Subgroups — #94739

Order 512.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13726	C₂	1	1	yes	maximal, characteristic
2	#7544	C₂	1	1	yes	maximal, characteristic
3	#7462	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₆₄	—	2	1	no	—
5	C₂×Q₆₄	C₄	1	2	yes	—
6	#2247	—	2	1	no	—
7	M₁₂₈	—	2	1	no	—
8	C₄×C₃₂	C₂²	1	1	yes	characteristic
9	#1373	—	2	1	no	—
10	C₂×Q₃₂	—	2	2	no	—
11	C₂×Q₃₂	—	4	1	no	—
12	#494	—	4	1	no	—
13	Q₆₄	—	4	2	no	—
14	Q₆₄	—	4	1	no	—
15	Q₆₄	—	4	1	no	—
16	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃₂	D₄	1	1	yes	characteristic
18	#364	—	4	1	no	—
19	C₄×C₁₆	D₄	1	1	yes	characteristic
20	C₆₄	—	4	1	no	—
21	C₂×Q₁₆	—	4	2	no	—
22	C₂×Q₁₆	—	8	1	no	—
23	C₄⋊Q₈	—	8	1	no	—
24	Q₃₂	—	4	2	no	—
25	Q₃₂	—	4	2	no	—
26	Q₃₂	—	8	2	no	—
27	M₃₂	—	8	1	no	—
28	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₁₆	SD₁₆	1	1	yes	characteristic
30	C₄×C₈	D₈	1	1	yes	characteristic
31	C₃₂	—	2	1	no	—
32	C₃₂	D₈	1	1	yes	characteristic
33	C₃₂	SD₁₆	1	1	yes	characteristic
34	C₂×Q₈	—	8	2	no	—
35	Q₁₆	—	8	2	no	—
36	Q₁₆	—	8	2	no	—
37	Q₁₆	—	16	1	no	—
38	Q₁₆	—	16	1	no	—
39	C₂×C₈	#102	1	1	yes	characteristic
40	C₂×C₈	D₁₆	1	1	yes	characteristic
41	C₄⋊C₄	—	16	2	no	—
42	C₄²	SD₃₂	1	1	yes	characteristic
43	C₁₆	—	2	1	no	—
44	C₁₆	—	16	1	no	—
45	C₁₆	#100	1	1	yes	characteristic
46	C₁₆	#102	1	1	yes	characteristic
47	Q₈	—	16	2	no	—
48	Q₈	—	16	2	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	16	2	no	—
51	C₂×C₄	#357	1	1	yes	characteristic
52	C₈	#331	1	1	yes	characteristic
53	C₈	#361	1	1	yes	characteristic
54	C₈	D₃₂	1	1	yes	characteristic
55	C₈	SD₆₄	1	1	yes	characteristic
56	C₂²	#1419	1	1	yes	characteristic
57	C₄	—	2	2	no	—
58	C₄	—	32	2	no	—
59	C₄	#1361	1	1	yes	characteristic
60	C₄	#1415	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	#7444	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#7444
commutator	C₂×C₃₂	C₂×C₄
Frattini, cosocle	C₄×C₃₂	C₂²
prime radical	C₂²	#1419
LCS	G ▷ C₂×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#13726	2
#7544	2
#7462	2