Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
168 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D4 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C42 | C23×C3 |
| socle | C22×C3 | C4×D4 |
| prime radical | C2×C3×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C3×C4×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#446 | C2 | semidirect |
| 4. C3×#334 | C2 | semidirect |
| 5. C3×Q8⋊C8 | C2 | semidirect |
| 7. C3×#322 | C2 | semidirect |
| 8. #1582 | C3 | direct (N × H) |
| 24. #446 | C2×C3 | semidirect |
| 27. #334 | C2×C3 | semidirect |
| 28. Q8⋊C8 | C2×C3 | semidirect |
| 30. #322 | C2×C3 | semidirect |
| 83. C3×C8 | SD16 | semidirect |
| 133. C8 | C3×SD16 | semidirect |
| 150. C3 | #1582 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#446 | 2 |
| C3×C4×SD16 | 2 |
| C3×C8×D4 | 2 |
| C3×#334 | 2 |
| C3×Q8⋊C8 | 2 |
| C3×D4⋊C8 | 2 |
| C3×#322 | 2 |
| #1582 | 3 |