Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
144 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 12 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⋊#6732 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3⋊#6728 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#6729 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #91870 | — | 3 | 1 | no | maximal |
| 5 | C3×SL(2,𝔽5) | C22 | 1 | 1 | yes | characteristic |
| 6 | #64197 | — | 5 | 1 | no | maximal, Hall |
| 7 | #6738 | — | 6 | 1 | no | maximal |
| 8 | #6732 | — | 3 | 1 | no | — |
| 9 | #6729 | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | #6728 | — | 3 | 1 | no | — |
| 11 | #3804 | — | 10 | 1 | no | maximal |
| 12 | C3⋊(#41⋊C3) | — | 5 | 1 | no | — |
| 13 | C3⋊#226 | — | 5 | 1 | no | — |
| 14 | C3×GL(2,𝔽3) | — | 5 | 1 | no | — |
| 15 | #1200 | — | 6 | 1 | no | — |
| 16 | C3⋊#170 | — | 6 | 1 | no | — |
| 17 | C3×#170 | — | 6 | 1 | no | — |
| 18 | SL(2,𝔽5) | C2×D3 | 1 | 1 | yes | characteristic |
| 19 | #985 | — | 15 | 1 | no | — |
| 20 | #916 | — | 15 | 1 | no | — |
| 21 | D5⋊C8 | — | 18 | 1 | no | — |
| 22 | #638 | — | 10 | 1 | no | — |
| 23 | C3×#67 | — | 10 | 1 | no | — |
| 24 | C2×C3×Dic3 | — | 10 | 1 | no | — |
| 25 | C3×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 26 | #627 | — | 10 | 1 | no | — |
| 27 | #626 | — | 10 | 1 | no | — |
| 28 | #624 | — | 10 | 1 | no | — |
| 29 | D3×Dic3 | — | 10 | 1 | no | — |
| 30 | C2×D15 | — | 6 | 1 | no | — |
| 31 | C3×Dic5 | — | 6 | 1 | no | — |
| 32 | C5×Dic3 | — | 6 | 1 | no | — |
| 33 | #239 | — | 5 | 1 | no | — |
| 34 | #237 | — | 15 | 1 | no | — |
| 35 | #235 | — | 30 | 1 | no | — |
| 36 | #41⋊C3 | — | 15 | 1 | no | — |
| 37 | GL(2,𝔽3) | — | 5 | 1 | no | — |
| 38 | #226 | — | 15 | 1 | no | — |
| 39 | #226 | — | 30 | 1 | no | — |
| 40 | C3×SD16 | — | 15 | 1 | no | — |
| 41 | #216 | — | 15 | 1 | no | — |
| 42 | #213 | — | 15 | 1 | no | — |
| 43 | #204 | — | 15 | 1 | no | — |
| 44 | D3×C8 | — | 15 | 1 | no | — |
| 45 | C4×D5 | — | 18 | 1 | no | — |
| 46 | #170 | — | 6 | 1 | no | — |
| 47 | #170 | — | 18 | 1 | no | — |
| 48 | C22×C32 | — | 10 | 1 | no | — |
| 49 | C2×C3⋊D3 | — | 10 | 1 | no | — |
| 50 | C2×C3×D3 | — | 10 | 1 | no | — |
| 51 | C3⋊Dic3 | — | 10 | 1 | no | — |
| 52 | C3×Dic3 | — | 10 | 1 | no | — |
| 53 | C3×Dic3 | — | 10 | 1 | no | — |
| 54 | C3×Dic3 | — | 10 | 1 | no | — |
| 55 | #135 | — | 45 | 1 | no | Sylow |
| 56 | C2×C3×C5 | — | 6 | 1 | no | — |
| 57 | D15 | — | 6 | 2 | no | — |
| 58 | C3×Q8 | — | 5 | 1 | no | — |
| 59 | C3×D4 | — | 15 | 1 | no | — |
| 60 | C2×C3×C4 | — | 30 | 1 | no | — |
| 61 | #67 | — | 10 | 1 | no | — |
| 62 | #67 | — | 30 | 1 | no | — |
| 63 | #67 | — | 30 | 1 | no | — |
| 64 | #67 | — | 60 | 1 | no | — |
| 65 | C2×Dic3 | — | 10 | 1 | no | — |
| 66 | D12 | — | 15 | 1 | no | — |
| 67 | D12 | — | 30 | 1 | no | — |
| 68 | C4×D3 | — | 15 | 1 | no | — |
| 69 | C4×D3 | — | 30 | 1 | no | — |
| 70 | C4×D3 | — | 30 | 1 | no | — |
| 71 | Dic6 | — | 15 | 1 | no | — |
| 72 | Dic6 | — | 30 | 1 | no | — |
| 73 | SL(2,𝔽3) | — | 5 | 1 | no | — |
