Subgroups — #483940

Order 512.

440 conjugacy classes of subgroups.

The subgroup lattice has 273 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33735	C₂	1	1	yes	maximal, characteristic
2	#17959	C₂	1	2	yes	maximal, complemented
3	#12083	C₂	1	1	yes	maximal, characteristic, complemented
4	#12082	C₂	1	2	yes	maximal, complemented
5	#12081	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×SD₃₂	C₂	1	2	yes	maximal, complemented
8	C₈×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	#12024	C₂	1	1	yes	maximal, characteristic, complemented
10	#11673	C₂	1	2	yes	maximal, complemented
11	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#3168	C₂²	1	2	yes	—
13	#2966	—	2	2	no	—
14	#2253	C₄	1	1	yes	characteristic
15	#2249	C₄	1	1	yes	characteristic
16	#2242	C₄	1	2	yes	—
17	#2217	C₄	1	2	yes	—
18	#2216	C₄	1	1	yes	characteristic
19	#2215	C₄	1	1	yes	characteristic
20	C₄×Q₃₂	C₂²	1	1	yes	characteristic
21	C₄×SD₃₂	C₂²	1	2	yes	—
22	C₄×D₁₆	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
25	#1591	C₂²	1	2	yes	—
26	#1590	—	2	2	no	—
27	#1589	C₂²	1	2	yes	—
28	C₈×Q₁₆	C₂²	1	2	yes	—
29	C₈×SD₁₆	—	2	2	no	—
30	C₈×D₈	C₂²	1	2	yes	—
31	#1568	C₂²	1	2	yes	—
32	#1494	—	2	2	no	—
33	C₂×C₈²	C₂²	1	1	yes	characteristic
34	#1372	C₂²	1	2	yes	—
35	#1371	C₂²	1	1	yes	characteristic
36	#1371	C₂²	1	1	yes	characteristic
37	#1332	C₂²	1	2	yes	—
38	#1332	C₂²	1	2	yes	—
39	#1331	C₂²	1	2	yes	—
40	#1331	C₂²	1	2	yes	—
41	C₈×C₁₆	C₂²	1	1	yes	characteristic
42	C₈×C₁₆	C₂²	1	1	yes	characteristic
43	C₈×C₁₆	C₂²	1	1	yes	characteristic
44	C₈×C₁₆	C₂²	1	1	yes	characteristic
45	#529	—	2	2	no	—
46	C₂×Q₃₂	C₂×C₄	1	1	yes	characteristic
47	C₂×SD₃₂	C₂×C₄	1	2	yes	—
48	C₂×D₁₆	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₁₆	—	2	1	no	—
50	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
52	#499	C₂×C₄	1	2	yes	—
53	#495	C₂×C₄	1	2	yes	—
54	#488	—	2	2	no	—
55	#467	C₂×C₄	1	2	yes	—
56	#466	C₂×C₄	1	2	yes	—
57	#465	—	2	2	no	—
58	#446	—	2	2	no	—
59	C₈×Q₈	—	2	2	no	—
60	C₄×Q₁₆	C₂³	1	2	yes	—
61	C₄×SD₁₆	—	2	2	no	—
62	C₄×D₈	C₂³	1	2	yes	—
63	#436	—	2	2	no	—
64	#435	—	4	2	no	—
65	C₈×D₄	—	2	2	no	—
66	C₈×D₄	—	4	2	no	—
67	#433	—	4	2	no	—
68	#432	—	2	2	no	—
69	#405	—	4	2	no	—
70	C₂×C₄×C₈	D₄	1	1	yes	characteristic
71	C₂×C₄×C₈	D₄	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
73	#367	C₂×C₄	1	2	yes	—
74	#366	C₂×C₄	1	1	yes	characteristic
75	#366	C₂×C₄	1	1	yes	characteristic
76	#358	C₂×C₄	1	2	yes	—
77	#358	C₂×C₄	1	2	yes	—
78	#357	C₂×C₄	1	2	yes	—
79	#357	C₂×C₄	1	2	yes	—
