Subgroups — #496718

Order 512.

591 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33418	C₂	1	1	yes	maximal, characteristic
2	C₂×#2150	C₂	1	1	yes	maximal, characteristic
3	C₈×#117	C₂	1	1	yes	maximal, characteristic
4	#13163	C₂	1	4	yes	maximal, complemented
5	#12914	C₂	1	4	yes	maximal
6	#12900	C₂	1	2	yes	maximal, complemented
7	#12883	C₂	1	2	yes	maximal, complemented
8	C₂²×M₃₂	C₂²	1	2	yes	—
9	C₂×#431	C₄	1	1	yes	characteristic
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₄×#117	C₄	1	1	yes	characteristic
12	#2163	—	2	1	no	—
13	#2163	—	2	1	no	—
14	#2162	—	2	1	no	—
15	#2162	—	2	1	no	—
16	C₂×#364	C₂²	1	8	yes	—
17	#2150	—	2	1	no	—
18	#2150	—	2	1	no	—
19	#2150	C₂²	1	1	yes	characteristic
20	#2150	C₂²	1	2	yes	—
21	#2150	C₂²	1	1	yes	characteristic
22	#2150	C₂²	1	2	yes	—
23	C₂×C₄⋊C₁₆	—	2	1	no	—
24	C₂×C₄⋊C₁₆	—	2	1	no	—
25	#2112	C₂²	1	2	yes	—
26	C₄×M₃₂	C₂²	1	2	yes	—
27	C₈×C₄⋊C₄	C₄	1	2	yes	—
28	C₈×C₄⋊C₄	C₂²	1	2	yes	—
29	C₈×C₂²⋊C₄	C₄	1	2	yes	—
30	C₈×C₂²⋊C₄	C₂²	1	2	yes	—
31	#1744	C₄	1	2	yes	—
32	C₄²×C₈	C₂²	1	2	yes	—
33	#1376	C₂²	1	8	yes	—
34	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₈	C₂³	1	1	yes	characteristic
36	C₂×#117	—	2	1	no	—
37	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
38	C₂×M₃₂	—	2	2	no	—
39	C₂×M₃₂	—	4	4	no	—
40	C₂×M₃₂	C₂³	1	2	yes	—
41	C₂×M₃₂	C₂³	1	2	yes	—
42	C₂×M₃₂	C₂³	1	4	yes	—
43	C₂²×C₁₆	—	2	2	no	—
44	#431	C₈	1	2	yes	—
45	#431	C₈	1	2	yes	—
46	#431	C₈	1	2	yes	—
47	#431	C₈	1	2	yes	—
48	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊C₈	C₂×C₄	1	2	yes	—
50	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
51	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
54	C₂×C₄×C₈	D₄	1	1	yes	characteristic
55	C₂×C₄×C₈	D₄	1	1	yes	characteristic
56	C₂×C₄×C₈	D₄	1	1	yes	characteristic
57	C₂×C₄×C₈	D₄	1	1	yes	characteristic
58	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
59	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
60	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
61	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂³	1	4	yes	—
63	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
65	C₄×C₄⋊C₄	—	2	1	no	—
66	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
67	C₄×C₂²⋊C₄	—	2	1	no	—
68	C₄×C₂²⋊C₄	C₂×C₄	1	2	yes	—
69	#376	—	2	1	no	—
70	C₄³	C₂×C₄	1	2	yes	—
71	#364	—	4	8	no	—
72	C₄⋊C₁₆	—	2	2	no	—
73	C₄⋊C₁₆	—	2	2	no	—
74	C₄⋊C₁₆	—	2	2	no	—
75	C₄⋊C₁₆	—	2	2	no	—
76	C₄⋊C₁₆	—	2	1	no	—
77	C₄⋊C₁₆	—	2	1	no	—
78	C₄⋊C₁₆	—	2	1	no	—
79	C₄⋊C₁₆	—	2	1	no	—
80	C₂²⋊C₁₆	—	2	2	no	—
81	C₂²⋊C₁₆	—	2	2	no	—
82	#346	—	2	2	no	—
83	C₄×C₁₆	—	2	2	no	—
84	#336	C₂×C₄	1	4	yes	—
85	#336	C₂×C₄	1	4	yes	—
86	C₂³×C₄	M₁₆	1	1	yes	characteristic
87	C₂³×C₄	M₁₆	1	1	yes	characteristic
88	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	—	2	2	no	—
90	C₂²×C₈	—	2	2	no	—
91	C₂²×C₈	—	4	2	no	—
92	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₈	C₂²×C₄	1	4	yes	—
100	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
103	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
104	C₂²×C₈	C₂⁴	1	1	yes	characteristic
105	#117	—	2	8	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₂²⋊C₄	—	2	2	no	—
108	C₂×C₄²	—	2	2	no	—
109	C₂×C₄²	—	2	2	no	—
110	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
115	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
116	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
117	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
118	C₂×C₄²	M₁₆	1	1	yes	characteristic
119	C₂×C₄²	M₁₆	1	1	yes	characteristic
120	C₂×C₄²	C₂²×C₄	1	4	yes	—
121	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
123	M₃₂	—	2	4	no	—
124	M₃₂	—	2	8	no	—
125	M₃₂	—	2	4	no	—
126	M₃₂	—	4	4	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	—	2	4	no	—
130	C₂×C₁₆	—	4	4	no	—
131	C₄⋊C₈	C₂×C₈	1	4	yes	—
132	C₄⋊C₈	C₂×C₈	1	4	yes	—
133	C₂²⋊C₈	C₂×C₈	1	4	yes	—
134	C₂²⋊C₈	C₂×C₈	1	4	yes	—
135	C₄×C₈	—	2	4	no	—
136	C₄×C₈	—	2	4	no	—
137	C₄×C₈	—	4	4	no	—
138	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
142	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₈	C₂×C₈	1	4	yes	—
147	C₄×C₈	C₂×C₈	1	4	yes	—
