Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1360 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 1360 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×A4 |
| commutator, cosocle | C23 | C24×C3 |
| Frattini | C2 | C24×A4 |
| Fitting | C23×#41 | C3 |
| socle | C24 | C23×C3 |
| hypercenter | C2×#41 | A4 |
| hypocenter | C22 | C2×C3×#41 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A4×#41 | C2 | direct (N × H) |
| 2. C2×Q8×A4 | C2 | semidirect |
| 3. C2×D4×A4 | C2 | semidirect |
| 4. C22×C4×A4 | C2 | semidirect |
| 5. C23×#41 | C3 | semidirect |
| 8. Q8×A4 | C22 | semidirect |
| 9. D4×A4 | C22 | semidirect |
| 12. C2×C4×A4 | C22 | semidirect |
| 18. C22×#41 | C2×C3 | semidirect |
| 19. C23×Q8 | C2×C3 | semidirect |
| 20. C23×D4 | C2×C3 | semidirect |
| 21. C24×C4 | C2×C3 | semidirect |
| 46. C2×#41 | A4 | direct (N × H) |
| 50. C22×Q8 | C22×C3 | semidirect |
| 55. C22×D4 | C22×C3 | semidirect |
| 63. C23×C4 | C22×C3 | semidirect |
| 66. C2×A4 | #41 | direct (N × H) |
| 86. #41 | C2×A4 | direct (N × H) |
| 93. C2×Q8 | C2×A4 | semidirect |
| 103. C2×D4 | C2×A4 | semidirect |
| 122. C22×C4 | C2×A4 | semidirect |
| 128. A4 | C2×#41 | direct (N × H) |
| 147. C23 | C3×#41 | semidirect |
| 152. Q8 | C22×A4 | semidirect |
| 156. D4 | C22×A4 | semidirect |
| 173. C2×C4 | C22×A4 | semidirect |
| 193. C22 | C2×C3×#41 | semidirect |
| 206. C2 | A4×#41 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×#41 | 2 |
| C2×Q8×A4 | 2 |
| C2×D4×A4 | 2 |
| C22×C4×A4 | 2 |
| C23×#41 | 3 |
| C2×C3×#41 | 4 |