Subgroups — #491487

Order 512.

550 conjugacy classes of subgroups.

The subgroup lattice has 263 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33736	C₂	1	2	yes	maximal, complemented
2	#33734	C₂	1	1	yes	maximal, characteristic, complemented
3	#33733	C₂	1	1	yes	maximal, characteristic
4	#19600	C₂	1	2	yes	maximal, complemented
5	#12522	C₂	1	1	yes	maximal, characteristic, complemented
6	#12518	C₂	1	1	yes	maximal, characteristic, complemented
7	#12511	C₂	1	2	yes	maximal, complemented
8	#12505	C₂	1	2	yes	maximal, complemented
9	#12504	C₂	1	1	yes	maximal, characteristic, complemented
10	#12503	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#3170	C₂²	1	2	yes	—
13	#3169	C₂²	1	2	yes	—
14	#3167	—	2	2	no	—
15	#3131	—	2	2	no	—
16	#2254	C₂²	1	2	yes	—
17	#2252	C₂²	1	1	yes	characteristic
18	#2252	C₂²	1	1	yes	characteristic
19	#2248	C₂²	1	2	yes	—
20	#2247	C₂²	1	1	yes	characteristic
21	#2246	C₂²	1	1	yes	characteristic
22	#2241	C₂²	1	2	yes	—
23	#2241	C₂²	1	2	yes	—
24	#2240	C₂²	1	2	yes	—
25	#2240	C₂²	1	2	yes	—
26	#2217	—	2	4	no	—
27	#2216	—	2	2	no	—
28	#2215	—	2	2	no	—
29	C₄×Q₃₂	—	2	2	no	—
30	C₄×SD₃₂	—	2	4	no	—
31	C₄×D₁₆	—	2	2	no	—
32	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
34	#1711	—	2	2	no	—
35	#1710	C₂²	1	2	yes	—
36	#1709	C₂²	1	2	yes	—
37	#1708	—	2	2	no	—
38	#1707	C₂²	1	2	yes	—
39	#1706	C₂²	1	2	yes	—
40	C₂×C₈²	C₂²	1	1	yes	characteristic
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	#533	—	2	2	no	—
46	#532	—	2	2	no	—
47	C₂×Q₃₂	—	2	2	no	—
48	C₂×SD₃₂	—	2	4	no	—
49	C₂×D₁₆	—	2	2	no	—
50	C₂²×C₁₆	D₄	1	2	yes	—
51	C₂²×C₁₆	D₄	1	2	yes	—
52	#500	C₂³	1	2	yes	—
53	#500	C₂³	1	2	yes	—
54	#498	—	2	2	no	—
55	#494	C₂³	1	2	yes	—
56	#493	C₂³	1	2	yes	—
57	#492	—	2	2	no	—
58	#488	—	4	4	no	—
59	#487	—	2	2	no	—
60	#487	—	2	2	no	—
61	#487	—	2	2	no	—
62	#487	—	2	2	no	—
63	#486	—	2	2	no	—
64	#486	—	2	2	no	—
65	#486	—	2	2	no	—
66	#486	—	2	2	no	—
67	#467	—	2	2	no	—
68	#467	—	2	2	no	—
69	#466	—	2	2	no	—
70	#466	—	2	2	no	—
71	#465	—	4	4	no	—
72	C₄×Q₁₆	—	2	2	no	—
73	C₄×Q₁₆	—	2	2	no	—
74	C₄×SD₁₆	—	4	4	no	—
75	C₄×D₈	—	2	2	no	—
76	C₄×D₈	—	2	2	no	—
77	#416	—	4	4	no	—
78	C₂×C₄×C₈	D₄	1	1	yes	characteristic
79	C₂×C₄×C₈	D₄	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
81	#367	—	2	4	no	—
82	#366	—	2	2	no	—
83	#366	—	2	2	no	—
84	#358	—	2	4	no	—
85	#358	—	2	4	no	—
86	#357	—	2	4	no	—
87	#357	—	2	4	no	—
88	C₄×C₁₆	D₄	1	2	yes	—
89	C₄×C₁₆	D₄	1	2	yes	—
90	C₄×C₁₆	C₂³	1	1	yes	characteristic
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	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×#41	—	4	2	no	—
99	C₂×Q₁₆	—	2	2	no	—
100	C₂×Q₁₆	—	2	2	no	—
101	C₂×SD₁₆	—	4	4	no	—
102	C₂×D₈	—	2	2	no	—
103	C₂×D₈	—	2	2	no	—
104	C₂²×C₈	D₈	1	2	yes	—
105	C₂²×C₈	D₈	1	2	yes	—
106	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
108	C₄⋊Q₈	—	2	2	no	—
109	C₄⋊Q₈	—	2	2	no	—
110	C₄⋊Q₈	—	2	2	no	—
111	#127	—	2	2	no	—
112	#125	—	4	2	no	—
113	#124	—	4	2	no	—
114	C₂²⋊Q₈	—	4	2	no	—
115	C₂²⋊Q₈	—	4	2	no	—
116	#121	—	4	2	no	—
117	#121	—	4	2	no	—
118	C₄×Q₈	—	4	2	no	—
119	C₄×Q₈	—	4	2	no	—
120	C₄×D₄	—	4	2	no	—
121	C₄×D₄	—	4	2	no	—
122	#117	—	4	2	no	—
123	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
124	Q₃₂	—	4	4	no	—
125	SD₃₂	—	4	8	no	—
126	D₁₆	—	4	4	no	—
127	C₂×C₁₆	—	2	4	no	—
128	C₂×C₁₆	—	2	4	no	—
129	C₂×C₁₆	C₂×D₄	1	2	yes	—
130	C₂×C₁₆	C₂×D₄	1	2	yes	—
131	C₂×C₁₆	C₂×D₄	1	2	yes	—
