Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1534 conjugacy classes of subgroups; only the 534 normal subgroups are shown.
This group has 1534 conjugacy classes of subgroups — too many to draw. Only its 534 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #62359 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#3391 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #33958 | C2 | 1 | 8 | yes | maximal, complemented |
| 4 | #33925 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | #33771 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | #33660 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #33496 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #3497 | — | 2 | 1 | no | — |
| 9 | C2×#550 | — | 2 | 1 | no | — |
| 10 | #3391 | — | 2 | 4 | no | — |
| 11 | #3391 | C22 | 1 | 8 | yes | — |
| 12 | #3391 | C22 | 1 | 4 | yes | complemented |
| 13 | Q8×D8 | C22 | 1 | 4 | yes | complemented |
| 14 | #3358 | — | 2 | 4 | no | — |
| 15 | #3245 | — | 2 | 3 | no | — |
| 16 | #3169 | C22 | 1 | 6 | yes | — |
| 17 | C2×#493 | C22 | 1 | 3 | yes | — |
| 18 | C2×#459 | — | 2 | 3 | no | — |
| 19 | C2×C8×Q8 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×C4×D8 | C22 | 1 | 3 | yes | — |
| 21 | #2246 | C22 | 1 | 12 | yes | complemented |
| 22 | #2240 | C22 | 1 | 12 | yes | complemented |
| 23 | #2232 | C22 | 1 | 4 | yes | — |
| 24 | #2224 | C22 | 1 | 12 | yes | complemented |
| 25 | #2206 | C22 | 1 | 12 | yes | complemented |
| 26 | #2206 | C22 | 1 | 24 | yes | complemented |
| 27 | #2184 | C22 | 1 | 12 | yes | — |
| 28 | Q8×C16 | C22 | 1 | 8 | yes | complemented |
| 29 | C4×D16 | C22 | 1 | 24 | yes | complemented |
| 30 | #2162 | C22 | 1 | 6 | yes | — |
| 31 | C2×2−1+4 | — | 4 | 1 | no | — |
| 32 | C22×#41 | — | 4 | 1 | no | — |
| 33 | C22×D8 | — | 2 | 3 | no | — |
| 34 | C22×D8 | C23 | 1 | 3 | yes | — |
| 35 | #550 | — | 2 | 4 | no | — |
| 36 | #550 | — | 2 | 8 | no | — |
| 37 | #550 | — | 4 | 4 | no | — |
| 38 | D4×Q8 | — | 2 | 4 | no | — |
| 39 | #548 | — | 4 | 4 | no | — |
| 40 | #537 | — | 4 | 3 | no | — |
| 41 | #532 | — | 2 | 6 | no | — |
| 42 | C2×#127 | — | 2 | 3 | no | — |
| 43 | C2×#121 | — | 4 | 3 | no | — |
| 44 | C2×C4×Q8 | D4 | 1 | 1 | yes | characteristic |
| 45 | C2×C4×D4 | — | 2 | 3 | no | — |
| 46 | C2×D16 | C23 | 1 | 24 | yes | — |
| 47 | #500 | C23 | 1 | 6 | yes | — |
| 48 | #493 | — | 2 | 6 | no | — |
| 49 | #493 | C23 | 1 | 12 | yes | — |
| 50 | #493 | C23 | 1 | 6 | yes | — |
| 51 | #486 | — | 2 | 6 | no | — |
| 52 | #474 | — | 2 | 12 | no | — |
| 53 | #466 | — | 2 | 6 | no | — |
