Subgroups — #493213

Order 512.

752 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2171	C₂	1	1	yes	maximal, characteristic
2	C₂×#2170	C₂	1	1	yes	maximal, characteristic
3	C₄×#504	C₂	1	2	yes	maximal, complemented
4	#20510	C₂	1	1	yes	maximal, characteristic
5	#12665	C₂	1	1	yes	maximal, characteristic, complemented
6	#12664	C₂	1	1	yes	maximal, characteristic, complemented
7	#12656	C₂	1	1	yes	maximal, characteristic, complemented
8	#12615	C₂	1	2	yes	maximal, complemented
9	#12614	C₂	1	2	yes	maximal, complemented
10	#12604	C₂	1	2	yes	maximal, complemented
11	#12539	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#504	C₂²	1	2	yes	—
13	C₂×#504	C₂²	1	2	yes	—
14	C₂×#444	C₄	1	1	yes	characteristic
15	C₂×#443	C₂²	1	1	yes	characteristic
16	#2943	C₄	1	1	yes	characteristic
17	C₄×#131	C₂²	1	2	yes	—
18	#2171	C₄	1	8	yes	complemented
19	#2170	C₄	1	8	yes	complemented
20	C₂×#364	C₂²	1	1	yes	characteristic
21	C₂×#364	C₂²	1	1	yes	characteristic
22	C₂×#350	C₂²	1	2	yes	—
23	C₂×#350	C₂²	1	2	yes	—
24	#2108	—	2	2	no	—
25	#2108	—	2	2	no	—
26	#2108	C₂²	1	2	yes	—
27	#2108	C₂²	1	2	yes	—
28	C₄×M₃₂	—	2	2	no	—
29	C₄×M₃₂	C₂²	1	2	yes	—
30	C₂×#346	C₂²	1	1	yes	characteristic
31	C₂×#346	C₂²	1	1	yes	characteristic
32	C₂×#346	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₁₆	—	2	2	no	—
34	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
37	#1778	C₄	1	1	yes	characteristic
38	#1773	C₄	1	1	yes	characteristic
39	#1772	C₂²	1	1	yes	characteristic
40	#1765	C₄	1	2	yes	—
41	#1764	C₂²	1	2	yes	—
42	#1759	C₄	1	2	yes	—
43	#1726	C₂²	1	1	yes	characteristic
44	#1379	C₂²	1	2	yes	—
45	#1379	C₂²	1	2	yes	—
46	#1376	C₂²	1	2	yes	—
47	C₂×#135	C₂×C₄	1	1	yes	characteristic
48	C₂×#131	C₂×C₄	1	2	yes	—
49	C₂×#131	C₂³	1	2	yes	—
50	C₄×#41	C₂×C₄	1	2	yes	—
51	#504	—	2	4	no	—
52	#504	—	2	4	no	—
53	#504	C₂×C₄	1	8	yes	complemented
54	#504	C₂×C₄	1	8	yes	complemented
55	C₂×M₃₂	—	2	2	no	—
56	C₂×M₃₂	—	2	2	no	—
57	C₂×M₃₂	C₂³	1	2	yes	—
58	C₂×M₃₂	C₂³	1	2	yes	—
59	C₂²×C₁₆	—	2	2	no	—
60	C₂²×C₁₆	—	2	2	no	—
61	C₂²×C₁₆	C₂³	1	1	yes	characteristic
62	C₂²×C₁₆	C₂³	1	1	yes	characteristic
63	C₂²×C₁₆	C₂³	1	1	yes	characteristic
64	C₂²×C₁₆	C₂³	1	1	yes	characteristic
65	#444	C₈	1	8	yes	—
66	#443	C₂×C₄	1	4	yes	—
67	#443	C₂×C₄	1	4	yes	—
68	#442	C₈	1	2	yes	—
69	#441	C₈	1	2	yes	—
70	#440	C₈	1	4	yes	—
71	C₂×#108	C₂×C₄	1	1	yes	characteristic
72	C₂×#108	C₂×C₄	1	1	yes	characteristic
73	#427	C₂×C₄	1	1	yes	characteristic
74	C₂×#104	C₂×C₄	1	2	yes	—
75	C₂×#104	C₂×C₄	1	2	yes	—
76	#416	C₂×C₄	1	2	yes	—
77	#405	—	2	2	no	—
78	#405	—	2	2	no	—
79	#405	C₂×C₄	1	2	yes	—
80	#405	C₂³	1	2	yes	—
81	C₄×M₁₆	—	2	2	no	—
82	C₄×M₁₆	C₂×C₄	1	2	yes	—
83	C₂×#97	C₂×C₄	1	1	yes	characteristic
84	C₂×#97	C₂×C₄	1	1	yes	characteristic
85	C₂×#97	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄×C₈	—	2	2	no	—
87	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
90	#376	C₂×C₄	1	1	yes	characteristic
91	#364	C₂×C₄	1	4	yes	complemented
92	#364	C₂×C₄	1	4	yes	complemented
93	#350	C₂×C₄	1	8	yes	complemented
