Subgroups — #123404

Order 512.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12521	C₂	1	4	yes	maximal
2	#12516	C₂	1	2	yes	maximal
3	C₁₆²	C₂	1	1	yes	maximal, characteristic
4	#2252	—	2	8	no	—
5	#2247	—	2	4	no	—
6	#1713	C₂²	1	4	yes	—
7	C₈×C₁₆	C₂²	1	1	yes	characteristic
8	C₈×C₁₆	C₂²	1	2	yes	—
9	C₂×Q₃₂	—	4	8	no	—
10	#500	—	2	8	no	—
11	#494	—	2	4	no	—
12	#366	—	4	16	no	—
13	C₄×C₁₆	D₄	1	2	yes	—
14	C₄×C₁₆	D₄	1	4	yes	—
15	C₈²	C₂³	1	1	yes	characteristic
16	C₂×Q₁₆	—	4	8	no	—
17	C₄⋊Q₈	—	4	4	no	—
18	Q₃₂	—	8	16	no	—
19	C₂×C₁₆	D₈	1	4	yes	—
20	C₂×C₁₆	D₈	1	8	yes	—
21	#107	—	4	16	no	—
22	C₄×C₈	C₂×D₄	1	1	yes	characteristic
23	C₄×C₈	C₂×D₄	1	2	yes	—
24	C₂×Q₈	—	8	4	no	—
25	Q₁₆	—	8	16	no	—
26	C₂×C₈	C₂×D₈	1	2	yes	—
27	C₂×C₈	C₂×D₈	1	4	yes	—
28	C₄⋊C₄	—	8	8	no	—
29	C₄²	#127	1	1	yes	characteristic
30	C₁₆	D₁₆	1	8	yes	—
31	C₁₆	Q₃₂	1	16	yes	complemented
32	Q₈	—	16	8	no	—
33	C₂×C₄	—	16	4	no	—
34	C₂×C₄	#493	1	1	yes	characteristic
35	C₂×C₄	#493	1	2	yes	—
36	C₈	C₂×D₁₆	1	4	yes	—
37	C₈	C₂×Q₃₂	1	8	yes	—
38	C₂²	#1712	1	1	yes	characteristic
39	C₄	—	32	4	no	—
40	C₄	#2246	1	2	yes	—
41	C₄	#2247	1	4	yes	—
42	C₂	#12515	1	1	yes	characteristic
43	C₂	#12516	1	2	yes	—

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

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

N	H ≅ G/N	type
31. C₁₆	Q₃₂	semidirect

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