Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
3304 conjugacy classes of subgroups.
The subgroup lattice has 192 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #3567 | C2 | 1 | 8 | yes | maximal, complemented |
| 2 | #3556 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #3548 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #3539 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | #3517 | C2 | 1 | 12 | yes | maximal, complemented |
| 6 | #3515 | C2 | 1 | 12 | yes | maximal, complemented |
| 7 | #3512 | C2 | 1 | 12 | yes | maximal, complemented |
| 8 | #3444 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | #564 | C22 | 1 | 4 | yes | complemented |
| 10 | #562 | C22 | 1 | 12 | yes | complemented |
| 11 | #560 | C22 | 1 | 16 | yes | — |
| 12 | #557 | C22 | 1 | 12 | yes | complemented |
| 13 | #554 | C22 | 1 | 12 | yes | complemented |
| 14 | #553 | C22 | 1 | 48 | yes | — |
| 15 | #553 | C22 | 1 | 24 | yes | — |
| 16 | #553 | C22 | 1 | 48 | yes | complemented |
| 17 | #552 | C22 | 1 | 24 | yes | — |
| 18 | #552 | C22 | 1 | 48 | yes | complemented |
| 19 | #550 | C22 | 1 | 24 | yes | — |
| 20 | #550 | C22 | 1 | 12 | yes | complemented |
| 21 | #547 | C22 | 1 | 24 | yes | — |
| 22 | #547 | C22 | 1 | 12 | yes | complemented |
| 23 | #541 | C22 | 1 | 12 | yes | — |
| 24 | #541 | C22 | 1 | 12 | yes | complemented |
| 25 | #540 | C22 | 1 | 12 | yes | — |
| 26 | #540 | C22 | 1 | 48 | yes | — |
| 27 | #540 | C22 | 1 | 12 | yes | complemented |
| 28 | #539 | C22 | 1 | 48 | yes | — |
| 29 | #537 | C22 | 1 | 24 | yes | — |
| 30 | #533 | C22 | 1 | 6 | yes | — |
| 31 | #533 | C22 | 1 | 3 | yes | — |
| 32 | #532 | C22 | 1 | 6 | yes | — |
| 33 | #532 | C22 | 1 | 3 | yes | — |
| 34 | #532 | C22 | 1 | 12 | yes | complemented |
| 35 | #529 | C22 | 1 | 12 | yes | — |
| 36 | #529 | C22 | 1 | 12 | yes | — |
| 37 | #529 | C22 | 1 | 12 | yes | — |
| 38 | #529 | C22 | 1 | 24 | yes | complemented |
| 39 | C2×#125 | C22 | 1 | 12 | yes | complemented |
| 40 | #525 | C22 | 1 | 6 | yes | — |
| 41 | #525 | C22 | 1 | 12 | yes | — |
| 42 | #520 | C22 | 1 | 24 | yes | complemented |
| 43 | C4×#41 | C22 | 1 | 6 | yes | — |
| 44 | C4×#41 | C22 | 1 | 12 | yes | — |
| 45 | C22×C42 | C22 | 1 | 1 | yes | characteristic |
| 46 | C2×#41 | C23 | 1 | 6 | yes | — |
| 47 | C2×#41 | C23 | 1 | 12 | yes | — |
| 48 | C2×#41 | C23 | 1 | 24 | yes | — |
| 49 | C23×C4 | C23 | 1 | 3 | yes | — |
| 50 | C4⋊Q8 | C23 | 1 | 6 | yes | — |
| 51 | C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 52 | C4⋊Q8 | C23 | 1 | 12 | yes | — |
| 53 | #127 | C23 | 1 | 3 | yes | — |
| 54 | #127 | C23 | 1 | 12 | yes | — |
| 55 | #127 | C23 | 1 | 16 | yes | — |
| 56 | #126 | C23 | 1 | 24 | yes | — |
| 57 | #126 | C23 | 1 | 48 | yes | — |
| 58 | #126 | C23 | 1 | 16 | yes | — |
| 59 | #126 | C23 | 1 | 16 | yes | — |
| 60 | #125 | C23 | 1 | 12 | yes | — |
| 61 | #125 | C23 | 1 | 48 | yes | — |
| 62 | #125 | C23 | 1 | 12 | yes | — |
