Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
592 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D4 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C24×C3 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×#103 | C2 | semidirect |
| 5. #1787 | C3 | direct (N × H) |
| 9. C22×C3×Q8 | C4 | semidirect |
| 20. C2×#103 | C2×C3 | semidirect |
| 48. C22×Q8 | C3×C4 | semidirect |
| 67. C3×Q8 | C22⋊C4 | semidirect |
| 108. C22×C3 | #103 | semidirect |
| 128. Q8 | C3×C22⋊C4 | semidirect |
| 153. C22 | C3×#103 | semidirect |
| 164. C3 | #1787 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×Q8 | 2 |
| C2×C3×#103 | 2 |
| C2×C3×C22⋊C8 | 2 |
| C3×#380 | 2 |
| #1787 | 3 |