Subgroups — #498774

Order 512.

414 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#125	C₂	1	1	yes	maximal, characteristic
2	#13160	C₂	1	2	yes	maximal
3	#13159	C₂	1	2	yes	maximal
4	#13105	C₂	1	2	yes	maximal
5	#13105	C₂	1	2	yes	maximal
6	#13104	C₂	1	2	yes	maximal
7	#13104	C₂	1	2	yes	maximal
8	#12879	C₂	1	1	yes	maximal, characteristic
9	#12879	C₂	1	1	yes	maximal, characteristic
10	C₄×#125	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
13	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
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	2	yes	—
18	C₂×C₄×C₁₆	C₂²	1	2	yes	—
19	#1939	C₄	1	1	yes	characteristic
20	#1939	C₄	1	1	yes	characteristic
21	#1916	C₄	1	2	yes	—
22	#1916	C₄	1	2	yes	—
23	#1831	C₄	1	1	yes	characteristic
24	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₈×C₄⋊C₄	C₂²	1	2	yes	—
26	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₈×C₄⋊C₄	C₂²	1	2	yes	—
28	C₄²×C₈	C₂²	1	1	yes	characteristic
29	#1374	C₂²	1	2	yes	—
30	#1374	C₂²	1	2	yes	—
31	#1374	C₂²	1	2	yes	—
32	#1374	C₂²	1	2	yes	—
33	#1374	C₂²	1	2	yes	—
34	#1374	C₂²	1	2	yes	—
35	#1374	C₂²	1	2	yes	—
36	#1374	C₂²	1	2	yes	—
37	C₂×#125	C₈	1	1	yes	characteristic
38	C₂²×C₁₆	C₂³	1	2	yes	—
39	C₂²×C₁₆	C₂³	1	2	yes	—
40	C₂²×C₁₆	C₂³	1	2	yes	—
41	C₂²×C₁₆	C₂³	1	2	yes	—
42	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₂×C₄×C₈	C₂×C₄	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	C₂³	1	2	yes	—
50	C₂×C₄×C₈	C₂³	1	2	yes	—
51	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	#389	C₈	1	1	yes	characteristic
55	#389	C₈	1	1	yes	characteristic
56	#387	C₈	1	2	yes	—
57	#387	C₈	1	2	yes	—
58	#382	C₈	1	1	yes	characteristic
59	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
60	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
61	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
62	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
63	C₄³	C₂×C₄	1	1	yes	characteristic
64	C₄⋊C₁₆	—	4	2	no	—
65	C₄⋊C₁₆	—	4	2	no	—
66	C₄⋊C₁₆	—	4	1	no	—
67	C₄⋊C₁₆	—	4	1	no	—
68	C₄⋊C₁₆	—	4	1	no	—
69	C₄⋊C₁₆	—	4	1	no	—
70	#346	—	4	2	no	—
71	C₄×C₁₆	—	4	2	no	—
72	#336	C₂×C₄	1	2	yes	—
73	#336	C₂×C₄	1	2	yes	—
74	#336	C₂×C₄	1	2	yes	—
75	#336	C₂×C₄	1	2	yes	—
76	#336	C₂×C₄	1	2	yes	—
77	#336	C₂×C₄	1	2	yes	—
78	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₈	C₂²×C₄	1	2	yes	—
80	C₂²×C₈	C₂²×C₄	1	2	yes	—
81	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₈	C₂⁴	1	1	yes	characteristic
84	#125	—	2	4	no	—
85	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
86	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
87	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
89	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
90	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
91	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
92	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
93	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
94	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
95	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
96	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
97	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄²	C₂²×C₄	1	2	yes	—
100	C₂×C₄²	C₂²×C₄	1	2	yes	—
101	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	2	2	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	2	no	—
113	C₂×C₁₆	—	2	2	no	—
114	C₄⋊C₈	—	4	1	no	—
115	C₄⋊C₈	—	4	1	no	—
116	C₄⋊C₈	—	4	2	no	—
117	C₄⋊C₈	—	4	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	1	no	—
121	C₄×C₈	—	2	1	no	—
122	C₄×C₈	—	2	1	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	1	no	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	—	4	1	no	—
129	C₄×C₈	—	4	1	no	—
130	C₄×C₈	—	4	1	no	—
131	#95	C₂×C₈	1	2	yes	—
132	#95	C₂×C₈	1	2	yes	—
133	#95	C₂×C₈	1	2	yes	—
134	#95	C₂×C₈	1	2	yes	—
135	#95	C₂×C₈	1	2	yes	—
136	#95	C₂×C₈	1	2	yes	—
137	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
138	C₂²×C₄	C₂²×C₈	1	2	yes	—
139	C₂²×C₄	C₂²×C₈	1	2	yes	—
140	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
141	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
142	C₂²×C₄	C₂²×C₈	1	2	yes	—
143	C₂²×C₄	C₂²×C₈	1	2	yes	—
144	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
145	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
146	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
147	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
165	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
166	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
167	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
168	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	M₃₂	1	4	yes	—
177	C₄²	—	2	1	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	4	1	no	—
191	C₁₆	—	4	2	no	—
192	C₁₆	—	4	2	no	—
193	C₁₆	—	4	2	no	—
194	C₁₆	—	4	2	no	—
195	C₂³	C₂³×C₈	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	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	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	C₂×M₃₂	1	2	yes	—
223	C₂×C₄	C₂×M₃₂	1	4	yes	—
224	C₂×C₄	#504	1	2	yes	—
225	C₂×C₄	#504	1	2	yes	—
226	C₂×C₄	#519	1	1	yes	characteristic
227	C₂×C₄	#520	1	1	yes	characteristic
228	C₂×C₄	#520	1	1	yes	characteristic
229	C₂×C₄	#568	1	1	yes	characteristic
230	C₂×C₄	#568	1	1	yes	characteristic
231	C₂×C₄	#568	1	1	yes	characteristic
232	C₈	—	4	1	no	—
233	C₈	—	4	1	no	—
234	C₈	—	4	1	no	—
235	C₈	—	4	2	no	—
236	C₈	—	4	2	no	—
237	C₈	#562	1	1	yes	characteristic
238	C₈	#563	1	1	yes	characteristic
239	C₈	#563	1	1	yes	characteristic
240	C₈	#564	1	1	yes	characteristic
241	C₂²	#2982	1	1	yes	characteristic
242	C₂²	#2988	1	1	yes	characteristic
243	C₂²	#2988	1	1	yes	characteristic
244	C₂²	C₂²×M₃₂	1	1	yes	characteristic
245	C₂²	C₂×#504	1	1	yes	characteristic
246	C₂²	C₂×#504	1	1	yes	characteristic
247	C₂²	#3407	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	4	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	1	no	—
253	C₄	—	4	1	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	2	no	—
258	C₄	#2382	1	1	yes	characteristic
259	C₄	#2987	1	1	yes	characteristic
260	C₄	#2995	1	1	yes	characteristic
261	C₄	#2995	1	1	yes	characteristic
262	C₂	#23369	1	1	yes	characteristic
263	C₂	#33497	1	1	yes	characteristic
264	C₂	#33500	1	1	yes	characteristic
265	C₂	#33504	1	1	yes	characteristic
266	C₂	#33505	1	1	yes	characteristic
267	C₂	#33506	1	1	yes	characteristic
268	C₂	#33506	1	1	yes	characteristic

	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
C₈×#125	2
#13160	2
#13159	2
#13105	2
#13104	2
#12879	2