Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
125 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 125 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D32 |
| commutator, socle, prime radical, cosocle | C22×C32 | C23 |
| Fitting | C2×C32×C4 | C22 |
| hypercenter | C2×C4 | D32 |
| hypocenter | C32 | #125 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 88. C32 | #125 | semidirect |
| 114. C3 | #889 | semidirect |
| 115. C3 | #890 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3750 | 2 |
| C3×C4⋊Dic3 | 2 |
| C3×Dic3⋊C4 | 2 |
| Dic3⋊Dic3 | 2 |
| Dic32 | 2 |
| #890 | 3 |
| #889 | 3 |