Subgroups — #498087

Order 512.

400 conjugacy classes of subgroups.

The subgroup lattice has 295 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20518	C₂	1	1	yes	maximal, characteristic
2	#13268	C₂	1	2	yes	maximal
3	#13107	C₂	1	2	yes	maximal
4	#13107	C₂	1	2	yes	maximal
5	#12897	C₂	1	1	yes	maximal, characteristic
6	#12896	C₂	1	1	yes	maximal, characteristic
7	#12884	C₂	1	1	yes	maximal, characteristic
8	#12666	C₂	1	2	yes	maximal
9	#12656	C₂	1	1	yes	maximal, characteristic
10	#12596	C₂	1	1	yes	maximal, characteristic
11	#12595	C₂	1	1	yes	maximal, characteristic
12	C₂×#446	C₄	1	1	yes	characteristic
13	C₂×#446	C₂²	1	1	yes	characteristic
14	#2295	C₄	1	1	yes	characteristic
15	C₂×#364	C₂²	1	2	yes	—
16	#2150	C₂²	1	2	yes	—
17	C₂×#347	C₂²	1	2	yes	—
18	C₂×#347	C₂²	1	2	yes	—
19	C₄×M₃₂	C₂²	1	2	yes	—
20	#1771	C₄	1	1	yes	characteristic
21	#1771	C₄	1	1	yes	characteristic
22	#1771	C₂²	1	1	yes	characteristic
23	#1771	C₂²	1	1	yes	characteristic
24	#1770	C₄	1	1	yes	characteristic
25	#1770	C₂²	1	1	yes	characteristic
26	#1768	C₄	1	1	yes	characteristic
27	#1768	C₄	1	1	yes	characteristic
28	#1767	C₄	1	1	yes	characteristic
29	#1726	C₂²	1	1	yes	characteristic
30	#1726	C₂²	1	1	yes	characteristic
31	C₄×#97	C₂²	1	1	yes	characteristic
32	#1377	C₂²	1	2	yes	—
33	#1377	C₂²	1	2	yes	—
34	#1377	C₂²	1	2	yes	—
35	#1377	C₂²	1	2	yes	—
36	#1376	C₂²	1	1	yes	characteristic
37	#1376	C₂²	1	1	yes	characteristic
38	#1376	C₂²	1	2	yes	—
39	#1376	C₂²	1	1	yes	characteristic
40	#1376	C₂²	1	1	yes	characteristic
41	#1375	C₂²	1	2	yes	—
42	#1375	C₂²	1	2	yes	—
43	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
44	C₂×M₃₂	C₂³	1	2	yes	—
45	C₂×M₃₂	C₂³	1	2	yes	—
46	C₂×M₃₂	C₂³	1	2	yes	—
47	C₂×M₃₂	C₂³	1	2	yes	—
48	#446	—	2	4	no	—
49	#446	—	2	2	no	—
50	#446	—	2	2	no	—
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×#97	C₂×C₄	1	1	yes	characteristic
58	C₂×#97	C₂×C₄	1	1	yes	characteristic
59	C₂×#97	C₂×C₄	1	1	yes	characteristic
60	C₂×#97	C₂×C₄	1	1	yes	characteristic
61	C₂×#97	C₂×C₄	1	1	yes	characteristic
62	C₂×#97	C₂×C₄	1	1	yes	characteristic
63	C₂×#97	C₂³	1	1	yes	characteristic
64	C₂×#97	C₂³	1	1	yes	characteristic
65	C₂×#97	C₂³	1	1	yes	characteristic
66	C₂×#97	C₂³	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂×C₄	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	#391	—	2	1	no	—
74	#389	—	2	1	no	—
75	#387	—	2	1	no	—
76	#387	—	2	1	no	—
77	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
78	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
79	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
80	#376	C₂×C₄	1	1	yes	characteristic
81	#376	C₂×C₄	1	1	yes	characteristic
82	C₄³	C₂×C₄	1	1	yes	characteristic
83	#364	—	2	4	no	—
84	C₄⋊C₁₆	—	2	4	no	—
85	#347	—	2	4	no	—
86	#347	—	2	4	no	—
87	#346	—	2	2	no	—
88	C₄×C₁₆	—	2	2	no	—
89	#336	C₂×C₄	1	1	yes	characteristic
90	#336	C₂×C₄	1	1	yes	characteristic
91	#336	C₂×C₄	1	1	yes	characteristic
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	C₂²×Q₈	M₁₆	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	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	C₂⁴	1	1	yes	characteristic
104	C₄×Q₈	—	2	2	no	—
105	C₄×Q₈	—	2	2	no	—
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₄⋊C₄	—	2	1	no	—
108	C₂×C₄⋊C₄	—	2	1	no	—
109	C₂×C₄⋊C₄	M₁₆	1	1	yes	characteristic
110	C₂×C₄⋊C₄	M₁₆	1	1	yes	characteristic
111	C₂×C₄⋊C₄	M₁₆	1	1	yes	characteristic
112	C₂×C₄²	—	2	1	no	—
113	C₂×C₄²	—	2	1	no	—
114	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
