Subgroups — #135751

Order 512.

215 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12901	C₂	1	2	yes	maximal
2	#12593	C₂	1	1	yes	maximal, characteristic
3	#12011	C₂	1	4	yes	maximal
4	C₂×#364	—	2	2	no	—
5	C₂×#364	C₂²	1	4	yes	—
6	#2150	—	2	2	no	—
7	C₄×M₃₂	C₂²	1	2	yes	—
8	#1726	C₂²	1	1	yes	characteristic
9	#1375	C₄	1	4	yes	—
10	#1373	—	2	8	no	—
11	C₂×M₃₂	—	2	4	no	—
12	C₂×M₃₂	C₂×C₄	1	4	yes	—
13	C₂×#97	C₂×C₄	1	2	yes	—
14	C₂×#97	Q₈	1	1	yes	characteristic
15	C₂×C₄×C₈	D₄	1	2	yes	—
16	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
17	#376	Q₈	1	1	yes	characteristic
18	#364	—	2	8	no	—
19	#364	—	4	4	no	—
20	C₄⋊C₁₆	—	4	4	no	—
21	#346	Q₈	1	4	yes	—
22	C₄×C₁₆	D₄	1	4	yes	—
23	C₂²×C₈	C₄⋊C₄	1	2	yes	—
24	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
26	C₂×C₄²	C₄⋊C₄	1	2	yes	—
27	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
28	M₃₂	—	2	8	no	—
29	M₃₂	—	4	4	no	—
30	M₃₂	—	4	4	no	—
31	C₂×C₁₆	—	4	4	no	—
32	C₂×C₁₆	C₄⋊C₄	1	8	yes	—
33	#97	—	2	4	no	—
34	#97	—	4	1	no	—
35	C₄×C₈	D₈	1	4	yes	—
36	C₄×C₈	Q₁₆	1	4	yes	—
37	C₄×C₈	C₂×D₄	1	2	yes	—
38	C₄×C₈	C₂×Q₈	1	2	yes	—
39	#95	—	4	1	no	—
40	C₂²×C₄	—	2	2	no	—
41	C₂²×C₄	—	2	1	no	—
42	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈	—	2	2	no	—
44	C₂×C₈	—	2	2	no	—
45	C₂×C₈	#107	1	4	yes	—
46	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
47	C₂×C₈	#127	1	1	yes	characteristic
48	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₈	#131	1	4	yes	—
50	C₂×C₈	C₂×D₈	1	2	yes	—
51	C₂×C₈	C₂×Q₁₆	1	2	yes	—
52	C₄²	—	2	2	no	—
53	C₄²	—	2	1	no	—
54	C₄²	#107	1	4	yes	—
55	C₄²	C₄⋊Q₈	1	2	yes	—
56	C₁₆	—	2	8	no	—
57	C₁₆	—	8	4	no	—
58	C₂³	#430	1	1	yes	characteristic
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	8	2	no	—
62	C₂×C₄	—	8	1	no	—
63	C₂×C₄	#384	1	1	yes	characteristic
64	C₂×C₄	#424	1	2	yes	—
65	C₂×C₄	C₂×#107	1	2	yes	—
66	C₂×C₄	#430	1	1	yes	characteristic
67	C₈	—	2	4	no	—
68	C₈	#424	1	4	yes	—
69	C₈	#493	1	1	yes	characteristic
70	C₈	#494	1	1	yes	characteristic
71	C₈	#500	1	2	yes	—
72	C₂²	—	4	1	no	—
73	C₂²	#1571	1	2	yes	—
74	C₂²	#1849	1	1	yes	characteristic
75	C₄	—	4	4	no	—
76	C₄	—	4	2	no	—
77	C₄	#1571	1	2	yes	—
78	C₄	#1843	1	1	yes	characteristic
79	C₄	#1845	1	1	yes	characteristic
80	C₂	—	2	2	no	—
81	C₂	—	2	1	no	—
82	C₂	#8810	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#493
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8810
prime radical	C₂³	#430
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#12901	2
#12593	2
#12011	2