Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
712 conjugacy classes of subgroups; only the 440 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 440 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C23 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C23 | C24×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×#382 | C2 | direct (N × H) |
| 5. C2×#382 | C3 | direct (N × H) |
| 11. C2×C3×C42 | C4 | semidirect |
| 16. #382 | C2×C3 | direct (N × H) |
| 30. C3×C42 | C2×C4 | semidirect |
| 36. C2×C42 | C3×C4 | semidirect |
| 66. C42 | C2×C3×C4 | semidirect |
| 93. C2×C3 | #382 | direct (N × H) |
| 109. C3 | C2×#382 | direct (N × H) |
| 110. C2 | C3×#382 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C4⋊C4 | 2 |
| C22×C3×C42 | 2 |
| C3×#382 | 2 |
| C2×C3×#95 | 2 |
| C2×#382 | 3 |