Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
6014 conjugacy classes of subgroups.
The subgroup lattice has 236 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Q8×2−1+4 | C2 | 1 | 2 | yes | maximal |
| 2 | #62472 | C2 | 1 | 10 | yes | maximal, complemented |
| 3 | #62360 | C2 | 1 | 5 | yes | maximal, complemented |
| 4 | #62100 | C2 | 1 | 20 | yes | maximal, complemented |
| 5 | #61821 | C2 | 1 | 10 | yes | maximal, complemented |
| 6 | Q16×#41 | C2 | 1 | 10 | yes | maximal, complemented |
| 7 | C2×Q8×Q16 | C2 | 1 | 5 | yes | maximal, complemented |
| 8 | C8×2−1+4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #3503 | — | 2 | 20 | no | — |
| 10 | #3503 | C22 | 1 | 20 | yes | — |
| 11 | #3502 | — | 2 | 10 | no | — |
| 12 | #3502 | C22 | 1 | 10 | yes | — |
| 13 | Q8×#41 | C22 | 1 | 20 | yes | — |
| 14 | C2×Q82 | C22 | 1 | 10 | yes | — |
| 15 | C4×2−1+4 | — | 2 | 2 | no | — |
| 16 | C4×2−1+4 | C22 | 1 | 1 | yes | characteristic |
| 17 | #3395 | C22 | 1 | 10 | yes | — |
| 18 | Q8×Q16 | C22 | 1 | 20 | yes | — |
| 19 | Q8×Q16 | C22 | 1 | 20 | yes | — |
| 20 | Q8×Q16 | C22 | 1 | 40 | yes | — |
| 21 | Q8×Q16 | C22 | 1 | 10 | yes | — |
| 22 | #3363 | C22 | 1 | 20 | yes | — |
| 23 | #3363 | C22 | 1 | 40 | yes | — |
| 24 | #3338 | C22 | 1 | 30 | yes | — |
| 25 | #3338 | C22 | 1 | 60 | yes | — |
| 26 | #3299 | C22 | 1 | 60 | yes | — |
| 27 | D4×Q16 | C22 | 1 | 10 | yes | — |
| 28 | #3249 | C22 | 1 | 30 | yes | — |
| 29 | #3230 | C22 | 1 | 60 | yes | — |
| 30 | #3170 | C22 | 1 | 30 | yes | — |
| 31 | C2×#500 | C22 | 1 | 15 | yes | — |
| 32 | #3104 | C22 | 1 | 60 | yes | — |
| 33 | C2×#477 | C22 | 1 | 30 | yes | — |
| 34 | #2988 | C22 | 1 | 10 | yes | — |
| 35 | #2982 | C22 | 1 | 5 | yes | — |
| 36 | C8×#41 | C22 | 1 | 10 | yes | — |
| 37 | C2×C8×Q8 | C22 | 1 | 5 | yes | — |
| 38 | C2×C4×Q16 | C22 | 1 | 15 | yes | — |
| 39 | C2×2−1+4 | D4 | 1 | 1 | yes | characteristic |
| 40 | C22×Q16 | C23 | 1 | 5 | yes | — |
| 41 | Q82 | — | 2 | 40 | no | — |
| 42 | Q82 | C23 | 1 | 20 | yes | — |
| 43 | Q82 | C23 | 1 | 40 | yes | — |
| 44 | #557 | — | 2 | 20 | no | — |
| 45 | #557 | — | 2 | 20 | no | — |
| 46 | #557 | — | 2 | 40 | no | — |
| 47 | #557 | C23 | 1 | 20 | yes | — |
| 48 | #554 | — | 2 | 60 | no | — |
| 49 | #554 | C23 | 1 | 60 | yes | — |
| 50 | D4×Q8 | — | 2 | 60 | no | — |
| 51 | D4×Q8 | C23 | 1 | 20 | yes | — |
| 52 | #544 | — | 2 | 60 | no | — |
| 53 | #544 | — | 2 | 30 | no | — |
| 54 | #533 | — | 2 | 60 | no | — |
| 55 | #533 | C23 | 1 | 60 | yes | — |
| 56 | C2×C4⋊Q8 | — | 2 | 30 | no | — |
| 57 | C2×C4⋊Q8 | C23 | 1 | 30 | yes | — |
| 58 | #520 | — | 2 | 20 | no | — |