121	M₃₂	—	2	2	no	—
122	M₃₂	—	2	2	no	—
123	M₃₂	—	2	2	no	—
124	M₃₂	—	2	2	no	—
125	M₃₂	—	2	2	no	—
126	M₃₂	—	2	2	no	—
127	M₃₂	—	2	2	no	—
128	M₃₂	#41	1	4	yes	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	#97	—	2	2	no	—
140	#97	—	2	2	no	—
141	#97	—	2	2	no	—
142	#97	—	2	1	no	—
143	#97	—	2	1	no	—
144	#97	—	2	1	no	—
145	#97	—	2	1	no	—
146	#97	—	2	2	no	—
147	#97	—	2	2	no	—
148	#97	—	4	1	no	—
149	#97	—	4	1	no	—
150	#97	M₁₆	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	—	4	1	no	—
156	C₄×C₈	#41	1	2	yes	—
157	C₄×C₈	#41	1	2	yes	—
158	#95	—	2	1	no	—
159	#95	—	2	1	no	—
160	#95	—	2	1	no	—
161	#95	—	2	1	no	—
162	#95	—	2	1	no	—
163	#95	—	2	1	no	—
164	#95	—	2	1	no	—
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	8	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
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	C₂×M₁₆	1	1	yes	characteristic
175	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
176	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
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	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	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	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	#117	1	2	yes	—
197	C₂×C₈	C₂×M₁₆	1	2	yes	—
198	C₂×C₈	C₂×M₁₆	1	2	yes	—
199	C₂×C₈	C₂×#41	1	1	yes	characteristic
200	C₂×C₈	C₂×#41	1	2	yes	—
201	C₂×C₈	C₂×#41	1	1	yes	characteristic
202	C₂×C₈	2₋¹⁺⁴	1	2	yes	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	8	1	no	—
207	C₄⋊C₄	—	8	1	no	—
208	C₄⋊C₄	—	8	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	2	1	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	2	1	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	8	1	no	—
221	C₄²	—	8	1	no	—
222	C₄²	#117	1	2	yes	—
223	C₄²	C₂×M₁₆	1	2	yes	—
224	C₄²	#131	1	2	yes	—
225	C₁₆	—	2	4	no	—
226	C₁₆	—	4	2	no	—
227	C₁₆	—	4	2	no	—
228	C₁₆	—	4	2	no	—
229	C₂³	C₂²×M₁₆	1	1	yes	characteristic
230	Q₈	—	8	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	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	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	#432	1	2	yes	—
254	C₂×C₄	#432	1	2	yes	—
255	C₂×C₄	C₂×#117	1	1	yes	characteristic
256	C₂×C₄	C₄×#41	1	1	yes	characteristic
257	C₂×C₄	#519	1	1	yes	characteristic
258	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
259	C₂×C₄	C₂×#131	1	1	yes	characteristic
260	C₂×C₄	#568	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	2	2	no	—
263	C₈	—	2	2	no	—
264	C₈	—	4	1	no	—
265	C₈	—	4	1	no	—
266	C₈	—	4	1	no	—
267	C₈	—	4	1	no	—
268	C₈	#552	1	2	yes	—
269	C₈	#556	1	2	yes	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	C₂×#432	1	1	yes	characteristic
273	C₂²	#2971	1	1	yes	characteristic
274	C₂²	#2984	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	2	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	1	no	—
282	C₄	—	8	1	no	—
283	C₄	—	8	1	no	—
284	C₄	—	8	1	no	—
285	C₄	#2356	1	1	yes	characteristic
286	C₄	#2925	1	1	yes	characteristic
287	C₄	#2965	1	1	yes	characteristic
288	C₄	#2983	1	1	yes	characteristic
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	#22832	1	1	yes	characteristic
292	C₂	#33426	1	1	yes	characteristic
293	C₂	#33427	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂³	C₂²×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1770 ◁ 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
#20518	2
#13268	2
#13107	2
#12897	2
#12896	2
#12884	2
#12666	2
#12656	2
#12596	2
#12595	2