Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1812 conjugacy classes of subgroups; only the 360 normal subgroups are shown.
This group has 1812 conjugacy classes of subgroups — too many to draw. Only its 360 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C3×D4 | C2 | semidirect |
| 2. C22×C3×C22⋊C4 | C2 | semidirect |
| 3. C3×#392 | C2 | direct (N × H) |
| 4. C2×C3×#95 | C2 | semidirect |
| 5. C2×#392 | C3 | direct (N × H) |
| 9. C22×C3×D4 | C22 | semidirect |
| 13. C2×C3×C22⋊C4 | C22 | semidirect |
| 14. C3×#95 | C22 | semidirect |
| 15. C23×D4 | C2×C3 | semidirect |
| 16. C22×C22⋊C4 | C2×C3 | semidirect |
| 17. #392 | C2×C3 | direct (N × H) |
| 18. C2×#95 | C2×C3 | semidirect |
| 23. C24×C3 | D4 | semidirect |
| 31. C22×C3×C4 | D4 | semidirect |
| 38. C22×D4 | C22×C3 | semidirect |
| 42. C2×C22⋊C4 | C22×C3 | semidirect |
| 43. #95 | C22×C3 | semidirect |
| 55. C23×C3 | C2×D4 | semidirect |
| 63. C2×C3×C4 | C2×D4 | semidirect |
| 68. C24 | C3×D4 | semidirect |
| 76. C22×C4 | C3×D4 | semidirect |
| 105. C23 | C2×C3×D4 | semidirect |
| 113. C2×C4 | C2×C3×D4 | semidirect |
| 115. C2×C3 | #392 | direct (N × H) |
| 133. C3 | C2×#392 | direct (N × H) |
| 135. C2 | C3×#392 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×D4 | 2 |
| C22×C3×C22⋊C4 | 2 |
| C3×#392 | 2 |
| C2×C3×#95 | 2 |
| C2×#392 | 3 |