80	C₄×C₁₆	—	2	1	no	—
81	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
82	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
83	C₄×C₁₆	C₂³	1	1	yes	characteristic
84	C₄×C₁₆	C₂³	1	1	yes	characteristic
85	C₄×C₁₆	C₂³	1	1	yes	characteristic
86	C₄×C₁₆	C₂³	1	1	yes	characteristic
87	#335	C₂³	1	2	yes	—
88	#335	C₂³	1	2	yes	—
89	#334	—	2	2	no	—
90	Q₈⋊C₈	—	2	2	no	—
91	Q₈⋊C₈	—	2	2	no	—
92	D₄⋊C₈	—	2	2	no	—
93	D₄⋊C₈	—	2	2	no	—
94	C₈²	D₄	1	1	yes	characteristic
95	C₈²	D₄	1	1	yes	characteristic
96	C₈²	C₂³	1	1	yes	characteristic
97	C₈²	C₂³	1	1	yes	characteristic
98	C₂×Q₁₆	C₂²×C₄	1	2	yes	—
99	C₂×SD₁₆	—	2	2	no	—
100	C₂×D₈	C₂²×C₄	1	2	yes	—
101	C₂×M₁₆	—	4	2	no	—
102	C₂²×C₈	—	2	1	no	—
103	C₂²×C₈	—	4	2	no	—
104	C₂²×C₈	D₈	1	1	yes	characteristic
105	C₂²×C₈	D₈	1	1	yes	characteristic
106	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₈	#41	1	1	yes	characteristic
108	#126	—	4	2	no	—
109	#125	—	2	2	no	—
110	#124	—	2	2	no	—
111	#123	—	4	2	no	—
112	C₂²⋊Q₈	—	2	2	no	—
113	#121	—	2	2	no	—
114	C₄×Q₈	—	2	2	no	—
115	C₄×D₄	—	2	2	no	—
116	C₄×D₄	—	4	2	no	—
117	#117	—	4	2	no	—
118	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
119	Q₃₂	—	2	2	no	—
120	SD₃₂	—	2	4	no	—
121	D₁₆	—	2	2	no	—
122	C₂×C₁₆	—	2	4	no	—
123	C₂×C₁₆	—	2	1	no	—
124	C₂×C₁₆	—	2	1	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
131	#107	C₂²×C₄	1	2	yes	—
132	#107	C₂²×C₄	1	2	yes	—
133	#106	—	2	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	4	2	no	—
137	C₄⋊C₈	—	4	2	no	—
138	#103	—	2	2	no	—
139	#103	—	2	2	no	—
140	#102	—	2	2	no	—
141	#102	—	2	2	no	—
142	C₂²⋊C₈	—	4	2	no	—
143	C₂²⋊C₈	—	4	2	no	—
144	C₂²⋊C₈	—	4	2	no	—
145	#97	—	4	2	no	—
146	#97	—	4	2	no	—
147	C₄×C₈	—	2	1	no	—
148	C₄×C₈	—	4	2	no	—
149	C₄×C₈	D₈	1	1	yes	characteristic
150	C₄×C₈	D₈	1	1	yes	characteristic
151	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
152	C₄×C₈	C₂×D₄	1	1	yes	characteristic
153	C₄×C₈	C₂×D₄	1	1	yes	characteristic
154	C₄×C₈	C₂×D₄	1	1	yes	characteristic
155	C₄×C₈	C₂×D₄	1	1	yes	characteristic
156	C₄×C₈	C₂×D₄	1	1	yes	characteristic
157	C₄×C₈	#41	1	1	yes	characteristic
158	C₄×C₈	C₂⁴	1	1	yes	characteristic
159	C₂×Q₈	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
165	C₂²×C₄	#135	1	1	yes	characteristic
166	Q₁₆	—	2	2	no	—
167	Q₁₆	—	2	2	no	—
168	SD₁₆	—	4	4	no	—
169	D₈	—	2	2	no	—
170	D₈	—	2	2	no	—
171	M₁₆	—	8	4	no	—
172	C₂×C₈	—	2	4	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	2	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	8	4	no	—
183	C₂×C₈	C₄×D₄	1	1	yes	characteristic
