Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
938 conjugacy classes of subgroups.
The subgroup lattice has 243 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #3548 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #3335 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #3332 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #3331 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #3284 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #3255 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #3253 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #3204 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #3167 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #3060 | C2 | 1 | 4 | yes | maximal, complemented |
| 11 | #2931 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C2×#135 | C22 | 1 | 4 | yes | — |
| 13 | C2×#131 | C22 | 1 | 2 | yes | — |
| 14 | #562 | — | 2 | 4 | no | — |
| 15 | #554 | — | 2 | 2 | no | — |
| 16 | #554 | C22 | 1 | 4 | yes | — |
| 17 | #550 | — | 2 | 2 | no | — |
| 18 | #550 | C22 | 1 | 4 | yes | — |
| 19 | #547 | C22 | 1 | 4 | yes | — |
| 20 | #547 | C22 | 1 | 4 | yes | — |
| 21 | #540 | — | 2 | 4 | no | — |
| 22 | #537 | — | 2 | 4 | no | — |
| 23 | #537 | C22 | 1 | 8 | yes | — |
| 24 | #533 | C22 | 1 | 2 | yes | — |
| 25 | #532 | C22 | 1 | 2 | yes | — |
| 26 | #532 | C22 | 1 | 1 | yes | characteristic |
| 27 | #498 | C22 | 1 | 2 | yes | — |
| 28 | #492 | C22 | 1 | 2 | yes | — |
| 29 | #487 | C22 | 1 | 2 | yes | — |
| 30 | #486 | C22 | 1 | 2 | yes | — |
| 31 | #483 | C22 | 1 | 8 | yes | — |
| 32 | #481 | C22 | 1 | 8 | yes | — |
| 33 | #476 | C22 | 1 | 4 | yes | — |
| 34 | #475 | C22 | 1 | 4 | yes | — |
| 35 | #470 | C22 | 1 | 4 | yes | — |
| 36 | #470 | C22 | 1 | 4 | yes | — |
| 37 | #469 | C22 | 1 | 4 | yes | — |
| 38 | #469 | C22 | 1 | 4 | yes | — |
| 39 | #465 | C22 | 1 | 8 | yes | — |
| 40 | #465 | C22 | 1 | 4 | yes | — |
| 41 | #465 | C22 | 1 | 4 | yes | — |
| 42 | #462 | C22 | 1 | 4 | yes | — |
| 43 | #459 | C22 | 1 | 4 | yes | — |
| 44 | #452 | C22 | 1 | 8 | yes | — |
| 45 | #449 | C22 | 1 | 8 | yes | — |
| 46 | C4×SD16 | C22 | 1 | 4 | yes | — |
| 47 | C4×SD16 | C22 | 1 | 4 | yes | — |
| 48 | #436 | C22 | 1 | 4 | yes | — |
| 49 | C8×D4 | C22 | 1 | 4 | yes | — |
| 50 | C2×#106 | C22 | 1 | 4 | yes | — |
| 51 | #423 | C22 | 1 | 1 | yes | characteristic |
| 52 | #417 | C22 | 1 | 8 | yes | — |
| 53 | #408 | C22 | 1 | 2 | yes | — |
| 54 | C2×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 55 | C2×#41 | — | 2 | 4 | no | — |
| 56 | C2×#41 | — | 2 | 2 | no | — |
| 57 | C2×#41 | C23 | 1 | 8 | yes | — |
