Finite Group DatabasebetaFinite Group DatabasebetaOrder 6480.
232 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 232 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 | #272×A5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | H(3,𝔽3)×A5 | C4 | 1 | 1 | yes | characteristic, complemented |
| 3 | A4×#1088 | — | 5 | 1 | no | maximal, Hall |
| 4 | C3×D3×A5 | — | 6 | 2 | no | — |
| 5 | D5×#1088 | — | 6 | 1 | no | maximal |
| 6 | C3×C4×A5 | — | 9 | 1 | no | maximal |
| 7 | A4×#272 | — | 5 | 1 | no | — |
| 8 | D3×#1088 | — | 10 | 1 | no | maximal |
| 9 | C32×A5 | — | 2 | 2 | no | — |
| 10 | D5×#272 | — | 6 | 1 | no | — |
| 11 | #10587382 | — | 6 | 1 | no | — |
| 12 | C5×#1088 | — | 6 | 1 | no | — |
| 13 | C22×#1088 | — | 5 | 1 | no | — |
| 14 | C2×C3×A5 | — | 9 | 1 | no | — |
| 15 | D3×A5 | — | 6 | 2 | no | — |
| 16 | A4×H(3,𝔽3) | — | 5 | 1 | no | — |
| 17 | D3×#272 | — | 10 | 1 | no | — |
| 18 | C3⋊#1088 | — | 10 | 1 | no | — |
| 19 | C3×#1088 | — | 10 | 1 | no | — |
| 20 | C5⋊#272 | — | 6 | 1 | no | — |
| 21 | C5×#272 | — | 6 | 1 | no | — |
| 22 | D5×H(3,𝔽3) | — | 6 | 1 | no | — |
| 23 | C4×A5 | — | 9 | 1 | no | — |
| 24 | C3×D3×A4 | — | 30 | 2 | no | — |
| 25 | C22×#272 | — | 5 | 1 | no | — |
| 26 | C2×#1088 | — | 15 | 1 | no | — |
| 27 | C2×#1088 | — | 15 | 1 | no | — |
| 28 | C3×D3×D5 | — | 36 | 2 | no | — |
| 29 | C3×A5 | — | 6 | 2 | no | — |
| 30 | C3×A5 | #157 | 1 | 1 | yes | characteristic |
| 31 | C3⋊#272 | — | 10 | 1 | no | — |
| 32 | C3×#272 | — | 10 | 1 | no | — |
| 33 | D3×H(3,𝔽3) | — | 10 | 1 | no | — |
| 34 | C3×C4×A4 | — | 45 | 1 | no | — |
| 35 | C5×H(3,𝔽3) | — | 6 | 1 | no | — |
| 36 | C2×A5 | — | 9 | 1 | no | — |
| 37 | C3×C4×D5 | — | 54 | 1 | no | — |
| 38 | C32×A4 | — | 10 | 2 | no | — |
| 39 | C3×D32 | — | 60 | 2 | no | — |
| 40 | C22×H(3,𝔽3) | — | 5 | 1 | no | — |
| 41 | C2×#272 | — | 15 | 1 | no | — |
| 42 | C2×#272 | — | 15 | 1 | no | — |
| 43 | C22⋊H(3,𝔽3) | — | 20 | 2 | no | — |
| 44 | #1088 | — | 15 | 1 | no | — |
| 45 | #1088 | A5 | 1 | 1 | yes | maximal normal, characteristic, direct product factor |
| 46 | C3×D15 | — | 36 | 2 | no | — |
| 47 | C3×C5×D3 | — | 36 | 2 | no | — |
| 48 | C32×D5 | — | 12 | 2 | no | — |
| 49 | C3×H(3,𝔽3) | — | 10 | 1 | no | Sylow |
| 50 | C22×C3×D3 | — | 30 | 2 | no | — |
| 51 | C2×C3×A4 | — | 45 | 1 | no | — |
| 52 | D3×A4 | — | 30 | 2 | no | — |
| 53 | D3×A4 | — | 30 | 4 | no | — |
| 54 | C3×C4×D3 | — | 90 | 1 | no | — |
