Subgroups — #127096

Order 512.

450 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	#33418	C₂	1	1	yes	maximal, characteristic
2	#33418	C₂	1	1	yes	maximal, characteristic
3	#20473	C₂	1	1	yes	maximal, characteristic
4	#7201	C₂	1	4	yes	maximal
5	C₂×#431	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₄×#117	C₄	1	1	yes	characteristic
8	#2163	—	2	1	no	—
9	#2163	—	2	1	no	—
10	#2163	—	2	1	no	—
11	#2163	C₄	1	2	yes	complemented
12	#2162	—	2	1	no	—
13	#2162	—	2	1	no	—
14	#2162	—	2	1	no	—
15	#2162	C₄	1	2	yes	complemented
16	#2150	—	2	1	no	—
17	#2150	C₄	1	2	yes	complemented
18	#2112	C₂²	1	2	yes	—
19	#2112	C₂²	1	2	yes	—
20	C₄×M₃₂	—	2	1	no	—
21	C₄×M₃₂	C₄	1	2	yes	complemented
22	#1770	—	2	1	no	—
23	#1770	—	2	1	no	—
24	#1752	C₄	1	2	yes	—
25	#1752	C₂²	1	2	yes	—
26	#1744	—	2	1	no	—
27	C₄×#97	—	2	1	no	—
28	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₈	C₂³	1	1	yes	characteristic
30	C₂×#117	C₈	1	2	yes	—
31	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂×M₃₂	—	2	2	no	—
33	C₂×M₃₂	C₂×C₄	1	4	yes	—
34	#431	—	2	1	no	—
35	#431	—	2	1	no	—
36	#431	C₈	1	2	yes	—
37	#431	C₈	1	2	yes	—
38	C₂×C₄⋊C₈	—	2	1	no	—
39	C₂×C₂²⋊C₈	C₂×C₄	1	2	yes	—
40	C₂×#97	—	2	2	no	—
41	C₂×#97	—	2	2	no	—
42	C₂×C₄×C₈	—	2	1	no	—
43	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	D₄	1	1	yes	characteristic
49	C₂×C₄×C₈	D₄	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	Q₈	1	1	yes	characteristic
54	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
55	C₄×C₄⋊C₄	—	2	1	no	—
56	C₄×C₄⋊C₄	—	2	1	no	—
57	C₄×C₂²⋊C₄	C₈	1	2	yes	—
58	C₄×C₂²⋊C₄	C₂×C₄	1	2	yes	—
59	#376	—	2	1	no	—
60	C₄³	—	2	1	no	—
61	C₄⋊C₁₆	—	2	2	no	—
62	C₄⋊C₁₆	—	2	2	no	—
63	C₄⋊C₁₆	—	2	2	no	—
64	C₄⋊C₁₆	—	2	2	no	—
65	C₂²⋊C₁₆	—	2	2	no	—
66	C₂²⋊C₁₆	—	2	2	no	—
67	C₂²⋊C₁₆	—	2	2	no	—
68	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
69	#346	—	2	2	no	—
70	#346	—	2	2	no	—
71	C₄×C₁₆	—	2	2	no	—
72	C₄×C₁₆	—	2	2	no	—
73	#336	—	2	2	no	—
74	#336	—	2	2	no	—
75	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
76	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
77	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₈	—	2	2	no	—
79	C₂²×C₈	—	4	2	no	—
80	C₂²×C₈	—	4	2	no	—
81	C₂²×C₈	C₄²	1	1	yes	characteristic
82	C₂²×C₈	C₄²	1	1	yes	characteristic
83	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₈	C₄⋊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	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
93	#117	—	4	4	no	—
94	C₂×C₄⋊C₄	—	2	2	no	—
95	C₂×C₂²⋊C₄	C₂×C₈	1	4	yes	—
96	C₂×C₄²	—	2	2	no	—
97	C₂×C₄²	—	2	2	no	—
98	C₂×C₄²	—	2	2	no	—
99	C₂×C₄²	C₄²	1	1	yes	characteristic
100	C₂×C₄²	C₄²	1	1	yes	characteristic
101	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
106	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
107	C₂×C₄²	M₁₆	1	1	yes	characteristic
108	C₂×C₄²	M₁₆	1	1	yes	characteristic
109	C₂×C₄²	M₁₆	1	1	yes	characteristic
110	C₂×C₄²	M₁₆	1	1	yes	characteristic
111	M₃₂	—	4	4	no	—
112	M₃₂	—	4	4	no	—
113	C₂×C₁₆	—	2	4	no	—
114	C₂×C₁₆	—	2	4	no	—
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₂²⋊C₈	—	2	2	no	—
118	C₂²⋊C₈	C₂×C₈	1	4	yes	—
119	#97	—	4	4	no	—
120	#97	—	4	4	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
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	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
