Subgroups — #493301

Order 512.

569 conjugacy classes of subgroups.

The subgroup lattice has 307 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	#20476	C₂	1	1	yes	maximal, characteristic
4	#12666	C₂	1	4	yes	maximal, complemented
5	#12657	C₂	1	4	yes	maximal, complemented
6	#12616	C₂	1	4	yes	maximal
7	C₂²×M₃₂	C₂²	1	2	yes	—
8	C₂×#432	C₄	1	1	yes	characteristic
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	#2283	C₄	1	1	yes	characteristic
11	#2163	C₄	1	2	yes	complemented
12	#2163	C₄	1	2	yes	complemented
13	#2162	C₄	1	2	yes	complemented
14	#2162	C₄	1	2	yes	complemented
15	#2150	C₄	1	2	yes	complemented
16	#2150	C₄	1	2	yes	complemented
17	#2150	C₂²	1	2	yes	—
18	#2150	C₂²	1	2	yes	—
19	#2150	C₂²	1	2	yes	—
20	C₂×C₄⋊C₁₆	C₄	1	2	yes	complemented
21	C₂×C₄⋊C₁₆	C₄	1	2	yes	complemented
22	#2112	C₂²	1	2	yes	—
23	C₂×#347	C₂²	1	8	yes	—
24	C₄×M₃₂	C₂²	1	2	yes	—
25	#1770	C₂²	1	2	yes	—
26	C₈×C₄⋊C₄	C₄	1	2	yes	—
27	#1752	C₂²	1	2	yes	—
28	C₈×C₂²⋊C₄	C₄	1	2	yes	—
29	#1746	C₄	1	2	yes	—
30	#1726	C₂²	1	2	yes	—
31	#1375	C₂²	1	8	yes	—
32	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₂×M₃₂	—	4	4	no	—
36	C₂×M₃₂	C₂×C₄	1	4	yes	—
37	C₂×M₃₂	C₂³	1	4	yes	—
38	C₂×M₃₂	C₂³	1	4	yes	—
39	C₂²×C₁₆	C₂×C₄	1	4	yes	—
40	#432	—	2	1	no	—
41	#432	—	2	1	no	—
42	#432	—	2	1	no	—
43	#432	—	2	1	no	—
44	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊C₈	C₂×C₄	1	2	yes	—
46	C₂×#97	C₂×C₄	1	2	yes	—
47	C₂×#97	C₂³	1	4	yes	—
48	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
49	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
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	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	C₂³	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
61	#383	—	2	1	no	—
62	#382	—	2	1	no	—
63	#381	—	2	1	no	—
64	#380	—	2	1	no	—
65	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
66	C₄×C₂²⋊C₄	C₂×C₄	1	2	yes	—
67	#376	C₂×C₄	1	2	yes	—
68	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
69	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
70	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
71	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
72	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
73	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
74	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
75	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
76	#347	—	4	8	no	—
77	#346	C₂×C₄	1	4	yes	—
78	C₄×C₁₆	C₂×C₄	1	4	yes	—
79	#336	C₂×C₄	1	4	yes	—
80	#336	C₂×C₄	1	4	yes	—
81	C₂³×C₄	M₁₆	1	1	yes	characteristic
82	C₂³×C₄	M₁₆	1	1	yes	characteristic
83	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₈	—	2	2	no	—
85	C₂²×C₈	—	2	2	no	—
86	C₂²×C₈	—	4	2	no	—
87	C₂²×C₈	C₄²	1	1	yes	characteristic
88	C₂²×C₈	C₄²	1	1	yes	characteristic
89	C₂²×C₈	C₄²	1	1	yes	characteristic
90	C₂²×C₈	C₄²	1	1	yes	characteristic
91	C₂²×C₈	C₂²×C₄	1	4	yes	—
92	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₈	C₂²×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	C₂⁴	1	1	yes	characteristic
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₂²⋊C₄	—	2	2	no	—
102	C₂×C₄²	C₄²	1	1	yes	characteristic
103	C₂×C₄²	C₄²	1	1	yes	characteristic
104	C₂×C₄²	C₄²	1	1	yes	characteristic
105	C₂×C₄²	C₄²	1	1	yes	characteristic
106	C₂×C₄²	M₁₆	1	1	yes	characteristic
107	C₂×C₄²	M₁₆	1	1	yes	characteristic
108	C₂×C₄²	M₁₆	1	1	yes	characteristic
109	C₂×C₄²	M₁₆	1	1	yes	characteristic
110	C₂×C₄²	M₁₆	1	1	yes	characteristic
111	C₂×C₄²	M₁₆	1	1	yes	characteristic
112	C₂×C₄²	C₂²×C₄	1	4	yes	—
113	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
115	M₃₂	—	2	4	no	—
116	M₃₂	—	2	8	no	—
117	M₃₂	—	2	4	no	—
118	M₃₂	—	4	4	no	—
119	C₂×C₁₆	—	4	4	no	—
120	C₂×C₁₆	C₂²×C₄	1	8	yes	—
121	C₂×C₁₆	C₂²×C₄	1	8	yes	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₂²⋊C₈	—	2	2	no	—
125	C₂²⋊C₈	—	2	2	no	—
126	#97	—	2	4	no	—
127	#97	—	2	4	no	—
128	#97	—	2	2	no	—
129	#97	—	2	2	no	—
130	C₄×C₈	—	4	4	no	—
131	C₄×C₈	C₄²	1	1	yes	characteristic
132	C₄×C₈	C₄²	1	1	yes	characteristic
133	C₄×C₈	C₄²	1	1	yes	characteristic
134	C₄×C₈	C₄²	1	1	yes	characteristic
135	C₄×C₈	C₄²	1	1	yes	characteristic
136	C₄×C₈	C₄²	1	1	yes	characteristic
137	C₄×C₈	C₄²	1	1	yes	characteristic
138	C₄×C₈	C₄²	1	1	yes	characteristic
139	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
140	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
141	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
