Finite Group DatabasebetaFinite Group DatabasebetaOrder 9720.
170 conjugacy classes of subgroups; only the 5 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 5 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 | C3×SL(2,ℤ/9) | — | 5 | 1 | no | maximal, Hall |
| 2 | #13108499 | — | 6 | 1 | no | maximal |
| 3 | C3⋊#66321 | — | 10 | 1 | no | maximal |
| 4 | C2×C34⋊C5 | — | 6 | 1 | no | — |
| 5 | C3×#6296 | — | 5 | 1 | no | — |
| 6 | SL(2,ℤ/9) | — | 5 | 3 | no | — |
| 7 | C2×C3×AΣL(1,𝔽27) | — | 10 | 1 | no | — |
| 8 | C34⋊C5 | — | 6 | 1 | no | — |
| 9 | C32×C3⋊Dic3 | — | 15 | 1 | no | — |
| 10 | C3⋊#1084 | — | 10 | 1 | no | — |
| 11 | #66321 | — | 30 | 3 | no | — |
| 12 | C3×AΣL(1,𝔽27) | — | 10 | 1 | no | Sylow |
| 13 | #6296 | — | 5 | 1 | no | — |
| 14 | C3×#627 | — | 45 | 1 | no | — |
| 15 | C2×C34 | A5 | 1 | 1 | yes | maximal normal, characteristic |
| 16 | C2×C3×3−1+2 | — | 20 | 1 | no | — |
| 17 | C2×C3×H(3,𝔽3) | — | 10 | 1 | no | — |
| 18 | C2×AΣL(1,𝔽27) | — | 10 | 3 | no | — |
| 19 | C2×AΣL(1,𝔽27) | — | 20 | 3 | no | — |
| 20 | SL(2,𝔽5) | — | 81 | 3 | no | maximal |
| 21 | C3×C3⋊Dic3 | — | 15 | 1 | no | — |
| 22 | C3×C3⋊Dic3 | — | 15 | 1 | no | — |
| 23 | C3×C3⋊Dic3 | — | 30 | 1 | no | — |
| 24 | C3×C3⋊Dic3 | — | 90 | 1 | no | — |
| 25 | C32×Dic3 | — | 45 | 2 | no | — |
| 26 | C32×Dic3 | — | 90 | 1 | no | — |
| 27 | #1084 | — | 30 | 3 | no | — |
| 28 | #1084 | — | 90 | 2 | no | — |
| 29 | C34 | SL(2,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 30 | C3×3−1+2 | — | 20 | 1 | no | — |
| 31 | C3×H(3,𝔽3) | — | 10 | 1 | no | — |
| 32 | AΣL(1,𝔽27) | — | 10 | 3 | no | — |
| 33 | AΣL(1,𝔽27) | — | 20 | 3 | no | — |
| 34 | C3×Dic6 | — | 135 | 1 | no | — |
| 35 | C3×SL(2,𝔽3) | — | 135 | 1 | no | — |
| 36 | #627 | — | 45 | 1 | no | — |
| 37 | C2×C33 | — | 5 | 1 | no | — |
| 38 | C2×C33 | — | 10 | 1 | no | — |
| 39 | C2×C33 | — | 10 | 1 | no | — |
| 40 | C2×C33 | — | 15 | 1 | no | — |
| 41 | C2×C33 | — | 30 | 1 | no | — |
| 42 | C2×3−1+2 | — | 20 | 3 | no | — |
| 43 | C2×3−1+2 | — | 60 | 2 | no | — |
| 44 | C2×H(3,𝔽3) | — | 10 | 3 | no | — |
| 45 | C2×H(3,𝔽3) | — | 30 | 2 | no | — |
| 46 | C2×C3×C9 | — | 60 | 1 | no | — |
| 47 | C32×C4 | — | 135 | 1 | no | — |
| 48 | C3⋊Dic3 | — | 15 | 1 | no | — |
| 49 | C3⋊Dic3 | — | 90 | 1 | no | — |
| 50 | C3×Dic3 | — | 270 | 3 | no | — |
| 51 | C3×Dic3 | — | 45 | 2 | no | — |
| 52 | C3×Dic3 | — | 45 | 2 | no | — |
| 53 | C3×Dic3 | — | 90 | 2 | no | — |
| 54 | C3×Dic3 | — | 90 | 1 | no | — |
| 55 | C3×Dic3 | — | 90 | 1 | no | — |