| 58 | C2×#41 | C23 | 1 | 2 | yes | — |
| 59 | #135 | — | 4 | 8 | no | — |
| 60 | C2×Q16 | C23 | 1 | 4 | yes | — |
| 61 | C2×SD16 | C23 | 1 | 4 | yes | — |
| 62 | C2×SD16 | C23 | 1 | 4 | yes | — |
| 63 | C2×D8 | C23 | 1 | 4 | yes | — |
| 64 | #131 | — | 4 | 4 | no | — |
| 65 | C2×M16 | C23 | 1 | 2 | yes | — |
| 66 | C2×M16 | C23 | 1 | 4 | yes | — |
| 67 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 68 | C22×C8 | C23 | 1 | 4 | yes | — |
| 69 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 70 | C4⋊Q8 | D4 | 1 | 1 | yes | characteristic |
| 71 | C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 72 | C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 73 | C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 74 | #127 | — | 2 | 4 | no | — |
| 75 | #127 | D4 | 1 | 1 | yes | characteristic |
| 76 | #127 | C23 | 1 | 2 | yes | — |
| 77 | #125 | — | 2 | 4 | no | — |
| 78 | #125 | — | 2 | 2 | no | — |
| 79 | #124 | — | 2 | 2 | no | — |
| 80 | #124 | D4 | 1 | 2 | yes | — |
| 81 | #123 | — | 2 | 8 | no | — |
| 82 | #123 | — | 2 | 8 | no | — |
| 83 | C22⋊Q8 | — | 2 | 8 | no | — |
| 84 | C22⋊Q8 | — | 2 | 2 | no | — |
| 85 | C22⋊Q8 | — | 2 | 4 | no | — |
| 86 | C22⋊Q8 | — | 2 | 8 | no | — |
| 87 | C22⋊Q8 | C23 | 1 | 4 | yes | — |
| 88 | C22⋊Q8 | C23 | 1 | 8 | yes | — |
| 89 | #121 | — | 2 | 8 | no | — |
| 90 | #121 | — | 2 | 8 | no | — |
| 91 | #121 | — | 2 | 4 | no | — |
| 92 | #121 | — | 2 | 8 | no | — |
| 93 | #121 | — | 2 | 2 | no | — |
| 94 | #121 | — | 2 | 8 | no | — |
| 95 | #121 | D4 | 1 | 4 | yes | — |
| 96 | #121 | C23 | 1 | 4 | yes | — |
| 97 | #121 | C23 | 1 | 8 | yes | — |
| 98 | C4×Q8 | — | 2 | 2 | no | — |
| 99 | C4×Q8 | C23 | 1 | 4 | yes | — |
| 100 | C4×D4 | — | 2 | 4 | no | — |
| 101 | C4×D4 | — | 2 | 4 | no | — |
| 102 | C4×D4 | — | 2 | 2 | no | — |
| 103 | C4×D4 | C23 | 1 | 4 | yes | — |
| 104 | C4×D4 | C23 | 1 | 4 | yes | — |
| 105 | #117 | — | 2 | 2 | no | — |
| 106 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 107 | C2×C4⋊C4 | C23 | 1 | 8 | yes | — |
| 108 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 109 | #106 | C23 | 1 | 4 | yes | — |
| 110 | #106 | C23 | 1 | 4 | yes | — |
| 111 | #106 | C23 | 1 | 4 | yes | — |
| 112 | #106 | C23 | 1 | 4 | yes | — |
| 113 | C4⋊C8 | C23 | 1 | 4 | yes | — |
| 114 | #103 | C23 | 1 | 4 | yes | — |
| 115 | #103 | C23 | 1 | 8 | yes | — |
| 116 | #103 | C23 | 1 | 4 | yes | — |
| 117 | #102 | C23 | 1 | 4 | yes | — |
| 118 | #102 | C23 | 1 | 8 | yes | — |
| 119 | #102 | C23 | 1 | 4 | yes | — |
| 120 | C22⋊C8 | C23 | 1 | 8 | yes | — |
| 121 | C4×C8 | C23 | 1 | 2 | yes | — |
| 122 | C4×C8 | C23 | 1 | 2 | yes | — |