| 55 | C2×C3×D5 | — | 54 | 1 | no | — |
| 56 | D3×D5 | — | 36 | 2 | no | — |
| 57 | A5 | #1088 | 1 | 1 | yes | characteristic, direct product factor |
| 58 | C3×C4×C5 | — | 54 | 1 | no | — |
| 59 | C3×Dic5 | — | 54 | 1 | no | — |
| 60 | C3×C3⋊D3 | — | 60 | 2 | no | — |
| 61 | C32×D3 | — | 20 | 2 | no | — |
| 62 | C32×D3 | — | 60 | 2 | no | — |
| 63 | C2×H(3,𝔽3) | — | 15 | 1 | no | — |
| 64 | #272 | — | 15 | 1 | no | — |
| 65 | #272 | — | 120 | 2 | no | — |
| 66 | #272 | C2×A5 | 1 | 1 | yes | characteristic |
| 67 | C22×C3×C4 | — | 45 | 1 | no | — |
| 68 | C4×A4 | — | 45 | 1 | no | — |
| 69 | C4×A4 | — | 45 | 2 | no | — |
| 70 | C32×C5 | — | 12 | 2 | no | — |
| 71 | C4×D5 | — | 54 | 1 | no | — |
| 72 | C22×C32 | — | 10 | 2 | no | — |
| 73 | C2×C3×D3 | — | 90 | 2 | no | — |
| 74 | C2×C3×D3 | — | 90 | 1 | no | — |
| 75 | C2×C3×D3 | — | 90 | 2 | no | — |
| 76 | C3×A4 | — | 5 | 1 | no | — |
| 77 | C3×A4 | — | 20 | 2 | no | — |
| 78 | C3×A4 | — | 30 | 2 | no | — |
| 79 | C3×A4 | — | 30 | 4 | no | — |
| 80 | D32 | — | 60 | 2 | no | — |
| 81 | C32×C4 | — | 90 | 1 | no | — |
| 82 | C3×Dic3 | — | 90 | 1 | no | — |
| 83 | C2×C3×C5 | — | 54 | 1 | no | — |
| 84 | D15 | — | 36 | 2 | no | — |
| 85 | C3×D5 | — | 6 | 1 | no | — |
| 86 | C3×D5 | — | 36 | 2 | no | — |
| 87 | C3×D5 | — | 54 | 1 | no | — |
| 88 | C5×D3 | — | 36 | 2 | no | — |
| 89 | C33 | — | 20 | 2 | no | — |
| 90 | H(3,𝔽3) | — | 40 | 2 | no | — |
| 91 | H(3,𝔽3) | C4×A5 | 1 | 1 | yes | characteristic, complemented |
| 92 | C23×C3 | — | 45 | 1 | no | — |
| 93 | C22×D3 | — | 30 | 2 | no | — |
| 94 | C2×A4 | — | 45 | 1 | no | — |
| 95 | C2×A4 | — | 45 | 2 | no | — |
| 96 | C2×C3×C4 | — | 135 | 1 | no | — |
| 97 | C2×C3×C4 | — | 135 | 1 | no | — |
| 98 | C4×D3 | — | 90 | 1 | no | — |
| 99 | C2×D5 | — | 54 | 1 | no | — |
| 100 | C4×C5 | — | 54 | 1 | no | — |
| 101 | Dic5 | — | 54 | 1 | no | — |
| 102 | C2×C32 | — | 30 | 2 | no | — |
| 103 | C2×C32 | — | 90 | 1 | no | — |
| 104 | C3⋊D3 | — | 60 | 2 | no | — |
| 105 | C3×D3 | — | 6 | 2 | no | — |
| 106 | C3×D3 | — | 10 | 1 | no | — |
| 107 | C3×D3 | — | 60 | 2 | no | — |
| 108 | C3×D3 | — | 60 | 2 | no | — |
| 109 | C3×D3 | — | 90 | 2 | no | — |
| 110 | C3×D3 | — | 90 | 1 | no | — |
| 111 | C3×D3 | — | 360 | 2 | no | — |
| 112 | C3×D3 | — | 120 | 2 | no | — |
| 113 | C22×C4 | — | 45 | 1 | no | Sylow |
| 114 | C3×C5 | — | 6 | 1 | no | — |
| 115 | C3×C5 | — | 36 | 2 | no | — |
