Finite Group DatabasebetaFinite Group DatabasebetaOrder 1080.
109 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 109 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C32×A5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | ΓL(2,𝔽4) | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 3 | C3×S5 | — | 3 | 1 | no | maximal |
| 4 | C3×C3⋊S4 | — | 5 | 1 | no | maximal, Hall |
| 5 | C3×ΓL(1,𝔽16) | — | 6 | 1 | no | maximal |
| 6 | C3×A5 | — | 2 | 1 | no | — |
| 7 | C3×A5 | D3 | 1 | 1 | yes | characteristic, complemented |
| 8 | C3×A5 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 9 | S5 | — | 3 | 1 | no | — |
| 10 | C32×A4 | — | 5 | 1 | no | — |
| 11 | C3×D32 | — | 10 | 1 | no | maximal |
| 12 | C32×D5 | — | 6 | 1 | no | — |
| 13 | C3⋊S4 | — | 5 | 1 | no | — |
| 14 | C3×S4 | — | 15 | 1 | no | — |
| 15 | C3×S4 | — | 15 | 2 | no | — |
| 16 | C3×#67 | — | 15 | 1 | no | — |
| 17 | ΓL(1,𝔽16) | — | 6 | 1 | no | — |
| 18 | C3×AGL(1,𝔽5) | — | 18 | 1 | no | — |
| 19 | A5 | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 20 | C3×C3⋊D3 | — | 10 | 1 | no | — |
| 21 | C32×D3 | — | 10 | 1 | no | — |
| 22 | C32×D3 | — | 10 | 1 | no | — |
| 23 | C32×C5 | — | 6 | 1 | no | — |
| 24 | C22×C32 | — | 5 | 1 | no | — |
| 25 | C2×C3×D3 | — | 15 | 1 | no | — |
| 26 | C2×C3×D3 | — | 30 | 1 | no | — |
| 27 | C3×A4 | — | 5 | 1 | no | — |
| 28 | C3×A4 | — | 5 | 1 | no | — |
| 29 | C3×A4 | — | 5 | 2 | no | — |
| 30 | C3×A4 | — | 10 | 1 | no | — |
| 31 | C3×A4 | — | 10 | 1 | no | — |
| 32 | C3×A4 | — | 10 | 2 | no | — |
| 33 | D32 | — | 10 | 1 | no | — |
| 34 | C3×Dic3 | — | 15 | 1 | no | — |
| 35 | C3×D5 | — | 6 | 1 | no | — |
| 36 | C3×D5 | — | 6 | 1 | no | — |
| 37 | C3×D5 | — | 12 | 1 | no | — |
| 38 | C33 | — | 10 | 1 | no | Sylow |
| 39 | S4 | — | 15 | 1 | no | — |
| 40 | S4 | — | 15 | 2 | no | — |
| 41 | C3×D4 | — | 45 | 1 | no | — |
| 42 | #67 | — | 15 | 1 | no | — |
| 43 | AGL(1,𝔽5) | — | 18 | 1 | no | — |
| 44 | C2×C32 | — | 15 | 1 | no | — |
| 45 | C2×C32 | — | 30 | 1 | no | — |
| 46 | C3⋊D3 | — | 10 | 1 | no | — |
| 47 | C3×D3 | — | 10 | 1 | no | — |
| 48 | C3×D3 | — | 10 | 1 | no | — |
| 49 | C3×D3 | — | 10 | 1 | no | — |
| 50 | C3×D3 | — | 10 | 1 | no | — |
| 51 | C3×D3 | — | 20 | 1 | no | — |
| 52 | C3×D3 | — | 20 | 1 | no | — |
| 53 | C3×D3 | — | 30 | 1 | no | — |
| 54 | C3×D3 | — | 60 | 1 | no | — |
| 55 | C3×C5 | — | 6 | 1 | no | — |
