Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
495 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 495 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C32×D8 |
| commutator, Frattini, cosocle | C3×C8 | C23×C32 |
| prime radical | C2×D8×H(3,𝔽3) | C2 |
| LCS | G ▷ C3×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C4×H(3,𝔽3) ◁ C2×C8×H(3,𝔽3) ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q16×H(3,𝔽3) | C2 | semidirect |
| 3. H(3,𝔽3)×SD32 | C2 | direct (N × H) |
| 4. C2×C16×H(3,𝔽3) | C2 | semidirect |
| 5. C2×C32×SD32 | C3 | semidirect |
| 9. Q16×H(3,𝔽3) | C22 | semidirect |
| 13. C16×H(3,𝔽3) | C22 | semidirect |
| 14. C2×C32×Q16 | C2×C3 | semidirect |
| 16. C32×SD32 | C2×C3 | semidirect |
| 17. C2×C32×C16 | C2×C3 | semidirect |
| 32. C32×Q16 | C22×C3 | semidirect |
| 36. C32×C16 | C22×C3 | semidirect |
| 60. C2×SD32 | H(3,𝔽3) | direct (N × H) |
| 62. C2×H(3,𝔽3) | SD32 | direct (N × H) |
| 86. C2×Q16 | C2×H(3,𝔽3) | semidirect |
| 88. SD32 | C2×H(3,𝔽3) | direct (N × H) |
| 89. C2×C16 | C2×H(3,𝔽3) | semidirect |
| 90. H(3,𝔽3) | C2×SD32 | direct (N × H) |
| 110. C2×C32 | C3×SD32 | semidirect |
| 115. Q16 | C22×H(3,𝔽3) | semidirect |
| 119. C16 | C22×H(3,𝔽3) | semidirect |
| 132. C32 | C2×C3×SD32 | semidirect |
| 157. C2 | H(3,𝔽3)×SD32 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q16×H(3,𝔽3) | 2 |
| C2×D8×H(3,𝔽3) | 2 |
| H(3,𝔽3)×SD32 | 2 |
| C2×C16×H(3,𝔽3) | 2 |
| C2×C32×SD32 | 3 |