94	#350	C₂×C₄	1	8	yes	complemented
95	#346	—	2	2	no	—
96	#346	—	2	2	no	—
97	#346	—	2	2	no	—
98	#346	—	2	2	no	—
99	#346	—	2	2	no	—
100	#346	C₂×C₄	1	4	yes	complemented
101	C₄×C₁₆	—	2	2	no	—
102	C₄×C₁₆	—	2	2	no	—
103	C₄×C₁₆	—	2	2	no	—
104	C₄×C₁₆	—	2	2	no	—
105	C₄×C₁₆	C₂×C₄	1	4	yes	complemented
106	C₄×C₁₆	D₄	1	2	yes	—
107	C₄×C₁₆	D₄	1	2	yes	—
108	#341	C₂×C₄	1	2	yes	—
109	#339	C₂×C₄	1	2	yes	—
110	#339	C₂×C₄	1	2	yes	—
111	C₂×#41	C₂²×C₄	1	2	yes	—
112	#135	C₄²	1	4	yes	—
113	#135	C₂×C₈	1	4	yes	—
114	C₂×Q₁₆	C₂×C₈	1	1	yes	characteristic
115	C₂×SD₁₆	C₂×C₈	1	2	yes	—
116	C₂×D₈	C₂×C₈	1	1	yes	characteristic
117	#131	—	2	4	no	—
118	#131	—	2	2	no	—
119	#131	—	2	2	no	—
120	#131	C₂×C₈	1	8	yes	—
121	#131	C₂²×C₄	1	4	yes	—
122	#131	C₂²×C₄	1	4	yes	—
123	C₂×M₁₆	—	2	2	no	—
124	C₂×M₁₆	—	2	2	no	—
125	C₂×M₁₆	C₂²×C₄	1	2	yes	—
126	C₂×M₁₆	C₂²×C₄	1	2	yes	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₈	C₂⁴	1	1	yes	characteristic
133	C₄×Q₈	C₂×C₈	1	4	yes	—
134	C₄×D₄	—	2	4	no	—
135	C₄×D₄	C₂×C₈	1	4	yes	—
136	#117	—	2	2	no	—
137	#117	C₂²×C₄	1	2	yes	—
138	C₂×C₄²	—	2	2	no	—
139	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
142	M₃₂	—	2	2	no	—
143	M₃₂	—	2	2	no	—
144	M₃₂	—	2	8	no	—
145	M₃₂	—	2	8	no	—
146	M₃₂	C₂²×C₄	1	4	yes	—
147	M₃₂	C₂²×C₄	1	4	yes	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	2	1	no	—
151	C₂×C₁₆	—	2	1	no	—
152	C₂×C₁₆	—	2	1	no	—
153	C₂×C₁₆	—	2	1	no	—
154	C₂×C₁₆	—	2	8	no	—
155	C₂×C₁₆	—	2	8	no	—
156	C₂×C₁₆	C₂²×C₄	1	2	yes	—
157	C₂×C₁₆	C₂²×C₄	1	2	yes	—
158	C₂×C₁₆	C₂²×C₄	1	2	yes	—
159	C₂×C₁₆	C₂²×C₄	1	2	yes	—
160	C₂×C₁₆	C₂×D₄	1	2	yes	—
161	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
162	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
163	C₂×C₁₆	#41	1	2	yes	—
164	C₂×C₁₆	#41	1	2	yes	—
165	#108	C₄²	1	4	yes	—
166	#108	C₂×C₈	1	4	yes	—
167	#107	C₂×C₈	1	2	yes	—
168	#106	C₂×C₈	1	2	yes	—
169	#104	C₄²	1	8	yes	—
170	#104	C₂×C₈	1	8	yes	—
171	#103	C₂×C₈	1	4	yes	—
172	#102	C₂×C₈	1	4	yes	—
173	#97	—	2	2	no	—
174	#97	—	2	2	no	—
175	#97	—	2	2	no	—
176	#97	—	2	2	no	—
177	#97	C₂×C₈	1	2	yes	—
178	#97	C₂×C₈	1	2	yes	—
179	#97	C₂×C₈	1	4	yes	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	—	2	2	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	C₂²×C₄	1	2	yes	—
186	C₄×C₈	C₂²×C₄	1	2	yes	—
187	C₄×C₈	C₂×D₄	1	2	yes	—
188	C₄×C₈	#41	1	2	yes	—
189	#95	—	2	2	no	—
190	#41	—	2	2	no	—
191	#41	—	2	2	no	—
192	#41	C₂×C₄²	1	4	yes	—
193	#41	C₂²×C₈	1	4	yes	—
194	C₂×Q₈	C₂²×C₈	1	2	yes	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	C₂²×C₈	1	2	yes	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	#131	1	2	yes	—
202	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
203	Q₁₆	C₄×C₈	1	4	yes	—
204	SD₁₆	C₄×C₈	1	8	yes	—
205	D₈	C₄×C₈	1	4	yes	—
206	M₁₆	—	2	2	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	2	8	no	—
211	M₁₆	C₂×C₄²	1	4	yes	—
212	M₁₆	C₂²×C₈	1	4	yes	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	8	no	—
220	C₂×C₈	C₂×C₄²	1	2	yes	—
221	C₂×C₈	C₄×D₄	1	2	yes	—
222	C₂×C₈	C₂²×C₈	1	2	yes	—
