Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1072 conjugacy classes of subgroups; only the 440 normal subgroups are shown.
This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 440 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 | D4 |
| commutator | C4 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C23×C3×D4 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ C23×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×C3×C8 | C2 | semidirect |
| 3. C22×C3×C4⋊C4 | C2 | semidirect |
| 4. C2×C3×#102 | C2 | direct (N × H) |
| 5. C22×#102 | C3 | direct (N × H) |
| 9. C22×C3×D4 | C4 | semidirect |
| 14. C22×C3×C8 | C22 | semidirect |
| 16. C2×C3×C4⋊C4 | C22 | semidirect |
| 17. C3×#102 | C22 | direct (N × H) |
| 19. C23×C8 | C2×C3 | semidirect |
| 20. C22×C4⋊C4 | C2×C3 | semidirect |
| 21. C2×#102 | C2×C3 | direct (N × H) |
| 30. C2×C3×D4 | C2×C4 | semidirect |
| 39. C2×C3×C8 | C23 | semidirect |
| 41. C3×C4⋊C4 | C23 | semidirect |
| 45. C22×D4 | C3×C4 | semidirect |
| 50. C22×C8 | C22×C3 | semidirect |
| 52. C2×C4⋊C4 | C22×C3 | semidirect |
| 53. #102 | C22×C3 | direct (N × H) |
| 65. C3×D4 | C22×C4 | semidirect |
| 82. C2×D4 | C2×C3×C4 | semidirect |
| 91. C2×C8 | C23×C3 | semidirect |
| 93. C4⋊C4 | C23×C3 | semidirect |
| 98. C22×C3 | #102 | direct (N × H) |
| 118. D4 | C22×C3×C4 | semidirect |
| 128. C2×C3 | C2×#102 | direct (N × H) |
| 136. C22 | C3×#102 | direct (N × H) |
| 145. C3 | C22×#102 | direct (N × H) |
| 147. C2 | C2×C3×#102 | 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 |
| C23×C3×C8 | 2 |
| C22×C3×C4⋊C4 | 2 |
| C2×C3×#102 | 2 |
| C22×#102 | 3 |