Subgroups — #123406

Order 512.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12516	C₂	1	1	yes	maximal, characteristic, complemented
2	#12515	C₂	1	1	yes	maximal, characteristic
3	#12509	C₂	1	4	yes	maximal, complemented
4	C₁₆²	C₂	1	1	yes	maximal, characteristic, complemented
5	#2247	—	2	2	no	—
6	#2246	—	2	2	no	—
7	#2240	—	2	8	no	—
8	#1713	C₂²	1	2	yes	—
9	#1712	C₂²	1	2	yes	—
10	C₈×C₁₆	C₂²	1	2	yes	—
11	C₈×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×Q₃₂	—	4	4	no	—
13	C₂×D₁₆	—	4	4	no	—
14	#500	—	2	4	no	—
15	#494	—	2	2	no	—
16	#493	—	2	4	no	—
17	#493	—	2	2	no	—
18	#357	—	4	16	no	—
19	C₄×C₁₆	D₄	1	4	yes	—
20	C₄×C₁₆	D₄	1	2	yes	—
21	C₈²	C₂³	1	1	yes	characteristic
22	C₂×Q₁₆	—	4	4	no	—
23	C₂×D₈	—	4	8	no	—
24	C₂×D₈	—	4	4	no	—
25	C₄⋊Q₈	—	4	2	no	—
26	#127	—	4	2	no	—
27	Q₃₂	—	8	8	no	—
28	D₁₆	—	8	8	no	—
29	C₂×C₁₆	D₈	1	8	yes	—
30	C₂×C₁₆	D₈	1	4	yes	—
31	#107	—	4	8	no	—
32	C₄×C₈	C₂×D₄	1	1	yes	characteristic
33	C₄×C₈	C₂×D₄	1	2	yes	—
34	C₂×Q₈	—	8	2	no	—
35	C₂×D₄	—	8	2	no	—
36	C₂×D₄	—	8	4	no	—
37	Q₁₆	—	8	8	no	—
38	D₈	—	8	8	no	—
39	D₈	—	8	16	no	—
40	C₂×C₈	C₂×D₈	1	2	yes	—
41	C₂×C₈	C₂×D₈	1	4	yes	—
42	C₄⋊C₄	—	8	4	no	—
43	C₄²	#127	1	1	yes	characteristic
44	C₁₆	—	2	8	no	—
45	C₁₆	D₁₆	1	8	yes	—
46	C₂³	—	16	2	no	—
47	Q₈	—	16	4	no	—
48	D₄	—	16	4	no	—
49	D₄	—	16	8	no	—
50	C₂×C₄	—	16	2	no	—
51	C₂×C₄	#493	1	1	yes	characteristic
52	C₂×C₄	#493	1	2	yes	—
53	C₈	C₂×D₁₆	1	4	yes	—
54	C₈	#508	1	8	yes	—
55	C₂²	—	32	2	no	—
56	C₂²	—	32	4	no	—
57	C₂²	#1712	1	1	yes	characteristic
58	C₄	—	32	2	no	—
59	C₄	#2246	1	2	yes	—
60	C₄	#2249	1	4	yes	—
61	C₂	—	64	2	no	—
62	C₂	#12515	1	1	yes	characteristic
63	C₂	#12518	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#1712
commutator, Frattini, cosocle	C₈²	C₂³
prime radical	#12515	C₂
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #12516	C₂	semidirect
3. #12509	C₂	semidirect
4. C₁₆²	C₂	semidirect

Maximal subgroups

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

subgroup	index
#12516	2
#12515	2
#12509	2
C₁₆²	2