| 54 | #466 | C23 | 1 | 4 | yes | — |
| 55 | #459 | — | 2 | 12 | no | — |
| 56 | #459 | — | 2 | 24 | no | — |
| 57 | C8×Q8 | D4 | 1 | 4 | yes | — |
| 58 | C8×Q8 | C23 | 1 | 4 | yes | — |
| 59 | C4×D8 | — | 2 | 6 | no | — |
| 60 | C4×D8 | — | 2 | 6 | no | — |
| 61 | C4×D8 | C23 | 1 | 12 | yes | — |
| 62 | C4×D8 | C23 | 1 | 12 | yes | — |
| 63 | C2×#107 | C23 | 1 | 1 | yes | characteristic |
| 64 | C2×C4⋊C8 | D4 | 1 | 3 | yes | — |
| 65 | C2×#102 | — | 2 | 3 | no | — |
| 66 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 67 | #366 | C23 | 1 | 8 | yes | — |
| 68 | C4⋊C16 | C23 | 1 | 12 | yes | — |
| 69 | #357 | C23 | 1 | 8 | yes | — |
| 70 | #357 | C23 | 1 | 24 | yes | — |
| 71 | C22⋊C16 | C23 | 1 | 4 | yes | — |
| 72 | C4×C16 | C23 | 1 | 12 | yes | — |
| 73 | 2−1+4 | — | 8 | 8 | no | — |
| 74 | C2×#41 | — | 4 | 4 | no | — |
| 75 | C2×#41 | — | 4 | 8 | no | — |
| 76 | C2×#41 | — | 4 | 6 | no | — |
| 77 | C2×#41 | — | 8 | 8 | no | — |
| 78 | C22×Q8 | — | 4 | 4 | no | — |
| 79 | C22×Q8 | D8 | 1 | 1 | yes | characteristic |
| 80 | C22×D4 | — | 2 | 3 | no | — |
| 81 | C22×D4 | — | 4 | 3 | no | — |
| 82 | C22×D4 | — | 4 | 3 | no | — |
| 83 | C23×C4 | — | 4 | 3 | no | — |
| 84 | C2×D8 | — | 2 | 3 | no | — |
| 85 | C2×D8 | — | 2 | 12 | no | — |
| 86 | C2×D8 | — | 2 | 6 | no | — |
| 87 | C2×D8 | — | 4 | 6 | no | — |
| 88 | C2×D8 | — | 8 | 6 | no | — |
| 89 | C2×D8 | C24 | 1 | 6 | yes | — |
| 90 | C2×D8 | C24 | 1 | 4 | yes | — |
| 91 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 92 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 93 | C4⋊Q8 | — | 2 | 6 | no | — |
| 94 | #127 | — | 2 | 12 | no | — |
| 95 | #127 | — | 2 | 6 | no | — |
| 96 | #127 | — | 4 | 6 | no | — |
| 97 | #124 | — | 4 | 6 | no | — |
| 98 | C22⋊Q8 | — | 4 | 12 | no | — |
| 99 | #121 | — | 2 | 4 | no | — |
| 100 | #121 | — | 4 | 6 | no | — |
| 101 | #121 | — | 4 | 24 | no | — |
| 102 | #121 | — | 4 | 12 | no | — |
| 103 | C4×Q8 | D8 | 1 | 4 | yes | — |
| 104 | C4×Q8 | C2×D4 | 1 | 4 | yes | — |
| 105 | C4×D4 | — | 2 | 12 | no | — |
| 106 | C4×D4 | — | 2 | 12 | no | — |
| 107 | C4×D4 | — | 4 | 6 | no | — |
| 108 | C4×D4 | — | 4 | 6 | no | — |
| 109 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 110 | C2×C4⋊C4 | D8 | 1 | 3 | yes | — |
| 111 | C2×C22⋊C4 | — | 4 | 3 | no | — |
| 112 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 113 | D16 | — | 4 | 24 | no | — |
| 114 | C2×C16 | C24 | 1 | 8 | yes | — |
| 115 | #107 | — | 2 | 1 | no | — |
| 116 | #107 | — | 2 | 6 | no | — |
| 117 | #107 | C24 | 1 | 4 | yes | — |
