Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
3557 conjugacy classes of subgroups.
The subgroup lattice has 184 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #3522 | C2 | 1 | 12 | yes | maximal, complemented |
| 2 | #3518 | C2 | 1 | 12 | yes | maximal, complemented |
| 3 | #3511 | C2 | 1 | 18 | yes | maximal, complemented |
| 4 | #3497 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #3483 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #3480 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | Q8×#41 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | C2×Q82 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C4×2−1+4 | C2 | 1 | 3 | yes | maximal, complemented |
| 10 | C2×2−1+4 | C22 | 1 | 3 | yes | — |
| 11 | #559 | C22 | 1 | 18 | yes | — |
| 12 | Q82 | C22 | 1 | 4 | yes | — |
| 13 | Q82 | C22 | 1 | 12 | yes | — |
| 14 | #557 | C22 | 1 | 6 | yes | — |
| 15 | #556 | C22 | 1 | 18 | yes | — |
| 16 | #556 | C22 | 1 | 36 | yes | — |
| 17 | #555 | C22 | 1 | 36 | yes | — |
| 18 | #554 | C22 | 1 | 36 | yes | — |
| 19 | #553 | C22 | 1 | 36 | yes | — |
| 20 | #550 | C22 | 1 | 6 | yes | — |
| 21 | #550 | C22 | 1 | 12 | yes | — |
| 22 | D4×Q8 | C22 | 1 | 12 | yes | — |
| 23 | #548 | C22 | 1 | 4 | yes | — |
| 24 | #548 | C22 | 1 | 24 | yes | — |
| 25 | #548 | C22 | 1 | 24 | yes | — |
| 26 | #547 | C22 | 1 | 36 | yes | — |
| 27 | #542 | C22 | 1 | 72 | yes | — |
| 28 | #537 | C22 | 1 | 18 | yes | — |
| 29 | #533 | C22 | 1 | 36 | yes | — |
| 30 | #532 | C22 | 1 | 18 | yes | — |
| 31 | #532 | C22 | 1 | 36 | yes | — |
| 32 | C2×C4⋊Q8 | C22 | 1 | 9 | yes | — |
| 33 | #529 | C22 | 1 | 36 | yes | — |
| 34 | #520 | C22 | 1 | 18 | yes | — |
| 35 | #520 | C22 | 1 | 12 | yes | — |
| 36 | #519 | C22 | 1 | 1 | yes | characteristic |
| 37 | #519 | C22 | 1 | 12 | yes | — |
| 38 | C4×#41 | C22 | 1 | 18 | yes | — |
| 39 | C4×#41 | C22 | 1 | 12 | yes | — |
| 40 | C4×#41 | C22 | 1 | 12 | yes | — |
| 41 | C2×C4×Q8 | C22 | 1 | 3 | yes | — |
| 42 | C2×C4×Q8 | C22 | 1 | 12 | yes | — |
| 43 | C2×C4×Q8 | C22 | 1 | 3 | yes | — |
| 44 | 2−1+4 | — | 2 | 24 | no | — |
| 45 | C2×#41 | C23 | 1 | 4 | yes | — |
| 46 | C2×#41 | C23 | 1 | 18 | yes | — |
| 47 | C2×#41 | C23 | 1 | 12 | yes | — |
| 48 | C2×#41 | C23 | 1 | 24 | yes | — |
| 49 | C22×Q8 | C23 | 1 | 1 | yes | characteristic |
| 50 | C22×Q8 | C23 | 1 | 12 | yes | — |
| 51 | C22×Q8 | C23 | 1 | 6 | yes | — |
| 52 | C22×Q8 | C23 | 1 | 1 | yes | characteristic |
| 53 | C4⋊Q8 | C23 | 1 | 36 | yes | — |
| 54 | C4⋊Q8 | C23 | 1 | 36 | yes | — |
| 55 | C4⋊Q8 | C23 | 1 | 18 | yes | — |
| 56 | #127 | C23 | 1 | 18 | yes | — |
| 57 | #126 | C23 | 1 | 72 | yes | — |
| 58 | #125 | C23 | 1 | 36 | yes | — |
| 59 | #124 | C23 | 1 | 12 | yes | — |
