Subgroups — #126839

Order 512.

459 conjugacy classes of subgroups.

The subgroup lattice has 287 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	#20816	C₂	1	1	yes	maximal, characteristic
3	C₂×#1564	C₂	1	1	yes	maximal, characteristic
4	#7174	C₂	1	4	yes	maximal, complemented
5	C₂×#431	—	2	1	no	—
6	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
7	#2285	—	2	1	no	—
8	#2163	—	2	1	no	—
9	#2163	—	2	1	no	—
10	#2162	C₄	1	2	yes	complemented
11	#2162	C₄	1	2	yes	complemented
12	#2150	—	2	1	no	—
13	#2112	—	2	1	no	—
14	C₄×M₃₂	C₂²	1	2	yes	—
15	#1844	C₄	1	2	yes	—
16	#1834	—	2	1	no	—
17	#1807	—	2	1	no	—
18	#1807	C₄	1	2	yes	—
19	C₄×#106	—	2	1	no	—
20	C₄×#106	C₂²	1	2	yes	—
21	#1564	—	2	1	no	—
22	#1564	—	2	1	no	—
23	#1564	C₄	1	2	yes	complemented
24	#1564	C₂²	1	2	yes	—
25	C₂×#335	—	2	1	no	—
26	C₂×#334	C₄	1	2	yes	complemented
27	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₈	D₄	1	1	yes	characteristic
29	C₂×#117	—	4	1	no	—
30	C₂²×C₄²	D₄	1	1	yes	characteristic
31	C₂×M₃₂	—	2	2	no	—
32	#431	—	2	2	no	—
33	#431	—	2	2	no	—
34	#431	—	4	1	no	—
35	#431	—	4	1	no	—
36	C₂×#106	—	2	2	no	—
37	C₂×#106	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₈	—	2	2	no	—
39	C₂×C₂²⋊C₈	—	4	1	no	—
40	C₂×C₄×C₈	—	4	1	no	—
41	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
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	1	yes	characteristic
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	D₄	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	Q₈	1	1	yes	characteristic
51	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
53	#384	—	2	2	no	—
54	#382	—	4	1	no	—
55	#380	—	2	2	no	—
56	#380	—	4	1	no	—
57	C₄×C₄⋊C₄	—	2	2	no	—
58	C₄×C₄⋊C₄	—	4	1	no	—
59	C₄⋊C₁₆	—	2	2	no	—
60	C₄⋊C₁₆	—	2	2	no	—
61	C₂²⋊C₁₆	—	2	2	no	—
62	C₂²⋊C₁₆	—	2	2	no	—
63	#346	—	2	2	no	—
64	C₄×C₁₆	C₂×C₄	1	4	yes	—
65	#340	—	2	2	no	—
66	#340	—	2	2	no	—
67	#335	—	2	2	no	—
68	#335	—	2	2	no	—
69	#334	—	2	2	no	—
70	#334	C₂×C₄	1	4	yes	—
71	C₂³×C₄	C₂²⋊C₄	1	1	yes	characteristic
72	C₂³×C₄	D₈	1	1	yes	characteristic
73	C₂³×C₄	SD₁₆	1	1	yes	characteristic
74	C₂²×C₈	—	2	4	no	—
75	C₂²×C₈	—	4	2	no	—
76	C₂²×C₈	—	4	2	no	—
77	C₂²×C₈	C₄²	1	1	yes	characteristic
78	C₂²×C₈	C₄²	1	1	yes	characteristic
79	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₈	C₄⋊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	C₂×D₄	1	1	yes	characteristic
89	#117	—	8	4	no	—
90	C₂×C₄⋊C₄	—	2	4	no	—
91	C₂×C₄⋊C₄	—	4	2	no	—
92	C₂×C₄⋊C₄	—	4	2	no	—
93	C₂×C₂²⋊C₄	—	4	2	no	—
94	C₂×C₄²	—	4	2	no	—
95	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄²	D₈	1	1	yes	characteristic
101	C₂×C₄²	SD₁₆	1	1	yes	characteristic
102	C₂×C₄²	SD₁₆	1	1	yes	characteristic
103	C₂×C₄²	SD₁₆	1	1	yes	characteristic
104	C₂×C₄²	Q₁₆	1	1	yes	characteristic
105	C₂×C₄²	Q₁₆	1	1	yes	characteristic
106	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
107	M₃₂	—	4	4	no	—
108	C₂×C₁₆	—	2	4	no	—
109	#106	—	2	8	no	—
110	#106	—	4	4	no	—
111	C₄⋊C₈	—	2	4	no	—
112	C₄⋊C₈	—	4	2	no	—
113	C₂²⋊C₈	—	4	2	no	—
114	C₂²⋊C₈	—	4	2	no	—
115	C₄×C₈	—	4	2	no	—
116	C₄×C₈	—	4	2	no	—
117	C₄×C₈	C₄²	1	1	yes	characteristic
118	C₄×C₈	C₄²	1	1	yes	characteristic
119	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
120	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
123	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
124	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
125	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
126	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
127	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
128	C₄×C₈	C₄⋊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	C₂×D₄	1	1	yes	characteristic
132	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
133	#95	—	4	2	no	—
134	#95	—	4	2	no	—
135	C₂⁴	#102	1	1	yes	characteristic
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	—	4	4	no	—
141	C₂²×C₄	#95	1	1	yes	characteristic
