Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
504 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 248 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C3 |
| socle, prime radical | C23×C3 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. #2399 | C3 | direct (N × H) |
| 35. C3×C42 | Q8 | semidirect |
| 72. C42 | C3×Q8 | semidirect |
| 105. C3 | #2399 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C4×Q8 | 2 |
| C3×#398 | 2 |
| C3×#395 | 2 |
| C3×C4×C4⋊C4 | 2 |
| C3×#376 | 2 |
| #2399 | 3 |