Subgroups — #498521

Order 512.

530 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2252	C₂	1	1	yes	maximal, characteristic
2	#23157	C₂	1	2	yes	maximal
3	#13275	C₂	1	2	yes	maximal
4	#13165	C₂	1	2	yes	maximal
5	#13109	C₂	1	4	yes	maximal
6	C₄×#366	C₂	1	2	yes	maximal
7	C₄×#366	C₂	1	1	yes	maximal, characteristic
8	C₁₆×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂×#500	C₂²	1	2	yes	—
10	#2365	—	2	2	no	—
11	#2252	C₄	1	8	yes	complemented
12	C₂×#366	C₂²	1	1	yes	characteristic
13	C₂×#366	C₂²	1	2	yes	—
14	C₂×#366	C₂²	1	2	yes	—
15	C₂×#366	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₁₆	C₂²	1	2	yes	—
18	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
19	#1988	—	2	2	no	—
20	#1942	C₂²	1	4	yes	—
21	#1918	—	2	4	no	—
22	C₄×#107	C₂²	1	4	yes	—
23	C₄×#107	C₂²	1	2	yes	—
24	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	#1380	C₂²	1	2	yes	—
26	#1380	C₂²	1	4	yes	—
27	#1380	C₂²	1	4	yes	—
28	#1374	C₂²	1	2	yes	—
29	C₂×C₄⋊Q₈	—	2	2	no	—
30	C₂×C₄×Q₈	—	4	2	no	—
31	C₂²×C₁₆	C₂³	1	2	yes	—
32	C₂²×C₁₆	C₂³	1	1	yes	characteristic
33	C₂²×C₁₆	C₂³	1	1	yes	characteristic
34	#500	—	2	4	no	—
35	#500	C₂×C₄	1	8	yes	—
36	C₂×#107	—	2	2	no	—
37	C₂×#107	C₂³	1	2	yes	—
38	C₂×#107	C₂³	1	4	yes	—
39	C₂×#107	C₂³	1	2	yes	—
40	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂³	1	2	yes	—
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	#391	—	4	2	no	—
44	#389	—	2	4	no	—
45	#387	—	4	4	no	—
46	C₄×C₄⋊C₄	—	2	4	no	—
47	C₄×C₄⋊C₄	—	2	2	no	—
48	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
49	#366	—	2	2	no	—
50	#366	C₂×C₄	1	8	yes	—
51	#366	C₂×C₄	1	8	yes	—
52	#366	Q₈	1	4	yes	complemented
53	C₄⋊C₁₆	—	2	2	no	—
54	C₄×C₁₆	—	2	4	no	—
55	C₄×C₁₆	C₂×C₄	1	4	yes	—
56	#340	—	2	2	no	—
57	#340	—	2	4	no	—
58	#340	—	2	4	no	—
59	#336	D₄	1	2	yes	—
60	C₂²×Q₈	—	4	2	no	—
61	C₂²×C₈	C₂×D₄	1	2	yes	—
62	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₈	C₂⁴	1	1	yes	characteristic
64	C₄⋊Q₈	—	2	8	no	—
65	C₄⋊Q₈	—	4	4	no	—
66	C₄×Q₈	—	8	8	no	—
67	C₂×C₄⋊C₄	—	2	4	no	—
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	2	2	no	—
70	C₂×C₄⋊C₄	—	4	2	no	—
71	C₂×C₄⋊C₄	—	4	4	no	—
72	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
73	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
74	C₂×C₄²	—	4	4	no	—
75	C₂×C₄²	—	4	2	no	—
76	C₂×C₄²	C₂×D₄	1	2	yes	—
77	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	—	2	1	no	—
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	2	2	no	—
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	C₂²×C₄	1	2	yes	—
84	C₂×C₁₆	C₂²×C₄	1	2	yes	—
85	C₂×C₁₆	C₂×Q₈	1	2	yes	—
86	C₂×C₁₆	#41	1	2	yes	—
87	#107	—	2	8	no	—
88	#107	—	2	4	no	—
89	#107	—	2	2	no	—
90	#107	—	2	2	no	—
91	#107	—	2	2	no	—
92	#107	C₂²×C₄	1	8	yes	—
93	#107	C₂×Q₈	1	4	yes	—
94	C₄⋊C₈	—	2	2	no	—
95	C₄×C₈	—	2	2	no	—
96	C₄×C₈	—	2	2	no	—
97	C₄×C₈	C₂²×C₄	1	2	yes	—
98	C₄×C₈	#41	1	2	yes	—
99	#95	—	4	2	no	—
100	#95	—	4	4	no	—
101	#95	—	4	4	no	—