142	C₂²×C₄	#95	1	1	yes	characteristic
143	C₂²×C₄	#102	1	1	yes	characteristic
144	C₂²×C₄	#102	1	1	yes	characteristic
145	C₂²×C₄	#102	1	1	yes	characteristic
146	C₂²×C₄	#102	1	1	yes	characteristic
147	C₂²×C₄	#102	1	1	yes	characteristic
148	C₂²×C₄	#103	1	1	yes	characteristic
149	C₂²×C₄	#103	1	1	yes	characteristic
150	C₂²×C₄	#103	1	1	yes	characteristic
151	C₂²×C₄	#103	1	1	yes	characteristic
152	C₂²×C₄	#106	1	1	yes	characteristic
153	C₂²×C₄	#106	1	1	yes	characteristic
154	C₂²×C₄	#107	1	1	yes	characteristic
155	C₂²×C₄	#107	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
158	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
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	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	4	no	—
171	C₂×C₈	—	8	4	no	—
172	C₂×C₈	#95	1	1	yes	characteristic
173	C₂×C₈	#95	1	1	yes	characteristic
174	C₂×C₈	#95	1	1	yes	characteristic
175	C₂×C₈	#95	1	1	yes	characteristic
176	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
177	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
178	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₈	#131	1	4	yes	—
181	C₄⋊C₄	—	4	4	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄⋊C₄	—	8	4	no	—
184	C₄⋊C₄	—	8	4	no	—
185	C₂²⋊C₄	—	8	4	no	—
186	C₄²	—	8	4	no	—
187	C₄²	#95	1	1	yes	characteristic
188	C₄²	#95	1	1	yes	characteristic
189	C₄²	#102	1	1	yes	characteristic
190	C₄²	#102	1	1	yes	characteristic
191	C₄²	#103	1	1	yes	characteristic
192	C₄²	#103	1	1	yes	characteristic
193	C₄²	#103	1	1	yes	characteristic
194	C₄²	#103	1	1	yes	characteristic
195	C₄²	#106	1	1	yes	characteristic
196	C₄²	#106	1	1	yes	characteristic
197	C₄²	#107	1	1	yes	characteristic
198	C₄²	#107	1	1	yes	characteristic
199	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
200	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
201	C₄²	C₂×SD₁₆	1	1	yes	characteristic
202	C₄²	C₂×Q₁₆	1	1	yes	characteristic
203	C₁₆	—	4	4	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	4	1	no	—
207	C₂³	#340	1	1	yes	characteristic
208	C₂³	#340	1	1	yes	characteristic
209	C₂³	C₂×#102	1	1	yes	characteristic
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	—	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	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	8	4	no	—
229	C₂×C₄	—	8	4	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	#340	1	1	yes	characteristic
232	C₂×C₄	#340	1	1	yes	characteristic
233	C₂×C₄	#340	1	1	yes	characteristic
234	C₂×C₄	#340	1	1	yes	characteristic
235	C₂×C₄	#340	1	1	yes	characteristic
236	C₂×C₄	#340	1	1	yes	characteristic
237	C₂×C₄	#365	1	4	yes	—
238	C₂×C₄	C₂×#95	1	1	yes	characteristic
239	C₂×C₄	#408	1	2	yes	—
240	C₂×C₄	C₂×#102	1	1	yes	characteristic
241	C₂×C₄	C₂×#103	1	1	yes	characteristic
242	C₂×C₄	C₂×#103	1	1	yes	characteristic
243	C₂×C₄	#421	1	2	yes	—
244	C₂×C₄	C₂×#106	1	1	yes	characteristic
245	C₂×C₄	C₂×#107	1	1	yes	characteristic
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	—	2	1	no	—
252	C₈	—	8	4	no	—
253	C₈	#405	1	2	yes	—
254	C₈	#424	1	2	yes	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	4	4	no	—
263	C₂²	#1383	1	2	yes	—
264	C₂²	#1480	1	2	yes	—
265	C₂²	C₂×#340	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	16	4	no	—
274	C₄	#1383	1	2	yes	—
275	C₄	#1384	1	2	yes	—
276	C₄	#1481	1	2	yes	—
277	C₄	#1729	1	1	yes	characteristic
278	C₄	#1733	1	1	yes	characteristic
279	C₄	C₂×#365	1	2	yes	—
280	C₂	—	2	2	no	—
281	C₂	—	2	2	no	—
282	C₂	—	4	1	no	—
283	C₂	#7893	1	1	yes	characteristic
284	C₂	C₂×#1383	1	1	yes	characteristic
285	C₂	C₂×#1384	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ 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. #7174	C₂	semidirect
10. #2162	C₄	semidirect
11. #2162	C₄	semidirect
23. #1564	C₄	semidirect
26. C₂×#334	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
#20816	2
C₂×#1564	2
#7174	2