Subgroups — #127167

Order 512.

472 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	C₂×#2162	C₂	1	1	yes	maximal, characteristic
2	C₂×#2162	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#7209	C₂	1	4	yes	maximal, complemented
5	C₂×#431	C₄	1	1	yes	characteristic
6	C₂×C₄×M₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	#2162	—	2	1	no	—
9	#2162	—	2	1	no	—
10	#2162	—	2	1	no	—
11	#2162	—	2	1	no	—
12	#2162	—	2	1	no	—
13	#2162	—	2	1	no	—
14	#2162	C₄	1	2	yes	—
15	#2162	C₂²	1	2	yes	—
16	C₂×C₄⋊C₁₆	—	2	1	no	—
17	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₁₆	C₄	1	2	yes	—
19	C₂×C₂²⋊C₁₆	C₄	1	2	yes	—
20	C₂×C₄×C₁₆	—	2	1	no	—
21	C₂×C₄×C₁₆	C₄	1	2	yes	—
22	C₈×M₁₆	—	2	1	no	—
23	C₈×M₁₆	—	2	1	no	—
24	C₈×M₁₆	C₄	1	2	yes	—
25	C₈×M₁₆	C₂²	1	2	yes	—
26	C₂×#322	—	2	1	no	—
27	C₂×C₈²	—	2	1	no	—
28	C₂²×M₁₆	C₈	1	2	yes	—
29	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₁₆	—	2	2	no	—
33	C₂²×C₁₆	C₂×C₄	1	4	yes	—
34	#431	—	2	1	no	—
35	#431	—	2	1	no	—
36	#431	C₈	1	2	yes	—
37	#431	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₈	—	2	1	no	—
39	C₂×C₂²⋊C₈	C₈	1	2	yes	—
40	C₄×M₁₆	—	2	1	no	—
41	C₄×M₁₆	—	2	1	no	—
42	C₄×M₁₆	C₈	1	2	yes	—
43	C₄×M₁₆	C₂×C₄	1	2	yes	—
44	C₂×#97	—	2	1	no	—
45	C₂×C₄×C₈	—	2	1	no	—
46	C₂×C₄×C₈	—	2	1	no	—
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	1	yes	characteristic
50	C₂×C₄×C₈	D₄	1	1	yes	characteristic
51	C₂×C₄×C₈	D₄	1	1	yes	characteristic
52	C₂×C₄×C₈	D₄	1	1	yes	characteristic
53	C₂×C₄×C₈	D₄	1	1	yes	characteristic
54	C₂×C₄×C₈	D₄	1	1	yes	characteristic
55	C₂×C₄×C₈	D₄	1	1	yes	characteristic
56	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
57	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
59	C₄⋊C₁₆	—	2	2	no	—
60	C₄⋊C₁₆	—	2	2	no	—
61	C₄⋊C₁₆	—	2	2	no	—
62	C₄⋊C₁₆	—	2	2	no	—
63	C₂²⋊C₁₆	—	2	2	no	—
64	C₂²⋊C₁₆	—	2	2	no	—
65	C₂²⋊C₁₆	—	2	2	no	—
66	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
67	C₄×C₁₆	—	2	2	no	—
68	C₄×C₁₆	—	2	2	no	—
69	C₄×C₁₆	—	2	2	no	—
70	C₄×C₁₆	—	2	2	no	—
71	#322	—	2	2	no	—
72	#322	—	2	2	no	—
73	C₈²	—	2	2	no	—
74	C₈²	—	2	2	no	—
75	C₂³×C₄	C₄²	1	1	yes	characteristic
76	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
77	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
78	C₂×M₁₆	—	2	2	no	—
79	C₂×M₁₆	—	2	4	no	—
80	C₂×M₁₆	C₁₆	1	8	yes	complemented
81	C₂×M₁₆	C₂×C₈	1	4	yes	—
82	C₂²×C₈	—	2	2	no	—
83	C₂²×C₈	—	2	2	no	—
84	C₂²×C₈	—	4	2	no	—
85	C₂²×C₈	—	4	2	no	—
86	C₂²×C₈	C₄²	1	1	yes	characteristic
87	C₂²×C₈	C₄²	1	1	yes	characteristic
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	C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄²	C₄²	1	1	yes	characteristic
99	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
104	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
105	C₂×C₄²	M₁₆	1	1	yes	characteristic
106	C₂×C₄²	M₁₆	1	1	yes	characteristic
107	C₂×C₄²	M₁₆	1	1	yes	characteristic
108	C₂×C₄²	M₁₆	1	1	yes	characteristic
109	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₁₆	—	2	4	no	—
111	C₂×C₁₆	—	2	4	no	—
112	C₂×C₁₆	—	4	4	no	—
113	C₂×C₁₆	—	4	4	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	2	2	no	—
116	C₂²⋊C₈	—	2	2	no	—
117	C₂²⋊C₈	C₂×C₈	1	4	yes	—
118	#97	—	2	2	no	—
119	#97	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
125	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
126	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
127	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
128	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
129	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
130	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
132	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
135	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
136	C₄×C₈	C₂×D₄	1	1	yes	characteristic
137	C₄×C₈	C₂×D₄	1	1	yes	characteristic
138	C₄×C₈	C₂×D₄	1	1	yes	characteristic
139	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
140	C₂⁴	C₄×C₈	1	1	yes	characteristic
141	C₂²×C₄	—	4	1	no	—
142	C₂²×C₄	—	4	1	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	#95	1	1	yes	characteristic
145	C₂²×C₄	#95	1	1	yes	characteristic
