Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
445 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 445 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C8 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C8 |
| Frattini | C8 | C22×S4 |
| Fitting | C22×#131 | D3 |
| socle | C23 | A4⋊C8 |
| prime radical | C22×#41⋊C3 | C4 |
| cosocle | #131⋊C3 | C23 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#4628 | C2 | direct (N × H) |
| 8. #4628 | C22 | direct (N × H) |
| 146. C22 | #4628 | direct (N × H) |
| 155. C2 | C2×#4628 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#131⋊C3 | 2 |
| C2×#4628 | 2 |
| C22×#350 | 3 |
| C22×C3⋊C16 | 4 |