| 60 | #124 | C23 | 1 | 36 | yes | — |
| 61 | #124 | C23 | 1 | 72 | yes | — |
| 62 | #123 | C23 | 1 | 72 | yes | — |
| 63 | C22⋊Q8 | C23 | 1 | 12 | yes | — |
| 64 | C22⋊Q8 | C23 | 1 | 72 | yes | — |
| 65 | C22⋊Q8 | C23 | 1 | 72 | yes | — |
| 66 | #121 | C23 | 1 | 36 | yes | — |
| 67 | #121 | C23 | 1 | 12 | yes | — |
| 68 | #121 | C23 | 1 | 36 | yes | — |
| 69 | #121 | C23 | 1 | 72 | yes | — |
| 70 | C4×Q8 | C23 | 1 | 4 | yes | — |
| 71 | C4×Q8 | C23 | 1 | 12 | yes | — |
| 72 | C4×Q8 | C23 | 1 | 18 | yes | — |
| 73 | C4×Q8 | C23 | 1 | 6 | yes | — |
| 74 | C4×Q8 | C23 | 1 | 36 | yes | — |
| 75 | C4×Q8 | C23 | 1 | 12 | yes | — |
| 76 | C4×Q8 | C23 | 1 | 18 | yes | — |
| 77 | C4×Q8 | C23 | 1 | 4 | yes | — |
| 78 | C4×Q8 | C23 | 1 | 4 | yes | — |
| 79 | C4×Q8 | C23 | 1 | 18 | yes | — |
| 80 | C4×Q8 | C23 | 1 | 24 | yes | — |
| 81 | C4×D4 | C23 | 1 | 12 | yes | — |
| 82 | C4×D4 | C23 | 1 | 18 | yes | — |
| 83 | C4×D4 | C23 | 1 | 36 | yes | — |
| 84 | C4×D4 | C23 | 1 | 18 | yes | — |
| 85 | C4×D4 | C23 | 1 | 36 | yes | — |
| 86 | C4×D4 | C23 | 1 | 36 | yes | — |
| 87 | C4×D4 | C23 | 1 | 36 | yes | — |
| 88 | C4×D4 | C23 | 1 | 36 | yes | — |
| 89 | #117 | C23 | 1 | 6 | yes | — |
| 90 | #117 | C23 | 1 | 36 | yes | — |
| 91 | #117 | C23 | 1 | 36 | yes | — |
| 92 | #117 | C23 | 1 | 36 | yes | — |
| 93 | C2×C4⋊C4 | C23 | 1 | 9 | yes | — |
| 94 | C2×C4⋊C4 | C23 | 1 | 36 | yes | — |
| 95 | C2×C4⋊C4 | C23 | 1 | 9 | yes | — |
| 96 | C2×C42 | C23 | 1 | 9 | yes | — |
| 97 | C2×C42 | C23 | 1 | 36 | yes | — |
| 98 | #41 | — | 2 | 48 | no | — |
| 99 | #41 | — | 2 | 16 | no | — |
| 100 | #41 | — | 2 | 72 | no | — |
| 101 | #41 | — | 8 | 24 | no | — |
| 102 | C2×Q8 | — | 2 | 48 | no | — |
| 103 | C2×Q8 | — | 2 | 4 | no | — |
| 104 | C2×Q8 | — | 8 | 6 | no | — |
| 105 | C2×Q8 | #41 | 1 | 8 | yes | complemented |
| 106 | C2×Q8 | C24 | 1 | 3 | yes | — |
| 107 | C2×Q8 | C24 | 1 | 18 | yes | — |
| 108 | C2×Q8 | C24 | 1 | 24 | yes | — |
| 109 | C2×Q8 | C24 | 1 | 4 | yes | — |
| 110 | C2×Q8 | C24 | 1 | 1 | yes | characteristic |
| 111 | C2×D4 | C24 | 1 | 12 | yes | — |
| 112 | C2×D4 | C24 | 1 | 12 | yes | — |
| 113 | C2×D4 | C24 | 1 | 18 | yes | — |
| 114 | C2×D4 | C24 | 1 | 36 | yes | — |
| 115 | C22×C4 | C24 | 1 | 3 | yes | — |
| 116 | C22×C4 | C24 | 1 | 12 | yes | — |
| 117 | C22×C4 | C24 | 1 | 36 | yes | — |
| 118 | C22×C4 | C24 | 1 | 9 | yes | — |
| 119 | C22×C4 | C24 | 1 | 36 | yes | — |
| 120 | C22×C4 | C24 | 1 | 3 | yes | — |
| 121 | C4⋊C4 | C24 | 1 | 6 | yes | — |
| 122 | C4⋊C4 | C24 | 1 | 36 | yes | — |
| 123 | C4⋊C4 | C24 | 1 | 36 | yes | — |
| 124 | C4⋊C4 | C24 | 1 | 18 | yes | — |
| 125 | C4⋊C4 | C24 | 1 | 18 | yes | — |
| 126 | C4⋊C4 | C24 | 1 | 6 | yes | — |
| 127 | C4⋊C4 | C24 | 1 | 12 | yes | — |
