Finite Group Databasebeta

Subgroups — #3776

Order 144.

30 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#627	C₂	1	2	yes	maximal
2	#622	C₂	1	1	yes	maximal, characteristic
3	C₃⋊Dic₃	C₂²	1	1	yes	characteristic
4	C₃×Dic₃	—	2	2	no	—
5	Dic₆	—	6	2	no	—
6	C₂×C₃²	D₄	1	1	yes	characteristic
7	Q₁₆	—	9	1	no	maximal, Sylow
8	C₃×C₄	—	6	2	no	—
9	Dic₃	—	2	2	no	—
10	Dic₃	—	6	2	no	—
11	C₃²	Q₁₆	1	1	yes	characteristic, complemented, Sylow
12	Q₈	—	9	2	no	—
13	C₈	—	9	1	no	—
14	C₂×C₃	—	2	2	no	—
15	C₄	—	6	2	no	—
16	C₄	—	9	1	no	—
17	C₃	—	2	2	no	—
18	C₂	GO⁺(4,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	GO⁺(4,𝔽₂)
commutator, cosocle	C₃⋊Dic₃	C₂²
Fitting, socle, prime radical	C₂×C₃²	D₄
hypocenter	C₃²	Q₁₆
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
11. C₃²	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#627	2
#622	2
Q₁₆	9