Finite Group DatabasebetaFinite Group DatabasebetaOrder 4320.
168 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 8 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,𝔽9) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #13088280 | — | 3 | 1 | no | maximal |
| 3 | C3⋊#6732 | — | 6 | 2 | no | maximal |
| 4 | SL(2,𝔽9) | D3 | 1 | 1 | yes | characteristic, complemented |
| 5 | #88924 | — | 10 | 1 | no | maximal |
| 6 | C3×SL(2,𝔽5) | — | 6 | 2 | no | — |
| 7 | #64198 | — | 15 | 2 | no | maximal |
| 8 | #6732 | — | 18 | 2 | no | — |
| 9 | #6289 | — | 10 | 1 | no | — |
| 10 | C3⋊#622 | — | 10 | 1 | no | — |
| 11 | C3×#622 | — | 10 | 1 | no | — |
| 12 | #3787 | — | 30 | 1 | no | — |
| 13 | C3⋊(#41⋊C3) | — | 15 | 2 | no | — |
| 14 | C3⋊GL(2,𝔽3) | — | 15 | 2 | no | — |
| 15 | C3×#226 | — | 15 | 2 | no | — |
| 16 | #1200 | — | 36 | 1 | no | — |
| 17 | SL(2,𝔽5) | — | 6 | 2 | no | — |
| 18 | C2×C3⋊(C3⋊D3) | — | 10 | 1 | no | — |
| 19 | C3×C3⋊Dic3 | — | 10 | 1 | no | — |
| 20 | C32×Dic3 | — | 10 | 1 | no | — |
| 21 | #985 | — | 45 | 2 | no | — |
| 22 | #919 | — | 45 | 1 | no | — |
| 23 | C4×C3⋊D3 | — | 30 | 1 | no | — |
| 24 | C3×SL(2,𝔽3) | — | 15 | 2 | no | — |
| 25 | #624 | — | 60 | 2 | no | — |
| 26 | #622 | — | 10 | 1 | no | — |
| 27 | #622 | — | 30 | 1 | no | — |
| 28 | C2×D15 | — | 36 | 1 | no | — |
| 29 | C3×Dic5 | — | 36 | 1 | no | — |
| 30 | C5×Dic3 | — | 36 | 1 | no | — |
| 31 | C2×C33 | — | 10 | 1 | no | — |
| 32 | C3⋊(C3⋊D3) | — | 10 | 2 | no | — |
| 33 | #239 | — | 15 | 2 | no | — |
| 34 | #41⋊C3 | — | 45 | 2 | no | — |
| 35 | GL(2,𝔽3) | — | 45 | 2 | no | — |
| 36 | GL(2,𝔽3) | — | 90 | 2 | no | — |
| 37 | #226 | — | 15 | 2 | no | — |
| 38 | C3×Q16 | — | 45 | 1 | no | — |
| 39 | #215 | — | 45 | 2 | no | — |
| 40 | D24 | — | 45 | 1 | no | — |
| 41 | D3×C8 | — | 45 | 1 | no | — |
| 42 | C4×D5 | — | 108 | 1 | no | — |
| 43 | C2×C3⋊D3 | — | 30 | 1 | no | — |
| 44 | C2×C3⋊D3 | — | 60 | 2 | no | — |
| 45 | C2×C3⋊D3 | — | 120 | 2 | no | — |
| 46 | C32×C4 | — | 30 | 1 | no | — |
| 47 | C3⋊Dic3 | — | 10 | 1 | no | — |
| 48 | C3×Dic3 | — | 20 | 2 | no | — |
| 49 | C3×Dic3 | — | 60 | 2 | no | — |
| 50 | #135 | — | 135 | 1 | no | Sylow |
| 51 | C2×C3×C5 | — | 36 | 1 | no | — |
| 52 | D15 | — | 36 | 2 | no | — |
| 53 | C33 | — | 10 | 1 | no | Sylow |
| 54 | C3×Q8 | — | 15 | 2 | no | — |
| 55 | D12 | — | 45 | 2 | no | — |
| 56 | C4×D3 | — | 45 | 1 | no | — |
