Subgroups — #491586

Order 512.

519 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33794	C₂	1	2	yes	maximal, complemented
2	#33733	C₂	1	1	yes	maximal, characteristic
3	#33726	C₂	1	1	yes	maximal, characteristic, complemented
4	#19646	C₂	1	2	yes	maximal, complemented
5	#12536	C₂	1	2	yes	maximal, complemented
6	#12532	C₂	1	2	yes	maximal, complemented
7	#12522	C₂	1	1	yes	maximal, characteristic, complemented
8	#12521	C₂	1	1	yes	maximal, characteristic, complemented
9	#12503	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	#3210	—	2	2	no	—
13	#3182	—	2	2	no	—
14	#3169	C₂²	1	2	yes	—
15	#3160	C₂²	1	2	yes	—
16	#2254	C₂²	1	2	yes	—
17	#2254	C₂²	1	2	yes	—
18	#2253	—	2	2	no	—
19	#2252	C₂²	1	2	yes	—
20	#2252	C₂²	1	1	yes	characteristic
21	#2252	C₂²	1	1	yes	characteristic
22	#2252	C₂²	1	1	yes	characteristic
23	#2246	C₂²	1	1	yes	characteristic
24	#2240	C₂²	1	2	yes	—
25	#2240	C₂²	1	2	yes	—
26	#2240	C₂²	1	2	yes	—
27	#2234	—	2	4	no	—
28	#2228	—	2	4	no	—
29	#2215	—	2	2	no	—
30	C₄×D₁₆	—	2	2	no	—
31	#2162	C₂²	1	1	yes	characteristic
32	#2158	—	2	2	no	—
33	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
34	#1720	—	2	2	no	—
35	#1714	—	2	2	no	—
36	#1713	C₂²	1	2	yes	—
37	#1710	C₂²	1	2	yes	—
38	#1706	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	#533	—	2	2	no	—
46	#532	—	2	2	no	—
47	C₂×D₁₆	—	2	2	no	—
48	C₂²×C₁₆	D₄	1	2	yes	—
49	#500	—	2	2	no	—
50	#500	C₂³	1	2	yes	—
51	#500	C₂³	1	2	yes	—
52	#500	C₂³	1	2	yes	—
53	#499	—	2	2	no	—
54	#499	—	2	2	no	—
55	#498	—	2	2	no	—
56	#494	—	2	2	no	—
57	#493	C₂³	1	2	yes	—
58	#492	—	2	2	no	—
59	#492	—	2	2	no	—
60	#487	—	2	2	no	—
61	#487	—	2	2	no	—
62	#486	—	2	2	no	—
63	#486	—	2	2	no	—
64	#486	—	2	2	no	—
65	#468	—	4	4	no	—
66	#466	—	2	2	no	—
67	#466	—	2	2	no	—
68	#464	—	4	4	no	—
69	#452	—	4	4	no	—
70	#440	—	4	4	no	—
71	C₄×D₈	—	2	2	no	—
72	C₄×D₈	—	2	2	no	—
73	#431	D₄	1	1	yes	characteristic
74	#427	—	2	2	no	—
75	#427	—	2	2	no	—
76	C₄×M₁₆	D₄	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
78	#367	—	2	4	no	—
79	#367	—	2	4	no	—
80	#366	—	2	2	no	—
81	#366	—	2	2	no	—
82	#366	—	2	2	no	—
83	#366	—	2	4	no	—
84	C₄⋊C₁₆	D₄	1	2	yes	—
85	#357	—	2	4	no	—
86	#357	—	2	4	no	—
87	#357	—	2	4	no	—
88	C₂²⋊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	#322	D₄	1	1	yes	characteristic
95	#322	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	—	4	4	no	—
100	C₂×SD₁₆	—	4	4	no	—
101	C₂×SD₁₆	—	4	4	no	—
102	C₂×D₈	—	2	2	no	—
103	C₂×D₈	—	2	2	no	—
104	C₂×M₁₆	D₈	1	2	yes	—
105	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
107	C₄⋊Q₈	—	2	2	no	—
108	C₄⋊Q₈	—	2	2	no	—
109	C₄⋊Q₈	—	2	2	no	—
110	#127	—	2	2	no	—
111	#125	—	4	2	no	—
112	#124	—	4	2	no	—
113	C₂²⋊Q₈	—	4	2	no	—
114	C₂²⋊Q₈	—	4	2	no	—
115	#121	—	4	2	no	—
116	#121	—	4	2	no	—
117	C₄×Q₈	—	4	2	no	—
118	C₄×Q₈	—	4	2	no	—
119	C₄×D₄	—	4	2	no	—
120	C₄×D₄	—	4	2	no	—
121	#117	—	4	2	no	—
122	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
123	D₁₆	—	4	4	no	—
124	C₂×C₁₆	—	2	4	no	—
125	C₂×C₁₆	C₂×D₄	1	2	yes	—
126	C₂×C₁₆	C₂×D₄	1	2	yes	—
127	C₂×C₁₆	C₂×D₄	1	2	yes	—
128	C₂×C₁₆	C₂×D₄	1	2	yes	—
129	#107	—	2	2	no	—
130	#107	—	2	2	no	—
131	#107	—	2	2	no	—
