Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
112 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 28 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 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C3×C4 |
| Frattini, cosocle | #124 | C22×C3 |
| prime radical | C3×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#124 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#354 | C2 | semidirect |
| 3. C3×#329 | C2 | semidirect |
| 4. #1404 | C3 | direct (N × H) |
| 5. C3×2−1+4 | C4 | semidirect |
| 9. C3×#101 | C4 | semidirect |
| 13. #354 | C2×C3 | semidirect |
| 14. #329 | C2×C3 | semidirect |
| 28. 2−1+4 | C3×C4 | semidirect |
| 32. #101 | C3×C4 | semidirect |
| 102. C3 | #1404 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#454 | 2 |
| C3×#354 | 2 |
| C3×#329 | 2 |
| #1404 | 3 |