Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
198 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 198 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 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#107 | C2 | semidirect |
| 4. C3×#343 | C2 | semidirect |
| 5. C3×#341 | C2 | semidirect |
| 6. #2078 | C3 | direct (N × H) |
| 16. C2×#107 | C2×C3 | semidirect |
| 19. #343 | C2×C3 | semidirect |
| 20. #341 | C2×C3 | semidirect |
| 127. C3 | #2078 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#107 | 2 |
| C3×#417 | 2 |
| C3×#408 | 2 |
| C3×#343 | 2 |
| C3×#341 | 2 |
| #2078 | 3 |