Subgroups — #491562

Order 512.

583 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	#33776	C₂	1	2	yes	maximal, complemented
2	#33734	C₂	1	1	yes	maximal, characteristic, complemented
3	#33713	C₂	1	1	yes	maximal, characteristic
4	#19645	C₂	1	2	yes	maximal, complemented
5	#12530	C₂	1	2	yes	maximal, complemented
6	#12526	C₂	1	2	yes	maximal, complemented
7	#12518	C₂	1	1	yes	maximal, characteristic, complemented
8	#12516	C₂	1	1	yes	maximal, characteristic, complemented
9	#12504	C₂	1	1	yes	maximal, characteristic, complemented
10	#12503	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₆×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#508	—	2	2	no	—
13	#3228	—	2	2	no	—
14	#3182	—	2	2	no	—
15	#3170	C₂²	1	2	yes	—
16	#3147	C₂²	1	2	yes	—
17	#2252	C₂²	1	1	yes	characteristic
18	#2249	—	2	2	no	—
19	#2248	C₂²	1	2	yes	—
20	#2248	C₂²	1	2	yes	—
21	#2247	C₂²	1	1	yes	characteristic
22	#2247	C₂²	1	1	yes	characteristic
23	#2247	C₂²	1	2	yes	—
24	#2246	C₂²	1	1	yes	characteristic
25	#2241	C₂²	1	2	yes	—
26	#2241	C₂²	1	2	yes	—
27	#2240	C₂²	1	2	yes	—
28	#2216	—	2	2	no	—
29	#2211	—	2	4	no	—
30	#2189	—	2	4	no	—
31	C₄×Q₃₂	—	2	2	no	—
32	#2162	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
34	#1719	—	2	2	no	—
35	#1714	—	2	2	no	—
36	#1713	C₂²	1	2	yes	—
37	#1709	C₂²	1	2	yes	—
38	#1707	C₂²	1	2	yes	—
39	#1706	C₂²	1	2	yes	—
40	C₈×M₁₆	C₂²	1	1	yes	characteristic
41	#1312	C₂²	1	1	yes	characteristic
42	#1312	C₂²	1	1	yes	characteristic
43	C₈×C₁₆	C₂²	1	1	yes	characteristic
44	C₈×C₁₆	C₂²	1	1	yes	characteristic
45	C₂×#135	—	2	2	no	—
46	C₂×#135	—	2	2	no	—
47	#533	—	2	2	no	—
48	#532	—	2	2	no	—
49	#508	—	4	8	no	—
50	C₂×Q₃₂	—	2	2	no	—
51	C₂×Q₃₂	—	2	2	no	—
52	C₂×Q₃₂	—	2	4	no	—
53	C₂×SD₃₂	—	2	4	no	—
54	C₂×SD₃₂	—	2	4	no	—
55	C₂×D₁₆	—	2	2	no	—
56	C₂²×C₁₆	D₄	1	2	yes	—
57	#500	—	2	2	no	—
58	#500	—	2	2	no	—
59	#500	C₂³	1	2	yes	—
60	#498	—	2	2	no	—
61	#495	—	2	2	no	—
62	#495	—	2	2	no	—
63	#494	C₂³	1	2	yes	—
64	#494	C₂³	1	2	yes	—
65	#493	C₂³	1	2	yes	—
66	#492	—	2	2	no	—
67	#492	—	2	2	no	—
68	#487	—	2	2	no	—
69	#487	—	2	2	no	—
70	#486	—	2	2	no	—
71	#486	—	2	2	no	—
72	#486	—	2	2	no	—
73	#482	—	4	4	no	—
74	#478	—	4	4	no	—
75	#468	—	4	4	no	—
76	#467	—	2	2	no	—
77	#467	—	2	2	no	—
78	#440	—	4	4	no	—
79	C₄×Q₁₆	—	2	2	no	—
80	C₄×Q₁₆	—	2	2	no	—
81	#431	D₄	1	1	yes	characteristic
82	C₄×M₁₆	D₄	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
84	#366	—	2	2	no	—
