Subgroups — #33301

Order 256.

1023 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493	C₂	1	12	yes	maximal, complemented
2	#2867	C₂	1	2	yes	maximal, complemented
3	C₄²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₈	C₂²	1	16	yes	—
5	C₂×#127	—	2	8	no	—
6	C₂×#127	C₂²	1	12	yes	—
7	#493	—	2	48	no	—
8	C₂×C₄×C₈	C₂²	1	6	yes	—
9	C₄³	C₂²	1	1	yes	characteristic
10	C₂²×D₄	—	2	24	no	—
11	C₂²×D₄	C₂³	1	8	yes	—
12	C₂×D₈	—	2	96	no	—
13	C₂²×C₈	C₂³	1	4	yes	—
14	#127	—	2	48	no	—
15	#127	—	4	32	no	—
16	C₂×C₄²	D₄	1	4	yes	—
17	C₂×C₄²	C₂³	1	3	yes	—
18	C₄×C₈	D₄	1	24	yes	complemented
19	C₂⁴	—	2	8	no	—
20	C₂×D₄	—	2	48	no	—
21	C₂×D₄	—	4	48	no	—
22	C₂×D₄	—	4	96	no	—
23	C₂²×C₄	C₂×D₄	1	6	yes	—
24	C₂²×C₄	C₂⁴	1	1	yes	characteristic
25	D₈	—	4	64	no	—
26	C₂×C₈	C₂×D₄	1	24	yes	—
27	C₄²	D₈	1	16	yes	complemented
28	C₄²	C₂×D₄	1	12	yes	—
29	C₂³	—	4	24	no	—
30	C₂³	—	4	32	no	—
31	C₂³	C₂²×D₄	1	1	yes	characteristic
32	D₄	—	4	32	no	—
33	D₄	—	8	96	no	—
34	C₂×C₄	#127	1	12	yes	—
35	C₂×C₄	C₂×D₈	1	24	yes	—
36	C₂×C₄	C₂²×D₄	1	6	yes	—
37	C₈	#127	1	16	yes	complemented
38	C₂²	—	8	8	no	—
39	C₂²	—	8	48	no	—
40	C₂²	C₂×#127	1	3	yes	—
41	C₂²	C₂²×D₈	1	4	yes	—
42	C₄	#493	1	24	yes	complemented
43	C₄	C₂×#127	1	4	yes	—
44	C₂	—	16	8	no	—
45	C₂	#2867	1	1	yes	characteristic
46	C₂	C₂×#493	1	6	yes	—

	subgroup	G/H
center, socle	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

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