Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
850 conjugacy classes of subgroups.
The subgroup lattice has 222 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #3550 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #3394 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | Q8×SD16 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #3373 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #3369 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #3263 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #3255 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #3253 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #3173 | C2 | 1 | 4 | yes | maximal, complemented |
| 10 | #3167 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #2960 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #563 | — | 2 | 4 | no | — |
| 13 | #559 | C22 | 1 | 4 | yes | — |
| 14 | Q82 | C22 | 1 | 4 | yes | — |
| 15 | #556 | — | 2 | 2 | no | — |
| 16 | #556 | C22 | 1 | 4 | yes | — |
| 17 | D4×Q8 | — | 2 | 2 | no | — |
| 18 | D4×Q8 | C22 | 1 | 4 | yes | — |
| 19 | #542 | — | 2 | 4 | no | — |
| 20 | #542 | C22 | 1 | 8 | yes | — |
| 21 | #536 | — | 2 | 4 | no | — |
| 22 | #533 | C22 | 1 | 2 | yes | — |
| 23 | #533 | C22 | 1 | 1 | yes | characteristic |
| 24 | #532 | C22 | 1 | 2 | yes | — |
| 25 | #498 | C22 | 1 | 2 | yes | — |
| 26 | #498 | C22 | 1 | 4 | yes | — |
| 27 | #495 | C22 | 1 | 4 | yes | — |
| 28 | #492 | C22 | 1 | 2 | yes | — |
| 29 | #489 | C22 | 1 | 8 | yes | — |
| 30 | #487 | C22 | 1 | 2 | yes | — |
| 31 | #486 | C22 | 1 | 2 | yes | — |
| 32 | #476 | C22 | 1 | 4 | yes | — |
| 33 | #476 | C22 | 1 | 8 | yes | — |
| 34 | #475 | C22 | 1 | 8 | yes | — |
| 35 | #475 | C22 | 1 | 4 | yes | — |
| 36 | #470 | C22 | 1 | 4 | yes | — |
| 37 | #469 | C22 | 1 | 4 | yes | — |
| 38 | #465 | C22 | 1 | 4 | yes | — |
| 39 | #465 | C22 | 1 | 4 | yes | — |
| 40 | #464 | C22 | 1 | 8 | yes | — |
| 41 | #463 | C22 | 1 | 8 | yes | — |
| 42 | #462 | C22 | 1 | 4 | yes | — |
| 43 | #459 | C22 | 1 | 4 | yes | — |
| 44 | #446 | C22 | 1 | 4 | yes | — |
| 45 | C8×Q8 | C22 | 1 | 4 | yes | — |
| 46 | #442 | C22 | 1 | 4 | yes | — |
| 47 | #441 | C22 | 1 | 4 | yes | — |
| 48 | C4×SD16 | C22 | 1 | 4 | yes | — |
| 49 | C4×SD16 | C22 | 1 | 4 | yes | — |
| 50 | C4×SD16 | C22 | 1 | 8 | yes | — |
| 51 | #424 | C22 | 1 | 2 | yes | — |
| 52 | #423 | C22 | 1 | 1 | yes | characteristic |
| 53 | #405 | C22 | 1 | 2 | yes | — |
| 54 | C2×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 55 | C2×#41 | — | 2 | 2 | no | — |
| 56 | C22×Q8 | — | 2 | 4 | no | — |
| 57 | C2×Q16 | C23 | 1 | 4 | yes | — |
| 58 | C2×SD16 | C23 | 1 | 4 | yes | — |
| 59 | C2×SD16 | C23 | 1 | 4 | yes | — |
| 60 | C2×D8 | C23 | 1 | 4 | yes | — |
| 61 | C2×M16 | C23 | 1 | 2 | yes | — |
| 62 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 63 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 64 | C4⋊Q8 | — | 2 | 8 | no | — |
| 65 | C4⋊Q8 | — | 2 | 4 | no | — |
| 66 | C4⋊Q8 | D4 | 1 | 2 | yes | — |
| 67 | C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 68 | C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 69 | C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 70 | C4⋊Q8 | C23 | 1 | 8 | yes | — |
| 71 | #127 | C23 | 1 | 2 | yes | — |
