Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1370 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 1370 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×A4 |
| commutator, socle, cosocle | C23 | C24×C3 |
| Fitting | C22×H(5,𝔽2) | C3 |
| hypercenter | H(5,𝔽2) | A4 |
| hypocenter | C22 | C3×H(5,𝔽2) |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A4×#41 | C2 | semidirect |
| 2. C2×D4×A4 | C2 | semidirect |
| 3. C22×H(5,𝔽2) | C3 | semidirect |
| 4. C23×A4 | C22 | semidirect |
| 6. Q8×A4 | C22 | semidirect |
| 7. D4×A4 | C22 | semidirect |
| 8. C2×C4×A4 | C22 | semidirect |
| 12. C22×#41 | C2×C3 | semidirect |
| 13. C23×D4 | C2×C3 | semidirect |
| 19. C25 | C22×C3 | semidirect |
| 26. H(5,𝔽2) | A4 | direct (N × H) |
| 31. C22×Q8 | C22×C3 | semidirect |
| 36. C22×D4 | C22×C3 | semidirect |
| 37. C23×C4 | C22×C3 | semidirect |
| 55. #41 | C2×A4 | semidirect |
| 67. C2×D4 | C2×A4 | semidirect |
| 74. A4 | H(5,𝔽2) | direct (N × H) |
| 86. C23 | C22×A4 | semidirect |
| 90. Q8 | C22×A4 | semidirect |
| 94. D4 | C22×A4 | semidirect |
| 100. C2×C4 | C22×A4 | semidirect |
| 113. C22 | C3×H(5,𝔽2) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×#41 | 2 |
| C2×D4×A4 | 2 |
| C22×H(5,𝔽2) | 3 |
| C3×H(5,𝔽2) | 4 |