄⋊C₄	C₈	1	1	yes	characteristic
78	C₄×C₄⋊C₄	C₈	1	1	yes	characteristic
79	C₄×C₄⋊C₄	C₈	1	1	yes	characteristic
80	C₄×C₄⋊C₄	C₈	1	1	yes	characteristic
81	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
82	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
83	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
84	C₄³	C₂×C₄	1	1	yes	characteristic
85	C₄³	C₂×C₄	1	1	yes	characteristic
86	C₄³	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₁₆	—	2	4	no	—
88	C₄⋊C₁₆	—	2	4	no	—
89	#346	—	2	4	no	—
90	#346	—	2	4	no	—
91	C₄×C₁₆	—	2	4	no	—
92	#336	C₂×C₄	1	1	yes	characteristic
93	#336	C₂×C₄	1	1	yes	characteristic
94	#336	C₂×C₄	1	1	yes	characteristic
95	#336	C₂×C₄	1	1	yes	characteristic
96	#336	C₂×C₄	1	1	yes	characteristic
97	#336	C₂×C₄	1	1	yes	characteristic
98	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
99	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₈	C₂⁴	1	1	yes	characteristic
107	C₄×Q₈	—	2	4	no	—
108	C₄×Q₈	—	2	4	no	—
109	C₄×Q₈	—	2	2	no	—
110	C₄×Q₈	—	2	2	no	—
111	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
112	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
113	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
116	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
117	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
118	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
120	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
121	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
122	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
123	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
124	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
125	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
126	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
127	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
128	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
129	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
130	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
131	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
132	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₁₆	—	2	2	no	—
140	C₂×C₁₆	—	2	2	no	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	#41	1	4	yes	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	M₁₆	1	4	yes	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	—	2	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	2	1	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	—	2	2	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	—	4	1	no	—
172	C₄×C₈	—	4	1	no	—
173	C₄×C₈	—	4	1	no	—
174	C₄×C₈	#41	1	2	yes	—
175	C₄×C₈	#41	1	2	yes	—
176	#95	C₂×C₈	1	1	yes	characteristic
177	#95	C₂×C₈	1	1	yes	characteristic
178	#95	C₂×C₈	1	1	yes	characteristic
179	#95	C₂×C₈	1	1	yes	characteristic
180	#95	C₂×C₈	1	1	yes	characteristic
181	#95	C₂×C₈	1	1	yes	characteristic
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
186	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
187	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
188	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
189	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
192	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
193	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
194	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
196	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
197	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
198	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
199	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	#117	1	2	yes	—
218	C₂×C₈	C₂×M₁₆	1	2	yes	—
219	C₂×C₈	C₂×M₁₆	1	2	yes	—
220	C₂×C₈	#131	1	2	yes	—
221	C₂×C₈	C₂×#41	1	1	yes	characteristic
222	C₂×C₈	C₂×#41	1	2	yes	—
223	C₂×C₈	C₂×#41	1	1	yes	characteristic
224	C₂×C₈	2₋¹⁺⁴	1	2	yes	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	M₃₂	1	4	yes	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	4	1	no	—
255	C₄²	—	4	1	no	—
256	C₄²	—	4	1	no	—
257	C₄²	M₃₂	1	4	yes	—
258	C₄²	#117	1	2	yes	—
259	C₄²	C₂×M₁₆	1	2	yes	—
260	C₁₆	—	2	4	no	—
261	C₁₆	—	4	2	no	—
262	C₁₆	—	4	2	no	—
263	C₁₆	—	4	2	no	—
264	C₂³	C₂³×C₈	1	1	yes	characteristic
265	Q₈	—	2	8	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	#431	1	2	yes	—
300	C₂×C₄	#431	1	2	yes	—
301	C₂×C₄	C₂×M₃₂	1	2	yes	—
302	C₂×C₄	C₂×M₃₂	1	2	yes	—
303	C₂×C₄	C₂×M₃₂	1	2	yes	—
304	C₂×C₄	C₂×M₃₂	1	2	yes	—
305	C₂×C₄	C₂×M₃₂	1	2	yes	—
306	C₂×C₄	C₂×M₃₂	1	2	yes	—
307	C₂×C₄	#504	1	2	yes	—
308	C₂×C₄	C₂×#117	1	1	yes	characteristic
309	C₂×C₄	C₄×#41	1	1	yes	characteristic
310	C₂×C₄	#519	1	1	yes	characteristic
311	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
312	C₂×C₄	C₂×#131	1	1	yes	characteristic
313	C₂×C₄	#568	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	2	2	no	—
316	C₈	—	2	2	no	—
317	C₈	—	2	2	no	—
318	C₈	—	4	1	no	—
319	C₈	—	4	1	no	—
320	C₈	—	4	1	no	—
321	C₈	#552	1	2	yes	—
322	C₈	#556	1	2	yes	—
323	C₂²	C₂×#431	1	1	yes	characteristic
324	C₂²	C₈×#41	1	1	yes	characteristic
325	C₂²	#2982	1	1	yes	characteristic
326	C₂²	C₂²×M₃₂	1	1	yes	characteristic
327	C₂²	C₂²×M₃₂	1	1	yes	characteristic
328	C₂²	C₂×#504	1	1	yes	characteristic
329	C₂²	#3407	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	#2356	1	1	yes	characteristic
345	C₄	#2925	1	1	yes	characteristic
346	C₄	#2970	1	1	yes	characteristic
347	C₄	#2984	1	1	yes	characteristic
348	C₂	#22828	1	1	yes	characteristic
349	C₂	#33418	1	1	yes	characteristic
350	C₂	#33423	1	1	yes	characteristic
351	C₂	#33479	1	1	yes	characteristic
352	C₂	#33484	1	1	yes	characteristic
353	C₂	#33497	1	1	yes	characteristic
354	C₂	#33498	1	1	yes	characteristic

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

Maximal subgroups

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

subgroup	index
C₄×C₈×Q₈	2
#13264	2
#13105	2
#13104	2
#12894	2
#12879	2
#12663	2
#12653	2
#12591	2
#12590	2