Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
797 conjugacy classes of subgroups.
The subgroup lattice has 217 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #2817 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #2789 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #2765 | C2 | 1 | 4 | yes | maximal |
| 4 | #2713 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #2608 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #2588 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #2550 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #2475 | C2 | 1 | 4 | yes | maximal |
| 9 | #2434 | C2 | 1 | 2 | yes | maximal |
| 10 | #2332 | C2 | 1 | 2 | yes | maximal |
| 11 | #2310 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | C23×Q8 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C4⋊Q8 | C22 | 1 | 2 | yes | — |
| 14 | C2×C4⋊Q8 | C22 | 1 | 2 | yes | — |
| 15 | C2×#125 | C22 | 1 | 4 | yes | — |
| 16 | C2×C22⋊Q8 | C22 | 1 | 2 | yes | — |
| 17 | C2×#117 | C22 | 1 | 2 | yes | — |
| 18 | #401 | C22 | 1 | 4 | yes | — |
| 19 | #400 | C22 | 1 | 4 | yes | — |
| 20 | #400 | C22 | 1 | 4 | yes | — |
| 21 | #400 | C22 | 1 | 4 | yes | — |
| 22 | #398 | C22 | 1 | 4 | yes | — |
| 23 | #398 | C22 | 1 | 4 | yes | — |
| 24 | #398 | C22 | 1 | 4 | yes | — |
| 25 | #397 | C22 | 1 | 4 | yes | — |
| 26 | #396 | C22 | 1 | 4 | yes | — |
| 27 | #395 | C22 | 1 | 4 | yes | — |
| 28 | #395 | C22 | 1 | 4 | yes | — |
| 29 | #395 | C22 | 1 | 4 | yes | — |
| 30 | #391 | C22 | 1 | 4 | yes | — |
| 31 | #391 | C22 | 1 | 4 | yes | — |
| 32 | #391 | C22 | 1 | 4 | yes | — |
| 33 | #391 | C22 | 1 | 2 | yes | — |
| 34 | #391 | C22 | 1 | 2 | yes | — |
| 35 | #389 | C22 | 1 | 4 | yes | — |
| 36 | #389 | C22 | 1 | 4 | yes | — |
| 37 | #389 | C22 | 1 | 4 | yes | — |
| 38 | #388 | C22 | 1 | 4 | yes | — |
| 39 | #388 | C22 | 1 | 4 | yes | — |
| 40 | #387 | C22 | 1 | 4 | yes | — |
| 41 | #387 | C22 | 1 | 4 | yes | — |
| 42 | #387 | C22 | 1 | 4 | yes | — |
| 43 | #387 | C22 | 1 | 4 | yes | — |
| 44 | #387 | C22 | 1 | 4 | yes | — |
| 45 | #387 | C22 | 1 | 4 | yes | — |
| 46 | #387 | C22 | 1 | 4 | yes | — |
| 47 | #387 | C22 | 1 | 4 | yes | — |
| 48 | #385 | C22 | 1 | 4 | yes | — |
| 49 | #385 | C22 | 1 | 4 | yes | — |
| 50 | #382 | C22 | 1 | 4 | yes | — |
| 51 | #381 | C22 | 1 | 1 | yes | characteristic |
| 52 | #381 | C22 | 1 | 1 | yes | characteristic |
| 53 | #381 | C22 | 1 | 2 | yes | — |
| 54 | C4×C4⋊C4 | C22 | 1 | 4 | yes | — |
| 55 | C4×C4⋊C4 | C22 | 1 | 4 | yes | — |
| 56 | C4×C22⋊C4 | C22 | 1 | 2 | yes | — |
| 57 | C22×Q8 | — | 2 | 4 | no | — |
| 58 | C22×Q8 | — | 4 | 4 | no | — |
| 59 | C22×Q8 | C23 | 1 | 2 | yes | — |
| 60 | C22×Q8 | C23 | 1 | 2 | yes | — |
| 61 | C22×Q8 | C23 | 1 | 2 | yes | — |
| 62 | C23×C4 | C23 | 1 | 1 | yes | characteristic |
| 63 | C23×C4 | C23 | 1 | 1 | yes | characteristic |
| 64 | C23×C4 | C23 | 1 | 1 | yes | characteristic |
| 65 | C4⋊Q8 | — | 4 | 4 | no | — |
| 66 | C4⋊Q8 | — | 4 | 4 | no | — |
| 67 | #125 | — | 4 | 8 | no | — |
| 68 | C22⋊Q8 | — | 2 | 8 | no | — |
| 69 | #117 | — | 4 | 4 | no | — |