| 123 | #41 | — | 4 | 2 | no | — |
| 124 | #41 | — | 4 | 2 | no | — |
| 125 | #41 | — | 4 | 8 | no | — |
| 126 | #41 | — | 4 | 8 | no | — |
| 127 | #41 | — | 8 | 4 | no | — |
| 128 | #41 | — | 8 | 8 | no | — |
| 129 | C2×Q8 | — | 2 | 2 | no | — |
| 130 | C2×Q8 | — | 2 | 2 | no | — |
| 131 | C2×Q8 | C2×D4 | 1 | 2 | yes | — |
| 132 | C2×Q8 | C24 | 1 | 4 | yes | — |
| 133 | C2×D4 | — | 2 | 4 | no | — |
| 134 | C2×D4 | — | 2 | 8 | no | — |
| 135 | C2×D4 | — | 2 | 8 | no | — |
| 136 | C2×D4 | — | 2 | 2 | no | — |
| 137 | C2×D4 | — | 2 | 2 | no | — |
| 138 | C2×D4 | C2×D4 | 1 | 4 | yes | — |
| 139 | C2×D4 | C2×D4 | 1 | 4 | yes | — |
| 140 | C2×D4 | C2×D4 | 1 | 2 | yes | — |
| 141 | C2×D4 | C24 | 1 | 4 | yes | — |
| 142 | C22×C4 | — | 2 | 8 | no | — |
| 143 | C22×C4 | — | 2 | 4 | no | — |
| 144 | C22×C4 | — | 2 | 8 | no | — |
| 145 | C22×C4 | C2×D4 | 1 | 2 | yes | — |
| 146 | C22×C4 | C24 | 1 | 4 | yes | — |
| 147 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 148 | Q16 | — | 4 | 4 | no | — |
| 149 | SD16 | — | 2 | 8 | no | — |
| 150 | SD16 | — | 4 | 4 | no | — |
| 151 | D8 | — | 4 | 4 | no | — |
| 152 | M16 | — | 4 | 4 | no | — |
| 153 | M16 | — | 4 | 2 | no | — |
| 154 | C2×C8 | — | 2 | 2 | no | — |
| 155 | C2×C8 | — | 2 | 8 | no | — |
| 156 | C2×C8 | — | 4 | 1 | no | — |
| 157 | C2×C8 | C24 | 1 | 2 | yes | — |
| 158 | C2×C8 | C24 | 1 | 1 | yes | characteristic |
| 159 | C2×C8 | C24 | 1 | 4 | yes | — |
| 160 | C2×C8 | C24 | 1 | 1 | yes | characteristic |
| 161 | C4⋊C4 | — | 2 | 2 | no | — |
| 162 | C4⋊C4 | — | 2 | 8 | no | — |
| 163 | C4⋊C4 | — | 2 | 8 | no | — |
| 164 | C4⋊C4 | — | 2 | 2 | no | — |
| 165 | C4⋊C4 | — | 2 | 2 | no | — |
| 166 | C4⋊C4 | — | 2 | 2 | no | — |
| 167 | C4⋊C4 | — | 2 | 4 | no | — |
| 168 | C4⋊C4 | — | 2 | 4 | no | — |
| 169 | C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 170 | C4⋊C4 | C24 | 1 | 4 | yes | — |
| 171 | C4⋊C4 | C24 | 1 | 4 | yes | — |
| 172 | C22⋊C4 | — | 2 | 4 | no | — |
| 173 | C22⋊C4 | — | 2 | 8 | no | — |
| 174 | C22⋊C4 | — | 2 | 8 | no | — |
| 175 | C22⋊C4 | — | 2 | 4 | no | — |
| 176 | C22⋊C4 | — | 2 | 4 | no | — |
| 177 | C22⋊C4 | C2×D4 | 1 | 8 | yes | — |
| 178 | C42 | — | 2 | 4 | no | — |
| 179 | C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 180 | C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 181 | C42 | C24 | 1 | 2 | yes | — |
| 182 | C23 | — | 2 | 4 | no | — |
| 183 | C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 184 | C23 | C22×D4 | 1 | 4 | yes | — |
| 185 | Q8 | — | 2 | 4 | no | — |
| 186 | Q8 | — | 2 | 4 | no | — |
| 187 | Q8 | — | 4 | 2 | no | — |
| 188 | Q8 | — | 4 | 4 | no | — |
