Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1036 conjugacy classes of subgroups; only the 328 normal subgroups are shown.
This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 328 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C3 |
| prime radical | C25×C3 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C3×Q8 | C2 | semidirect |
| 3. C3×#393 | C2 | direct (N × H) |
| 4. C2×C3×#95 | C2 | semidirect |
| 5. C2×#393 | C3 | direct (N × H) |
| 8. C22×C3×Q8 | C22 | semidirect |
| 13. C3×#95 | C22 | semidirect |
| 14. C23×Q8 | C2×C3 | semidirect |
| 16. #393 | C2×C3 | direct (N × H) |
| 17. C2×#95 | C2×C3 | semidirect |
| 20. C24×C3 | Q8 | semidirect |
| 34. C22×Q8 | C22×C3 | semidirect |
| 39. #95 | C22×C3 | semidirect |
| 47. C23×C3 | C2×Q8 | semidirect |
| 59. C24 | C3×Q8 | semidirect |
| 90. C23 | C2×C3×Q8 | semidirect |
| 101. C2×C3 | #393 | direct (N × H) |
| 117. C3 | C2×#393 | direct (N × H) |
| 119. C2 | C3×#393 | 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 |
| C22×C3×C22⋊C4 | 2 |
| C3×#393 | 2 |
| C2×C3×#95 | 2 |
| C2×#393 | 3 |