| 56 | C3×C5 | — | 6 | 1 | no | — |
| 57 | C3×C5 | — | 12 | 1 | no | — |
| 58 | C22×C3 | — | 5 | 1 | no | — |
| 59 | C22×C3 | — | 5 | 1 | no | — |
| 60 | C22×C3 | — | 10 | 1 | no | — |
| 61 | C22×C3 | — | 45 | 1 | no | — |
| 62 | C2×D3 | — | 15 | 1 | no | — |
| 63 | C2×D3 | — | 30 | 1 | no | — |
| 64 | A4 | — | 5 | 1 | no | — |
| 65 | A4 | — | 5 | 2 | no | — |
| 66 | A4 | — | 10 | 1 | no | — |
| 67 | A4 | — | 10 | 2 | no | — |
| 68 | C3×C4 | — | 45 | 1 | no | — |
| 69 | Dic3 | — | 15 | 1 | no | — |
| 70 | D5 | — | 6 | 1 | no | — |
| 71 | C32 | — | 10 | 1 | no | — |
| 72 | C32 | — | 10 | 1 | no | — |
| 73 | C32 | — | 20 | 1 | no | — |
| 74 | C32 | — | 20 | 1 | no | — |
| 75 | C32 | — | 20 | 1 | no | — |
| 76 | C32 | — | 40 | 1 | no | — |
| 77 | C32 | S5 | 1 | 1 | yes | characteristic, complemented |
| 78 | D4 | — | 45 | 1 | no | Sylow |
| 79 | C2×C3 | — | 15 | 1 | no | — |
| 80 | C2×C3 | — | 15 | 1 | no | — |
| 81 | C2×C3 | — | 30 | 1 | no | — |
| 82 | C2×C3 | — | 30 | 1 | no | — |
| 83 | C2×C3 | — | 30 | 1 | no | — |
| 84 | C2×C3 | — | 60 | 1 | no | — |
| 85 | D3 | — | 10 | 1 | no | — |
| 86 | D3 | — | 10 | 1 | no | — |
| 87 | D3 | — | 30 | 1 | no | — |
| 88 | D3 | — | 60 | 1 | no | — |
| 89 | C5 | — | 6 | 1 | no | Sylow |
| 90 | C22 | — | 5 | 1 | no | — |
| 91 | C22 | — | 45 | 1 | no | — |
| 92 | C4 | — | 45 | 1 | no | — |
| 93 | C3 | — | 2 | 1 | no | — |
| 94 | C3 | — | 10 | 1 | no | — |
| 95 | C3 | — | 20 | 1 | no | — |
| 96 | C3 | — | 20 | 1 | no | — |
| 97 | C3 | — | 40 | 1 | no | — |
| 98 | C3 | C3×S5 | 1 | 1 | yes | characteristic, complemented |
| 99 | C3 | ΓL(2,𝔽4) | 1 | 1 | yes | characteristic, direct product factor |
| 100 | C2 | — | 15 | 1 | no | — |
| 101 | C2 | — | 30 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | ΓL(2,𝔽4) |
| commutator, cosocle, hypocenter | C3×A5 | C2×C3 |
| Frattini | C1 | G |
| Fitting | C32 | S5 |
| socle | C32×A5 | C2 |
| LCS | G ▷ C3×A5 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×A5 | C2 | semidirect |
| 2. ΓL(2,𝔽4) | C3 | direct (N × H) |
| 7. C3×A5 | D3 | semidirect |
| 8. C3×A5 | C2×C3 | semidirect |
| 19. A5 | C3×D3 | semidirect |
| 77. C32 | S5 | semidirect |
| 98. C3 | C3×S5 | semidirect |
| 99. C3 | ΓL(2,𝔽4) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×A5 | 2 |
| ΓL(2,𝔽4) | 3 |
| C3×S5 | 3 |
| C3×C3⋊S4 | 5 |
| C3×ΓL(1,𝔽16) | 6 |
| C3×D32 | 10 |