Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
7146 conjugacy classes of subgroups.
The subgroup lattice has 174 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×#380 | C2 | 1 | 6 | yes | maximal |
| 2 | C23×#95 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #14353 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 4 | C2×#1444 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | C25×C4 | C22 | 1 | 3 | yes | — |
| 6 | C23×C4⋊C4 | C22 | 1 | 12 | yes | — |
| 7 | C23×C22⋊C4 | C22 | 1 | 4 | yes | — |
| 8 | C2×#380 | — | 2 | 48 | no | — |
| 9 | C2×#380 | C22 | 1 | 48 | yes | — |
| 10 | C22×#95 | C4 | 1 | 16 | yes | complemented |
| 11 | C22×#95 | C22 | 1 | 12 | yes | — |
| 12 | C22×#95 | C22 | 1 | 4 | yes | — |
| 13 | C22×#95 | C22 | 1 | 8 | yes | — |
| 14 | #1444 | C22 | 1 | 64 | yes | complemented |
| 15 | C26 | C23 | 1 | 1 | yes | characteristic |
| 16 | C24×C4 | — | 4 | 24 | no | — |
| 17 | C24×C4 | C2×C4 | 1 | 24 | yes | — |
| 18 | C24×C4 | D4 | 1 | 24 | yes | — |
| 19 | C24×C4 | Q8 | 1 | 24 | yes | — |
| 20 | C24×C4 | C23 | 1 | 8 | yes | — |
| 21 | C24×C4 | C23 | 1 | 6 | yes | — |
| 22 | C24×C4 | C23 | 1 | 12 | yes | — |
| 23 | C22×C4⋊C4 | — | 2 | 48 | no | — |
| 24 | C22×C4⋊C4 | — | 4 | 48 | no | — |
| 25 | C22×C4⋊C4 | C23 | 1 | 48 | yes | — |
| 26 | C22×C22⋊C4 | — | 2 | 48 | no | — |
| 27 | C22×C22⋊C4 | C23 | 1 | 16 | yes | — |
| 28 | #380 | — | 2 | 192 | no | — |
| 29 | C2×#95 | — | 2 | 48 | no | — |
| 30 | C2×#95 | C2×C4 | 1 | 32 | yes | — |
| 31 | C2×#95 | C2×C4 | 1 | 64 | yes | complemented |
| 32 | C2×#95 | C23 | 1 | 48 | yes | — |
| 33 | C2×#95 | C23 | 1 | 16 | yes | — |
| 34 | C25 | — | 2 | 12 | no | — |
| 35 | C25 | — | 2 | 4 | no | — |
| 36 | C25 | C22×C4 | 1 | 4 | yes | — |
| 37 | C25 | C2×D4 | 1 | 12 | yes | — |
| 38 | C25 | C2×Q8 | 1 | 12 | yes | — |
| 39 | C25 | C24 | 1 | 2 | yes | — |
| 40 | C25 | C24 | 1 | 1 | yes | characteristic |
| 41 | C23×C4 | — | 2 | 48 | no | — |
| 42 | C23×C4 | — | 2 | 48 | no | — |
| 43 | C23×C4 | — | 2 | 48 | no | — |
| 44 | C23×C4 | — | 2 | 16 | no | — |
| 45 | C23×C4 | — | 4 | 96 | no | — |
| 46 | C23×C4 | — | 4 | 96 | no | — |
| 47 | C23×C4 | — | 4 | 96 | no | — |
| 48 | C23×C4 | — | 4 | 48 | no | — |
| 49 | C23×C4 | C4⋊C4 | 1 | 192 | yes | complemented |
| 50 | C23×C4 | C22×C4 | 1 | 24 | yes | — |
| 51 | C23×C4 | C22×C4 | 1 | 48 | yes | — |
| 52 | C23×C4 | C2×D4 | 1 | 24 | yes | — |
| 53 | C23×C4 | C2×D4 | 1 | 48 | yes | — |
| 54 | C23×C4 | C2×Q8 | 1 | 24 | yes | — |
| 55 | C23×C4 | C2×Q8 | 1 | 48 | yes | — |
| 56 | C23×C4 | C24 | 1 | 16 | yes | — |