| 189 | Q8 | — | 8 | 2 | no | — |
| 190 | D4 | — | 2 | 8 | no | — |
| 191 | D4 | — | 2 | 4 | no | — |
| 192 | D4 | — | 2 | 4 | no | — |
| 193 | D4 | — | 4 | 4 | no | — |
| 194 | D4 | — | 4 | 2 | no | — |
| 195 | D4 | — | 4 | 4 | no | — |
| 196 | D4 | — | 8 | 2 | no | — |
| 197 | D4 | — | 8 | 8 | no | — |
| 198 | D4 | — | 8 | 4 | no | — |
| 199 | D4 | — | 8 | 8 | no | — |
| 200 | C2×C4 | — | 2 | 4 | no | — |
| 201 | C2×C4 | — | 2 | 1 | no | — |
| 202 | C2×C4 | — | 2 | 4 | no | — |
| 203 | C2×C4 | — | 2 | 1 | no | — |
| 204 | C2×C4 | — | 2 | 4 | no | — |
| 205 | C2×C4 | — | 2 | 4 | no | — |
| 206 | C2×C4 | — | 2 | 4 | no | — |
| 207 | C2×C4 | — | 2 | 4 | no | — |
| 208 | C2×C4 | — | 4 | 8 | no | — |
| 209 | C2×C4 | — | 8 | 8 | no | — |
| 210 | C2×C4 | — | 8 | 8 | no | — |
| 211 | C2×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 212 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 213 | C2×C4 | C22×D4 | 1 | 4 | yes | — |
| 214 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 215 | C2×C4 | C25 | 1 | 1 | yes | characteristic |
| 216 | C8 | — | 2 | 1 | no | — |
| 217 | C8 | — | 2 | 4 | no | — |
| 218 | C8 | — | 2 | 2 | no | — |
| 219 | C8 | H(5,𝔽2) | 1 | 2 | yes | — |
| 220 | C22 | — | 2 | 1 | no | — |
| 221 | C22 | — | 2 | 4 | no | — |
| 222 | C22 | — | 2 | 2 | no | — |
| 223 | C22 | — | 2 | 8 | no | — |
| 224 | C22 | — | 4 | 4 | no | — |
| 225 | C22 | — | 4 | 4 | no | — |
| 226 | C22 | — | 4 | 4 | no | — |
| 227 | C22 | C23×D4 | 1 | 1 | yes | characteristic |
| 228 | C4 | — | 2 | 4 | no | — |
| 229 | C4 | — | 2 | 1 | no | — |
| 230 | C4 | — | 2 | 2 | no | — |
| 231 | C4 | — | 4 | 4 | no | — |
| 232 | C4 | — | 4 | 4 | no | — |
| 233 | C4 | — | 4 | 4 | no | — |
| 234 | C4 | #577 | 1 | 4 | yes | — |
| 235 | C4 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 236 | C4 | #585 | 1 | 1 | yes | characteristic |
| 237 | C2 | — | 4 | 1 | no | — |
| 238 | C2 | — | 4 | 4 | no | — |
| 239 | C2 | — | 8 | 4 | no | — |
| 240 | C2 | #3496 | 1 | 1 | yes | characteristic |
| 241 | C2 | C2×#577 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3548 | C2 | semidirect |
| 2. #3335 | C2 | semidirect |
| 3. #3332 | C2 | semidirect |
| 4. #3331 | C2 | semidirect |
| 5. #3284 | C2 | semidirect |
| 6. #3255 | C2 | semidirect |
| 7. #3253 | C2 | semidirect |
| 8. #3204 | C2 | semidirect |
| 9. #3167 | C2 | semidirect |
| 10. #3060 | C2 | semidirect |
| 11. #2931 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3548 | 2 |
| #3335 | 2 |
| #3332 | 2 |
| #3331 | 2 |
| #3284 | 2 |
| #3255 | 2 |
| #3253 | 2 |
| #3204 | 2 |
| #3167 | 2 |
| #3060 | 2 |
| #2931 | 2 |