| 74 | SL(2,𝔽3) | — | 10 | 1 | no | — |
| 75 | C3×C8 | — | 15 | 1 | no | — |
| 76 | C3⋊C8 | — | 15 | 1 | no | — |
| 77 | C2×D5 | — | 18 | 1 | no | — |
| 78 | C4×C5 | — | 18 | 1 | no | — |
| 79 | Dic5 | — | 6 | 1 | no | — |
| 80 | C2×C32 | — | 10 | 1 | no | — |
| 81 | C2×C32 | — | 20 | 1 | no | — |
| 82 | C3⋊D3 | — | 20 | 1 | no | — |
| 83 | C3×D3 | — | 20 | 1 | no | — |
| 84 | #41 | — | 15 | 1 | no | — |
| 85 | #41 | — | 45 | 1 | no | — |
| 86 | Q16 | — | 45 | 1 | no | — |
| 87 | SD16 | — | 15 | 1 | no | — |
| 88 | SD16 | — | 45 | 1 | no | — |
| 89 | D8 | — | 45 | 1 | no | — |
| 90 | C2×C8 | — | 45 | 1 | no | — |
| 91 | C3×C5 | — | 6 | 1 | no | — |
| 92 | C22×C3 | — | 10 | 1 | no | — |
| 93 | C22×C3 | — | 10 | 1 | no | — |
| 94 | C22×C3 | — | 20 | 1 | no | — |
| 95 | C2×D3 | — | 10 | 1 | no | — |
| 96 | C2×D3 | — | 15 | 1 | no | — |
| 97 | C2×D3 | — | 30 | 1 | no | — |
| 98 | C2×D3 | — | 60 | 1 | no | — |
| 99 | C3×C4 | — | 15 | 1 | no | — |
| 100 | C3×C4 | — | 30 | 1 | no | — |
| 101 | C3×C4 | — | 30 | 1 | no | — |
| 102 | Dic3 | — | 10 | 1 | no | — |
| 103 | Dic3 | — | 10 | 1 | no | — |
| 104 | Dic3 | — | 30 | 1 | no | — |
| 105 | Dic3 | — | 60 | 1 | no | — |
| 106 | Dic3 | S5 | 1 | 1 | yes | characteristic |
| 107 | C2×C5 | — | 6 | 1 | no | — |
| 108 | D5 | — | 18 | 2 | no | — |
| 109 | C32 | — | 10 | 1 | no | Sylow |
| 110 | Q8 | — | 5 | 1 | no | — |
| 111 | Q8 | — | 45 | 1 | no | — |
| 112 | D4 | — | 15 | 1 | no | — |
| 113 | D4 | — | 45 | 1 | no | — |
| 114 | D4 | — | 90 | 1 | no | — |
| 115 | C2×C4 | — | 30 | 1 | no | — |
| 116 | C2×C4 | — | 45 | 1 | no | — |
| 117 | C8 | — | 15 | 1 | no | — |
| 118 | C8 | — | 45 | 1 | no | — |
| 119 | C2×C3 | — | 10 | 1 | no | — |
| 120 | C2×C3 | — | 20 | 1 | no | — |
| 121 | C2×C3 | — | 20 | 1 | no | — |
| 122 | C2×C3 | — | 20 | 1 | no | — |
| 123 | C2×C3 | — | 40 | 1 | no | — |
| 124 | C2×C3 | C2×S5 | 1 | 1 | yes | characteristic |
| 125 | D3 | — | 20 | 1 | no | — |
| 126 | D3 | — | 30 | 1 | no | — |
| 127 | D3 | — | 60 | 1 | no | — |
| 128 | D3 | — | 120 | 1 | no | — |
| 129 | C5 | — | 6 | 1 | no | Sylow |
| 130 | C22 | — | 10 | 1 | no | — |
| 131 | C22 | — | 45 | 1 | no | — |
| 132 | C4 | — | 3 | 1 | no | — |
| 133 | C4 | — | 15 | 1 | no | — |
| 134 | C4 | — | 30 | 1 | no | — |
| 135 | C3 | — | 10 | 1 | no | — |
| 136 | C3 | — | 20 | 1 | no | — |
| 137 | C3 | #91870 | 1 | 1 | yes | characteristic, complemented |
| 138 | C2 | — | 20 | 1 | no | — |
| 139 | C2 | — | 90 | 1 | no | — |
| 140 | C2 | D3×S5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×S5 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽5) | C22 |
| Fitting, socle | C2×C3 | C2×S5 |
| LCS | G ▷ C3×SL(2,𝔽5) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3⋊#6732 | C2 | semidirect |
| 2. C3⋊#6728 | C2 | semidirect |
| 3. C3×#6729 | C2 | semidirect |
| 9. #6729 | D3 | semidirect |
| 137. C3 | #91870 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊#6732 | 2 |
| C3⋊#6728 | 2 |
| C3×#6729 | 2 |
| #91870 | 3 |
| #64197 | 5 |
| #6738 | 6 |
| #3804 | 10 |