| 116 | C22×C3 | — | 5 | 1 | no | — |
| 117 | C22×C3 | — | 30 | 2 | no | — |
| 118 | C22×C3 | — | 135 | 1 | no | — |
| 119 | C22×C3 | — | 135 | 1 | no | — |
| 120 | C2×D3 | — | 90 | 2 | no | — |
| 121 | C2×D3 | — | 90 | 1 | no | — |
| 122 | C2×D3 | — | 90 | 2 | no | — |
| 123 | A4 | — | 5 | 1 | no | — |
| 124 | A4 | — | 5 | 2 | no | — |
| 125 | A4 | — | 60 | 2 | no | — |
| 126 | C3×C4 | — | 9 | 1 | no | — |
| 127 | C3×C4 | — | 90 | 1 | no | — |
| 128 | C3×C4 | — | 135 | 1 | no | — |
| 129 | C3×C4 | — | 180 | 1 | no | — |
| 130 | Dic3 | — | 90 | 1 | no | — |
| 131 | C2×C5 | — | 54 | 1 | no | — |
| 132 | D5 | — | 6 | 1 | no | — |
| 133 | D5 | — | 54 | 1 | no | — |
| 134 | C32 | — | 2 | 2 | no | — |
| 135 | C32 | — | 10 | 1 | no | — |
| 136 | C32 | — | 40 | 2 | no | — |
| 137 | C32 | — | 60 | 2 | no | — |
| 138 | C32 | — | 120 | 2 | no | — |
| 139 | C23 | — | 45 | 1 | no | — |
| 140 | C2×C4 | — | 135 | 1 | no | — |
| 141 | C2×C4 | — | 135 | 1 | no | — |
| 142 | C2×C3 | — | 9 | 1 | no | — |
| 143 | C2×C3 | — | 15 | 1 | no | — |
| 144 | C2×C3 | — | 90 | 1 | no | — |
| 145 | C2×C3 | — | 90 | 2 | no | — |
| 146 | C2×C3 | — | 135 | 1 | no | — |
| 147 | C2×C3 | — | 180 | 1 | no | — |
| 148 | D3 | — | 6 | 2 | no | — |
| 149 | D3 | — | 10 | 1 | no | — |
| 150 | D3 | — | 90 | 2 | no | — |
| 151 | D3 | — | 90 | 1 | no | — |
| 152 | D3 | — | 360 | 2 | no | — |
| 153 | C5 | — | 6 | 1 | no | Sylow |
| 154 | C22 | — | 5 | 1 | no | — |
| 155 | C22 | — | 135 | 1 | no | — |
| 156 | C22 | — | 135 | 1 | no | — |
| 157 | C4 | — | 9 | 1 | no | — |
| 158 | C4 | — | 135 | 1 | no | — |
| 159 | C3 | — | 6 | 2 | no | — |
| 160 | C3 | — | 10 | 1 | no | — |
| 161 | C3 | — | 20 | 1 | no | — |
| 162 | C3 | — | 120 | 2 | no | — |
| 163 | C3 | #157×A5 | 1 | 1 | yes | characteristic |
| 164 | C2 | — | 9 | 1 | no | — |
| 165 | C2 | — | 15 | 1 | no | — |
| 166 | C2 | — | 135 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | #157×A5 |
| commutator, hypocenter | H(3,𝔽3)×A5 | C4 |
| Fitting | H(3,𝔽3) | C4×A5 |
| socle | C3×A5 | #157 |
| prime radical | #272×A5 | C2 |
| cosocle | #272 | C2×A5 |
| LCS | G ▷ H(3,𝔽3)×A5 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. H(3,𝔽3)×A5 | C4 | semidirect |
| 45. #1088 | A5 | direct (N × H) |
| 57. A5 | #1088 | direct (N × H) |
| 91. H(3,𝔽3) | C4×A5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #272×A5 | 2 |
| A4×#1088 | 5 |
| D5×#1088 | 6 |
| C3×C4×A5 | 9 |
| D3×#1088 | 10 |