Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
3134 conjugacy classes of subgroups.
The subgroup lattice has 175 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×C43 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C22×#336 | C2 | 1 | 6 | yes | maximal |
| 3 | C24×C8 | C22 | 1 | 4 | yes | — |
| 4 | C23×C42 | C4 | 1 | 4 | yes | — |
| 5 | C23×C42 | C22 | 1 | 3 | yes | — |
| 6 | C2×C43 | — | 2 | 12 | no | — |
| 7 | C2×#336 | — | 2 | 12 | no | — |
| 8 | C2×#336 | — | 2 | 12 | no | — |
| 9 | C2×#336 | C4 | 1 | 24 | yes | complemented |
| 10 | C24×C4 | C2×C4 | 1 | 6 | yes | — |
| 11 | C24×C4 | C23 | 1 | 1 | yes | characteristic |
| 12 | C23×C8 | — | 2 | 12 | no | — |
| 13 | C23×C8 | — | 2 | 12 | no | — |
| 14 | C23×C8 | — | 2 | 24 | no | — |
| 15 | C23×C8 | C2×C4 | 1 | 24 | yes | — |
| 16 | C22×C42 | — | 2 | 24 | no | — |
| 17 | C22×C42 | — | 2 | 24 | no | — |
| 18 | C22×C42 | — | 2 | 12 | no | — |
| 19 | C22×C42 | — | 2 | 12 | no | — |
| 20 | C22×C42 | C8 | 1 | 16 | yes | complemented |
| 21 | C22×C42 | C2×C4 | 1 | 12 | yes | — |
| 22 | C22×C42 | D4 | 1 | 6 | yes | — |
| 23 | C22×C42 | D4 | 1 | 12 | yes | — |
| 24 | C22×C42 | Q8 | 1 | 6 | yes | — |
| 25 | C43 | — | 2 | 32 | no | — |
| 26 | #336 | — | 2 | 48 | no | — |
| 27 | C25 | C22×C4 | 1 | 1 | yes | characteristic |
| 28 | C23×C4 | — | 2 | 12 | no | — |
| 29 | C23×C4 | — | 2 | 12 | no | — |
| 30 | C23×C4 | — | 2 | 24 | no | — |
| 31 | C23×C4 | C42 | 1 | 12 | yes | — |
| 32 | C23×C4 | C22⋊C4 | 1 | 12 | yes | — |
| 33 | C23×C4 | C4⋊C4 | 1 | 12 | yes | — |
| 34 | C23×C4 | C2×C8 | 1 | 24 | yes | — |
| 35 | C23×C4 | M16 | 1 | 24 | yes | — |
| 36 | C23×C4 | C22×C4 | 1 | 6 | yes | — |
| 37 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 38 | C23×C4 | C2×D4 | 1 | 6 | yes | — |
| 39 | C23×C4 | C2×Q8 | 1 | 3 | yes | — |
| 40 | C23×C4 | #41 | 1 | 12 | yes | — |
| 41 | C22×C8 | — | 2 | 24 | no | — |
| 42 | C22×C8 | — | 2 | 24 | no | — |
| 43 | C22×C8 | — | 2 | 48 | no | — |
| 44 | C22×C8 | — | 2 | 48 | no | — |
| 45 | C22×C8 | — | 4 | 4 | no | — |
| 46 | C22×C8 | — | 4 | 24 | no | — |
| 47 | C22×C8 | — | 4 | 12 | no | — |
| 48 | C22×C8 | — | 4 | 12 | no | — |
| 49 | C22×C8 | — | 4 | 8 | no | — |
| 50 | C22×C8 | — | 4 | 4 | no | — |
| 51 | C22×C8 | C42 | 1 | 16 | yes | complemented |
| 52 | C2×C42 | — | 2 | 32 | no | — |
| 53 | C2×C42 | — | 2 | 96 | no | — |
| 54 | C2×C42 | — | 2 | 96 | no | — |
| 55 | C2×C42 | — | 2 | 48 | no | — |
| 56 | C2×C42 | — | 2 | 24 | no | — |
| 57 | C2×C42 | — | 2 | 24 | no | — |
| 58 | C2×C42 | — | 2 | 24 | no | — |