| 57 | C4×D3 | — | 180 | 2 | no | — |
| 58 | SL(2,𝔽3) | — | 15 | 2 | no | — |
| 59 | SL(2,𝔽3) | — | 30 | 2 | no | — |
| 60 | C3×C8 | — | 45 | 1 | no | — |
| 61 | C3⋊C8 | — | 45 | 1 | no | — |
| 62 | C2×D5 | — | 108 | 1 | no | — |
| 63 | C4×C5 | — | 108 | 1 | no | — |
| 64 | Dic5 | — | 36 | 1 | no | — |
| 65 | C2×C32 | — | 10 | 1 | no | — |
| 66 | C2×C32 | — | 20 | 2 | no | — |
| 67 | C2×C32 | — | 40 | 2 | no | — |
| 68 | C3⋊D3 | — | 30 | 2 | no | — |
| 69 | C3⋊D3 | — | 60 | 4 | no | — |
| 70 | C3⋊D3 | — | 120 | 4 | no | — |
| 71 | #41 | — | 45 | 2 | no | — |
| 72 | Q16 | — | 45 | 1 | no | — |
| 73 | SD16 | — | 135 | 2 | no | — |
| 74 | D8 | — | 135 | 1 | no | — |
| 75 | C2×C8 | — | 135 | 1 | no | — |
| 76 | C3×C5 | — | 36 | 1 | no | — |
| 77 | C2×D3 | — | 45 | 1 | no | — |
| 78 | C2×D3 | — | 360 | 2 | no | — |
| 79 | C2×D3 | — | 180 | 2 | no | — |
| 80 | C3×C4 | — | 45 | 1 | no | — |
| 81 | C3×C4 | — | 60 | 2 | no | — |
| 82 | Dic3 | — | 60 | 2 | no | — |
| 83 | Dic3 | A6 | 1 | 1 | yes | maximal normal, characteristic |
| 84 | C2×C5 | — | 36 | 1 | no | — |
| 85 | D5 | — | 108 | 2 | no | — |
| 86 | C32 | — | 10 | 1 | no | — |
| 87 | C32 | — | 20 | 2 | no | — |
| 88 | C32 | — | 40 | 2 | no | — |
| 89 | Q8 | — | 15 | 2 | no | — |
| 90 | D4 | — | 135 | 2 | no | — |
| 91 | C2×C4 | — | 135 | 1 | no | — |
| 92 | C8 | — | 45 | 1 | no | — |
| 93 | C8 | — | 135 | 1 | no | — |
| 94 | C2×C3 | — | 20 | 2 | no | — |
| 95 | C2×C3 | — | 40 | 2 | no | — |
| 96 | C2×C3 | C2×A6 | 1 | 1 | yes | characteristic |
| 97 | D3 | — | 90 | 1 | no | — |
| 98 | D3 | — | 360 | 4 | no | — |
| 99 | D3 | — | 180 | 4 | no | — |
| 100 | C5 | — | 36 | 1 | no | Sylow |
| 101 | C22 | — | 135 | 1 | no | — |
| 102 | C4 | — | 3 | 1 | no | — |
| 103 | C4 | — | 45 | 1 | no | — |
| 104 | C3 | — | 20 | 2 | no | — |
| 105 | C3 | — | 40 | 2 | no | — |
| 106 | C3 | #13088280 | 1 | 1 | yes | characteristic, complemented |
| 107 | C2 | — | 270 | 1 | no | — |
| 108 | C2 | D3×A6 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×A6 |
| commutator, hypocenter | C3×SL(2,𝔽9) | C2 |
| Fitting, socle, cosocle | C2×C3 | C2×A6 |
| LCS | G ▷ C3×SL(2,𝔽9) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×SL(2,𝔽9) | C2 | semidirect |
| 4. SL(2,𝔽9) | D3 | semidirect |
| 106. C3 | #13088280 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×SL(2,𝔽9) | 2 |
| #13088280 | 3 |
| C3⋊#6732 | 6 |
| #88924 | 10 |
| #64198 | 15 |