146	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
147	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
148	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
149	C₂²×C₄	#97	1	1	yes	characteristic
150	C₂²×C₄	#97	1	1	yes	characteristic
151	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
152	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
153	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
154	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
155	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
156	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
157	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
158	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₁₆	1	4	yes	—
160	C₂²×C₄	M₃₂	1	4	yes	—
161	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
162	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
163	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
164	M₁₆	—	2	8	no	—
165	M₁₆	—	2	4	no	—
166	M₁₆	C₂×C₁₆	1	8	yes	complemented
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	4	4	no	—
183	C₂×C₈	#95	1	1	yes	characteristic
184	C₂×C₈	#95	1	1	yes	characteristic
185	C₂×C₈	#95	1	1	yes	characteristic
186	C₂×C₈	#95	1	1	yes	characteristic
187	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
188	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
189	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₄²	#95	1	1	yes	characteristic
192	C₄²	#95	1	1	yes	characteristic
193	C₄²	#97	1	1	yes	characteristic
194	C₄²	#97	1	1	yes	characteristic
195	C₄²	C₂²⋊C₈	1	1	yes	characteristic
196	C₄²	C₂²⋊C₈	1	1	yes	characteristic
197	C₄²	C₂²⋊C₈	1	1	yes	characteristic
198	C₄²	C₂²⋊C₈	1	1	yes	characteristic
199	C₄²	C₄⋊C₈	1	1	yes	characteristic
200	C₄²	C₄⋊C₈	1	1	yes	characteristic
201	C₄²	C₄⋊C₈	1	1	yes	characteristic
202	C₄²	C₄⋊C₈	1	1	yes	characteristic
203	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
204	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
205	C₄²	C₂×M₁₆	1	1	yes	characteristic
206	C₄²	C₂×M₁₆	1	1	yes	characteristic
207	C₁₆	—	4	4	no	—
208	C₁₆	—	4	4	no	—
209	C₂³	—	4	1	no	—
210	C₂³	—	4	1	no	—
211	C₂³	#336	1	1	yes	characteristic
212	C₂³	#336	1	1	yes	characteristic
213	C₂³	C₄×C₁₆	1	2	yes	—
214	C₂³	#346	1	2	yes	—
215	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	#336	1	1	yes	characteristic
232	C₂×C₄	#336	1	1	yes	characteristic
233	C₂×C₄	#336	1	1	yes	characteristic
234	C₂×C₄	#336	1	1	yes	characteristic
235	C₂×C₄	#336	1	1	yes	characteristic
236	C₂×C₄	#336	1	1	yes	characteristic
237	C₂×C₄	C₄×C₁₆	1	2	yes	—
238	C₂×C₄	#346	1	2	yes	—
239	C₂×C₄	C₂²⋊C₁₆	1	4	yes	—
240	C₂×C₄	C₂²⋊C₁₆	1	4	yes	—
241	C₂×C₄	C₄⋊C₁₆	1	4	yes	—
242	C₂×C₄	C₄⋊C₁₆	1	4	yes	—
243	C₂×C₄	C₂×#95	1	1	yes	characteristic
244	C₂×C₄	C₂×#97	1	1	yes	characteristic
245	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
246	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
247	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
249	C₂×C₄	C₂²×C₁₆	1	2	yes	—
250	C₂×C₄	C₂×M₃₂	1	2	yes	—
251	C₈	—	2	8	no	—
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	4	4	no	—
259	C₈	#344	1	2	yes	—
260	C₈	#413	1	2	yes	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	#1374	1	2	yes	—
270	C₂²	#1374	1	2	yes	—
271	C₂²	C₂×#336	1	1	yes	characteristic
272	C₂²	C₂×C₄×C₁₆	1	1	yes	characteristic
273	C₂²	C₂×#346	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	#1374	1	2	yes	—
281	C₄	#1374	1	2	yes	—
282	C₄	#1378	1	2	yes	—
283	C₄	#1742	1	1	yes	characteristic
284	C₄	C₂×#344	1	1	yes	characteristic
285	C₄	C₂×C₂²⋊C₁₆	1	2	yes	—
286	C₄	C₂×C₄⋊C₁₆	1	2	yes	—
287	C₂	—	2	1	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	#7932	1	1	yes	characteristic
292	C₂	C₂×#1374	1	1	yes	characteristic
293	C₂	C₂×#1378	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²⋊C₄
commutator	C₂²	C₂×C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₄×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #7209	C₂	semidirect
80. C₂×M₁₆	C₁₆	semidirect
166. M₁₆	C₂×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2162	2
C₂×C₈×M₁₆	2
#7209	2