| 56 | C3×Dic3 | — | 90 | 1 | no | — |
| 57 | C3×Dic3 | — | 90 | 1 | no | — |
| 58 | C33 | — | 5 | 1 | no | — |
| 59 | C33 | — | 10 | 1 | no | — |
| 60 | C33 | — | 10 | 1 | no | — |
| 61 | C33 | — | 15 | 1 | no | — |
| 62 | C33 | — | 30 | 1 | no | — |
| 63 | 3−1+2 | — | 20 | 3 | no | — |
| 64 | 3−1+2 | — | 60 | 2 | no | — |
| 65 | H(3,𝔽3) | — | 10 | 3 | no | — |
| 66 | H(3,𝔽3) | — | 30 | 2 | no | — |
| 67 | C3×C9 | — | 60 | 1 | no | — |
| 68 | C3×Q8 | — | 135 | 1 | no | — |
| 69 | Dic6 | — | 135 | 1 | no | — |
| 70 | SL(2,𝔽3) | — | 135 | 3 | no | — |
| 71 | Dic5 | — | 486 | 1 | no | — |
| 72 | C2×C32 | — | 10 | 1 | no | — |
| 73 | C2×C32 | — | 10 | 1 | no | — |
| 74 | C2×C32 | — | 10 | 2 | no | — |
| 75 | C2×C32 | — | 15 | 1 | no | — |
| 76 | C2×C32 | — | 15 | 1 | no | — |
| 77 | C2×C32 | — | 30 | 1 | no | — |
| 78 | C2×C32 | — | 30 | 1 | no | — |
| 79 | C2×C32 | — | 30 | 3 | no | — |
| 80 | C2×C32 | — | 90 | 1 | no | — |
| 81 | C2×C32 | — | 180 | 1 | no | — |
| 82 | C2×C9 | — | 60 | 3 | no | — |
| 83 | C3×C4 | — | 270 | 1 | no | — |
| 84 | C3×C4 | — | 135 | 1 | no | — |
| 85 | C3×C4 | — | 135 | 1 | no | — |
| 86 | Dic3 | — | 270 | 3 | no | — |
| 87 | Dic3 | — | 45 | 2 | no | — |
| 88 | Dic3 | — | 90 | 1 | no | — |
| 89 | C2×C5 | — | 486 | 1 | no | — |
| 90 | C32 | — | 10 | 1 | no | — |
| 91 | C32 | — | 10 | 1 | no | — |
| 92 | C32 | — | 10 | 2 | no | — |
| 93 | C32 | — | 15 | 1 | no | — |
| 94 | C32 | — | 15 | 1 | no | — |
| 95 | C32 | — | 30 | 1 | no | — |
| 96 | C32 | — | 30 | 1 | no | — |
| 97 | C32 | — | 30 | 3 | no | — |
| 98 | C32 | — | 90 | 1 | no | — |
| 99 | C32 | — | 180 | 1 | no | — |
| 100 | C9 | — | 60 | 3 | no | — |
| 101 | Q8 | — | 135 | 1 | no | Sylow |
| 102 | C2×C3 | — | 5 | 1 | no | — |
| 103 | C2×C3 | — | 10 | 1 | no | — |
| 104 | C2×C3 | — | 10 | 1 | no | — |
| 105 | C2×C3 | — | 15 | 1 | no | — |
| 106 | C2×C3 | — | 90 | 3 | no | — |
| 107 | C5 | — | 486 | 1 | no | Sylow |
| 108 | C4 | — | 135 | 1 | no | — |
| 109 | C3 | — | 5 | 1 | no | — |
| 110 | C3 | — | 10 | 1 | no | — |
| 111 | C3 | — | 10 | 1 | no | — |
| 112 | C3 | — | 15 | 1 | no | — |
| 113 | C3 | — | 90 | 3 | no | — |
| 114 | C2 | #72209608 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #72209608 |
| commutator, hypocenter | G | C1 |
| Fitting, socle, cosocle | C2×C34 | A5 |
| LCS | G | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 29. C34 | SL(2,𝔽5) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×SL(2,ℤ/9) | 5 |
| #13108499 | 6 |
| C3⋊#66321 | 10 |
| SL(2,𝔽5) | 81 |