184	C₂×C₈	C₄×D₄	1	1	yes	characteristic
185	C₂×C₈	C₂×D₈	1	1	yes	characteristic
186	C₂×C₈	C₂×D₈	1	1	yes	characteristic
187	C₂×C₈	C₂×D₈	1	1	yes	characteristic
188	C₂×C₈	C₂×D₈	1	1	yes	characteristic
189	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
190	C₂×C₈	C₂×#41	1	1	yes	characteristic
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₂²⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	—	4	2	no	—
200	C₂²⋊C₄	—	4	2	no	—
201	C₂²⋊C₄	—	4	2	no	—
202	C₄²	—	4	2	no	—
203	C₄²	C₄×D₄	1	1	yes	characteristic
204	C₄²	C₂×D₈	1	1	yes	characteristic
205	C₄²	#135	1	1	yes	characteristic
206	C₄²	C₂²×D₄	1	1	yes	characteristic
207	C₁₆	—	2	2	no	—
208	C₁₆	—	2	2	no	—
209	C₁₆	—	2	1	no	—
210	C₁₆	—	2	1	no	—
211	C₁₆	#131	1	2	yes	—
212	C₁₆	#131	1	2	yes	—
213	C₂³	—	4	2	no	—
214	C₂³	C₄×D₈	1	1	yes	characteristic
215	Q₈	—	4	2	no	—
216	Q₈	—	4	2	no	—
217	D₄	—	4	2	no	—
218	D₄	—	4	2	no	—
219	D₄	—	8	4	no	—
220	C₂×C₄	—	2	2	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	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	C₄×D₈	1	1	yes	characteristic
231	C₂×C₄	C₄×D₈	1	1	yes	characteristic
232	C₂×C₄	C₄×D₈	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
235	C₂×C₄	C₂×#135	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	2	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	8	2	no	—
243	C₈	—	8	2	no	—
244	C₈	#443	1	2	yes	—
245	C₈	#508	1	2	yes	—
246	C₈	#508	1	2	yes	—
247	C₈	C₂×#131	1	1	yes	characteristic
248	C₈	C₂×#131	1	1	yes	characteristic
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	8	2	no	—
252	C₂²	—	8	2	no	—
253	C₂²	—	8	2	no	—
254	C₂²	#2178	1	2	yes	—
255	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	8	2	no	—
261	C₄	—	8	2	no	—
262	C₄	—	8	2	no	—
263	C₄	#2931	1	1	yes	characteristic
264	C₄	C₂×#443	1	1	yes	characteristic
265	C₄	C₂×#508	1	1	yes	characteristic
266	C₄	C₂×#508	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	16	2	no	—
269	C₂	#17956	1	1	yes	characteristic
270	C₂	C₄×#508	1	1	yes	characteristic
271	C₂	C₂×#2178	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#2215	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #17959	C₂	semidirect
3. #12083	C₂	semidirect
4. #12082	C₂	semidirect
5. #12081	C₂	semidirect
6. C₈×Q₃₂	C₂	semidirect
7. C₈×SD₃₂	C₂	semidirect
8. C₈×D₁₆	C₂	semidirect
9. #12024	C₂	semidirect
10. #11673	C₂	semidirect
11. C₂×C₈×C₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33735	2
#17959	2
#12083	2
#12082	2
#12081	2
C₈×Q₃₂	2
C₈×SD₃₂	2
C₈×D₁₆	2
#12024	2
#11673	2
C₂×C₈×C₁₆	2