| 57 | C23×C4 | C24 | 1 | 12 | yes | — |
| 58 | C2×C4⋊C4 | — | 2 | 96 | no | — |
| 59 | C2×C4⋊C4 | — | 4 | 96 | no | — |
| 60 | C2×C4⋊C4 | — | 4 | 192 | no | — |
| 61 | C2×C22⋊C4 | — | 2 | 96 | no | — |
| 62 | C2×C22⋊C4 | — | 4 | 64 | no | — |
| 63 | #95 | — | 2 | 96 | no | — |
| 64 | #95 | C22×C4 | 1 | 64 | yes | — |
| 65 | C24 | — | 2 | 24 | no | — |
| 66 | C24 | — | 2 | 8 | no | — |
| 67 | C24 | — | 2 | 48 | no | — |
| 68 | C24 | — | 2 | 48 | no | — |
| 69 | C24 | — | 2 | 48 | no | — |
| 70 | C24 | — | 2 | 48 | no | — |
| 71 | C24 | — | 2 | 12 | no | — |
| 72 | C24 | — | 2 | 4 | no | — |
| 73 | C24 | C2×C4⋊C4 | 1 | 48 | yes | — |
| 74 | C24 | #127 | 1 | 16 | yes | — |
| 75 | C24 | C4⋊Q8 | 1 | 48 | yes | — |
| 76 | C24 | C23×C4 | 1 | 4 | yes | — |
| 77 | C24 | C23×C4 | 1 | 2 | yes | — |
| 78 | C24 | C22×D4 | 1 | 12 | yes | — |
| 79 | C24 | C22×D4 | 1 | 6 | yes | — |
| 80 | C24 | C22×Q8 | 1 | 12 | yes | — |
| 81 | C24 | C22×Q8 | 1 | 6 | yes | — |
| 82 | C24 | H(5,𝔽2) | 1 | 12 | yes | — |
| 83 | C24 | 2−1+4 | 1 | 4 | yes | — |
| 84 | C24 | C25 | 1 | 1 | yes | characteristic |
| 85 | C22×C4 | — | 2 | 96 | no | — |
| 86 | C22×C4 | — | 2 | 96 | no | — |
| 87 | C22×C4 | — | 2 | 96 | no | — |
| 88 | C22×C4 | — | 2 | 48 | no | — |
| 89 | C22×C4 | — | 2 | 48 | no | — |
| 90 | C22×C4 | — | 2 | 48 | no | — |
| 91 | C22×C4 | — | 2 | 16 | no | — |
| 92 | C22×C4 | — | 4 | 384 | no | — |
| 93 | C22×C4 | — | 4 | 96 | no | — |
| 94 | C22×C4 | — | 4 | 96 | no | — |
| 95 | C22×C4 | — | 4 | 96 | no | — |
| 96 | C22×C4 | — | 4 | 32 | no | — |
| 97 | C22×C4 | — | 4 | 96 | no | — |
| 98 | C22×C4 | — | 4 | 96 | no | — |
| 99 | C22×C4 | C2×C4⋊C4 | 1 | 96 | yes | — |
| 100 | C22×C4 | C2×C4⋊C4 | 1 | 192 | yes | complemented |
| 101 | C22×C4 | C23×C4 | 1 | 24 | yes | — |
| 102 | C22×C4 | C22×D4 | 1 | 24 | yes | — |
| 103 | C22×C4 | C22×Q8 | 1 | 24 | yes | — |
| 104 | C4⋊C4 | — | 4 | 192 | no | — |
| 105 | C22⋊C4 | — | 4 | 64 | no | — |
| 106 | C23 | — | 2 | 96 | no | — |
| 107 | C23 | — | 2 | 96 | no | — |
| 108 | C23 | — | 2 | 64 | no | — |
| 109 | C23 | — | 2 | 12 | no | — |
| 110 | C23 | — | 2 | 4 | no | — |
| 111 | C23 | — | 2 | 192 | no | — |
| 112 | C23 | — | 2 | 24 | no | — |
| 113 | C23 | — | 2 | 24 | no | — |
| 114 | C23 | — | 2 | 24 | no | — |
| 115 | C23 | — | 2 | 24 | no | — |
| 116 | C23 | #384 | 1 | 64 | yes | — |
| 117 | C23 | C22×C4⋊C4 | 1 | 24 | yes | — |
| 118 | C23 | C22×C4⋊C4 | 1 | 12 | yes | — |
| 119 | C23 | #518 | 1 | 12 | yes | — |
| 120 | C23 | #519 | 1 | 4 | yes | — |
| 121 | C23 | C2×#127 | 1 | 8 | yes | — |