| 118 | #107 | C24 | 1 | 2 | yes | — |
| 119 | C4⋊C8 | C2×D4 | 1 | 12 | yes | — |
| 120 | #102 | — | 2 | 12 | no | — |
| 121 | #102 | — | 2 | 4 | no | — |
| 122 | #102 | — | 2 | 12 | no | — |
| 123 | C4×C8 | C2×D4 | 1 | 6 | yes | — |
| 124 | C4×C8 | C24 | 1 | 6 | yes | — |
| 125 | C24 | — | 4 | 3 | no | — |
| 126 | #41 | — | 8 | 32 | no | — |
| 127 | #41 | — | 8 | 16 | no | — |
| 128 | #41 | — | 8 | 24 | no | — |
| 129 | C2×Q8 | — | 4 | 6 | no | — |
| 130 | C2×Q8 | — | 8 | 16 | no | — |
| 131 | C2×Q8 | D16 | 1 | 8 | yes | complemented |
| 132 | C2×Q8 | C2×D8 | 1 | 4 | yes | — |
| 133 | C2×D4 | — | 2 | 4 | no | — |
| 134 | C2×D4 | — | 2 | 6 | no | — |
| 135 | C2×D4 | — | 4 | 6 | no | — |
| 136 | C2×D4 | — | 4 | 6 | no | — |
| 137 | C2×D4 | — | 4 | 4 | no | — |
| 138 | C2×D4 | — | 4 | 3 | no | — |
| 139 | C2×D4 | — | 4 | 6 | no | — |
| 140 | C2×D4 | — | 4 | 12 | no | — |
| 141 | C2×D4 | — | 4 | 6 | no | — |
| 142 | C2×D4 | — | 4 | 12 | no | — |
| 143 | C2×D4 | — | 8 | 12 | no | — |
| 144 | C2×D4 | — | 16 | 6 | no | — |
| 145 | C22×C4 | — | 4 | 1 | no | — |
| 146 | C22×C4 | — | 4 | 12 | no | — |
| 147 | C22×C4 | — | 4 | 12 | no | — |
| 148 | C22×C4 | — | 8 | 12 | no | — |
| 149 | C22×C4 | C2×D8 | 1 | 3 | yes | — |
| 150 | C22×C4 | C2×D8 | 1 | 3 | yes | — |
| 151 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 152 | D8 | — | 2 | 8 | no | — |
| 153 | D8 | — | 2 | 12 | no | — |
| 154 | D8 | — | 4 | 6 | no | — |
| 155 | D8 | — | 8 | 6 | no | — |
| 156 | D8 | — | 8 | 12 | no | — |
| 157 | C2×C8 | — | 2 | 2 | no | — |
| 158 | C2×C8 | — | 4 | 3 | no | — |
| 159 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 160 | C2×C8 | C22×D4 | 1 | 6 | yes | — |
| 161 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 162 | C4⋊C4 | — | 2 | 2 | no | — |
| 163 | C4⋊C4 | — | 2 | 4 | no | — |
| 164 | C4⋊C4 | — | 4 | 1 | no | — |
| 165 | C4⋊C4 | — | 4 | 6 | no | — |
| 166 | C4⋊C4 | C2×D8 | 1 | 12 | yes | — |
| 167 | C22⋊C4 | — | 4 | 12 | no | — |
| 168 | C22⋊C4 | — | 4 | 12 | no | — |
| 169 | C22⋊C4 | — | 4 | 4 | no | — |
| 170 | C42 | C2×D8 | 1 | 6 | yes | — |
| 171 | C42 | C22×D4 | 1 | 6 | yes | — |
| 172 | C16 | — | 2 | 8 | no | — |
| 173 | C23 | — | 4 | 4 | no | — |
| 174 | C23 | — | 4 | 6 | no | — |
| 175 | C23 | — | 8 | 6 | no | — |
| 176 | C23 | — | 8 | 6 | no | — |
| 177 | C23 | — | 8 | 6 | no | — |
| 178 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 179 | Q8 | — | 8 | 12 | no | — |
| 180 | Q8 | C2×D16 | 1 | 16 | yes | — |
| 181 | D4 | — | 4 | 6 | no | — |
| 182 | D4 | — | 4 | 4 | no | — |
| 183 | D4 | — | 4 | 6 | no | — |
