Subgroups — #12523

Order 256.

131 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2254	C₂	1	4	yes	maximal
2	#1713	C₂	1	2	yes	maximal
3	C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#500	—	2	2	no	—
5	#500	C₂²	1	4	yes	—
6	#494	—	2	2	no	—
7	#367	—	2	8	no	—
8	C₄×C₁₆	C₂²	1	2	yes	—
9	C₈²	C₂²	1	1	yes	characteristic
10	C₂×Q₁₆	—	4	4	no	—
11	C₄⋊Q₈	—	2	4	no	—
12	C₂×C₁₆	D₄	1	4	yes	—
13	#107	—	2	8	no	—
14	#107	—	4	4	no	—
15	C₄×C₈	D₄	1	2	yes	—
16	C₄×C₈	C₂³	1	1	yes	characteristic
17	C₂×Q₈	—	4	4	no	—
18	Q₁₆	—	8	8	no	—
19	C₂×C₈	D₈	1	4	yes	—
20	C₂×C₈	C₂×D₄	1	2	yes	—
21	C₄⋊C₄	—	4	4	no	—
22	C₄⋊C₄	—	4	4	no	—
23	C₄²	C₂×D₄	1	1	yes	characteristic
24	C₁₆	Q₁₆	1	8	yes	complemented
25	Q₈	—	8	8	no	—
26	C₂×C₄	—	8	4	no	—
27	C₂×C₄	#127	1	1	yes	characteristic
28	C₂×C₄	C₂×D₈	1	2	yes	—
29	C₈	SD₃₂	1	8	yes	—
30	C₈	C₂×Q₁₆	1	4	yes	—
31	C₂²	#493	1	1	yes	characteristic
32	C₄	—	16	4	no	—
33	C₄	#494	1	2	yes	—
34	C₄	C₂×SD₃₂	1	4	yes	—
35	C₂	#1713	1	1	yes	characteristic
36	C₂	#2248	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
24. C₁₆	Q₁₆	semidirect

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