Subgroups — #497166

Order 512.

418 conjugacy classes of subgroups.

The subgroup lattice has 259 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	#13160	C₂	1	2	yes	maximal
5	#13159	C₂	1	2	yes	maximal
6	#13105	C₂	1	2	yes	maximal
7	#13104	C₂	1	2	yes	maximal
8	#12899	C₂	1	1	yes	maximal, characteristic
9	#12662	C₂	1	1	yes	maximal, characteristic
10	C₁₆×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
11	#12592	C₂	1	1	yes	maximal, characteristic
12	#2402	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₂×#346	C₂²	1	1	yes	characteristic
20	C₂×#346	C₂²	1	2	yes	—
21	C₂×C₄×C₁₆	C₂²	1	2	yes	—
22	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
23	#1940	C₄	1	1	yes	characteristic
24	#1940	C₄	1	1	yes	characteristic
25	#1917	C₄	1	2	yes	—
26	#1917	C₄	1	2	yes	—
27	#1832	C₄	1	1	yes	characteristic
28	C₈×C₄⋊C₄	C₂²	1	2	yes	—
29	C₈×C₄⋊C₄	C₂²	1	2	yes	—
30	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
31	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	#1726	C₂²	1	1	yes	characteristic
33	#1374	C₂²	1	1	yes	characteristic
34	#1374	C₂²	1	1	yes	characteristic
35	#1374	C₂²	1	1	yes	characteristic
36	#1374	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
43	C₂²×C₁₆	C₂³	1	1	yes	characteristic
44	C₂²×C₁₆	C₂³	1	2	yes	—
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	2	yes	—
55	C₂×C₄×C₈	C₂³	1	2	yes	—
56	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
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	2	no	—
69	C₄⋊C₁₆	—	2	2	no	—
70	C₄⋊C₁₆	—	4	2	no	—
71	C₄⋊C₁₆	—	4	2	no	—
72	C₄⋊C₁₆	—	4	2	no	—
73	C₄⋊C₁₆	—	4	2	no	—
74	#346	—	2	2	no	—
75	#346	—	4	2	no	—
76	C₄×C₁₆	—	4	2	no	—
77	C₄×C₁₆	Q₈	1	4	yes	—
78	#336	C₂×C₄	1	2	yes	—
79	#336	C₂×C₄	1	2	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	C₂²×C₈	C₂²×C₄	1	2	yes	—
85	C₂²×C₈	C₂²×C₄	1	2	yes	—
86	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	C₂⁴	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
91	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
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×C₄²	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄²	C₂²×C₄	1	2	yes	—
99	C₂×C₄²	C₂²×C₄	1	2	yes	—
100	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	—	2	1	no	—
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	2	no	—
108	C₂×C₁₆	—	2	2	no	—
109	C₂×C₁₆	—	2	2	no	—
110	C₂×C₁₆	—	2	2	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	2	1	no	—
113	C₂×C₁₆	—	2	1	no	—
114	C₂×C₁₆	—	2	2	no	—
115	C₂×C₁₆	—	2	1	no	—
116	C₂×C₁₆	—	2	1	no	—
117	C₂×C₁₆	C₂×Q₈	1	2	yes	—
118	C₂×C₁₆	C₂×Q₈	1	2	yes	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	4	2	no	—
122	C₄⋊C₈	—	4	2	no	—
123	#97	—	2	2	no	—
124	#97	—	4	1	no	—
125	#97	—	4	1	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	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	2	no	—
132	C₄×C₈	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₄×C₈	C₂×Q₈	1	2	yes	—
135	C₄×C₈	#41	1	2	yes	—
136	#95	C₂×C₈	1	2	yes	—
137	#95	C₂×C₈	1	2	yes	—
138	#95	C₂×C₈	1	1	yes	characteristic
139	#95	C₂×C₈	1	1	yes	characteristic
140	#95	C₂×C₈	1	2	yes	—
141	#95	C₂×C₈	1	2	yes	—
142	#95	C₂×C₈	1	2	yes	—
143	#95	C₂×C₈	1	2	yes	—
144	#95	C₂×C₈	1	1	yes	characteristic
145	#95	C₂×C₈	1	1	yes	characteristic
146	C₂²×C₄	C₂²×C₈	1	2	yes	—
147	C₂²×C₄	C₂²×C₈	1	2	yes	—
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	1	yes	characteristic
152	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
153	C₂²×C₄	C₂²×C₈	1	2	yes	—
154	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	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	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	C₄×Q₈	1	2	yes	—
170	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
171	C₂×C₈	C₂×#41	1	1	yes	characteristic
172	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
173	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
174	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
175	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	C₄×Q₈	1	2	yes	—
191	C₄²	#131	1	2	yes	—
192	C₁₆	—	2	2	no	—
193	C₁₆	—	2	2	no	—
194	C₁₆	—	4	1	no	—
195	C₁₆	—	4	2	no	—
196	C₁₆	—	4	2	no	—
197	C₁₆	—	4	1	no	—
198	C₂³	C₂³×C₈	1	1	yes	characteristic
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
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	C₈×Q₈	1	2	yes	—
219	C₂×C₄	C₈×Q₈	1	2	yes	—
220	C₂×C₄	#504	1	4	yes	—
221	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
222	C₂×C₄	#520	1	1	yes	characteristic
223	C₂×C₄	#520	1	1	yes	characteristic
224	C₂×C₄	C₂×#131	1	1	yes	characteristic
225	C₂×C₄	#568	1	1	yes	characteristic
226	C₂×C₄	#568	1	1	yes	characteristic
227	C₈	—	2	2	no	—
228	C₈	—	4	1	no	—
229	C₈	—	4	1	no	—
230	C₈	—	4	2	no	—
231	C₈	—	4	2	no	—
232	C₈	#540	1	1	yes	characteristic
233	C₈	#541	1	1	yes	characteristic
234	C₈	#544	1	1	yes	characteristic
235	C₈	#544	1	1	yes	characteristic
236	C₂²	C₂×C₈×Q₈	1	1	yes	characteristic
237	C₂²	#2988	1	1	yes	characteristic
238	C₂²	#2988	1	1	yes	characteristic
239	C₂²	C₂×#504	1	2	yes	—
240	C₂²	#3407	1	2	yes	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	4	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	2	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	#2330	1	1	yes	characteristic
250	C₄	#2961	1	1	yes	characteristic
251	C₄	#2993	1	1	yes	characteristic
252	C₄	#2993	1	1	yes	characteristic
253	C₂	#22299	1	1	yes	characteristic
254	C₂	#33474	1	1	yes	characteristic
255	C₂	#33475	1	1	yes	characteristic
256	C₂	#33506	1	2	yes	—
257	C₂	#33507	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
#13159	2
#13105	2
#13104	2
#12899	2
#12662	2
C₁₆×C₄⋊C₄	2
#12592	2