139	#95	—	2	2	no	—
140	#95	—	2	2	no	—
141	C₂⁴	C₂²×C₈	1	1	yes	characteristic
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	4	1	no	—
145	C₂²×C₄	—	4	1	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
148	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
149	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
150	C₂²×C₄	C₄×C₈	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×C₄²	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
163	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
164	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
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	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	4	1	no	—
173	C₂×C₈	—	4	1	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	#95	1	1	yes	characteristic
181	C₂×C₈	#95	1	1	yes	characteristic
182	C₂×C₈	#95	1	1	yes	characteristic
183	C₂×C₈	#95	1	1	yes	characteristic
184	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₄⋊C₄	—	4	4	no	—
189	C₂²⋊C₄	—	4	4	no	—
190	C₄²	—	4	2	no	—
191	C₄²	—	4	2	no	—
192	C₄²	—	4	4	no	—
193	C₄²	—	4	4	no	—
194	C₄²	#95	1	1	yes	characteristic
195	C₄²	#95	1	1	yes	characteristic
196	C₄²	#95	1	1	yes	characteristic
197	C₄²	#95	1	1	yes	characteristic
198	C₄²	#97	1	1	yes	characteristic
199	C₄²	#97	1	1	yes	characteristic
200	C₄²	#97	1	1	yes	characteristic
201	C₄²	#97	1	1	yes	characteristic
202	C₄²	C₂²⋊C₈	1	1	yes	characteristic
203	C₄²	C₂²⋊C₈	1	1	yes	characteristic
204	C₄²	C₂²⋊C₈	1	1	yes	characteristic
205	C₄²	C₂²⋊C₈	1	1	yes	characteristic
206	C₄²	C₄⋊C₈	1	1	yes	characteristic
207	C₄²	C₄⋊C₈	1	1	yes	characteristic
208	C₄²	C₄⋊C₈	1	1	yes	characteristic
209	C₄²	C₄⋊C₈	1	1	yes	characteristic
210	C₁₆	—	4	4	no	—
211	C₁₆	—	4	4	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
216	C₂³	C₂×C₂²⋊C₈	1	1	yes	characteristic
217	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	#336	1	1	yes	characteristic
242	C₂×C₄	#336	1	1	yes	characteristic
243	C₂×C₄	#336	1	1	yes	characteristic
244	C₂×C₄	#336	1	1	yes	characteristic
245	C₂×C₄	#336	1	1	yes	characteristic
246	C₂×C₄	#336	1	1	yes	characteristic
247	C₂×C₄	#336	1	1	yes	characteristic
248	C₂×C₄	#336	1	1	yes	characteristic
249	C₂×C₄	#349	1	4	yes	—
250	C₂×C₄	C₂×#95	1	1	yes	characteristic
251	C₂×C₄	C₂×#97	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	4	4	no	—
261	C₈	#344	1	2	yes	—
262	C₈	#410	1	2	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	4	4	no	—
272	C₂²	C₂×#336	1	1	yes	characteristic
273	C₂²	C₂×#349	1	2	yes	—
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	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	8	4	no	—
283	C₄	#1377	1	2	yes	—
284	C₄	#1377	1	2	yes	—
285	C₄	#1378	1	2	yes	—
286	C₄	#1740	1	1	yes	characteristic
287	C₄	C₂×#344	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	4	1	no	—
290	C₂	—	4	1	no	—
291	C₂	#7924	1	1	yes	characteristic
292	C₂	C₂×#1377	1	1	yes	characteristic
293	C₂	C₂×#1378	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#95
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
11. #2163	C₄	semidirect
15. #2162	C₄	semidirect
17. #2150	C₄	semidirect
21. C₄×M₃₂	C₄	semidirect

Maximal subgroups

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

subgroup	index
#33418	2
#20473	2
#7201	2