102	#95	D₈	1	2	yes	—
103	C₂×Q₈	—	4	8	no	—
104	C₂×Q₈	—	8	4	no	—
105	C₂×Q₈	—	8	4	no	—
106	C₂²×C₄	—	4	2	no	—
107	C₂²×C₄	—	4	4	no	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	C₂×D₈	1	2	yes	—
110	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₈	1	2	yes	—
112	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	C₄×D₄	1	2	yes	—
120	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₈	C₂×#41	1	1	yes	characteristic
123	C₂×C₈	C₂×#41	1	1	yes	characteristic
124	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
125	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	2	8	no	—
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	4	4	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	4	8	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	8	8	no	—
135	C₄⋊C₄	D₁₆	1	4	yes	—
136	C₄²	—	2	2	no	—
137	C₄²	—	2	2	no	—
138	C₄²	—	8	8	no	—
139	C₄²	—	8	4	no	—
140	C₄²	C₄×D₄	1	2	yes	—
141	C₄²	#135	1	2	yes	—
142	C₁₆	—	2	2	no	—
143	C₁₆	—	4	2	no	—
144	C₁₆	C₄×Q₈	1	4	yes	complemented
145	C₂³	C₂²×D₈	1	1	yes	characteristic
146	Q₈	—	8	16	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	4	8	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	4	no	—
158	C₂×C₄	—	8	4	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	C₄×D₈	1	2	yes	—
164	C₂×C₄	C₄×D₈	1	2	yes	—
165	C₂×C₄	C₂×D₁₆	1	4	yes	—
166	C₂×C₄	C₂×D₁₆	1	2	yes	—
167	C₂×C₄	#508	1	2	yes	—
168	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂×C₄	#547	1	1	yes	characteristic
170	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
171	C₂×C₄	C₂×#135	1	1	yes	characteristic
172	C₂×C₄	#578	1	1	yes	characteristic
173	C₂×C₄	#578	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	4	2	no	—
176	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
177	C₈	#519	1	1	yes	characteristic
178	C₈	#556	1	1	yes	characteristic
179	C₈	#557	1	1	yes	characteristic
180	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
181	C₂²	#3334	1	1	yes	characteristic
182	C₂²	Q₈×D₈	1	1	yes	characteristic
183	C₂²	C₂²×D₁₆	1	1	yes	characteristic
184	C₂²	C₂×#508	1	1	yes	characteristic
185	C₂²	#3417	1	1	yes	characteristic
186	C₂²	#3417	1	1	yes	characteristic
187	C₄	—	2	4	no	—
188	C₄	—	2	2	no	—
189	C₄	—	4	2	no	—
190	C₄	—	8	4	no	—
191	C₄	—	8	8	no	—
192	C₄	—	16	2	no	—
193	C₄	C₄×D₁₆	1	4	yes	—
194	C₄	#2365	1	1	yes	characteristic
195	C₄	#2950	1	1	yes	characteristic
196	C₄	#3342	1	1	yes	characteristic
197	C₄	#3383	1	1	yes	characteristic
198	C₂	#23157	1	1	yes	characteristic
199	C₂	C₂×C₄×D₁₆	1	1	yes	characteristic
200	C₂	#33458	1	1	yes	characteristic
201	C₂	#33901	1	1	yes	characteristic
202	C₂	#33909	1	1	yes	characteristic
203	C₂	Q₈×D₁₆	1	1	yes	characteristic
204	C₂	#33950	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
11. #2252	C₄	semidirect
52. #366	Q₈	semidirect
144. C₁₆	C₄×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2252	2
#23157	2
#13275	2
#13165	2
#13109	2
C₄×#366	2
C₁₆×C₄⋊C₄	2