Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1072 conjugacy classes of subgroups; only the 520 normal subgroups are shown.
This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 520 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C4 | C22 |
| commutator | C2 | C23×C3×C8 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C24×C3 | C2×C4 |
| prime radical | C25×C3 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×C3×C8 | C2 | semidirect |
| 3. C2×C3×C22⋊C8 | C2 | direct (N × H) |
| 4. C22×C22⋊C8 | C3 | direct (N × H) |
| 12. C22×C3×C8 | C22 | semidirect |
| 13. C3×C22⋊C8 | C22 | direct (N × H) |
| 15. C23×C8 | C2×C3 | semidirect |
| 16. C2×C22⋊C8 | C2×C3 | direct (N × H) |
| 19. C24×C3 | C8 | semidirect |
| 32. C2×C3×C8 | C23 | semidirect |
| 40. C22×C8 | C22×C3 | semidirect |
| 41. C22⋊C8 | C22×C3 | direct (N × H) |
| 47. C23×C3 | C2×C8 | semidirect |
| 62. C24 | C3×C8 | semidirect |
| 75. C2×C8 | C23×C3 | semidirect |
| 79. C22×C3 | C22⋊C8 | direct (N × H) |
| 82. C22×C3 | C22×C8 | semidirect |
| 93. C23 | C2×C3×C8 | semidirect |
| 107. C2×C3 | C2×C22⋊C8 | direct (N × H) |
| 114. C22 | C3×C22⋊C8 | direct (N × H) |
| 117. C22 | C22×C3×C8 | semidirect |
| 123. C3 | C22×C22⋊C8 | direct (N × H) |
| 125. C2 | C2×C3×C22⋊C8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3×C4 | 2 |
| C23×C3×C8 | 2 |
| C2×C3×C22⋊C8 | 2 |
| C22×C22⋊C8 | 3 |