85	C₄⋊C₁₆	D₄	1	2	yes	—
86	#358	—	2	4	no	—
87	#358	—	2	4	no	—
88	#357	—	2	4	no	—
89	C₂²⋊C₁₆	D₄	1	2	yes	—
90	C₄×C₁₆	D₄	1	2	yes	—
91	C₄×C₁₆	C₂³	1	1	yes	characteristic
92	C₄×C₁₆	C₂³	1	1	yes	characteristic
93	C₄×C₁₆	C₂³	1	1	yes	characteristic
94	C₄×C₁₆	C₂³	1	1	yes	characteristic
95	#322	D₄	1	1	yes	characteristic
96	#322	D₄	1	1	yes	characteristic
97	C₈²	C₂³	1	1	yes	characteristic
98	C₈²	C₂³	1	1	yes	characteristic
99	C₂×#41	—	4	2	no	—
100	#135	—	4	8	no	—
101	#135	—	8	4	no	—
102	C₂×Q₁₆	—	2	2	no	—
103	C₂×Q₁₆	—	2	2	no	—
104	C₂×Q₁₆	—	2	2	no	—
105	C₂×Q₁₆	—	2	2	no	—
106	C₂×SD₁₆	—	4	2	no	—
107	C₂×SD₁₆	—	4	2	no	—
108	C₂×SD₁₆	—	4	4	no	—
109	C₂×D₈	—	2	2	no	—
110	C₂×D₈	—	2	2	no	—
111	C₂×M₁₆	D₈	1	2	yes	—
112	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
114	C₄⋊Q₈	—	2	2	no	—
115	C₄⋊Q₈	—	2	2	no	—
116	C₄⋊Q₈	—	2	2	no	—
117	#127	—	2	2	no	—
118	#125	—	4	2	no	—
119	#124	—	4	2	no	—
120	C₂²⋊Q₈	—	4	2	no	—
121	C₂²⋊Q₈	—	4	2	no	—
122	#121	—	4	2	no	—
123	#121	—	4	2	no	—
124	C₄×Q₈	—	4	2	no	—
125	C₄×Q₈	—	4	2	no	—
126	C₄×D₄	—	4	2	no	—
127	C₄×D₄	—	4	2	no	—
128	#117	—	4	2	no	—
129	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
130	Q₃₂	—	4	4	no	—
131	Q₃₂	—	4	4	no	—
132	Q₃₂	—	8	4	no	—
133	SD₃₂	—	4	8	no	—
134	SD₃₂	—	8	4	no	—
135	D₁₆	—	4	4	no	—
136	C₂×C₁₆	—	2	4	no	—
137	C₂×C₁₆	C₂×D₄	1	2	yes	—
138	C₂×C₁₆	C₂×D₄	1	2	yes	—
139	C₂×C₁₆	C₂×D₄	1	2	yes	—
140	C₂×C₁₆	C₂×D₄	1	2	yes	—
141	#107	—	2	2	no	—
142	#107	—	2	2	no	—
143	#107	—	4	4	no	—
144	#107	—	4	4	no	—
145	#106	—	4	4	no	—
146	C₄⋊C₈	D₈	1	2	yes	—
147	#103	—	4	2	no	—
148	#103	—	4	2	no	—
149	#103	—	4	4	no	—
150	#102	—	4	4	no	—
151	#102	—	4	4	no	—
152	#102	—	4	4	no	—
153	C₂²⋊C₈	D₈	1	2	yes	—
154	#97	D₈	1	2	yes	—
155	C₄×C₈	C₂×D₄	1	1	yes	characteristic
156	C₄×C₈	C₂×D₄	1	1	yes	characteristic
157	C₄×C₈	C₂×D₄	1	1	yes	characteristic
158	C₄×C₈	C₂×D₄	1	1	yes	characteristic
159	C₄×C₈	C₂×D₄	1	1	yes	characteristic
160	C₄×C₈	C₂×D₄	1	1	yes	characteristic
161	C₄×C₈	C₂×D₄	1	1	yes	characteristic
162	C₄×C₈	C₂⁴	1	1	yes	characteristic
163	#41	—	8	4	no	—
164	#41	—	8	4	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂×D₄	—	8	2	no	—
172	C₂²×C₄	—	8	2	no	—
173	C₂²×C₄	#127	1	1	yes	characteristic
174	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
175	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
176	Q₁₆	—	4	4	no	—
177	Q₁₆	—	4	4	no	—
178	Q₁₆	—	4	4	no	—
179	Q₁₆	—	4	4	no	—
180	SD₁₆	—	8	4	no	—
181	SD₁₆	—	8	4	no	—
182	SD₁₆	—	8	8	no	—
