Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
184 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C211 | #121 |
| commutator | C2×C4 | C22×C4×C211 |
| Frattini, cosocle | C22×C4 | C23×C211 |
| socle | C2×C211 | #389 |
| prime radical | C23×C211 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C211 ◁ C22×C4×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 14. #108×C211 | C4 | semidirect |
| 60. C211 | #1953 | direct (N × H) |
| 61. #1953 | C211 | direct (N × H) |
| 75. #108 | C4×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #430×C211 | 2 |
| C2×#108×C211 | 2 |
| #405×C211 | 2 |
| #344×C211 | 2 |
| #1953 | 211 |