Subgroups — #473

Order 64.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂	#121
commutator, Frattini, cosocle	C₂×C₄	C₂³
prime radical	#41	C₂²
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.