Subgroups — #496084

Order 512.

448 conjugacy classes of subgroups.

The subgroup lattice has 261 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21676	C₂	1	1	yes	maximal, characteristic
2	#13160	C₂	1	1	yes	maximal, characteristic
3	#13160	C₂	1	1	yes	maximal, characteristic
4	#13104	C₂	1	2	yes	maximal
5	#13104	C₂	1	2	yes	maximal
6	#12880	C₂	1	1	yes	maximal, characteristic
7	#12663	C₂	1	2	yes	maximal
8	#12662	C₂	1	2	yes	maximal
9	#12662	C₂	1	1	yes	maximal, characteristic
10	C₁₆×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
11	#12539	C₂	1	1	yes	maximal, characteristic
12	#2402	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
14	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂×#346	C₂²	1	1	yes	characteristic
18	C₂×#346	C₂²	1	2	yes	—
19	C₂×#346	C₂²	1	1	yes	characteristic
20	C₂×#346	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₁₆	C₂²	1	2	yes	—
24	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
25	#1940	C₄	1	1	yes	characteristic
26	#1940	C₄	1	1	yes	characteristic
27	#1917	C₄	1	2	yes	—
28	#1917	C₄	1	2	yes	—
29	#1832	C₄	1	1	yes	characteristic
30	C₈×C₄⋊C₄	C₂²	1	2	yes	—
31	C₈×C₄⋊C₄	C₂²	1	2	yes	—
32	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
33	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	#1726	C₂²	1	1	yes	characteristic
35	#1374	C₂²	1	2	yes	—
36	#1374	C₂²	1	2	yes	—
37	#1374	C₂²	1	2	yes	—
38	#1374	C₂²	1	2	yes	—
39	#1374	C₂²	1	2	yes	—
40	#1374	C₂²	1	2	yes	—
41	C₂²×C₁₆	C₂³	1	2	yes	—
42	C₂²×C₁₆	C₂³	1	2	yes	—
43	C₂²×C₁₆	C₂³	1	1	yes	characteristic
44	C₂²×C₁₆	C₂³	1	1	yes	characteristic
45	C₂²×C₁₆	C₂³	1	1	yes	characteristic
46	C₂²×C₁₆	C₂³	1	1	yes	characteristic
47	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
51	C₂×#97	C₂×C₄	1	1	yes	characteristic
52	C₂×#97	C₂×C₄	1	1	yes	characteristic
53	C₂×#97	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂³	1	2	yes	—
57	C₂×C₄×C₈	C₂³	1	2	yes	—
58	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
59	#400	C₈	1	2	yes	—
60	#400	C₈	1	2	yes	—
61	#398	C₈	1	2	yes	—
62	#398	C₈	1	2	yes	—
63	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
64	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
65	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
66	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
67	#376	C₂×C₄	1	1	yes	characteristic
68	C₄⋊C₁₆	—	2	4	no	—
69	C₄⋊C₁₆	—	2	4	no	—
70	C₄⋊C₁₆	—	2	2	no	—
71	C₄⋊C₁₆	—	2	2	no	—
72	#346	—	2	2	no	—
73	#346	—	2	2	no	—
74	#346	—	2	4	no	—
75	#346	—	2	2	no	—
76	C₄×C₁₆	—	2	2	no	—
77	C₄×C₁₆	—	2	2	no	—
78	C₄×C₁₆	—	2	4	no	—
79	C₄×C₁₆	Q₈	1	4	yes	—
80	#336	C₂×C₄	1	2	yes	—
81	#336	C₂×C₄	1	2	yes	—
82	#336	C₂×C₄	1	2	yes	—
83	#336	C₂×C₄	1	2	yes	—
84	#336	C₂×C₄	1	2	yes	—
85	#336	C₂×C₄	1	2	yes	—
86	C₂²×C₈	C₂²×C₄	1	2	yes	—
87	C₂²×C₈	C₂²×C₄	1	2	yes	—
88	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₈	C₂⁴	1	1	yes	characteristic
92	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
93	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
94	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
95	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
96	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
97	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
98	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄²	C₂²×C₄	1	2	yes	—
101	C₂×C₄²	C₂²×C₄	1	2	yes	—
102	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₁₆	—	2	1	no	—
