Subgroups — #135771

Order 512.

282 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12888	C₂	1	4	yes	maximal
2	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#8816	C₂	1	2	yes	maximal
4	C₂×C₄×C₁₆	C₂²	1	2	yes	—
5	#1845	C₂²	1	4	yes	—
6	#1834	—	2	4	no	—
7	C₂×C₈²	C₂²	1	1	yes	characteristic
8	#1380	—	2	8	no	—
9	C₈×C₁₆	C₄	1	4	yes	complemented
10	C₂²×C₁₆	D₄	1	4	yes	—
11	C₂×#107	—	2	8	no	—
12	C₂×C₄×C₈	D₄	1	2	yes	—
13	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
14	#384	—	2	4	no	—
15	C₄×C₁₆	C₂×C₄	1	4	yes	—
16	C₄×C₁₆	Q₈	1	4	yes	—
17	#340	—	4	8	no	—
18	C₈²	C₂×C₄	1	2	yes	—
19	C₈²	Q₈	1	2	yes	—
20	C₂²×C₈	D₈	1	4	yes	—
21	C₂²×C₈	C₂×D₄	1	2	yes	—
22	C₂×C₄⋊C₄	—	4	4	no	—
23	C₂×C₄⋊C₄	—	4	8	no	—
24	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
25	C₂×C₁₆	—	2	8	no	—
26	C₂×C₁₆	C₄⋊C₄	1	8	yes	—
27	#107	—	4	16	no	—
28	C₄×C₈	C₄⋊C₄	1	4	yes	—
29	C₄×C₈	Q₁₆	1	4	yes	—
30	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
32	C₄×C₈	C₂×Q₈	1	2	yes	—
33	C₂²×C₄	—	8	4	no	—
34	C₂²×C₄	#127	1	1	yes	characteristic
35	C₂²×C₄	C₂×D₈	1	2	yes	—
36	C₂×C₈	—	2	8	no	—
37	C₂×C₈	#107	1	8	yes	—
38	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
39	C₂×C₈	C₄⋊Q₈	1	2	yes	—
40	C₂×C₈	#135	1	8	yes	—
41	C₄⋊C₄	—	8	8	no	—
42	C₄⋊C₄	—	8	16	no	—
43	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₄²	C₄⋊Q₈	1	1	yes	characteristic
45	C₄²	C₂×Q₁₆	1	2	yes	—
46	C₁₆	—	2	8	no	—
47	C₂³	#493	1	1	yes	characteristic
48	C₂×C₄	—	16	4	no	—
49	C₂×C₄	—	16	8	no	—
50	C₂×C₄	#384	1	1	yes	characteristic
51	C₂×C₄	C₂×#107	1	2	yes	—
52	C₂×C₄	#494	1	1	yes	characteristic
53	C₂×C₄	#495	1	4	yes	—
54	C₂×C₄	#500	1	2	yes	—
55	C₂×C₄	#508	1	8	yes	—
56	C₈	—	2	8	no	—
57	C₈	#427	1	4	yes	—
58	C₈	#499	1	4	yes	—
59	C₂²	#1709	1	2	yes	—
60	C₂²	#1845	1	1	yes	characteristic
61	C₂²	#2249	1	4	yes	—
62	C₄	—	32	4	no	—
63	C₄	#1710	1	2	yes	—
64	C₄	#1846	1	2	yes	—
65	C₄	#2158	1	4	yes	—
66	C₄	#2253	1	4	yes	—
67	C₂	#8816	1	1	yes	characteristic
68	C₂	#12511	1	4	yes	—
69	C₂	#12907	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	#493
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
9. C₈×C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
#12888	2
C₂×C₈×C₁₆	2
#8816	2