Subgroups — #151730

Order 512.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13731	C₂	1	4	yes	maximal
2	#12521	C₂	1	2	yes	maximal
3	C₈×C₃₂	C₂	1	1	yes	maximal, characteristic
4	#2252	—	2	2	no	—
5	#2252	C₂²	1	4	yes	—
6	#1713	—	2	2	no	—
7	#1423	—	2	8	no	—
8	C₄×C₃₂	C₂²	1	2	yes	—
9	C₈×C₁₆	C₂²	1	1	yes	characteristic
10	#500	—	2	4	no	—
11	#500	—	4	2	no	—
12	#494	—	4	2	no	—
13	C₂×C₃₂	D₄	1	4	yes	—
14	#366	—	2	8	no	—
15	#366	—	4	4	no	—
16	C₄×C₁₆	D₄	1	1	yes	characteristic
17	C₄×C₁₆	C₂³	1	1	yes	characteristic
18	C₈²	D₄	1	1	yes	characteristic
19	C₂×Q₁₆	—	8	4	no	—
20	C₄⋊Q₈	—	4	4	no	—
21	C₂×C₁₆	D₈	1	2	yes	—
22	C₂×C₁₆	C₂×D₄	1	2	yes	—
23	#107	—	4	4	no	—
24	#107	—	4	4	no	—
25	#107	—	8	4	no	—
26	C₄×C₈	D₈	1	1	yes	characteristic
27	C₄×C₈	D₈	1	1	yes	characteristic
28	C₄×C₈	C₂×D₄	1	1	yes	characteristic
29	C₃₂	Q₁₆	1	8	yes	complemented
30	C₂×Q₈	—	8	4	no	—
31	Q₁₆	—	16	8	no	—
32	C₂×C₈	D₁₆	1	2	yes	—
33	C₂×C₈	D₁₆	1	2	yes	—
34	C₂×C₈	#127	1	1	yes	characteristic
35	C₂×C₈	C₂×D₈	1	1	yes	characteristic
36	C₄⋊C₄	—	8	4	no	—
37	C₄⋊C₄	—	8	4	no	—
38	C₄²	C₂×D₈	1	1	yes	characteristic
39	C₁₆	Q₃₂	1	4	yes	—
40	C₁₆	C₂×Q₁₆	1	4	yes	—
41	Q₈	—	16	8	no	—
42	C₂×C₄	—	16	4	no	—
43	C₂×C₄	#493	1	1	yes	characteristic
44	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
45	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
46	C₈	D₃₂	1	4	yes	—
47	C₈	Q₆₄	1	4	yes	—
48	C₈	#494	1	2	yes	—
49	C₈	C₂×Q₃₂	1	2	yes	—
50	C₂²	#2246	1	1	yes	characteristic
51	C₄	—	32	4	no	—
52	C₄	#1713	1	1	yes	characteristic
53	C₄	#2247	1	1	yes	characteristic
54	C₄	C₂×D₃₂	1	2	yes	—
55	C₄	C₂×Q₆₄	1	2	yes	—
56	C₂	#12521	1	1	yes	characteristic
57	C₂	#13725	1	1	yes	characteristic
58	C₂	#13726	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#2246
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
29. C₃₂	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#13731	2
#12521	2
C₈×C₃₂	2