Subgroups — #495287

Order 512.

558 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33421	C₂	1	2	yes	maximal, complemented
2	#20510	C₂	1	1	yes	maximal, characteristic
3	#13196	C₂	1	1	yes	maximal, characteristic
4	#13195	C₂	1	1	yes	maximal, characteristic
5	#13162	C₂	1	1	yes	maximal, characteristic, complemented
6	#13161	C₂	1	1	yes	maximal, characteristic, complemented
7	#13125	C₂	1	2	yes	maximal, complemented
8	#13124	C₂	1	2	yes	maximal, complemented
9	#12656	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#350	C₂	1	2	yes	maximal, complemented
11	#12539	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#444	C₄	1	1	yes	characteristic
13	C₂×#443	C₂²	1	1	yes	characteristic
14	#2943	C₄	1	1	yes	characteristic
15	C₄×#131	C₂²	1	2	yes	—
16	#2164	—	2	4	no	—
17	#2163	—	2	4	no	—
18	C₂×#364	C₂²	1	1	yes	characteristic
19	C₂×#364	C₂²	1	1	yes	characteristic
20	#2151	C₂²	1	2	yes	—
21	#2151	C₂²	1	2	yes	—
22	C₂×#350	C₂²	1	2	yes	—
23	C₂×#350	C₂²	1	2	yes	—
24	#2113	C₂²	1	2	yes	—
25	#2113	C₂²	1	2	yes	—
26	C₄×M₃₂	C₂²	1	2	yes	—
27	C₂×#346	C₂²	1	1	yes	characteristic
28	C₂×#346	C₂²	1	1	yes	characteristic
29	C₂×#346	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
33	#1778	C₄	1	1	yes	characteristic
34	#1773	C₄	1	1	yes	characteristic
35	#1772	C₂²	1	1	yes	characteristic
36	#1765	C₄	1	2	yes	—
37	#1764	C₂²	1	2	yes	—
38	#1759	C₄	1	2	yes	—
39	#1726	C₂²	1	1	yes	characteristic
40	#1379	C₂²	1	2	yes	—
41	#1379	C₂²	1	2	yes	—
42	#1376	C₂²	1	2	yes	—
43	C₂×#135	C₂×C₄	1	1	yes	characteristic
44	C₂×#131	C₂×C₄	1	2	yes	—
45	C₂×#131	C₂³	1	2	yes	—
46	C₄×#41	C₂×C₄	1	2	yes	—
47	C₂×M₃₂	C₂³	1	2	yes	—
48	C₂×M₃₂	C₂³	1	2	yes	—
49	C₂²×C₁₆	C₂³	1	1	yes	characteristic
50	C₂²×C₁₆	C₂³	1	1	yes	characteristic
51	C₂²×C₁₆	C₂³	1	1	yes	characteristic
52	C₂²×C₁₆	C₂³	1	1	yes	characteristic
53	#444	—	2	4	no	—
54	#443	—	2	2	no	—
55	#443	—	2	2	no	—
56	#442	C₈	1	2	yes	—
57	#441	C₈	1	2	yes	—
58	#440	C₈	1	4	yes	—
59	C₂×#108	C₂×C₄	1	1	yes	characteristic
60	C₂×#108	C₂×C₄	1	1	yes	characteristic
61	#427	C₂×C₄	1	1	yes	characteristic
62	C₂×#104	C₂×C₄	1	2	yes	—
63	C₂×#104	C₂×C₄	1	2	yes	—
64	#416	C₂×C₄	1	2	yes	—
65	#405	—	2	2	no	—
66	#405	—	2	2	no	—
67	#405	C₂×C₄	1	2	yes	—
68	#405	C₂³	1	2	yes	—
69	C₄×M₁₆	—	2	2	no	—
70	C₄×M₁₆	C₂×C₄	1	2	yes	—
71	C₂×#97	C₂×C₄	1	1	yes	characteristic
72	C₂×#97	C₂×C₄	1	1	yes	characteristic
73	C₂×#97	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄×C₈	—	2	2	no	—
75	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
78	#376	C₂×C₄	1	1	yes	characteristic
79	#364	—	2	2	no	—
80	#364	—	2	2	no	—
81	C₄⋊C₁₆	—	2	4	no	—
82	C₄⋊C₁₆	—	2	4	no	—
83	#350	—	2	4	no	—