| 70 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 71 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 72 | C2×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 73 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 74 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 75 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 76 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 77 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 78 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 79 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 80 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 81 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 82 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 83 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 84 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 85 | C2×C22⋊C4 | C23 | 1 | 2 | yes | — |
| 86 | C2×C22⋊C4 | C23 | 1 | 2 | yes | — |
| 87 | C2×C42 | C23 | 1 | 4 | yes | — |
| 88 | C2×C42 | C23 | 1 | 4 | yes | — |
| 89 | C2×C42 | C23 | 1 | 4 | yes | — |
| 90 | C2×C42 | C23 | 1 | 2 | yes | — |
| 91 | C2×C42 | C23 | 1 | 2 | yes | — |
| 92 | #95 | C23 | 1 | 4 | yes | — |
| 93 | #95 | C23 | 1 | 4 | yes | — |
| 94 | #95 | C23 | 1 | 4 | yes | — |
| 95 | #95 | C23 | 1 | 4 | yes | — |
| 96 | #95 | C23 | 1 | 4 | yes | — |
| 97 | #95 | C23 | 1 | 4 | yes | — |
| 98 | #95 | C23 | 1 | 4 | yes | — |
| 99 | #95 | C23 | 1 | 4 | yes | — |
| 100 | #95 | C23 | 1 | 4 | yes | — |
| 101 | #95 | C23 | 1 | 4 | yes | — |
| 102 | #95 | C23 | 1 | 4 | yes | — |
| 103 | #95 | C23 | 1 | 4 | yes | — |
| 104 | #95 | C23 | 1 | 4 | yes | — |
| 105 | #95 | C23 | 1 | 4 | yes | — |
| 106 | #95 | C23 | 1 | 4 | yes | — |
| 107 | #95 | C23 | 1 | 4 | yes | — |
| 108 | #95 | C23 | 1 | 4 | yes | — |
| 109 | C24 | C24 | 1 | 1 | yes | characteristic |
| 110 | C2×Q8 | — | 2 | 4 | no | — |
| 111 | C2×Q8 | — | 2 | 4 | no | — |
| 112 | C2×Q8 | — | 2 | 4 | no | — |
| 113 | C2×Q8 | — | 2 | 4 | no | — |
| 114 | C2×Q8 | — | 2 | 4 | no | — |
| 115 | C2×Q8 | — | 4 | 2 | no | — |
| 116 | C2×Q8 | — | 4 | 2 | no | — |
| 117 | C2×Q8 | — | 4 | 2 | no | — |
| 118 | C2×Q8 | — | 4 | 2 | no | — |
| 119 | C2×Q8 | — | 4 | 16 | no | — |
| 120 | C22×C4 | — | 2 | 2 | no | — |
| 121 | C22×C4 | — | 2 | 4 | no | — |
| 122 | C22×C4 | — | 2 | 2 | no | — |
| 123 | C22×C4 | — | 4 | 2 | no | — |
| 124 | C22×C4 | — | 4 | 2 | no | — |
| 125 | C22×C4 | #41 | 1 | 4 | yes | — |
| 126 | C22×C4 | C24 | 1 | 4 | yes | — |
| 127 | C22×C4 | C24 | 1 | 4 | yes | — |
| 128 | C22×C4 | C24 | 1 | 4 | yes | — |
| 129 | C22×C4 | C24 | 1 | 4 | yes | — |
| 130 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 131 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 132 | C22×C4 | C24 | 1 | 4 | yes | — |
| 133 | C22×C4 | C24 | 1 | 2 | yes | — |
| 134 | C22×C4 | C24 | 1 | 2 | yes | — |
| 135 | C22×C4 | C24 | 1 | 4 | yes | — |
| 136 | C4⋊C4 | — | 2 | 4 | no | — |
| 137 | C4⋊C4 | — | 2 | 8 | no | — |
| 138 | C4⋊C4 | — | 2 | 8 | no | — |
| 139 | C4⋊C4 | — | 2 | 8 | no | — |
| 140 | C4⋊C4 | — | 2 | 8 | no | — |
| 141 | C4⋊C4 | — | 2 | 8 | no | — |