183	D₈	—	4	4	no	—
184	D₈	—	4	4	no	—
185	M₁₆	—	2	4	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	#127	1	1	yes	characteristic
189	C₂×C₈	#127	1	1	yes	characteristic
190	C₂×C₈	C₂×D₈	1	2	yes	—
191	C₂×C₈	C₂×D₈	1	2	yes	—
192	C₂×C₈	C₂×D₈	1	2	yes	—
193	C₂×C₈	C₂×D₈	1	2	yes	—
194	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	8	2	no	—
203	C₄⋊C₄	—	8	2	no	—
204	C₂²⋊C₄	—	8	2	no	—
205	C₂²⋊C₄	—	8	2	no	—
206	C₂²⋊C₄	—	8	2	no	—
207	C₄²	—	8	2	no	—
208	C₄²	#127	1	1	yes	characteristic
209	C₄²	C₂×D₈	1	1	yes	characteristic
210	C₄²	C₂×D₈	1	1	yes	characteristic
211	C₄²	C₂²×D₄	1	1	yes	characteristic
212	C₁₆	—	2	2	no	—
213	C₁₆	—	2	2	no	—
214	C₁₆	—	2	2	no	—
215	C₁₆	AGL(1,ℤ/8)	1	4	yes	—
216	C₂³	—	8	2	no	—
217	C₂³	#493	1	1	yes	characteristic
218	Q₈	—	8	2	no	—
219	Q₈	—	8	2	no	—
220	Q₈	—	8	2	no	—
221	Q₈	—	8	2	no	—
222	Q₈	—	8	4	no	—
223	D₄	—	8	2	no	—
224	D₄	—	8	2	no	—
225	D₄	—	8	4	no	—
226	D₄	—	8	4	no	—
227	D₄	—	16	4	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	—	16	4	no	—
236	C₂×C₄	#493	1	1	yes	characteristic
237	C₂×C₄	#493	1	1	yes	characteristic
238	C₂×C₄	#493	1	1	yes	characteristic
239	C₂×C₄	C₂×#127	1	1	yes	characteristic
240	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₈	—	2	2	no	—
247	C₈	#508	1	4	yes	complemented
248	C₈	C₂×AGL(1,ℤ/8)	1	2	yes	—
249	C₈	#576	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	16	2	no	—
254	C₂²	—	16	2	no	—
255	C₂²	—	16	2	no	—
256	C₂²	C₂×#493	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	16	2	no	—
261	C₄	—	16	2	no	—
262	C₄	—	16	2	no	—
263	C₄	#3151	1	1	yes	characteristic
264	C₄	#3155	1	1	yes	characteristic
265	C₄	C₂×#508	1	2	yes	—
266	C₄	#3417	1	2	yes	—
267	C₂	—	4	1	no	—
268	C₂	—	32	2	no	—
269	C₂	#19629	1	1	yes	characteristic
270	C₂	#33713	1	1	yes	characteristic
271	C₂	#33721	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33713	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ 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
1. #33776	C₂	semidirect
2. #33734	C₂	semidirect
4. #19645	C₂	semidirect
5. #12530	C₂	semidirect
6. #12526	C₂	semidirect
7. #12518	C₂	semidirect
8. #12516	C₂	semidirect
9. #12504	C₂	semidirect
10. #12503	C₂	semidirect
11. C₁₆×M₁₆	C₂	semidirect
247. C₈	#508	semidirect

Maximal subgroups

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

subgroup	index
#33776	2
#33734	2
#33713	2
#19645	2
#12530	2
#12526	2
#12518	2
#12516	2
#12504	2
#12503	2
C₁₆×M₁₆	2