Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
398 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 70 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 | C2×Q8 | C23×C3 |
| Frattini, socle | C22 | C23×A4 |
| Fitting | D4×#41 | C3 |
| hypercenter | C4×D4 | A4 |
| hypocenter | Q8 | C2×C3×D4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C4 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#41⋊C3 | C2 | semidirect |
| 2. C4⋊(#41⋊C3) | C2 | semidirect |
| 3. D4×SL(2,𝔽3) | C2 | semidirect |
| 4. #548⋊C3 | C2 | semidirect |
| 5. C4×#41⋊C3 | C2 | semidirect |
| 6. D4×#41 | C3 | semidirect |
| 9. C2×#41⋊C3 | C22 | semidirect |
| 10. C22×SL(2,𝔽3) | C22 | semidirect |
| 12. C4×SL(2,𝔽3) | C22 | semidirect |
| 13. C4×SL(2,𝔽3) | C22 | semidirect |
| 14. C22×#41 | C2×C3 | semidirect |
| 15. #550 | C2×C3 | semidirect |
| 17. #548 | C2×C3 | semidirect |
| 23. C4×#41 | C2×C3 | semidirect |
| 28. #41⋊C3 | D4 | direct (N × H) |
| 77. SL(2,𝔽3) | C2×D4 | semidirect |
| 86. #41 | C3×D4 | semidirect |
| 153. D4 | #41⋊C3 | direct (N × H) |
| 186. C22 | C2×#41⋊C3 | semidirect |
| 196. C4 | C2×#41⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#41⋊C3 | 2 |
| C4⋊(#41⋊C3) | 2 |
| D4×SL(2,𝔽3) | 2 |
| #548⋊C3 | 2 |
| C4×#41⋊C3 | 2 |
| D4×#41 | 3 |
| C3×C4×D4 | 4 |