Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
123 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 123 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C8 | D3×A4 |
| commutator, hypocenter | C3×Q8 | C3×C16 |
| Fitting, cosocle | C3×C8×Q8 | C2×C3 |
| socle | C22×C3 | C8×A4 |
| prime radical | C2×C3×SL(2,𝔽3) | C8 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C2×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Q8×C3⋊C16 | C3 | semidirect |
| 24. C3×SL(2,𝔽3) | C16 | semidirect |
| 41. C3⋊C16 | SL(2,𝔽3) | direct (N × H) |
| 51. C3×Q8 | C3×C16 | semidirect |
| 58. SL(2,𝔽3) | C3⋊C16 | direct (N × H) |
| 90. Q8 | C3×C3⋊C16 | semidirect |
| 114. C3 | C16×SL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C8×SL(2,𝔽3) | 2 |
| Q8×C3⋊C16 | 3 |
| C16×SL(2,𝔽3) | 3 |
| C2×C3×C3⋊C16 | 4 |