| 122 | C23 | C2×#127 | 1 | 4 | yes | — |
| 123 | C23 | C2×C4⋊Q8 | 1 | 24 | yes | — |
| 124 | C23 | C2×C4⋊Q8 | 1 | 12 | yes | — |
| 125 | C23 | #534 | 1 | 24 | yes | — |
| 126 | C23 | #537 | 1 | 24 | yes | — |
| 127 | C23 | #543 | 1 | 24 | yes | — |
| 128 | C23 | #544 | 1 | 24 | yes | — |
| 129 | C23 | C24×C4 | 1 | 1 | yes | characteristic |
| 130 | C23 | C23×D4 | 1 | 3 | yes | — |
| 131 | C23 | C23×Q8 | 1 | 3 | yes | — |
| 132 | C23 | C2×H(5,𝔽2) | 1 | 6 | yes | — |
| 133 | C23 | C2×2−1+4 | 1 | 2 | yes | — |
| 134 | C2×C4 | — | 2 | 192 | no | — |
| 135 | C2×C4 | — | 2 | 48 | no | — |
| 136 | C2×C4 | — | 2 | 48 | no | — |
| 137 | C2×C4 | — | 2 | 48 | no | — |
| 138 | C2×C4 | — | 4 | 192 | no | — |
| 139 | C2×C4 | — | 4 | 16 | no | — |
| 140 | C2×C4 | — | 4 | 48 | no | — |
| 141 | C2×C4 | — | 4 | 48 | no | — |
| 142 | C2×C4 | — | 8 | 64 | no | — |
| 143 | C2×C4 | C22×C4⋊C4 | 1 | 48 | yes | — |
| 144 | C22 | — | 2 | 24 | no | — |
| 145 | C22 | — | 2 | 24 | no | — |
| 146 | C22 | — | 2 | 16 | no | — |
| 147 | C22 | — | 2 | 48 | no | — |
| 148 | C22 | — | 2 | 128 | no | — |
| 149 | C22 | C2×#384 | 1 | 16 | yes | — |
| 150 | C22 | C2×#384 | 1 | 8 | yes | — |
| 151 | C22 | #2315 | 1 | 24 | yes | — |
| 152 | C22 | #2316 | 1 | 24 | yes | — |
| 153 | C22 | #2656 | 1 | 16 | yes | — |
| 154 | C22 | #2659 | 1 | 48 | yes | — |
| 155 | C22 | C23×C4⋊C4 | 1 | 3 | yes | — |
| 156 | C22 | C2×#518 | 1 | 3 | yes | — |
| 157 | C22 | C2×#519 | 1 | 1 | yes | characteristic |
| 158 | C22 | C22×#127 | 1 | 1 | yes | characteristic |
| 159 | C22 | C22×C4⋊Q8 | 1 | 3 | yes | — |
| 160 | C22 | C2×#534 | 1 | 6 | yes | — |
| 161 | C22 | C2×#537 | 1 | 6 | yes | — |
| 162 | C22 | C2×#543 | 1 | 6 | yes | — |
| 163 | C22 | C2×#544 | 1 | 6 | yes | — |
| 164 | C4 | — | 2 | 48 | no | — |
| 165 | C4 | — | 8 | 16 | no | — |
| 166 | C2 | — | 2 | 16 | no | — |
| 167 | C2 | #14353 | 1 | 16 | yes | direct product factor |
| 168 | C2 | C22×#384 | 1 | 1 | yes | characteristic |
| 169 | C2 | C2×#2315 | 1 | 3 | yes | — |
| 170 | C2 | C2×#2316 | 1 | 3 | yes | — |
| 171 | C2 | C2×#2656 | 1 | 2 | yes | — |
| 172 | C2 | C2×#2659 | 1 | 6 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #14353 | C2 | direct (N × H) |
| 4. C2×#1444 | C2 | semidirect |
| 10. C22×#95 | C4 | semidirect |
| 14. #1444 | C22 | semidirect |
| 31. C2×#95 | C2×C4 | semidirect |
| 49. C23×C4 | C4⋊C4 | semidirect |
| 100. C22×C4 | C2×C4⋊C4 | semidirect |
| 167. C2 | #14353 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#380 | 2 |
| C23×#95 | 2 |
| #14353 | 2 |
| C2×#1444 | 2 |