Subgroups — #126813

Order 512.

503 conjugacy classes of subgroups.

The subgroup lattice has 304 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	C₂×#2150	C₂	1	1	yes	maximal, characteristic
3	C₈×#117	C₂	1	1	yes	maximal, characteristic
4	#7169	C₂	1	4	yes	maximal, complemented
5	C₂²×M₃₂	—	2	1	no	—
6	C₂×#431	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₄×#117	C₄	1	1	yes	characteristic
9	#2163	C₄	1	2	yes	complemented
10	#2163	C₄	1	2	yes	complemented
11	#2162	—	2	1	no	—
12	#2162	—	2	1	no	—
13	#2150	—	2	1	no	—
14	#2150	—	2	1	no	—
15	#2150	—	2	1	no	—
16	#2150	C₄	1	2	yes	complemented
17	#2150	C₂²	1	2	yes	—
18	C₂×C₄⋊C₁₆	—	2	1	no	—
19	C₂×C₄⋊C₁₆	C₄	1	2	yes	complemented
20	#2112	—	2	1	no	—
21	C₄×M₃₂	C₂²	1	2	yes	—
22	C₈×C₄⋊C₄	C₄	1	2	yes	—
23	C₈×C₄⋊C₄	C₂²	1	2	yes	—
24	C₈×C₂²⋊C₄	—	2	1	no	—
25	C₈×C₂²⋊C₄	—	2	1	no	—
26	#1744	—	2	1	no	—
27	C₄²×C₈	—	2	1	no	—
28	C₂³×C₈	D₄	1	1	yes	characteristic
29	C₂³×C₈	D₄	1	1	yes	characteristic
30	C₂×#117	—	2	1	no	—
31	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂×M₃₂	—	2	2	no	—
33	C₂×M₃₂	—	2	2	no	—
34	C₂×M₃₂	—	2	2	no	—
35	C₂×M₃₂	—	4	4	no	—
36	C₂²×C₁₆	—	2	2	no	—
37	#431	—	2	1	no	—
38	#431	—	2	1	no	—
39	#431	C₈	1	2	yes	—
40	#431	C₈	1	2	yes	—
41	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₈	—	2	1	no	—
43	C₂×C₄×C₈	—	2	1	no	—
44	C₂×C₄×C₈	—	2	2	no	—
45	C₂×C₄×C₈	—	2	2	no	—
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	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂×C₄	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	Q₈	1	1	yes	characteristic
56	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
58	C₄×C₄⋊C₄	—	2	1	no	—
59	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
60	C₄×C₂²⋊C₄	—	2	1	no	—
61	C₄×C₂²⋊C₄	C₈	1	2	yes	—
62	#376	C₈	1	2	yes	—
63	C₄³	—	2	1	no	—
64	C₄⋊C₁₆	—	2	2	no	—
65	C₄⋊C₁₆	—	2	2	no	—
66	C₄⋊C₁₆	—	2	2	no	—
67	C₄⋊C₁₆	—	2	2	no	—
68	C₄⋊C₁₆	—	2	2	no	—
69	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
70	C₂²⋊C₁₆	—	2	2	no	—
71	C₂²⋊C₁₆	—	2	2	no	—
72	#346	C₂×C₄	1	4	yes	—
73	C₄×C₁₆	—	2	2	no	—
74	#336	—	2	2	no	—
75	#336	—	2	2	no	—
76	C₂³×C₄	C₂²⋊C₄	1	1	yes	characteristic
77	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
78	C₂³×C₄	M₁₆	1	1	yes	characteristic
79	C₂²×C₈	—	2	2	no	—
80	C₂²×C₈	—	2	2	no	—
81	C₂²×C₈	—	2	2	no	—
82	C₂²×C₈	—	4	2	no	—
83	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
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	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
95	#117	—	2	8	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₂²⋊C₄	—	2	2	no	—
98	C₂×C₄²	—	2	2	no	—
99	C₂×C₄²	—	2	2	no	—
100	C₂×C₄²	—	2	2	no	—
101	C₂×C₄²	C₄²	1	1	yes	characteristic
102	C₂×C₄²	C₄²	1	1	yes	characteristic
103	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
107	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
108	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
109	C₂×C₄²	M₁₆	1	1	yes	characteristic
110	C₂×C₄²	M₁₆	1	1	yes	characteristic
111	C₂×C₄²	M₁₆	1	1	yes	characteristic
112	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
113	M₃₂	—	4	4	no	—
114	M₃₂	—	4	4	no	—
115	M₃₂	—	4	4	no	—
116	C₂×C₁₆	—	2	4	no	—
117	C₂×C₁₆	—	2	4	no	—
118	C₂×C₁₆	—	4	4	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	C₂×C₈	1	4	yes	—
121	C₂²⋊C₈	—	2	2	no	—
122	C₂²⋊C₈	—	2	2	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	4	4	no	—
126	C₄×C₈	—	4	2	no	—
127	C₄×C₈	—	4	2	no	—
128	C₄×C₈	C₄²	1	1	yes	characteristic
129	C₄×C₈	C₄²	1	1	yes	characteristic
130	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
131	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
132	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
133	C₄×C₈	C₂²⋊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	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
141	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
142	C₄×C₈	C₂×D₄	1	1	yes	characteristic
