Subgroups — C₂²⋊(C₂²⋊C₉) #3768

Order 144.

61 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃	C₃	1	1	yes	maximal, characteristic
2	C₂²⋊C₉	—	4	5	no	maximal
3	C₂³×C₃	—	3	5	no	—
4	C₂⁴	C₉	1	1	yes	characteristic, complemented, Sylow
5	C₂²×C₃	—	3	10	no	—
6	C₂²×C₃	A₄	1	5	yes	—
7	C₉	—	16	1	no	Sylow
8	C₂³	—	3	5	no	—
9	C₂×C₃	—	3	5	no	—
10	C₂²	—	3	10	no	—
11	C₂²	C₂²⋊C₉	1	5	yes	complemented
12	C₃	C₂²⋊A₄	1	1	yes	characteristic
13	C₂	—	3	5	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₂²⋊A₄
commutator, hypocenter	C₂⁴	C₉
Fitting, socle, prime radical, cosocle	C₂⁴×C₃	C₃
LCS	G ▷ C₂⁴
UCS	C₁ ◁ C₃

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

N	H ≅ G/N	type
4. C₂⁴	C₉	semidirect
11. C₂²	C₂²⋊C₉	semidirect

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

subgroup	index
C₂⁴×C₃	3
C₂²⋊C₉	4