| 59 | C2×C42 | — | 2 | 24 | no | — |
| 60 | C2×C42 | C22⋊C4 | 1 | 24 | yes | — |
| 61 | C2×C42 | C4⋊C4 | 1 | 24 | yes | — |
| 62 | C24 | C2×C42 | 1 | 3 | yes | — |
| 63 | C24 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 64 | C24 | C2×C4⋊C4 | 1 | 3 | yes | — |
| 65 | C24 | #117 | 1 | 6 | yes | — |
| 66 | C24 | C22×C8 | 1 | 4 | yes | — |
| 67 | C24 | C2×M16 | 1 | 12 | yes | — |
| 68 | C22×C4 | — | 2 | 24 | no | — |
| 69 | C22×C4 | — | 2 | 12 | no | — |
| 70 | C22×C4 | — | 2 | 24 | no | — |
| 71 | C22×C4 | — | 2 | 12 | no | — |
| 72 | C22×C4 | — | 2 | 24 | no | — |
| 73 | C22×C4 | — | 2 | 48 | no | — |
| 74 | C22×C4 | — | 2 | 24 | no | — |
| 75 | C22×C4 | — | 2 | 48 | no | — |
| 76 | C22×C4 | — | 2 | 24 | no | — |
| 77 | C22×C4 | — | 2 | 48 | no | — |
| 78 | C22×C4 | — | 2 | 48 | no | — |
| 79 | C22×C4 | #95 | 1 | 24 | yes | — |
| 80 | C22×C4 | C4×C8 | 1 | 24 | yes | — |
| 81 | C22×C4 | #97 | 1 | 24 | yes | — |
| 82 | C22×C4 | C22⋊C8 | 1 | 48 | yes | — |
| 83 | C22×C4 | C4⋊C8 | 1 | 48 | yes | — |
| 84 | C22×C4 | C2×C42 | 1 | 4 | yes | — |
| 85 | C22×C4 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 86 | C22×C4 | C2×C4⋊C4 | 1 | 6 | yes | — |
| 87 | C22×C4 | #117 | 1 | 12 | yes | — |
| 88 | C22×C4 | C4×D4 | 1 | 12 | yes | — |
| 89 | C22×C4 | C4×D4 | 1 | 24 | yes | — |
| 90 | C22×C4 | C4×Q8 | 1 | 12 | yes | — |
| 91 | C22×C4 | #120 | 1 | 4 | yes | — |
| 92 | C22×C4 | #121 | 1 | 12 | yes | — |
| 93 | C22×C4 | C22⋊Q8 | 1 | 12 | yes | — |
| 94 | C22×C4 | #123 | 1 | 12 | yes | — |
| 95 | C22×C4 | #124 | 1 | 6 | yes | — |
| 96 | C22×C4 | #125 | 1 | 6 | yes | — |
| 97 | C22×C4 | #126 | 1 | 8 | yes | — |
| 98 | C22×C4 | #127 | 1 | 1 | yes | characteristic |
| 99 | C22×C4 | C4⋊Q8 | 1 | 3 | yes | — |
| 100 | C2×C8 | — | 2 | 48 | no | — |
| 101 | C2×C8 | — | 4 | 24 | no | — |
| 102 | C2×C8 | — | 4 | 48 | no | — |
| 103 | C2×C8 | — | 4 | 24 | no | — |
| 104 | C42 | — | 2 | 128 | no | — |
| 105 | C42 | — | 2 | 48 | no | — |
| 106 | C42 | — | 2 | 48 | no | — |
| 107 | C23 | C43 | 1 | 1 | yes | characteristic |
| 108 | C23 | C2×#95 | 1 | 3 | yes | — |
| 109 | C23 | #376 | 1 | 3 | yes | — |
| 110 | C23 | C4×C22⋊C4 | 1 | 6 | yes | — |
| 111 | C23 | C4×C4⋊C4 | 1 | 6 | yes | — |
| 112 | C23 | #379 | 1 | 1 | yes | characteristic |
| 113 | C23 | #380 | 1 | 6 | yes | — |
| 114 | C23 | #381 | 1 | 3 | yes | — |
| 115 | C23 | #382 | 1 | 3 | yes | — |
| 116 | C23 | #383 | 1 | 2 | yes | — |
| 117 | C23 | #384 | 1 | 1 | yes | characteristic |
| 118 | C23 | C2×C4×C8 | 1 | 6 | yes | — |
| 119 | C23 | C2×#97 | 1 | 6 | yes | — |
| 120 | C23 | C4×M16 | 1 | 12 | yes | — |
