Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
224 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C8 | C22 |
| commutator | C2 | C2×C3×C4×C8 |
| Frattini, cosocle | C42 | C23×C3 |
| socle | C22×C3 | C2×C42 |
| prime radical | C23×C3 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C3×#322 | C2 | semidirect |
| 5. C3×C82 | C2 | semidirect |
| 6. C8×M16 | C3 | direct (N × H) |
| 21. #322 | C2×C3 | semidirect |
| 22. C82 | C2×C3 | semidirect |
| 26. C3×M16 | C8 | direct (N × H) |
| 59. C3×C8 | C2×C8 | semidirect |
| 60. C3×C8 | M16 | direct (N × H) |
| 64. M16 | C3×C8 | direct (N × H) |
| 96. C8 | C2×C3×C8 | semidirect |
| 97. C8 | C3×M16 | direct (N × H) |
| 112. C3 | C8×M16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#431 | 2 |
| C3×C4×M16 | 2 |
| C2×C3×C4×C8 | 2 |
| C3×#322 | 2 |
| C3×C82 | 2 |
| C8×M16 | 3 |