Finite Group DatabasebetaFinite Group DatabasebetaOrder 480.
100 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×SL(2,𝔽5) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #6732 | C2 | 1 | 2 | yes | maximal, direct product factor |
| 3 | SL(2,𝔽5) | C22 | 1 | 1 | yes | characteristic, complemented |
| 4 | C2×#41⋊C3 | — | 5 | 1 | no | maximal, Hall |
| 5 | C2×C4×D5 | — | 6 | 1 | no | maximal |
| 6 | C2×C4×D3 | — | 10 | 1 | no | maximal |
| 7 | #41⋊C3 | — | 5 | 2 | no | — |
| 8 | C2×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 9 | C22×D5 | — | 6 | 1 | no | — |
| 10 | C2×C4×C5 | — | 6 | 1 | no | — |
| 11 | C2×Dic5 | — | 6 | 1 | no | — |
| 12 | C4×D5 | — | 6 | 2 | no | — |
| 13 | C4×D5 | — | 6 | 2 | no | — |
| 14 | C2×#41 | — | 5 | 1 | no | Sylow |
| 15 | C22×D3 | — | 10 | 1 | no | — |
| 16 | C2×C3×C4 | — | 10 | 1 | no | — |
| 17 | C2×Dic3 | — | 10 | 1 | no | — |
| 18 | C4×D3 | — | 10 | 2 | no | — |
| 19 | C4×D3 | — | 10 | 2 | no | — |
| 20 | SL(2,𝔽3) | — | 5 | 1 | no | — |
| 21 | C22×C5 | — | 6 | 1 | no | — |
| 22 | C2×D5 | — | 6 | 2 | no | — |
| 23 | C2×D5 | — | 6 | 4 | no | — |
| 24 | C4×C5 | — | 6 | 2 | no | — |
| 25 | Dic5 | — | 6 | 1 | no | — |
| 26 | Dic5 | — | 6 | 1 | no | — |
| 27 | #41 | — | 5 | 2 | no | — |
| 28 | #41 | — | 15 | 2 | no | — |
| 29 | C2×Q8 | — | 5 | 1 | no | — |
| 30 | C2×D4 | — | 15 | 1 | no | — |
| 31 | C22×C4 | — | 15 | 1 | no | — |
| 32 | C22×C3 | — | 10 | 1 | no | — |
| 33 | C2×D3 | — | 10 | 2 | no | — |
| 34 | C2×D3 | — | 10 | 4 | no | — |
| 35 | C3×C4 | — | 10 | 2 | no | — |
| 36 | Dic3 | — | 10 | 1 | no | — |
| 37 | Dic3 | — | 10 | 1 | no | — |
| 38 | C2×C5 | — | 6 | 2 | no | — |
| 39 | C2×C5 | — | 6 | 1 | no | — |
| 40 | D5 | — | 6 | 4 | no | — |
| 41 | C23 | — | 15 | 1 | no | — |
| 42 | Q8 | — | 5 | 1 | no | — |
| 43 | Q8 | — | 15 | 1 | no | — |
| 44 | D4 | — | 15 | 2 | no | — |
| 45 | D4 | — | 15 | 2 | no | — |
| 46 | C2×C4 | — | 15 | 1 | no | — |
| 47 | C2×C4 | — | 15 | 2 | no | — |
| 48 | C2×C4 | — | 15 | 2 | no | — |
| 49 | C2×C4 | A5 | 1 | 1 | yes | maximal normal, characteristic |
| 50 | C2×C3 | — | 10 | 2 | no | — |
| 51 | C2×C3 | — | 10 | 1 | no | — |
| 52 | D3 | — | 10 | 4 | no | — |
| 53 | C5 | — | 6 | 1 | no | Sylow |
| 54 | C22 | — | 15 | 2 | no | — |
| 55 | C22 | — | 30 | 2 | no | — |
| 56 | C22 | C2×A5 | 1 | 1 | yes | characteristic |
| 57 | C4 | — | 15 | 1 | no | — |
| 58 | C4 | — | 15 | 1 | no | — |
| 59 | C4 | C2×A5 | 1 | 2 | yes | — |
| 60 | C3 | — | 10 | 1 | no | Sylow |
| 61 | C2 | — | 30 | 2 | no | — |
| 62 | C2 | #6732 | 1 | 2 | yes | direct product factor |
| 63 | C2 | C22×A5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2×C4 | A5 |
| commutator, hypocenter | SL(2,𝔽5) | C22 |
| Frattini, cosocle | C2 | C22×A5 |
| socle | C22 | C2×A5 |
| LCS | G ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×SL(2,𝔽5) | C2 | semidirect |
| 2. #6732 | C2 | direct (N × H) |
| 3. SL(2,𝔽5) | C22 | semidirect |
| 62. C2 | #6732 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×SL(2,𝔽5) | 2 |
| #6732 | 2 |
| C2×#41⋊C3 | 5 |
| C2×C4×D5 | 6 |
| C2×C4×D3 | 10 |