Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
234 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| prime radical | C2×C3×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#500 | C2 | semidirect |
| 4. C3×#462 | C2 | semidirect |
| 5. C3×#332 | C2 | semidirect |
| 6. C3×Q8⋊C8 | C2 | semidirect |
| 8. #1649 | C3 | direct (N × H) |
| 27. #500 | C2×C3 | semidirect |
| 29. #462 | C2×C3 | semidirect |
| 30. #332 | C2×C3 | semidirect |
| 31. Q8⋊C8 | C2×C3 | semidirect |
| 84. C3×D4 | Q16 | semidirect |
| 147. D4 | C3×Q16 | semidirect |
| 180. C3 | #1649 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×D4×Q8 | 2 |
| C3×#500 | 2 |
| C3×#474 | 2 |
| C3×#462 | 2 |
| C3×#332 | 2 |
| C3×Q8⋊C8 | 2 |
| C3×D4⋊C8 | 2 |
| #1649 | 3 |