132	C₂×C₁₆	C₂×D₄	1	2	yes	—
133	#107	—	2	2	no	—
134	#107	—	2	2	no	—
135	#107	—	2	2	no	—
136	#107	—	2	2	no	—
137	#106	—	4	4	no	—
138	#103	—	4	2	no	—
139	#103	—	4	2	no	—
140	#103	—	4	4	no	—
141	#103	—	4	4	no	—
142	#103	—	4	4	no	—
143	#102	—	4	2	no	—
144	#102	—	4	2	no	—
145	#102	—	4	4	no	—
146	#102	—	4	4	no	—
147	#102	—	4	4	no	—
148	C₄×C₈	D₈	1	2	yes	—
149	C₄×C₈	D₈	1	2	yes	—
150	C₄×C₈	C₂×D₄	1	1	yes	characteristic
151	C₄×C₈	C₂×D₄	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	C₂⁴	1	1	yes	characteristic
158	#41	—	8	8	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂×D₄	—	4	2	no	—
163	C₂×D₄	—	4	2	no	—
164	C₂×D₄	—	4	2	no	—
165	C₂×D₄	—	8	2	no	—
166	C₂²×C₄	—	8	2	no	—
167	C₂²×C₄	#127	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
169	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
170	Q₁₆	—	4	4	no	—
171	Q₁₆	—	4	4	no	—
172	SD₁₆	—	8	8	no	—
173	D₈	—	4	4	no	—
174	D₈	—	4	4	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	#127	1	1	yes	characteristic
180	C₂×C₈	#127	1	1	yes	characteristic
181	C₂×C₈	C₂×D₈	1	2	yes	—
182	C₂×C₈	C₂×D₈	1	2	yes	—
183	C₂×C₈	C₂×D₈	1	2	yes	—
184	C₂×C₈	C₂×D₈	1	2	yes	—
185	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	8	2	no	—
194	C₄⋊C₄	—	8	2	no	—
195	C₂²⋊C₄	—	8	2	no	—
196	C₂²⋊C₄	—	8	2	no	—
197	C₂²⋊C₄	—	8	2	no	—
198	C₄²	—	8	2	no	—
199	C₄²	#127	1	1	yes	characteristic
200	C₄²	C₂×D₈	1	1	yes	characteristic
201	C₄²	C₂×D₈	1	1	yes	characteristic
202	C₄²	C₂²×D₄	1	1	yes	characteristic
203	C₁₆	—	2	2	no	—
204	C₁₆	—	2	2	no	—
205	C₁₆	#135	1	4	yes	—
206	C₁₆	#135	1	4	yes	—
207	C₂³	—	8	2	no	—
208	C₂³	#493	1	1	yes	characteristic
209	Q₈	—	8	2	no	—
210	Q₈	—	8	2	no	—
211	Q₈	—	8	4	no	—
212	Q₈	—	8	4	no	—
213	D₄	—	8	2	no	—
214	D₄	—	8	2	no	—
215	D₄	—	8	4	no	—
216	D₄	—	8	4	no	—
217	D₄	—	16	4	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	—	16	4	no	—
225	C₂×C₄	#493	1	1	yes	characteristic
226	C₂×C₄	#493	1	1	yes	characteristic
227	C₂×C₄	#493	1	1	yes	characteristic
228	C₂×C₄	#508	1	4	yes	—
229	C₂×C₄	C₂×#127	1	1	yes	characteristic
230	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
231	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	2	no	—
235	C₈	—	2	2	no	—
236	C₈	—	2	2	no	—
237	C₈	#508	1	4	yes	—
238	C₈	C₂×#135	1	2	yes	—
239	C₈	C₂×#135	1	2	yes	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	16	2	no	—
243	C₂²	—	16	2	no	—
244	C₂²	—	16	2	no	—
245	C₂²	#2249	1	2	yes	—
246	C₂²	C₂×#493	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	16	2	no	—
250	C₄	—	16	2	no	—
251	C₄	—	16	2	no	—
252	C₄	#2249	1	2	yes	—
253	C₄	#3147	1	1	yes	characteristic
254	C₄	#3147	1	1	yes	characteristic
255	C₄	C₂×#508	1	2	yes	—
256	C₄	C₂×#508	1	2	yes	—
257	C₂	—	2	2	no	—
258	C₂	—	32	2	no	—
259	C₂	#19603	1	1	yes	characteristic
260	C₂	C₂×#2249	1	1	yes	characteristic
261	C₂	#33713	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#493
prime radical	#33733	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ 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
1. #33736	C₂	semidirect
2. #33734	C₂	semidirect
4. #19600	C₂	semidirect
5. #12522	C₂	semidirect
6. #12518	C₂	semidirect
7. #12511	C₂	semidirect
8. #12505	C₂	semidirect
9. #12504	C₂	semidirect
10. #12503	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
#33736	2
#33734	2
#33733	2
#19600	2
#12522	2
#12518	2
#12511	2
#12505	2
#12504	2
#12503	2
C₂×C₈×C₁₆	2