Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
168 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 84 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 | C2×C4 | C22×C4×C211 |
| Frattini, cosocle | C42 | C23×C211 |
| socle | C22×C211 | C4×D4 |
| prime radical | C2×D4×C211 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C211 ◁ C4×C8×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #446×C211 | C2 | semidirect |
| 4. #335×C211 | C2 | semidirect |
| 5. Q8⋊C8×C211 | C2 | semidirect |
| 7. #322×C211 | C2 | semidirect |
| 77. C211 | #1585 | direct (N × H) |
| 78. #1585 | C211 | direct (N × H) |
| 79. #446 | C2×C211 | semidirect |
| 82. #335 | C2×C211 | semidirect |
| 83. Q8⋊C8 | C2×C211 | semidirect |
| 85. #322 | C2×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #446×C211 | 2 |
| C4×SD16×C211 | 2 |
| C8×D4×C211 | 2 |
| #335×C211 | 2 |
| Q8⋊C8×C211 | 2 |
| D4⋊C8×C211 | 2 |
| #322×C211 | 2 |
| #1585 | 211 |