| 128 | C4⋊C4 | C24 | 1 | 12 | yes | — |
| 129 | C4⋊C4 | C24 | 1 | 36 | yes | — |
| 130 | C22⋊C4 | C24 | 1 | 4 | yes | — |
| 131 | C22⋊C4 | C24 | 1 | 36 | yes | — |
| 132 | C22⋊C4 | C24 | 1 | 12 | yes | — |
| 133 | C22⋊C4 | C24 | 1 | 72 | yes | — |
| 134 | C22⋊C4 | C24 | 1 | 24 | yes | — |
| 135 | C42 | C24 | 1 | 12 | yes | — |
| 136 | C42 | C24 | 1 | 12 | yes | — |
| 137 | C42 | C24 | 1 | 18 | yes | — |
| 138 | C42 | C24 | 1 | 36 | yes | — |
| 139 | C42 | C24 | 1 | 18 | yes | — |
| 140 | C23 | C25 | 1 | 12 | yes | — |
| 141 | C23 | C25 | 1 | 1 | yes | characteristic |
| 142 | Q8 | — | 2 | 36 | no | — |
| 143 | Q8 | — | 2 | 6 | no | — |
| 144 | Q8 | — | 4 | 24 | no | — |
| 145 | Q8 | C2×#41 | 1 | 16 | yes | — |
| 146 | Q8 | 2−1+4 | 1 | 4 | yes | complemented |
| 147 | D4 | — | 2 | 24 | no | — |
| 148 | D4 | — | 2 | 36 | no | — |
| 149 | D4 | — | 2 | 24 | no | — |
| 150 | D4 | — | 4 | 36 | no | — |
| 151 | C2×C4 | — | 2 | 72 | no | — |
| 152 | C2×C4 | — | 2 | 6 | no | — |
| 153 | C2×C4 | — | 2 | 24 | no | — |
| 154 | C2×C4 | — | 4 | 9 | no | — |
| 155 | C2×C4 | — | 4 | 36 | no | — |
| 156 | C2×C4 | C2×#41 | 1 | 12 | yes | — |
| 157 | C2×C4 | C25 | 1 | 4 | yes | — |
| 158 | C2×C4 | C25 | 1 | 3 | yes | — |
| 159 | C2×C4 | C25 | 1 | 4 | yes | — |
| 160 | C2×C4 | C25 | 1 | 12 | yes | — |
| 161 | C2×C4 | C25 | 1 | 18 | yes | — |
| 162 | C2×C4 | C25 | 1 | 6 | yes | — |
| 163 | C2×C4 | C25 | 1 | 3 | yes | — |
| 164 | C22 | — | 2 | 1 | no | — |
| 165 | C22 | — | 2 | 1 | no | — |
| 166 | C22 | — | 2 | 12 | no | — |
| 167 | C22 | — | 2 | 12 | no | — |
| 168 | C22 | — | 2 | 12 | no | — |
| 169 | C22 | C22×#41 | 1 | 2 | yes | — |
| 170 | C22 | C26 | 1 | 1 | yes | characteristic |
| 171 | C4 | — | 2 | 12 | no | — |
| 172 | C4 | — | 2 | 4 | no | — |
| 173 | C4 | — | 2 | 3 | no | — |
| 174 | C4 | — | 2 | 4 | no | — |
| 175 | C4 | — | 2 | 18 | no | — |
| 176 | C4 | C22×#41 | 1 | 12 | yes | — |
| 177 | C4 | C2×2−1+4 | 1 | 6 | yes | — |
| 178 | C2 | — | 2 | 2 | no | — |
| 179 | C2 | — | 4 | 12 | no | — |
| 180 | C2 | C23×#41 | 1 | 1 | yes | characteristic |
| 181 | C2 | C22×2−1+4 | 1 | 1 | yes | characteristic |
| 182 | C2 | H(7,𝔽2) | 1 | 1 | yes | characteristic |
| 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. #3522 | C2 | semidirect |
| 2. #3518 | C2 | semidirect |
| 3. #3511 | C2 | semidirect |
| 4. #3497 | C2 | semidirect |
| 5. #3483 | C2 | semidirect |
| 6. #3480 | C2 | semidirect |
| 7. Q8×#41 | C2 | semidirect |
| 8. C2×Q82 | C2 | semidirect |
| 9. C4×2−1+4 | C2 | semidirect |
| 105. C2×Q8 | #41 | semidirect |
| 146. Q8 | 2−1+4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3522 | 2 |
| #3518 | 2 |
| #3511 | 2 |
| #3497 | 2 |
| #3483 | 2 |
| #3480 | 2 |
| Q8×#41 | 2 |
| C2×Q82 | 2 |
| C4×2−1+4 | 2 |