104	C₂×C₁₆	—	2	1	no	—
105	C₂×C₁₆	—	2	1	no	—
106	C₂×C₁₆	—	2	1	no	—
107	C₂×C₁₆	—	2	1	no	—
108	C₂×C₁₆	—	2	1	no	—
109	C₂×C₁₆	—	2	2	no	—
110	C₂×C₁₆	—	2	2	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	2	2	no	—
113	C₂×C₁₆	—	2	2	no	—
114	C₂×C₁₆	—	2	2	no	—
115	C₂×C₁₆	C₂×Q₈	1	2	yes	—
116	C₂×C₁₆	C₂×Q₈	1	2	yes	—
117	C₂×C₁₆	#41	1	2	yes	—
118	C₂×C₁₆	#41	1	2	yes	—
119	C₂×C₁₆	#41	1	2	yes	—
120	C₂×C₁₆	#41	1	2	yes	—
121	C₄⋊C₈	—	2	4	no	—
122	C₄⋊C₈	—	2	4	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	#97	—	2	2	no	—
126	#97	—	2	2	no	—
127	#97	—	2	2	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	—	2	1	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	—	2	2	no	—
136	C₄×C₈	C₂×Q₈	1	2	yes	—
137	C₄×C₈	#41	1	2	yes	—
138	#95	C₂×C₈	1	2	yes	—
139	#95	C₂×C₈	1	2	yes	—
140	#95	C₂×C₈	1	1	yes	characteristic
141	#95	C₂×C₈	1	1	yes	characteristic
142	#95	C₂×C₈	1	2	yes	—
143	#95	C₂×C₈	1	2	yes	—
144	#95	C₂×C₈	1	2	yes	—
145	#95	C₂×C₈	1	2	yes	—
146	#95	C₂×C₈	1	1	yes	characteristic
147	#95	C₂×C₈	1	1	yes	characteristic
148	C₂²×C₄	C₂²×C₈	1	2	yes	—
149	C₂²×C₄	C₂²×C₈	1	2	yes	—
150	C₂²×C₄	C₂²×C₈	1	2	yes	—
151	C₂²×C₄	C₂²×C₈	1	2	yes	—
152	C₂²×C₄	C₂²×C₈	1	2	yes	—
153	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
154	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
155	C₂²×C₄	C₂²×C₈	1	2	yes	—
156	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	C₄×Q₈	1	2	yes	—
170	C₂×C₈	#131	1	2	yes	—
171	C₂×C₈	#131	1	2	yes	—
172	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
173	C₂×C₈	C₂×#41	1	1	yes	characteristic
174	C₂×C₈	C₂×#41	1	1	yes	characteristic
175	C₂×C₈	C₂×#41	1	1	yes	characteristic
176	C₂×C₈	C₂×#41	1	1	yes	characteristic
177	C₂×C₈	C₂×#41	1	1	yes	characteristic
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	C₄×Q₈	1	2	yes	—
193	C₄²	#131	1	2	yes	—
194	C₁₆	—	2	2	no	—
195	C₁₆	—	2	2	no	—
196	C₁₆	—	2	2	no	—
197	C₁₆	—	4	2	no	—
198	C₁₆	—	4	2	no	—
199	C₁₆	#125	1	4	yes	—
200	C₂³	C₂³×C₈	1	1	yes	characteristic
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	C₈×Q₈	1	2	yes	—
221	C₂×C₄	C₈×Q₈	1	2	yes	—
222	C₂×C₄	#504	1	4	yes	—
223	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
224	C₂×C₄	C₄×#41	1	1	yes	characteristic
225	C₂×C₄	C₄×#41	1	1	yes	characteristic
226	C₂×C₄	C₂×#131	1	1	yes	characteristic
227	C₂×C₄	C₂×#131	1	1	yes	characteristic
228	C₂×C₄	C₂×#131	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	—	2	2	no	—
231	C₈	—	2	2	no	—
232	C₈	—	4	2	no	—
233	C₈	—	4	2	no	—
234	C₈	C₂×#125	1	1	yes	characteristic
235	C₈	#529	1	1	yes	characteristic
236	C₈	#529	1	1	yes	characteristic
237	C₈	#533	1	1	yes	characteristic
238	C₂²	C₂×C₈×Q₈	1	1	yes	characteristic
239	C₂²	C₈×#41	1	1	yes	characteristic
240	C₂²	C₈×#41	1	1	yes	characteristic
241	C₂²	C₂×#504	1	2	yes	—
242	C₂²	#3407	1	2	yes	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	4	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₄	C₄×#125	1	1	yes	characteristic
252	C₄	#2960	1	1	yes	characteristic
253	C₄	#2966	1	1	yes	characteristic
254	C₄	#2966	1	1	yes	characteristic
255	C₂	C₈×#125	1	1	yes	characteristic
256	C₂	#33474	1	1	yes	characteristic
257	C₂	#33475	1	1	yes	characteristic
258	C₂	#33482	1	2	yes	—
259	C₂	#33483	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ 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
#21676	2
#13160	2
#13104	2
#12880	2
#12663	2
#12662	2
C₁₆×C₄⋊C₄	2
#12539	2