| 121 | C23 | C2×C22⋊C8 | 1 | 12 | yes | — |
| 122 | C23 | #407 | 1 | 12 | yes | — |
| 123 | C23 | C2×C4⋊C8 | 1 | 12 | yes | — |
| 124 | C23 | #423 | 1 | 12 | yes | — |
| 125 | C23 | #431 | 1 | 24 | yes | — |
| 126 | C23 | #432 | 1 | 24 | yes | — |
| 127 | C2×C4 | — | 2 | 96 | no | — |
| 128 | C2×C4 | — | 2 | 48 | no | — |
| 129 | C2×C4 | — | 2 | 24 | no | — |
| 130 | C2×C4 | — | 2 | 24 | no | — |
| 131 | C2×C4 | — | 2 | 48 | no | — |
| 132 | C2×C4 | — | 2 | 48 | no | — |
| 133 | C2×C4 | — | 2 | 48 | no | — |
| 134 | C2×C4 | #336 | 1 | 48 | yes | complemented |
| 135 | C2×C4 | C4×C22⋊C4 | 1 | 12 | yes | — |
| 136 | C2×C4 | C4×C4⋊C4 | 1 | 12 | yes | — |
| 137 | C2×C4 | #385 | 1 | 12 | yes | — |
| 138 | C2×C4 | #386 | 1 | 12 | yes | — |
| 139 | C2×C4 | #387 | 1 | 24 | yes | — |
| 140 | C2×C4 | #388 | 1 | 24 | yes | — |
| 141 | C2×C4 | #389 | 1 | 12 | yes | — |
| 142 | C2×C4 | #390 | 1 | 6 | yes | — |
| 143 | C2×C4 | #391 | 1 | 6 | yes | — |
| 144 | C8 | — | 4 | 16 | no | — |
| 145 | C22 | C4×#95 | 1 | 3 | yes | — |
| 146 | C22 | C22⋊#95 | 1 | 3 | yes | — |
| 147 | C22 | #1434 | 1 | 6 | yes | — |
| 148 | C22 | C4⋊#95 | 1 | 3 | yes | — |
| 149 | C22 | C42×C8 | 1 | 1 | yes | characteristic |
| 150 | C22 | C4×#97 | 1 | 3 | yes | — |
| 151 | C22 | C2×#336 | 1 | 6 | yes | — |
| 152 | C22 | #1728 | 1 | 6 | yes | — |
| 153 | C22 | #1744 | 1 | 6 | yes | — |
| 154 | C22 | #1745 | 1 | 6 | yes | — |
| 155 | C22 | C4×C22⋊C8 | 1 | 12 | yes | — |
| 156 | C22 | #1749 | 1 | 12 | yes | — |
| 157 | C22 | C4×C4⋊C8 | 1 | 12 | yes | — |
| 158 | C22 | #1767 | 1 | 12 | yes | — |
| 159 | C22 | #1779 | 1 | 4 | yes | — |
| 160 | C22 | #1797 | 1 | 24 | yes | — |
| 161 | C22 | #1817 | 1 | 12 | yes | — |
| 162 | C22 | #1831 | 1 | 12 | yes | — |
| 163 | C22 | #1839 | 1 | 8 | yes | — |
| 164 | C22 | #1842 | 1 | 4 | yes | — |
| 165 | C4 | — | 2 | 48 | no | — |
| 166 | C4 | #1436 | 1 | 12 | yes | — |
| 167 | C4 | #1437 | 1 | 4 | yes | — |
| 168 | C2 | #7554 | 1 | 1 | yes | characteristic |
| 169 | C2 | C4×#336 | 1 | 3 | yes | — |
| 170 | C2 | #10693 | 1 | 3 | yes | — |
| 171 | C2 | #10712 | 1 | 6 | yes | — |
| 172 | C2 | #10795 | 1 | 12 | yes | — |
| 173 | C2 | #10832 | 1 | 6 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C24×C4 | C23 |
| commutator | C22 | C42×C8 |
| socle, prime radical | C25 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. C2×#336 | C4 | semidirect |
| 20. C22×C42 | C8 | semidirect |
| 51. C22×C8 | C42 | semidirect |
| 134. C2×C4 | #336 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C43 | 2 |
| C22×#336 | 2 |