142	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
143	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
144	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
145	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
146	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
147	#95	—	2	2	no	—
148	#95	—	2	2	no	—
149	#95	—	2	2	no	—
150	#95	—	2	2	no	—
151	C₂⁴	C₂×M₁₆	1	1	yes	characteristic
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	#97	1	1	yes	characteristic
158	C₂²×C₄	#97	1	1	yes	characteristic
159	C₂²×C₄	#97	1	1	yes	characteristic
160	C₂²×C₄	#97	1	1	yes	characteristic
161	C₂²×C₄	#97	1	1	yes	characteristic
162	C₂²×C₄	#97	1	1	yes	characteristic
163	C₂²×C₄	#97	1	1	yes	characteristic
164	C₂²×C₄	#97	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
167	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
168	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
169	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
170	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
171	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
172	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
173	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	—	4	1	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
192	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
193	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
194	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
195	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
196	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
197	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
198	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
199	C₂×C₈	H(5,𝔽₂)	1	2	yes	—
200	C₂×C₈	2₋¹⁺⁴	1	2	yes	—
201	C₄⋊C₄	—	4	4	no	—
202	C₂²⋊C₄	—	4	4	no	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	#97	1	1	yes	characteristic
206	C₄²	#97	1	1	yes	characteristic
207	C₄²	#97	1	1	yes	characteristic
208	C₄²	#97	1	1	yes	characteristic
209	C₄²	#97	1	1	yes	characteristic
210	C₄²	#97	1	1	yes	characteristic
211	C₄²	#97	1	1	yes	characteristic
212	C₄²	#97	1	1	yes	characteristic
213	C₄²	C₂×C₄²	1	1	yes	characteristic
214	C₄²	C₂×C₄²	1	1	yes	characteristic
215	C₄²	C₂×C₄²	1	1	yes	characteristic
216	C₄²	C₂×C₄²	1	1	yes	characteristic
217	C₄²	C₂×M₁₆	1	1	yes	characteristic
218	C₄²	C₂×M₁₆	1	1	yes	characteristic
219	C₄²	C₂×M₁₆	1	1	yes	characteristic
220	C₄²	C₂×M₁₆	1	1	yes	characteristic
221	C₁₆	—	2	8	no	—
222	C₁₆	—	2	8	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	4	1	no	—
226	C₂³	C₂×#97	1	1	yes	characteristic
227	C₂³	C₂×#97	1	1	yes	characteristic
228	C₂³	C₂²×M₁₆	1	1	yes	characteristic
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	C₂×#97	1	1	yes	characteristic
252	C₂×C₄	C₂×#97	1	1	yes	characteristic
253	C₂×C₄	C₂×#97	1	1	yes	characteristic
254	C₂×C₄	C₂×#97	1	1	yes	characteristic
255	C₂×C₄	C₂×#97	1	1	yes	characteristic
256	C₂×C₄	C₂×#97	1	1	yes	characteristic
257	C₂×C₄	C₂×#97	1	1	yes	characteristic
258	C₂×C₄	C₂×#97	1	1	yes	characteristic
259	C₂×C₄	C₂×#97	1	1	yes	characteristic
260	C₂×C₄	C₂×#97	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
262	C₂×C₄	#520	1	2	yes	—
263	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
264	C₂×C₄	#568	1	2	yes	—
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	2	1	no	—
271	C₈	—	4	4	no	—
272	C₈	#518	1	1	yes	characteristic
273	C₈	#519	1	1	yes	characteristic
274	C₈	#520	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	4	4	no	—
283	C₂²	#2110	1	2	yes	—
284	C₂²	C₂²×#97	1	1	yes	characteristic
285	C₂²	#2989	1	2	yes	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	8	4	no	—
296	C₄	#2110	1	2	yes	—
297	C₄	#2274	1	1	yes	characteristic
298	C₄	#2873	1	1	yes	characteristic
299	C₄	#2989	1	2	yes	—
300	C₂	—	2	2	no	—
301	C₂	—	2	2	no	—
302	C₂	—	4	1	no	—
303	C₂	#20547	1	1	yes	characteristic
304	C₂	C₂×#2110	1	1	yes	characteristic
305	C₂	C₂×#2110	1	1	yes	characteristic

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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. #12666	C₂	semidirect
5. #12657	C₂	semidirect
11. #2163	C₄	semidirect
12. #2163	C₄	semidirect
13. #2162	C₄	semidirect
14. #2162	C₄	semidirect
15. #2150	C₄	semidirect
16. #2150	C₄	semidirect
20. C₂×C₄⋊C₁₆	C₄	semidirect
21. C₂×C₄⋊C₁₆	C₄	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
#20476	2
#12666	2
#12657	2
#12616	2