| 142 | C4⋊C4 | — | 2 | 8 | no | — |
| 143 | C4⋊C4 | — | 2 | 8 | no | — |
| 144 | C4⋊C4 | — | 4 | 4 | no | — |
| 145 | C4⋊C4 | — | 4 | 4 | no | — |
| 146 | C4⋊C4 | — | 4 | 4 | no | — |
| 147 | C4⋊C4 | — | 4 | 4 | no | — |
| 148 | C4⋊C4 | — | 4 | 4 | no | — |
| 149 | C22⋊C4 | — | 2 | 8 | no | — |
| 150 | C22⋊C4 | — | 2 | 8 | no | — |
| 151 | C22⋊C4 | — | 4 | 2 | no | — |
| 152 | C22⋊C4 | — | 4 | 2 | no | — |
| 153 | C42 | — | 2 | 8 | no | — |
| 154 | C42 | — | 4 | 4 | no | — |
| 155 | C42 | — | 4 | 4 | no | — |
| 156 | C42 | — | 4 | 2 | no | — |
| 157 | C42 | — | 4 | 2 | no | — |
| 158 | C23 | — | 2 | 2 | no | — |
| 159 | C23 | — | 2 | 4 | no | — |
| 160 | C23 | C2×#41 | 1 | 2 | yes | — |
| 161 | C23 | C25 | 1 | 1 | yes | characteristic |
| 162 | Q8 | — | 4 | 8 | no | — |
| 163 | Q8 | — | 4 | 8 | no | — |
| 164 | Q8 | — | 8 | 4 | no | — |
| 165 | C2×C4 | — | 2 | 8 | no | — |
| 166 | C2×C4 | — | 2 | 2 | no | — |
| 167 | C2×C4 | — | 2 | 4 | no | — |
| 168 | C2×C4 | — | 2 | 2 | no | — |
| 169 | C2×C4 | — | 2 | 2 | no | — |
| 170 | C2×C4 | — | 2 | 2 | no | — |
| 171 | C2×C4 | — | 2 | 4 | no | — |
| 172 | C2×C4 | — | 2 | 4 | no | — |
| 173 | C2×C4 | — | 2 | 4 | no | — |
| 174 | C2×C4 | — | 2 | 1 | no | — |
| 175 | C2×C4 | — | 2 | 1 | no | — |
| 176 | C2×C4 | — | 2 | 4 | no | — |
| 177 | C2×C4 | — | 2 | 4 | no | — |
| 178 | C2×C4 | — | 2 | 4 | no | — |
| 179 | C2×C4 | — | 2 | 4 | no | — |
| 180 | C2×C4 | — | 2 | 4 | no | — |
| 181 | C2×C4 | — | 2 | 2 | no | — |
| 182 | C2×C4 | — | 2 | 2 | no | — |
| 183 | C2×C4 | — | 2 | 4 | no | — |
| 184 | C2×C4 | — | 2 | 4 | no | — |
| 185 | C2×C4 | — | 2 | 4 | no | — |
| 186 | C2×C4 | — | 2 | 4 | no | — |
| 187 | C2×C4 | — | 2 | 4 | no | — |
| 188 | C2×C4 | — | 2 | 4 | no | — |
| 189 | C2×C4 | — | 2 | 4 | no | — |
| 190 | C2×C4 | — | 2 | 4 | no | — |
| 191 | C2×C4 | — | 4 | 4 | no | — |
| 192 | C2×C4 | — | 4 | 4 | no | — |
| 193 | C2×C4 | C2×#41 | 1 | 4 | yes | — |
| 194 | C2×C4 | H(5,𝔽2) | 1 | 4 | yes | — |
| 195 | C2×C4 | 2−1+4 | 1 | 8 | yes | — |
| 196 | C22 | — | 2 | 2 | no | — |
| 197 | C22 | — | 2 | 4 | no | — |
| 198 | C22 | — | 4 | 4 | no | — |
| 199 | C22 | C22×#41 | 1 | 1 | yes | characteristic |
| 200 | C22 | C2×H(5,𝔽2) | 1 | 2 | yes | — |
| 201 | C22 | C2×2−1+4 | 1 | 4 | yes | — |
| 202 | C4 | — | 2 | 4 | no | — |
| 203 | C4 | — | 2 | 4 | no | — |
| 204 | C4 | — | 2 | 8 | no | — |
| 205 | C4 | — | 4 | 1 | no | — |
| 206 | C4 | — | 4 | 1 | no | — |
| 207 | C4 | — | 4 | 4 | no | — |
| 208 | C4 | — | 4 | 4 | no | — |
| 209 | C4 | — | 4 | 4 | no | — |
| 210 | C4 | — | 4 | 4 | no | — |
| 211 | C4 | — | 4 | 4 | no | — |
| 212 | C2 | — | 4 | 2 | no | — |
| 213 | C2 | #3490 | 1 | 1 | yes | characteristic |
| 214 | C2 | #3509 | 1 | 2 | yes | — |
| 215 | C2 | #3552 | 1 | 4 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2789 | C2 | semidirect |
| 4. #2713 | C2 | semidirect |
| 6. #2588 | C2 | semidirect |
| 7. #2550 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2817 | 2 |
| #2789 | 2 |
| #2765 | 2 |
| #2713 | 2 |
| #2608 | 2 |
| #2588 | 2 |
| #2550 | 2 |
| #2475 | 2 |
| #2434 | 2 |
| #2332 | 2 |
| #2310 | 2 |