| 59 | #520 | D4 | 1 | 10 | yes | — |
| 60 | #519 | — | 2 | 10 | no | — |
| 61 | #519 | D4 | 1 | 5 | yes | — |
| 62 | C4×#41 | — | 2 | 20 | no | — |
| 63 | C4×#41 | C23 | 1 | 10 | yes | — |
| 64 | C2×C4×Q8 | — | 2 | 10 | no | — |
| 65 | C2×C4×Q8 | C23 | 1 | 30 | yes | — |
| 66 | C2×C4×Q8 | C23 | 1 | 5 | yes | — |
| 67 | #500 | C23 | 1 | 30 | yes | — |
| 68 | #500 | C23 | 1 | 60 | yes | — |
| 69 | #494 | C23 | 1 | 10 | yes | — |
| 70 | #487 | C23 | 1 | 60 | yes | — |
| 71 | #484 | C23 | 1 | 120 | yes | — |
| 72 | #477 | C23 | 1 | 60 | yes | — |
| 73 | #477 | C23 | 1 | 120 | yes | — |
| 74 | #477 | C23 | 1 | 60 | yes | — |
| 75 | #467 | C23 | 1 | 20 | yes | — |
| 76 | #467 | C23 | 1 | 60 | yes | — |
| 77 | #462 | C23 | 1 | 20 | yes | — |
| 78 | #462 | C23 | 1 | 60 | yes | — |
| 79 | #451 | C23 | 1 | 40 | yes | — |
| 80 | C8×Q8 | C23 | 1 | 20 | yes | — |
| 81 | C8×Q8 | C23 | 1 | 10 | yes | — |
| 82 | C8×Q8 | C23 | 1 | 10 | yes | — |
| 83 | C4×Q16 | C23 | 1 | 60 | yes | — |
| 84 | C4×Q16 | C23 | 1 | 30 | yes | — |
| 85 | C4×Q16 | C23 | 1 | 60 | yes | — |
| 86 | C4×Q16 | C23 | 1 | 10 | yes | — |
| 87 | C8×D4 | C23 | 1 | 10 | yes | — |
| 88 | C8×D4 | C23 | 1 | 30 | yes | — |
| 89 | #431 | C23 | 1 | 30 | yes | — |
| 90 | C2×#107 | C23 | 1 | 30 | yes | — |
| 91 | C2×#107 | C23 | 1 | 15 | yes | — |
| 92 | C2×C4⋊C8 | C23 | 1 | 15 | yes | — |
| 93 | C2×#103 | C23 | 1 | 30 | yes | — |
| 94 | C2×C4×C8 | C23 | 1 | 15 | yes | — |
| 95 | 2−1+4 | Q16 | 1 | 16 | yes | direct product factor |
| 96 | C2×#41 | C2×D4 | 1 | 10 | yes | — |
| 97 | C22×Q8 | — | 2 | 30 | no | — |
| 98 | C22×Q8 | C2×D4 | 1 | 5 | yes | — |
| 99 | C22×Q8 | C24 | 1 | 10 | yes | — |
| 100 | C2×Q16 | — | 2 | 20 | no | — |
| 101 | C2×Q16 | C24 | 1 | 10 | yes | — |
| 102 | C2×Q16 | C24 | 1 | 5 | yes | — |
| 103 | C2×Q16 | C24 | 1 | 20 | yes | — |
| 104 | C2×Q16 | C24 | 1 | 1 | yes | characteristic |
| 105 | C22×C8 | C24 | 1 | 5 | yes | — |
| 106 | C22×C8 | C24 | 1 | 15 | yes | — |
| 107 | C4⋊Q8 | — | 2 | 60 | no | — |
| 108 | C4⋊Q8 | — | 2 | 120 | no | — |
| 109 | C4⋊Q8 | — | 2 | 20 | no | — |
| 110 | C4⋊Q8 | — | 2 | 60 | no | — |
| 111 | C4⋊Q8 | C24 | 1 | 20 | yes | — |
| 112 | C4⋊Q8 | C24 | 1 | 60 | yes | — |
| 113 | #125 | — | 2 | 60 | no | — |
| 114 | #125 | — | 2 | 60 | no | — |
| 115 | #125 | — | 2 | 60 | no | — |
| 116 | C22⋊Q8 | — | 2 | 60 | no | — |
| 117 | C22⋊Q8 | — | 2 | 40 | no | — |
| 118 | C22⋊Q8 | — | 2 | 120 | no | — |
| 119 | C22⋊Q8 | C24 | 1 | 40 | yes | — |
| 120 | C4×Q8 | — | 2 | 20 | no | — |
| 121 | C4×Q8 | — | 2 | 40 | no | — |
| 122 | C4×Q8 | — | 2 | 20 | no | — |
| 123 | C4×Q8 | — | 2 | 60 | no | — |
