Finite Group Databasebeta

← C₂²×3₋¹⁺² #1104

Subgroups — C₂²×3₋¹⁺² #1104

Order 108.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×3₋¹⁺²	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₃²	C₃	1	1	yes	maximal, characteristic
3	C₂²×C₉	C₃	1	3	yes	maximal, complemented
4	3₋¹⁺²	C₂²	1	1	yes	characteristic, direct product factor, Sylow
5	C₂×C₃²	C₂×C₃	1	3	yes	—
6	C₂×C₉	C₂×C₃	1	9	yes	complemented
7	C₂²×C₃	—	3	1	no	—
8	C₂²×C₃	C₃²	1	1	yes	characteristic
9	C₃²	C₂²×C₃	1	1	yes	characteristic
10	C₉	C₂²×C₃	1	3	yes	complemented
11	C₂×C₃	—	3	3	no	—
12	C₂×C₃	C₂×C₃²	1	3	yes	—
13	C₂²	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
14	C₃	—	3	1	no	—
15	C₃	C₂²×C₃²	1	1	yes	characteristic
16	C₂	C₂×3₋¹⁺²	1	3	yes	direct product factor

	subgroup	G/H
center, socle	C₂²×C₃	C₃²
commutator, Frattini, cosocle	C₃	C₂²×C₃²
prime radical	C₂²×C₃²	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×3₋¹⁺²	C₂	direct (N × H)
3. C₂²×C₉	C₃	semidirect
4. 3₋¹⁺²	C₂²	direct (N × H)
6. C₂×C₉	C₂×C₃	semidirect
10. C₉	C₂²×C₃	semidirect
13. C₂²	3₋¹⁺²	direct (N × H)
16. C₂	C₂×3₋¹⁺²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×3₋¹⁺²	2
C₂²×C₃²	3
C₂²×C₉	3