| 72 | #126 | — | 2 | 8 | no | — |
| 73 | #126 | — | 2 | 8 | no | — |
| 74 | #125 | — | 2 | 2 | no | — |
| 75 | #125 | D4 | 1 | 2 | yes | — |
| 76 | #124 | — | 2 | 4 | no | — |
| 77 | #124 | — | 2 | 8 | no | — |
| 78 | #124 | — | 2 | 2 | no | — |
| 79 | #124 | C23 | 1 | 8 | yes | — |
| 80 | #123 | — | 2 | 8 | no | — |
| 81 | #123 | — | 2 | 8 | no | — |
| 82 | C22⋊Q8 | — | 2 | 4 | no | — |
| 83 | C22⋊Q8 | — | 2 | 2 | no | — |
| 84 | C22⋊Q8 | — | 2 | 4 | no | — |
| 85 | C22⋊Q8 | — | 2 | 8 | no | — |
| 86 | C22⋊Q8 | — | 2 | 8 | no | — |
| 87 | C22⋊Q8 | D4 | 1 | 4 | yes | — |
| 88 | C22⋊Q8 | C23 | 1 | 4 | yes | — |
| 89 | #121 | — | 2 | 2 | no | — |
| 90 | #121 | C23 | 1 | 4 | yes | — |
| 91 | C4×Q8 | — | 2 | 4 | no | — |
| 92 | C4×Q8 | — | 2 | 2 | no | — |
| 93 | C4×Q8 | C23 | 1 | 4 | yes | — |
| 94 | C4×Q8 | C23 | 1 | 8 | yes | — |
| 95 | C4×Q8 | C23 | 1 | 4 | yes | — |
| 96 | C4×D4 | — | 2 | 4 | no | — |
| 97 | C4×D4 | — | 2 | 2 | no | — |
| 98 | C4×D4 | C23 | 1 | 4 | yes | — |
| 99 | C4×D4 | C23 | 1 | 8 | yes | — |
| 100 | #117 | — | 2 | 2 | no | — |
| 101 | #117 | — | 2 | 4 | no | — |
| 102 | #117 | C23 | 1 | 2 | yes | — |
| 103 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 104 | #106 | C23 | 1 | 4 | yes | — |
| 105 | #106 | C23 | 1 | 4 | yes | — |
| 106 | #106 | C23 | 1 | 4 | yes | — |
| 107 | #106 | C23 | 1 | 4 | yes | — |
| 108 | C4⋊C8 | C23 | 1 | 4 | yes | — |
| 109 | C4⋊C8 | C23 | 1 | 8 | yes | — |
| 110 | #103 | C23 | 1 | 4 | yes | — |
| 111 | #103 | C23 | 1 | 8 | yes | — |
| 112 | #103 | C23 | 1 | 4 | yes | — |
| 113 | #102 | C23 | 1 | 4 | yes | — |
| 114 | #102 | C23 | 1 | 8 | yes | — |
| 115 | #102 | C23 | 1 | 4 | yes | — |
| 116 | #97 | C23 | 1 | 4 | yes | — |
| 117 | C4×C8 | C23 | 1 | 2 | yes | — |
| 118 | C4×C8 | C23 | 1 | 2 | yes | — |
| 119 | C4×C8 | C23 | 1 | 4 | yes | — |
| 120 | #41 | — | 8 | 4 | no | — |
| 121 | C2×Q8 | — | 2 | 2 | no | — |
| 122 | C2×Q8 | — | 2 | 8 | no | — |
| 123 | C2×Q8 | — | 2 | 2 | no | — |
| 124 | C2×Q8 | — | 8 | 8 | no | — |
| 125 | C2×Q8 | C2×D4 | 1 | 4 | yes | — |
| 126 | C2×Q8 | C24 | 1 | 4 | yes | — |
| 127 | C2×D4 | — | 2 | 4 | no | — |
| 128 | C2×D4 | — | 2 | 2 | no | — |
| 129 | C2×D4 | — | 2 | 2 | no | — |
| 130 | C2×D4 | C24 | 1 | 4 | yes | — |
| 131 | C22×C4 | — | 2 | 4 | no | — |
| 132 | C22×C4 | — | 2 | 8 | no | — |
| 133 | C22×C4 | C2×D4 | 1 | 2 | yes | — |
| 134 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 135 | Q16 | — | 4 | 4 | no | — |
| 136 | SD16 | — | 2 | 8 | no | — |
| 137 | SD16 | — | 4 | 4 | no | — |
| 138 | D8 | — | 4 | 4 | no | — |
| 139 | M16 | — | 4 | 2 | no | — |
| 140 | C2×C8 | — | 2 | 2 | no | — |
| 141 | C2×C8 | — | 4 | 1 | no | — |
| 142 | C2×C8 | C24 | 1 | 1 | yes | characteristic |
| 143 | C2×C8 | C24 | 1 | 2 | yes | — |
| 144 | C2×C8 | C24 | 1 | 4 | yes | — |
| 145 | C2×C8 | C24 | 1 | 1 | yes | characteristic |
| 146 | C4⋊C4 | — | 2 | 2 | no | — |
| 147 | C4⋊C4 | — | 2 | 2 | no | — |
| 148 | C4⋊C4 | — | 2 | 2 | no | — |
| 149 | C4⋊C4 | — | 2 | 2 | no | — |
| 150 | C4⋊C4 | — | 2 | 4 | no | — |
| 151 | C4⋊C4 | — | 2 | 8 | no | — |
| 152 | C4⋊C4 | — | 2 | 8 | no | — |
| 153 | C4⋊C4 | — | 2 | 4 | no | — |
