Subgroups — #471

Order 64.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#137	C₂	1	1	yes	maximal, characteristic, complemented
2	AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
3	C₂×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#131	C₂	1	1	yes	maximal, characteristic, complemented
5	#108	C₂	1	1	yes	maximal, characteristic, complemented
6	#101	C₂	1	1	yes	maximal, characteristic, complemented
7	#100	C₂	1	1	yes	maximal, characteristic, complemented
8	#41	C₂²	1	1	yes	characteristic
9	C₂×Q₈	C₂²	1	1	yes	characteristic
10	C₂×D₄	C₂²	1	1	yes	characteristic
11	Q₁₆	—	2	1	no	—
12	SD₁₆	—	2	1	no	—
13	SD₁₆	—	2	1	no	—
14	SD₁₆	—	2	1	no	—
15	SD₁₆	—	2	1	no	—
16	D₈	—	2	1	no	—
17	M₁₆	—	2	1	no	—
18	M₁₆	C₂²	1	1	yes	characteristic
19	M₁₆	C₂²	1	1	yes	characteristic
20	M₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₈	—	2	1	no	—
22	C₂×C₈	C₂²	1	1	yes	characteristic
23	C₂³	—	2	1	no	—
24	Q₈	—	2	1	no	—
25	Q₈	—	2	1	no	—
26	Q₈	D₄	1	1	yes	characteristic
27	D₄	—	2	1	no	—
28	D₄	—	2	1	no	—
29	D₄	—	2	1	no	—
30	D₄	D₄	1	1	yes	characteristic
31	C₂×C₄	—	2	1	no	—
32	C₂×C₄	—	2	1	no	—
33	C₂×C₄	C₂³	1	1	yes	characteristic
34	C₈	—	2	1	no	—
35	C₈	—	2	1	no	—
36	C₈	—	2	1	no	—
37	C₈	D₄	1	1	yes	characteristic
38	C₈	D₄	1	1	yes	characteristic
39	C₂²	—	2	1	no	—
40	C₂²	—	4	1	no	—
41	C₂²	—	8	1	no	—
42	C₂²	#41	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	—	4	1	no	—
45	C₄	C₂×D₄	1	1	yes	characteristic
46	C₄	C₂×D₄	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	—	4	1	no	—
49	C₂	—	8	1	no	—
50	C₂	#121	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#121
commutator, Frattini, cosocle	C₂×C₄	C₂³
prime radical	AGL(1,ℤ/8)	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×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.