Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
348 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 160 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 | C22 | C22×C3×C8 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C23×C3 | C22×C4 |
| prime radical | C24×C3 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×C4⋊C8 | C2 | semidirect |
| 5. C3×#336 | C2 | semidirect |
| 6. #1850 | C3 | direct (N × H) |
| 20. C2×C4⋊C8 | C2×C3 | semidirect |
| 23. #336 | C2×C3 | semidirect |
| 42. C3×C22⋊C4 | C8 | semidirect |
| 73. C2×C3×C4 | C2×C8 | semidirect |
| 99. C22⋊C4 | C3×C8 | semidirect |
| 106. C22×C3 | C4⋊C8 | semidirect |
| 134. C2×C4 | C2×C3×C8 | semidirect |
| 152. C22 | C3×C4⋊C8 | semidirect |
| 164. C3 | #1850 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C8 | 2 |
| C2×C3×C4⋊C8 | 2 |
| C2×C3×C22⋊C8 | 2 |
| C3×C4×C22⋊C4 | 2 |
| C3×#336 | 2 |
| #1850 | 3 |