223	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
224	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₈	1	2	yes	—
226	C₂×C₈	C₂²×C₈	1	2	yes	—
227	C₂×C₈	#131	1	2	yes	—
228	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
229	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
230	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₈	C₂×#41	1	1	yes	characteristic
232	C₂×C₈	C₂×#41	1	1	yes	characteristic
233	C₂×C₈	C₂×#41	1	1	yes	characteristic
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	C₂²×C₈	1	4	yes	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	2	4	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	4	no	—
242	C₄²	C₂×C₄²	1	2	yes	—
243	C₄²	C₄×D₄	1	2	yes	—
244	C₄²	C₂²×C₈	1	2	yes	—
245	C₄²	C₂²×C₈	1	2	yes	—
246	C₁₆	—	2	4	no	—
247	C₁₆	—	2	4	no	—
248	C₁₆	—	2	2	no	—
249	C₁₆	—	2	2	no	—
250	C₁₆	C₄×D₄	1	2	yes	—
251	C₁₆	C₄×D₄	1	4	yes	—
252	C₁₆	C₄×D₄	1	2	yes	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×#131	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	Q₈	C₂×C₄×C₈	1	4	yes	—
257	D₄	—	2	4	no	—
258	D₄	—	2	4	no	—
259	D₄	—	2	2	no	—
260	D₄	C₂×C₄×C₈	1	4	yes	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	8	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	C₈×D₄	1	4	yes	—
272	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
273	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
274	C₂×C₄	C₄×#41	1	1	yes	characteristic
275	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
276	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
277	C₂×C₄	C₂×#131	1	1	yes	characteristic
278	C₈	—	2	4	no	—
279	C₈	—	2	2	no	—
280	C₈	—	2	4	no	—
281	C₈	C₂×C₄×C₈	1	2	yes	—
282	C₈	C₂×C₄×C₈	1	2	yes	—
283	C₈	C₈×D₄	1	4	yes	—
284	C₈	C₂×C₄×D₄	1	1	yes	characteristic
285	C₈	C₂×C₄×D₄	1	1	yes	characteristic
286	C₈	C₄×#41	1	1	yes	characteristic
287	C₈	C₄×#41	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	C₄×#131	1	1	yes	characteristic
292	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
293	C₂²	C₈×#41	1	1	yes	characteristic
294	C₄	—	2	4	no	—
295	C₄	—	2	4	no	—
296	C₄	—	4	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	4	no	—
299	C₄	C₄²×D₄	1	1	yes	characteristic
300	C₄	C₂²×C₄×C₈	1	1	yes	characteristic
301	C₄	C₂×C₈×D₄	1	1	yes	characteristic
302	C₄	C₈×#41	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	4	4	no	—
306	C₂	C₄×C₈×D₄	1	1	yes	characteristic
307	C₂	#33441	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₄×#504	C₂	semidirect
5. #12665	C₂	semidirect
6. #12664	C₂	semidirect
7. #12656	C₂	semidirect
8. #12615	C₂	semidirect
9. #12614	C₂	semidirect
10. #12604	C₂	semidirect
11. #12539	C₂	semidirect
18. #2171	C₄	semidirect
19. #2170	C₄	semidirect
53. #504	C₂×C₄	semidirect
54. #504	C₂×C₄	semidirect
91. #364	C₂×C₄	semidirect
92. #364	C₂×C₄	semidirect
93. #350	C₂×C₄	semidirect
94. #350	C₂×C₄	semidirect
100. #346	C₂×C₄	semidirect
105. C₄×C₁₆	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2171	2
C₂×#2170	2
C₄×#504	2
#20510	2
#12665	2
#12664	2
#12656	2
#12615	2
#12614	2
#12604	2
#12539	2