132	#107	—	2	2	no	—
133	#107	—	2	2	no	—
134	#107	—	2	2	no	—
135	#107	—	4	4	no	—
136	#106	—	4	2	no	—
137	#106	—	4	2	no	—
138	C₄⋊C₈	D₈	1	2	yes	—
139	#103	—	4	4	no	—
140	#103	—	4	4	no	—
141	#102	—	4	2	no	—
142	#102	—	4	2	no	—
143	#102	—	4	4	no	—
144	#102	—	4	4	no	—
145	C₂²⋊C₈	D₈	1	2	yes	—
146	#97	D₈	1	2	yes	—
147	C₄×C₈	C₂×D₄	1	1	yes	characteristic
148	C₄×C₈	C₂×D₄	1	1	yes	characteristic
149	C₄×C₈	C₂×D₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
155	#41	—	16	4	no	—
156	C₂×Q₈	—	4	2	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×D₄	—	4	2	no	—
160	C₂×D₄	—	4	2	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	8	2	no	—
163	C₂²×C₄	—	8	2	no	—
164	C₂²×C₄	#127	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
167	Q₁₆	—	16	4	no	—
168	SD₁₆	—	8	8	no	—
169	SD₁₆	—	16	4	no	—
170	D₈	—	4	4	no	—
171	D₈	—	4	4	no	—
172	M₁₆	—	2	4	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	#127	1	1	yes	characteristic
176	C₂×C₈	#127	1	1	yes	characteristic
177	C₂×C₈	C₂×D₈	1	2	yes	—
178	C₂×C₈	C₂×D₈	1	2	yes	—
179	C₂×C₈	C₂×D₈	1	2	yes	—
180	C₂×C₈	C₂×D₈	1	2	yes	—
181	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	8	2	no	—
190	C₄⋊C₄	—	8	2	no	—
191	C₂²⋊C₄	—	8	2	no	—
192	C₂²⋊C₄	—	8	2	no	—
193	C₂²⋊C₄	—	8	2	no	—
194	C₄²	—	8	2	no	—
195	C₄²	#127	1	1	yes	characteristic
196	C₄²	C₂×D₈	1	1	yes	characteristic
197	C₄²	C₂×D₈	1	1	yes	characteristic
198	C₄²	C₂²×D₄	1	1	yes	characteristic
199	C₁₆	—	2	2	no	—
200	C₁₆	—	2	2	no	—
201	C₁₆	—	2	2	no	—
202	C₁₆	#137	1	4	yes	—
203	C₂³	—	8	2	no	—
204	C₂³	#493	1	1	yes	characteristic
205	Q₈	—	8	4	no	—
206	Q₈	—	8	4	no	—
207	Q₈	—	8	4	no	—
208	D₄	—	8	2	no	—
209	D₄	—	8	4	no	—
210	D₄	—	8	2	no	—
211	D₄	—	8	4	no	—
212	D₄	—	16	4	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	—	16	4	no	—
221	C₂×C₄	#493	1	1	yes	characteristic
222	C₂×C₄	#493	1	1	yes	characteristic
223	C₂×C₄	#493	1	1	yes	characteristic
224	C₂×C₄	C₂×#127	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
226	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
227	C₈	—	2	1	no	—
228	C₈	—	2	2	no	—
229	C₈	—	2	1	no	—
230	C₈	—	2	2	no	—
231	C₈	—	2	2	no	—
232	C₈	#508	1	4	yes	—
233	C₈	C₂×#137	1	2	yes	—
234	C₈	#578	1	2	yes	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	16	2	no	—
239	C₂²	—	16	2	no	—
240	C₂²	—	16	2	no	—
241	C₂²	C₂×#493	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	16	2	no	—
246	C₄	—	16	2	no	—
247	C₄	—	16	2	no	—
248	C₄	#3152	1	1	yes	characteristic
249	C₄	#3156	1	1	yes	characteristic
250	C₄	C₂×#508	1	2	yes	—
251	C₄	#3415	1	2	yes	—
252	C₂	—	4	1	no	—
253	C₂	—	32	2	no	—
254	C₂	#19630	1	1	yes	characteristic
255	C₂	#33713	1	1	yes	characteristic
256	C₂	#33720	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33733	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. #33794	C₂	semidirect
3. #33726	C₂	semidirect
4. #19646	C₂	semidirect
5. #12536	C₂	semidirect
6. #12532	C₂	semidirect
7. #12522	C₂	semidirect
8. #12521	C₂	semidirect
9. #12503	C₂	semidirect
10. #12503	C₂	semidirect
11. C₁₆×M₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33794	2
#33733	2
#33726	2
#19646	2
#12536	2
#12532	2
#12522	2
#12521	2
#12503	2
C₁₆×M₁₆	2