| 154 | C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 155 | C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 156 | C4⋊C4 | C2×D4 | 1 | 8 | yes | — |
| 157 | C4⋊C4 | C24 | 1 | 4 | yes | — |
| 158 | C4⋊C4 | C24 | 1 | 4 | yes | — |
| 159 | C22⋊C4 | — | 2 | 4 | no | — |
| 160 | C22⋊C4 | — | 2 | 8 | no | — |
| 161 | C22⋊C4 | — | 2 | 4 | no | — |
| 162 | C22⋊C4 | — | 2 | 8 | no | — |
| 163 | C22⋊C4 | — | 2 | 4 | no | — |
| 164 | C22⋊C4 | — | 2 | 8 | no | — |
| 165 | C22⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 166 | C42 | — | 2 | 4 | no | — |
| 167 | C42 | — | 2 | 8 | no | — |
| 168 | C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 169 | C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 170 | C42 | C24 | 1 | 2 | yes | — |
| 171 | C42 | C24 | 1 | 4 | yes | — |
| 172 | C23 | — | 2 | 4 | no | — |
| 173 | C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 174 | Q8 | — | 2 | 8 | no | — |
| 175 | Q8 | — | 2 | 4 | no | — |
| 176 | Q8 | — | 2 | 4 | no | — |
| 177 | Q8 | — | 4 | 4 | no | — |
| 178 | Q8 | — | 8 | 8 | no | — |
| 179 | Q8 | — | 8 | 2 | no | — |
| 180 | D4 | — | 2 | 4 | no | — |
| 181 | D4 | — | 2 | 4 | no | — |
| 182 | D4 | — | 4 | 4 | no | — |
| 183 | D4 | — | 8 | 2 | no | — |
| 184 | D4 | — | 8 | 4 | no | — |
| 185 | C2×C4 | — | 2 | 4 | no | — |
| 186 | C2×C4 | — | 2 | 1 | no | — |
| 187 | C2×C4 | — | 2 | 1 | no | — |
| 188 | C2×C4 | — | 2 | 4 | no | — |
| 189 | C2×C4 | — | 2 | 4 | no | — |
| 190 | C2×C4 | — | 2 | 4 | no | — |
| 191 | C2×C4 | — | 4 | 8 | no | — |
| 192 | C2×C4 | — | 8 | 8 | no | — |
| 193 | C2×C4 | C22×D4 | 1 | 8 | yes | — |
| 194 | C2×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 195 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 196 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 197 | C2×C4 | C25 | 1 | 1 | yes | characteristic |
| 198 | C8 | — | 2 | 1 | no | — |
| 199 | C8 | — | 2 | 2 | no | — |
| 200 | C8 | — | 2 | 4 | no | — |
| 201 | C8 | 2−1+4 | 1 | 2 | yes | — |
| 202 | C22 | — | 2 | 1 | no | — |
| 203 | C22 | — | 2 | 2 | no | — |
| 204 | C22 | — | 4 | 4 | no | — |
| 205 | C22 | — | 4 | 4 | no | — |
| 206 | C22 | — | 4 | 4 | no | — |
| 207 | C22 | C23×D4 | 1 | 1 | yes | characteristic |
| 208 | C4 | — | 2 | 1 | no | — |
| 209 | C4 | — | 2 | 2 | no | — |
| 210 | C4 | — | 2 | 8 | no | — |
| 211 | C4 | — | 4 | 4 | no | — |
| 212 | C4 | — | 4 | 4 | no | — |
| 213 | C4 | — | 4 | 4 | no | — |
| 214 | C4 | #577 | 1 | 4 | yes | — |
| 215 | C4 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 216 | C4 | #585 | 1 | 1 | yes | characteristic |
| 217 | C2 | — | 4 | 1 | no | — |
| 218 | C2 | — | 8 | 4 | no | — |
| 219 | C2 | #3497 | 1 | 1 | yes | characteristic |
| 220 | C2 | C2×#577 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3253 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3550 | C2 | semidirect |
| 2. #3394 | C2 | semidirect |
| 3. Q8×SD16 | C2 | semidirect |
| 4. #3373 | C2 | semidirect |
| 5. #3369 | C2 | semidirect |
| 6. #3263 | C2 | semidirect |
| 7. #3255 | C2 | semidirect |
| 9. #3173 | C2 | semidirect |
| 10. #3167 | C2 | semidirect |
| 11. #2960 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3550 | 2 |
| #3394 | 2 |
| Q8×SD16 | 2 |
| #3373 | 2 |
| #3369 | 2 |
| #3263 | 2 |
| #3255 | 2 |
| #3253 | 2 |
| #3173 | 2 |
| #3167 | 2 |
| #2960 | 2 |