84	#350	—	2	4	no	—
85	C₂²⋊C₁₆	—	2	4	no	—
86	C₂²⋊C₁₆	—	2	4	no	—
87	#346	—	2	2	no	—
88	#346	—	2	2	no	—
89	#346	—	2	2	no	—
90	#346	D₄	1	4	yes	—
91	C₄×C₁₆	—	2	2	no	—
92	C₄×C₁₆	—	2	2	no	—
93	C₄×C₁₆	—	2	2	no	—
94	C₄×C₁₆	—	2	2	no	—
95	#341	C₂×C₄	1	2	yes	—
96	#339	C₂×C₄	1	2	yes	—
97	#339	C₂×C₄	1	2	yes	—
98	C₂×#41	C₂²×C₄	1	2	yes	—
99	#135	—	2	2	no	—
100	#135	—	2	2	no	—
101	C₂×Q₁₆	C₂×C₈	1	1	yes	characteristic
102	C₂×SD₁₆	C₂×C₈	1	2	yes	—
103	C₂×D₈	C₂×C₈	1	1	yes	characteristic
104	#131	—	2	4	no	—
105	#131	—	2	4	no	—
106	#131	—	2	2	no	—
107	#131	—	2	2	no	—
108	#131	—	2	2	no	—
109	#131	—	2	2	no	—
110	C₂×M₁₆	—	2	2	no	—
111	C₂×M₁₆	—	2	2	no	—
112	C₂×M₁₆	C₂²×C₄	1	2	yes	—
113	C₂×M₁₆	C₂²×C₄	1	2	yes	—
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	—	2	2	no	—
116	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₈	C₂⁴	1	1	yes	characteristic
120	C₄×Q₈	C₂×C₈	1	4	yes	—
121	C₄×D₄	—	2	4	no	—
122	C₄×D₄	C₂×C₈	1	4	yes	—
123	#117	—	2	2	no	—
124	#117	C₂²×C₄	1	2	yes	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
129	M₃₂	—	2	2	no	—
130	M₃₂	—	2	2	no	—
131	M₃₂	—	2	2	no	—
132	M₃₂	—	2	2	no	—
133	C₂×C₁₆	—	2	1	no	—
134	C₂×C₁₆	—	2	1	no	—
135	C₂×C₁₆	—	2	1	no	—
136	C₂×C₁₆	—	2	1	no	—
137	C₂×C₁₆	—	2	1	no	—
138	C₂×C₁₆	—	2	1	no	—
139	C₂×C₁₆	—	2	1	no	—
140	C₂×C₁₆	—	2	1	no	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	C₂×D₄	1	2	yes	—
144	C₂×C₁₆	C₂×D₄	1	2	yes	—
145	C₂×C₁₆	#41	1	2	yes	—
146	C₂×C₁₆	#41	1	2	yes	—
147	#108	—	2	2	no	—
148	#108	—	2	2	no	—
149	#107	C₂×C₈	1	2	yes	—
150	#106	C₂×C₈	1	2	yes	—
151	#104	—	2	4	no	—
152	#104	—	2	4	no	—
153	#103	C₂×C₈	1	4	yes	—
154	#102	C₂×C₈	1	4	yes	—
155	#97	—	2	2	no	—
156	#97	—	2	2	no	—
157	#97	—	2	2	no	—
158	#97	—	2	2	no	—
159	#97	C₂×C₈	1	2	yes	—
160	#97	C₂×C₈	1	2	yes	—
161	#97	M₁₆	1	4	yes	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	—	2	1	no	—
165	C₄×C₈	—	2	1	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	C₂×D₄	1	2	yes	—
168	C₄×C₈	#41	1	2	yes	—
169	C₄×C₈	#41	1	2	yes	—
170	C₄×C₈	#41	1	2	yes	—
171	#95	—	2	2	no	—
172	#41	—	2	2	no	—
173	#41	—	2	2	no	—
174	#41	—	2	2	no	—
175	#41	—	2	2	no	—
176	C₂×Q₈	C₂²×C₈	1	2	yes	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	C₂²×C₈	1	2	yes	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	#131	1	2	yes	—
184	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
185	Q₁₆	—	2	2	no	—
186	SD₁₆	—	2	4	no	—
187	D₈	—	2	2	no	—
188	M₁₆	—	2	2	no	—
189	M₁₆	—	2	2	no	—
190	M₁₆	—	2	2	no	—
191	M₁₆	—	2	2	no	—
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	M₁₆	—	4	4	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	#117	1	2	yes	—
203	C₂×C₈	C₄×D₄	1	2	yes	—
