Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
820 conjugacy classes of subgroups; only the 392 normal subgroups are shown.
This group has 820 conjugacy classes of subgroups — too many to draw. Only its 392 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C24 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C23 | C24×C3 |
| prime radical | C24×C3 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×C4×Q8 | C2 | semidirect |
| 3. C3×#391 | C2 | semidirect |
| 6. Q8×C22⋊C4 | C3 | direct (N × H) |
| 8. C22×C3×Q8 | C4 | semidirect |
| 20. C2×C4×Q8 | C2×C3 | semidirect |
| 21. #391 | C2×C3 | semidirect |
| 38. C3×C22⋊C4 | Q8 | direct (N × H) |
| 41. C22×Q8 | C3×C4 | semidirect |
| 59. C3×Q8 | C22⋊C4 | direct (N × H) |
| 70. C2×C3×C4 | C2×Q8 | semidirect |
| 85. C22⋊C4 | C3×Q8 | direct (N × H) |
| 91. C22×C3 | C4×Q8 | semidirect |
| 108. Q8 | C3×C22⋊C4 | direct (N × H) |
| 119. C2×C4 | C2×C3×Q8 | semidirect |
| 130. C22 | C3×C4×Q8 | semidirect |
| 140. C3 | Q8×C22⋊C4 | 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 |
| C2×C3×C4×Q8 | 2 |
| C3×#391 | 2 |
| C3×#380 | 2 |
| C3×C4×C22⋊C4 | 2 |
| Q8×C22⋊C4 | 3 |