| 63 | #125 | C23 | 1 | 12 | yes | — |
| 64 | #125 | C23 | 1 | 6 | yes | — |
| 65 | #125 | C23 | 1 | 48 | yes | — |
| 66 | #124 | C23 | 1 | 12 | yes | — |
| 67 | #124 | C23 | 1 | 6 | yes | — |
| 68 | #123 | C23 | 1 | 24 | yes | — |
| 69 | #123 | C23 | 1 | 12 | yes | — |
| 70 | #123 | C23 | 1 | 48 | yes | — |
| 71 | #123 | C23 | 1 | 48 | yes | — |
| 72 | #123 | C23 | 1 | 48 | yes | — |
| 73 | C22⋊Q8 | C23 | 1 | 12 | yes | — |
| 74 | C22⋊Q8 | C23 | 1 | 12 | yes | — |
| 75 | C22⋊Q8 | C23 | 1 | 12 | yes | — |
| 76 | C22⋊Q8 | C23 | 1 | 24 | yes | — |
| 77 | C22⋊Q8 | C23 | 1 | 48 | yes | — |
| 78 | #121 | C23 | 1 | 12 | yes | — |
| 79 | #121 | C23 | 1 | 12 | yes | — |
| 80 | #121 | C23 | 1 | 12 | yes | — |
| 81 | #121 | C23 | 1 | 24 | yes | — |
| 82 | #121 | C23 | 1 | 48 | yes | — |
| 83 | #121 | C23 | 1 | 48 | yes | — |
| 84 | #121 | C23 | 1 | 48 | yes | — |
| 85 | #121 | C23 | 1 | 48 | yes | — |
| 86 | #120 | C23 | 1 | 12 | yes | — |
| 87 | C4×Q8 | C23 | 1 | 12 | yes | — |
| 88 | C4×Q8 | C23 | 1 | 12 | yes | — |
| 89 | C4×Q8 | C23 | 1 | 12 | yes | — |
| 90 | C4×Q8 | C23 | 1 | 24 | yes | — |
| 91 | C4×D4 | C23 | 1 | 12 | yes | — |
| 92 | C4×D4 | C23 | 1 | 24 | yes | — |
| 93 | C4×D4 | C23 | 1 | 24 | yes | — |
| 94 | C4×D4 | C23 | 1 | 24 | yes | — |
| 95 | C4×D4 | C23 | 1 | 12 | yes | — |
| 96 | C4×D4 | C23 | 1 | 12 | yes | — |
| 97 | C4×D4 | C23 | 1 | 48 | yes | — |
| 98 | C4×D4 | C23 | 1 | 24 | yes | — |
| 99 | C4×D4 | C23 | 1 | 48 | yes | — |
| 100 | #117 | C23 | 1 | 12 | yes | — |
| 101 | #117 | C23 | 1 | 6 | yes | — |
| 102 | #117 | C23 | 1 | 12 | yes | — |
| 103 | #117 | C23 | 1 | 24 | yes | — |
| 104 | #117 | C23 | 1 | 48 | yes | — |
| 105 | C2×C4⋊C4 | C23 | 1 | 24 | yes | — |
| 106 | C2×C4⋊C4 | C23 | 1 | 48 | yes | — |
| 107 | C2×C42 | C23 | 1 | 12 | yes | — |
| 108 | C2×C42 | C23 | 1 | 3 | yes | — |
| 109 | C2×C42 | C23 | 1 | 6 | yes | — |
| 110 | C2×C42 | C23 | 1 | 2 | yes | — |
| 111 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 112 | C24 | C24 | 1 | 1 | yes | characteristic |
| 113 | #41 | — | 4 | 12 | no | — |
| 114 | #41 | — | 4 | 24 | no | — |
| 115 | #41 | — | 4 | 48 | no | — |
| 116 | C2×Q8 | C24 | 1 | 12 | yes | — |
| 117 | C2×Q8 | C24 | 1 | 6 | yes | — |
| 118 | C2×D4 | C24 | 1 | 12 | yes | — |
| 119 | C2×D4 | C24 | 1 | 12 | yes | — |
| 120 | C2×D4 | C24 | 1 | 24 | yes | — |
| 121 | C2×D4 | C24 | 1 | 6 | yes | — |
| 122 | C2×D4 | C24 | 1 | 48 | yes | — |
| 123 | C22×C4 | — | 4 | 6 | no | — |
| 124 | C22×C4 | C24 | 1 | 6 | yes | — |
| 125 | C22×C4 | C24 | 1 | 3 | yes | — |
| 126 | C22×C4 | C24 | 1 | 12 | yes | — |
| 127 | C22×C4 | C24 | 1 | 3 | yes | — |
| 128 | C22×C4 | C24 | 1 | 6 | yes | — |
| 129 | C22×C4 | C24 | 1 | 24 | yes | — |
| 130 | C22×C4 | C24 | 1 | 48 | yes | — |
| 131 | C4⋊C4 | C24 | 1 | 24 | yes | — |
| 132 | C4⋊C4 | C24 | 1 | 12 | yes | — |
| 133 | C4⋊C4 | C24 | 1 | 24 | yes | — |
| 134 | C4⋊C4 | C24 | 1 | 12 | yes | — |
| 135 | C4⋊C4 | C24 | 1 | 48 | yes | — |
| 136 | C4⋊C4 | C24 | 1 | 48 | yes | — |
