Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
348 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 104 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 | Q8×#41 | C3 |
| hypercenter | C2×#41 | A4 |
| hypocenter | Q8 | C3×#41 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×SL(2,𝔽3) | C2 | semidirect |
| 2. D4×SL(2,𝔽3) | C2 | semidirect |
| 3. C2×C4×SL(2,𝔽3) | C2 | semidirect |
| 4. Q8×#41 | C3 | semidirect |
| 9. Q82 | C2×C3 | semidirect |
| 12. D4×Q8 | C2×C3 | semidirect |
| 15. C2×C4×Q8 | C2×C3 | semidirect |
| 47. SL(2,𝔽3) | #41 | direct (N × H) |
| 48. #41 | SL(2,𝔽3) | direct (N × H) |
| 81. Q8 | C2×SL(2,𝔽3) | semidirect |
| 82. Q8 | C3×#41 | semidirect |
| 83. D4 | C2×SL(2,𝔽3) | semidirect |
| 91. C2×C4 | C2×SL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×SL(2,𝔽3) | 2 |
| D4×SL(2,𝔽3) | 2 |
| C2×C4×SL(2,𝔽3) | 2 |
| Q8×#41 | 3 |
| C2×C3×#41 | 4 |