Subgroups — #12516

Order 256.

155 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

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

N	H ≅ G/N	type
27. C₈	Q₃₂	semidirect

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