| 137 | C4⋊C4 | C24 | 1 | 24 | yes | — |
| 138 | C4⋊C4 | C24 | 1 | 12 | yes | — |
| 139 | C22⋊C4 | C24 | 1 | 24 | yes | — |
| 140 | C22⋊C4 | C24 | 1 | 24 | yes | — |
| 141 | C22⋊C4 | C24 | 1 | 12 | yes | — |
| 142 | C22⋊C4 | C24 | 1 | 24 | yes | — |
| 143 | C22⋊C4 | C24 | 1 | 48 | yes | — |
| 144 | C22⋊C4 | C24 | 1 | 24 | yes | — |
| 145 | C42 | C24 | 1 | 3 | yes | — |
| 146 | C42 | C24 | 1 | 3 | yes | — |
| 147 | C42 | C24 | 1 | 6 | yes | — |
| 148 | C42 | C24 | 1 | 3 | yes | — |
| 149 | C42 | C24 | 1 | 1 | yes | characteristic |
| 150 | C42 | C24 | 1 | 24 | yes | — |
| 151 | C42 | C24 | 1 | 12 | yes | — |
| 152 | C42 | C24 | 1 | 16 | yes | — |
| 153 | C23 | — | 4 | 3 | no | — |
| 154 | C23 | C25 | 1 | 1 | yes | characteristic |
| 155 | C23 | C25 | 1 | 2 | yes | — |
| 156 | C23 | C25 | 1 | 12 | yes | — |
| 157 | Q8 | — | 2 | 24 | no | — |
| 158 | Q8 | — | 4 | 6 | no | — |
| 159 | D4 | — | 2 | 24 | no | — |
| 160 | D4 | — | 4 | 12 | no | — |
| 161 | D4 | — | 4 | 6 | no | — |
| 162 | D4 | — | 4 | 24 | no | — |
| 163 | D4 | — | 4 | 48 | no | — |
| 164 | C2×C4 | — | 2 | 12 | no | — |
| 165 | C2×C4 | — | 2 | 6 | no | — |
| 166 | C2×C4 | — | 2 | 24 | no | — |
| 167 | C2×C4 | — | 4 | 3 | no | — |
| 168 | C2×C4 | — | 4 | 6 | no | — |
| 169 | C2×C4 | — | 4 | 24 | no | — |
| 170 | C2×C4 | — | 4 | 48 | no | — |
| 171 | C2×C4 | C25 | 1 | 6 | yes | — |
| 172 | C2×C4 | C25 | 1 | 3 | yes | — |
| 173 | C2×C4 | C25 | 1 | 24 | yes | — |
| 174 | C2×C4 | C25 | 1 | 12 | yes | — |
| 175 | C2×C4 | C25 | 1 | 3 | yes | — |
| 176 | C22 | — | 2 | 6 | no | — |
| 177 | C22 | — | 2 | 3 | no | — |
| 178 | C22 | — | 2 | 12 | no | — |
| 179 | C22 | — | 2 | 24 | no | — |
| 180 | C22 | — | 4 | 4 | no | — |
| 181 | C22 | C26 | 1 | 1 | yes | characteristic |
| 182 | C4 | — | 2 | 6 | no | — |
| 183 | C4 | — | 2 | 12 | no | — |
| 184 | C4 | — | 2 | 3 | no | — |
| 185 | C4 | — | 2 | 24 | no | — |
| 186 | C4 | #585 | 1 | 6 | yes | — |
| 187 | C2 | — | 4 | 12 | no | — |
| 188 | C2 | — | 4 | 2 | no | — |
| 189 | C2 | — | 4 | 1 | no | — |
| 190 | C2 | C2×#585 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3567 | C2 | semidirect |
| 2. #3556 | C2 | semidirect |
| 3. #3548 | C2 | semidirect |
| 4. #3539 | C2 | semidirect |
| 5. #3517 | C2 | semidirect |
| 6. #3515 | C2 | semidirect |
| 7. #3512 | C2 | semidirect |
| 8. #3444 | C2 | semidirect |
| 9. #564 | C22 | semidirect |
| 10. #562 | C22 | semidirect |
| 12. #557 | C22 | semidirect |
| 13. #554 | C22 | semidirect |
| 16. #553 | C22 | semidirect |
| 18. #552 | C22 | semidirect |
| 20. #550 | C22 | semidirect |
| 22. #547 | C22 | semidirect |
| 24. #541 | C22 | semidirect |
| 27. #540 | C22 | semidirect |
| 34. #532 | C22 | semidirect |
| 38. #529 | C22 | semidirect |
| 39. C2×#125 | C22 | semidirect |
| 42. #520 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3567 | 2 |
| #3556 | 2 |
| #3548 | 2 |
| #3539 | 2 |
| #3517 | 2 |
| #3515 | 2 |
| #3512 | 2 |
| #3444 | 2 |