Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
278 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #393 |
| commutator, Frattini, cosocle | C24 | C23×C3 |
| prime radical | C22×C3×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C24×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#543 | C2 | semidirect |
| 3. C3×#342 | C2 | semidirect |
| 4. #2032 | C3 | direct (N × H) |
| 10. #543 | C2×C3 | semidirect |
| 12. #342 | C2×C3 | semidirect |
| 13. C24×C3 | Q8 | semidirect |
| 20. C22×C3×C4 | Q8 | semidirect |
| 45. C24 | C3×Q8 | semidirect |
| 52. C22×C4 | C3×Q8 | semidirect |
| 99. C3 | #2032 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#543 | 2 |
| C2×C3×#99 | 2 |
| C3×#342 | 2 |
| #2032 | 3 |