| 124 | C4×Q8 | — | 2 | 60 | no | — |
| 125 | C4×Q8 | C2×D4 | 1 | 20 | yes | — |
| 126 | C4×Q8 | C2×D4 | 1 | 10 | yes | — |
| 127 | C4×Q8 | C24 | 1 | 20 | yes | — |
| 128 | C4×Q8 | C24 | 1 | 60 | yes | — |
| 129 | C4×Q8 | C24 | 1 | 10 | yes | — |
| 130 | C4×D4 | — | 2 | 20 | no | — |
| 131 | C4×D4 | — | 2 | 60 | no | — |
| 132 | C4×D4 | C2×D4 | 1 | 30 | yes | — |
| 133 | C4×D4 | C24 | 1 | 10 | yes | — |
| 134 | #117 | — | 2 | 60 | no | — |
| 135 | #117 | C2×D4 | 1 | 30 | yes | — |
| 136 | C2×C4⋊C4 | — | 2 | 30 | no | — |
| 137 | C2×C4⋊C4 | — | 2 | 30 | no | — |
| 138 | C2×C4⋊C4 | C2×D4 | 1 | 15 | yes | — |
| 139 | C2×C4⋊C4 | C24 | 1 | 30 | yes | — |
| 140 | C2×C42 | — | 2 | 30 | no | — |
| 141 | C2×C42 | C24 | 1 | 15 | yes | — |
| 142 | #107 | C24 | 1 | 10 | yes | — |
| 143 | #107 | C24 | 1 | 60 | yes | — |
| 144 | #107 | C24 | 1 | 30 | yes | — |
| 145 | C4⋊C8 | C24 | 1 | 30 | yes | — |
| 146 | C4⋊C8 | C24 | 1 | 10 | yes | — |
| 147 | #103 | C24 | 1 | 20 | yes | — |
| 148 | #103 | C24 | 1 | 60 | yes | — |
| 149 | #103 | C24 | 1 | 40 | yes | — |
| 150 | C22⋊C8 | C24 | 1 | 20 | yes | — |
| 151 | C4×C8 | C24 | 1 | 10 | yes | — |
| 152 | C4×C8 | C24 | 1 | 30 | yes | — |
| 153 | #41 | C2×Q16 | 1 | 80 | yes | complemented |
| 154 | C2×Q8 | — | 2 | 10 | no | — |
| 155 | C2×Q8 | — | 2 | 60 | no | — |
| 156 | C2×Q8 | — | 2 | 20 | no | — |
| 157 | C2×Q8 | — | 4 | 10 | no | — |
| 158 | C2×Q8 | — | 16 | 30 | no | — |
| 159 | C2×Q8 | C2×Q16 | 1 | 40 | yes | complemented |
| 160 | C2×Q8 | C22×D4 | 1 | 10 | yes | — |
| 161 | C2×Q8 | C25 | 1 | 10 | yes | — |
| 162 | C2×Q8 | C25 | 1 | 2 | yes | — |
| 163 | C2×D4 | C22×D4 | 1 | 10 | yes | — |
| 164 | C22×C4 | — | 2 | 10 | no | — |
| 165 | C22×C4 | — | 2 | 30 | no | — |
| 166 | C22×C4 | C22×D4 | 1 | 15 | yes | — |
| 167 | C22×C4 | C22×D4 | 1 | 15 | yes | — |
| 168 | C22×C4 | C25 | 1 | 5 | yes | — |
| 169 | Q16 | — | 2 | 20 | no | — |
| 170 | Q16 | — | 2 | 10 | no | — |
| 171 | Q16 | — | 4 | 20 | no | — |
| 172 | Q16 | 2−1+4 | 1 | 4 | yes | direct product factor |
| 173 | C2×C8 | — | 2 | 10 | no | — |
| 174 | C2×C8 | — | 4 | 15 | no | — |
| 175 | C2×C8 | C25 | 1 | 5 | yes | — |
| 176 | C2×C8 | C25 | 1 | 10 | yes | — |
| 177 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 178 | C4⋊C4 | — | 2 | 20 | no | — |
| 179 | C4⋊C4 | — | 2 | 60 | no | — |
| 180 | C4⋊C4 | — | 2 | 60 | no | — |
| 181 | C4⋊C4 | — | 2 | 30 | no | — |
| 182 | C4⋊C4 | — | 2 | 20 | no | — |
| 183 | C4⋊C4 | — | 2 | 20 | no | — |
| 184 | C4⋊C4 | — | 2 | 40 | no | — |
| 185 | C4⋊C4 | C22×D4 | 1 | 30 | yes | — |
| 186 | C4⋊C4 | C22×D4 | 1 | 10 | yes | — |