204	C₂×C₈	C₂²×C₈	1	2	yes	—
205	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
206	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
207	C₂×C₈	C₂×M₁₆	1	2	yes	—
208	C₂×C₈	C₂×M₁₆	1	2	yes	—
209	C₂×C₈	#131	1	2	yes	—
210	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₈	C₂×#41	1	1	yes	characteristic
212	C₂×C₈	C₂×#41	1	1	yes	characteristic
213	C₂×C₈	C₂×#41	1	1	yes	characteristic
214	C₂×C₈	C₂×#41	1	1	yes	characteristic
215	C₂×C₈	C₂×#41	1	1	yes	characteristic
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	C₂²×C₈	1	4	yes	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₂²⋊C₄	—	2	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	4	no	—
224	C₄²	#117	1	2	yes	—
225	C₄²	C₄×D₄	1	2	yes	—
226	C₄²	C₂²×C₈	1	2	yes	—
227	C₄²	C₂×M₁₆	1	2	yes	—
228	C₁₆	—	2	2	no	—
229	C₁₆	—	2	2	no	—
230	C₁₆	—	2	2	no	—
231	C₁₆	—	2	2	no	—
232	C₁₆	—	4	2	no	—
233	C₁₆	—	4	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	C₂×#131	1	1	yes	characteristic
236	Q₈	—	2	2	no	—
237	Q₈	—	2	2	no	—
238	D₄	—	2	2	no	—
239	D₄	—	2	2	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	8	4	no	—
252	C₂×C₄	C₈×D₄	1	4	yes	—
253	C₂×C₄	C₂×#117	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
255	C₂×C₄	C₄×#41	1	1	yes	characteristic
256	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
257	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
258	C₂×C₄	C₂×#131	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	4	2	no	—
261	C₈	—	4	2	no	—
262	C₈	#431	1	2	yes	—
263	C₈	#431	1	2	yes	—
264	C₈	#436	1	4	yes	—
265	C₈	#529	1	1	yes	characteristic
266	C₈	#529	1	1	yes	characteristic
267	C₈	#532	1	1	yes	characteristic
268	C₈	#532	1	1	yes	characteristic
269	C₂²	—	2	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	4	no	—
272	C₂²	#2922	1	1	yes	characteristic
273	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
274	C₂²	#2965	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	4	2	no	—
277	C₄	—	4	2	no	—
278	C₄	—	4	2	no	—
279	C₄	—	4	4	no	—
280	C₄	#2293	1	1	yes	characteristic
281	C₄	C₂×#431	1	1	yes	characteristic
282	C₄	C₂×#436	1	1	yes	characteristic
283	C₄	C₈×#41	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	8	2	no	—
287	C₂	#21264	1	1	yes	characteristic
288	C₂	#33442	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #33421	C₂	semidirect
5. #13162	C₂	semidirect
6. #13161	C₂	semidirect
7. #13125	C₂	semidirect
8. #13124	C₂	semidirect
9. #12656	C₂	semidirect
10. C₄×#350	C₂	semidirect
11. #12539	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33421	2
#20510	2
#13196	2
#13195	2
#13162	2
#13161	2
#13125	2
#13124	2
#12656	2
C₄×#350	2
#12539	2