| 184 | D4 | — | 4 | 4 | no | — |
| 185 | D4 | — | 8 | 6 | no | — |
| 186 | D4 | — | 8 | 8 | no | — |
| 187 | D4 | — | 8 | 24 | no | — |
| 188 | D4 | — | 8 | 12 | no | — |
| 189 | D4 | — | 16 | 6 | no | — |
| 190 | D4 | — | 16 | 12 | no | — |
| 191 | C2×C4 | — | 2 | 1 | no | — |
| 192 | C2×C4 | — | 2 | 1 | no | — |
| 193 | C2×C4 | — | 4 | 3 | no | — |
| 194 | C2×C4 | — | 4 | 2 | no | — |
| 195 | C2×C4 | — | 4 | 4 | no | — |
| 196 | C2×C4 | — | 8 | 2 | no | — |
| 197 | C2×C4 | — | 8 | 24 | no | — |
| 198 | C2×C4 | — | 8 | 24 | no | — |
| 199 | C2×C4 | C2×D16 | 1 | 12 | yes | — |
| 200 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 201 | C2×C4 | C22×D8 | 1 | 6 | yes | — |
| 202 | C2×C4 | C22×D8 | 1 | 6 | yes | — |
| 203 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 204 | C8 | — | 2 | 6 | no | — |
| 205 | C8 | — | 2 | 1 | no | — |
| 206 | C8 | #585 | 1 | 2 | yes | — |
| 207 | C22 | — | 2 | 1 | no | — |
| 208 | C22 | — | 2 | 1 | no | — |
| 209 | C22 | — | 8 | 4 | no | — |
| 210 | C22 | — | 8 | 6 | no | — |
| 211 | C22 | — | 8 | 6 | no | — |
| 212 | C22 | — | 8 | 4 | no | — |
| 213 | C22 | — | 8 | 6 | no | — |
| 214 | C22 | — | 8 | 4 | no | — |
| 215 | C22 | — | 16 | 12 | no | — |
| 216 | C22 | C22×D16 | 1 | 2 | yes | — |
| 217 | C22 | C23×D8 | 1 | 1 | yes | characteristic |
| 218 | C4 | — | 2 | 6 | no | — |
| 219 | C4 | — | 2 | 1 | no | — |
| 220 | C4 | — | 8 | 2 | no | — |
| 221 | C4 | — | 8 | 4 | no | — |
| 222 | C4 | C22×D16 | 1 | 12 | yes | — |
| 223 | C4 | #3487 | 1 | 1 | yes | characteristic |
| 224 | C4 | #3585 | 1 | 1 | yes | characteristic |
| 225 | C2 | — | 2 | 2 | no | — |
| 226 | C2 | — | 16 | 4 | no | — |
| 227 | C2 | — | 16 | 6 | no | — |
| 228 | C2 | #61880 | 1 | 1 | yes | characteristic |
| 229 | C2 | C23×D16 | 1 | 1 | yes | characteristic |
| 230 | C2 | #62636 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #62359 | C2 | semidirect |
| 2. C2×#3391 | C2 | semidirect |
| 3. #33958 | C2 | semidirect |
| 4. #33925 | C2 | semidirect |
| 5. #33771 | C2 | semidirect |
| 6. #33660 | C2 | semidirect |
| 7. #33496 | C2 | semidirect |
| 12. #3391 | C22 | semidirect |
| 13. Q8×D8 | C22 | semidirect |
| 21. #2246 | C22 | semidirect |
| 22. #2240 | C22 | semidirect |
| 24. #2224 | C22 | semidirect |
| 25. #2206 | C22 | semidirect |
| 26. #2206 | C22 | semidirect |
| 28. Q8×C16 | C22 | semidirect |
| 29. C4×D16 | C22 | semidirect |
| 131. C2×Q8 | D16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62359 | 2 |
| C2×#3391 | 2 |
| #33958 | 2 |
| #33925 | 2 |
| #33771 | 2 |
| #33660 | 2 |
| #33496 | 2 |