143	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
144	#95	—	2	2	no	—
145	#95	C₂×C₈	1	4	yes	—
146	C₂⁴	C₂²⋊C₈	1	1	yes	characteristic
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	#95	1	1	yes	characteristic
153	C₂²×C₄	#95	1	1	yes	characteristic
154	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
155	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
156	C₂²×C₄	#97	1	1	yes	characteristic
157	C₂²×C₄	#97	1	1	yes	characteristic
158	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
159	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
160	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
161	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
162	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
164	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
165	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
166	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
169	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	#95	1	1	yes	characteristic
187	C₂×C₈	#95	1	1	yes	characteristic
188	C₂×C₈	#95	1	1	yes	characteristic
189	C₂×C₈	#95	1	1	yes	characteristic
190	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
191	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
192	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
193	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
194	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
195	C₂×C₈	#137	1	2	yes	—
196	C₄⋊C₄	—	2	8	no	—
197	C₂²⋊C₄	—	2	8	no	—
198	C₄²	—	2	8	no	—
199	C₄²	—	4	4	no	—
200	C₄²	—	4	2	no	—
201	C₄²	—	4	2	no	—
202	C₄²	#95	1	1	yes	characteristic
203	C₄²	#95	1	1	yes	characteristic
204	C₄²	C₄×C₈	1	1	yes	characteristic
205	C₄²	C₄×C₈	1	1	yes	characteristic
206	C₄²	#97	1	1	yes	characteristic
207	C₄²	#97	1	1	yes	characteristic
208	C₄²	C₂²⋊C₈	1	1	yes	characteristic
209	C₄²	C₂²⋊C₈	1	1	yes	characteristic
210	C₄²	C₄⋊C₈	1	1	yes	characteristic
211	C₄²	C₄⋊C₈	1	1	yes	characteristic
212	C₄²	C₄⋊C₈	1	1	yes	characteristic
213	C₄²	C₄⋊C₈	1	1	yes	characteristic
214	C₄²	C₂×C₄²	1	1	yes	characteristic
215	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
216	C₄²	C₂²×C₈	1	1	yes	characteristic
217	C₄²	C₂×M₁₆	1	1	yes	characteristic
218	C₁₆	—	4	4	no	—
219	C₁₆	—	4	4	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	4	1	no	—
223	C₂³	#336	1	1	yes	characteristic
224	C₂³	#336	1	1	yes	characteristic
225	C₂³	C₂×C₂²⋊C₈	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	#336	1	1	yes	characteristic
248	C₂×C₄	#336	1	1	yes	characteristic
249	C₂×C₄	#336	1	1	yes	characteristic
250	C₂×C₄	#336	1	1	yes	characteristic
251	C₂×C₄	#336	1	1	yes	characteristic
252	C₂×C₄	#336	1	1	yes	characteristic
253	C₂×C₄	#364	1	4	yes	—
254	C₂×C₄	C₂×#95	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
256	C₂×C₄	C₂×#97	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
258	C₂×C₄	#417	1	2	yes	—
259	C₂×C₄	#421	1	2	yes	—
260	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	4	4	no	—
269	C₈	#418	1	1	yes	characteristic
270	C₈	#419	1	1	yes	characteristic
271	C₈	#428	1	2	yes	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	4	4	no	—
280	C₂²	#1376	1	2	yes	—
281	C₂²	#1477	1	2	yes	—
282	C₂²	C₂×#336	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	8	no	—
291	C₄	#1376	1	2	yes	—
292	C₄	#1378	1	2	yes	—
293	C₄	#1477	1	2	yes	—
294	C₄	#1733	1	1	yes	characteristic
295	C₄	#1736	1	1	yes	characteristic
296	C₄	C₂×#364	1	2	yes	—
297	C₂	—	2	2	no	—
298	C₂	—	2	2	no	—
299	C₂	—	4	1	no	—
300	C₂	#7891	1	1	yes	characteristic
301	C₂	C₂×#1376	1	1	yes	characteristic
302	C₂	C₂×#1378	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×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. #7169	C₂	semidirect
9. #2163	C₄	semidirect
10. #2163	C₄	semidirect
16. #2150	C₄	semidirect
19. C₂×C₄⋊C₁₆	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
C₂×#2150	2
C₈×#117	2
#7169	2