| 187 | C4⋊C4 | C25 | 1 | 20 | yes | — |
| 188 | C22⋊C4 | — | 2 | 40 | no | — |
| 189 | C22⋊C4 | C22×D4 | 1 | 20 | yes | — |
| 190 | C42 | — | 2 | 20 | no | — |
| 191 | C42 | — | 2 | 60 | no | — |
| 192 | C42 | C22×D4 | 1 | 30 | yes | — |
| 193 | C42 | C25 | 1 | 10 | yes | — |
| 194 | C23 | C23×D4 | 1 | 5 | yes | — |
| 195 | Q8 | — | 2 | 2 | no | — |
| 196 | Q8 | — | 2 | 10 | no | — |
| 197 | Q8 | — | 2 | 10 | no | — |
| 198 | Q8 | — | 4 | 20 | no | — |
| 199 | Q8 | — | 8 | 60 | no | — |
| 200 | Q8 | C22×Q16 | 1 | 40 | yes | — |
| 201 | Q8 | C2×2−1+4 | 1 | 4 | yes | — |
| 202 | D4 | C22×Q16 | 1 | 40 | yes | — |
| 203 | C2×C4 | — | 2 | 2 | no | — |
| 204 | C2×C4 | — | 2 | 5 | no | — |
| 205 | C2×C4 | — | 2 | 5 | no | — |
| 206 | C2×C4 | — | 2 | 10 | no | — |
| 207 | C2×C4 | — | 2 | 20 | no | — |
| 208 | C2×C4 | — | 4 | 15 | no | — |
| 209 | C2×C4 | — | 4 | 20 | no | — |
| 210 | C2×C4 | — | 8 | 30 | no | — |
| 211 | C2×C4 | C22×Q16 | 1 | 60 | yes | — |
| 212 | C2×C4 | C23×D4 | 1 | 5 | yes | — |
| 213 | C2×C4 | C23×D4 | 1 | 10 | yes | — |
| 214 | C2×C4 | C23×D4 | 1 | 10 | yes | — |
| 215 | C2×C4 | C26 | 1 | 1 | yes | characteristic |
| 216 | C8 | — | 2 | 5 | no | — |
| 217 | C8 | — | 2 | 10 | no | — |
| 218 | C8 | C2×2−1+4 | 1 | 2 | yes | — |
| 219 | C22 | — | 2 | 5 | no | — |
| 220 | C22 | — | 2 | 5 | no | — |
| 221 | C22 | C23×Q16 | 1 | 10 | yes | — |
| 222 | C22 | C24×D4 | 1 | 1 | yes | characteristic |
| 223 | C4 | — | 2 | 5 | no | — |
| 224 | C4 | — | 2 | 4 | no | — |
| 225 | C4 | — | 2 | 10 | no | — |
| 226 | C4 | — | 4 | 10 | no | — |
| 227 | C4 | — | 4 | 20 | no | — |
| 228 | C4 | C23×Q16 | 1 | 20 | yes | — |
| 229 | C4 | C22×2−1+4 | 1 | 1 | yes | characteristic |
| 230 | C4 | 2−1+6 | 1 | 1 | yes | characteristic |
| 231 | C2 | — | 2 | 10 | no | — |
| 232 | C2 | D4×2−1+4 | 1 | 1 | yes | characteristic |
| 233 | C2 | C24×Q16 | 1 | 1 | yes | characteristic |
| 234 | C2 | #63091 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C24×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×2−1+4 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×2−1+4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #62472 | C2 | semidirect |
| 3. #62360 | C2 | semidirect |
| 4. #62100 | C2 | semidirect |
| 5. #61821 | C2 | semidirect |
| 6. Q16×#41 | C2 | semidirect |
| 7. C2×Q8×Q16 | C2 | semidirect |
| 95. 2−1+4 | Q16 | direct (N × H) |
| 153. #41 | C2×Q16 | semidirect |
| 159. C2×Q8 | C2×Q16 | semidirect |
| 172. Q16 | 2−1+4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×2−1+4 | 2 |
| #62472 | 2 |
| #62360 | 2 |
| #62100 | 2 |
| #61821 | 2 |
| Q16×#41 | 2 |
| C2×Q8×Q16 | 2 |
| C8×2−1+4 | 2 |