Subgroups — #3415

Order 128.

185 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#576	C₂	1	2	yes	maximal, complemented
2	#509	C₂	1	6	yes	maximal, complemented
3	#508	C₂	1	3	yes	maximal, complemented
4	C₂×D₁₆	C₂	1	3	yes	maximal, complemented
5	#504	C₂	1	1	yes	maximal, characteristic, complemented
6	H(5,𝔽₂)	—	2	2	no	—
7	AGL(1,ℤ/8)	—	2	6	no	—
8	#135	C₂²	1	6	yes	complemented
9	C₂×D₈	C₂²	1	6	yes	complemented
10	#131	C₂²	1	1	yes	characteristic
11	Q₃₂	C₂²	1	1	yes	characteristic, complemented
12	SD₃₂	C₂²	1	6	yes	complemented
13	D₁₆	C₂²	1	3	yes	complemented
14	D₁₆	C₂²	1	6	yes	complemented
15	M₃₂	C₂²	1	3	yes	complemented
16	C₂×C₁₆	C₂²	1	3	yes	complemented
17	#41	—	2	6	no	—
18	#41	—	4	2	no	—
19	#41	D₄	1	1	yes	characteristic
20	C₂×D₄	—	2	6	no	—
21	C₂×D₄	—	4	6	no	—
22	Q₁₆	C₂³	1	2	yes	—
23	SD₁₆	—	2	6	no	—
24	D₈	—	2	6	no	—
25	D₈	C₂³	1	6	yes	—
26	M₁₆	D₄	1	3	yes	—
27	C₂×C₈	C₂³	1	3	yes	—
28	C₁₆	C₂³	1	1	yes	characteristic
29	C₁₆	C₂³	1	3	yes	—
30	C₂³	—	4	6	no	—
31	Q₈	—	2	2	no	—
32	Q₈	D₈	1	1	yes	characteristic
33	D₄	—	2	6	no	—
34	D₄	—	4	6	no	—
35	D₄	—	4	6	no	—
36	D₄	D₈	1	3	yes	—
37	C₂×C₄	—	4	6	no	—
38	C₂×C₄	C₂×D₄	1	3	yes	—
39	C₈	C₂×D₄	1	3	yes	—
40	C₈	C₂⁴	1	1	yes	characteristic
41	C₂²	—	4	6	no	—
42	C₂²	—	16	6	no	—
43	C₂²	C₂×D₈	1	3	yes	—
44	C₄	—	4	2	no	—
45	C₄	C₂×D₈	1	3	yes	—
46	C₄	C₂²×D₄	1	1	yes	characteristic
47	C₂	—	2	3	no	—
48	C₂	—	8	6	no	—
49	C₂	C₂²×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁴
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ G
LECS	G ▷ C₈ ▷ 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.