Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
372 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 216 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×C211 | C2×D4 |
| commutator | C4 | C2×C42×C211 |
| Frattini, cosocle | C22×C4 | C23×C211 |
| socle | C22×C211 | C2×C4⋊C4 |
| prime radical | C24×C211 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C211 ◁ C23×C4×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #341×C211 | C2 | semidirect |
| 58. C211 | #1738 | direct (N × H) |
| 59. #1738 | C211 | direct (N × H) |
| 62. #341 | C2×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×M16×C211 | 2 |
| #341×C211 | 2 |
| #1738 | 211 |