148	C₄×C₈	C₂×D₄	1	1	yes	characteristic
149	C₄×C₈	C₂×D₄	1	1	yes	characteristic
150	C₄×C₈	C₂×D₄	1	1	yes	characteristic
151	C₄×C₈	C₂×D₄	1	1	yes	characteristic
152	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
153	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
154	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
155	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
156	#95	—	2	2	no	—
157	#95	—	2	2	no	—
158	C₂⁴	C₂×M₁₆	1	1	yes	characteristic
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
165	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
166	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
167	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
168	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
169	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
170	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
171	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
175	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
176	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
177	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
178	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
179	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
180	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	4	1	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
200	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
201	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
203	C₂×C₈	C₂²×C₈	1	8	yes	—
204	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
206	C₂×C₈	H(5,𝔽₂)	1	2	yes	—
207	C₂×C₈	2₋¹⁺⁴	1	2	yes	—
208	C₄⋊C₄	—	2	8	no	—
209	C₂²⋊C₄	—	2	8	no	—
210	C₄²	—	2	8	no	—
211	C₄²	—	2	4	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	4	4	no	—
214	C₄²	C₄⋊C₈	1	1	yes	characteristic
215	C₄²	C₄⋊C₈	1	1	yes	characteristic
216	C₄²	C₄⋊C₈	1	1	yes	characteristic
217	C₄²	C₄⋊C₈	1	1	yes	characteristic
218	C₄²	C₄⋊C₈	1	1	yes	characteristic
219	C₄²	C₄⋊C₈	1	1	yes	characteristic
220	C₄²	C₄⋊C₈	1	1	yes	characteristic
221	C₄²	C₄⋊C₈	1	1	yes	characteristic
222	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
223	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
224	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
225	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
226	C₄²	C₂²×C₈	1	1	yes	characteristic
227	C₄²	C₂²×C₈	1	1	yes	characteristic
228	C₄²	C₂²×C₈	1	1	yes	characteristic
229	C₄²	C₂²×C₈	1	1	yes	characteristic
230	C₁₆	—	4	4	no	—
231	C₁₆	—	4	2	no	—
232	C₁₆	—	4	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	4	1	no	—
236	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
237	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
238	C₂³	C₂²×M₁₆	1	1	yes	characteristic
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	#364	1	4	yes	complemented
261	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
268	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
272	C₂×C₄	#520	1	2	yes	—
273	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
274	C₂×C₄	#568	1	2	yes	—
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	—	2	8	no	—
282	C₈	#535	1	1	yes	characteristic
283	C₈	#536	1	1	yes	characteristic
284	C₈	#544	1	2	yes	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	4	4	no	—
293	C₂²	C₂×#364	1	2	yes	—
294	C₂²	C₂²×C₄⋊C₈	1	1	yes	characteristic
295	C₂²	#2989	1	2	yes	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	8	no	—
304	C₄	C₂×#364	1	2	yes	—
305	C₄	C₂×#364	1	2	yes	—
306	C₄	#2153	1	2	yes	—
307	C₄	#2317	1	1	yes	characteristic
308	C₄	#2905	1	1	yes	characteristic
309	C₄	#2988	1	2	yes	—
310	C₂	—	2	2	no	—
311	C₂	—	2	2	no	—
312	C₂	—	4	1	no	—
313	C₂	#21972	1	1	yes	characteristic
314	C₂	C₂²×#364	1	1	yes	characteristic
315	C₂	C₂×#2153	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₄⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #13163	C₂	semidirect
6. #12900	C₂	semidirect
7. #12883	C₂	semidirect
260. C₂×C₄	#364	semidirect

Maximal subgroups

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

subgroup	index
#33418	2
C₂×#2150	2
C₈×#117	2
#13163	2
#12914	2
#12900	2
#12883	2