Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
315 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 315 conjugacy classes of subgroups — too many to draw. Only its 54 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 | D82 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #12509 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #12445 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #12426 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #7096 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #7068 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #7068 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | D4×D8 | — | 2 | 1 | no | — |
| 9 | D4×D8 | — | 2 | 1 | no | — |
| 10 | #2240 | — | 2 | 2 | no | — |
| 11 | #2224 | — | 2 | 1 | no | — |
| 12 | #2208 | — | 2 | 1 | no | — |
| 13 | #1713 | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | #1712 | C22 | 1 | 1 | yes | characteristic |
| 15 | #1667 | — | 2 | 1 | no | — |
| 16 | #1667 | — | 2 | 1 | no | — |
| 17 | #1619 | — | 2 | 2 | no | — |
| 18 | C8×D8 | C22 | 1 | 1 | yes | characteristic |
| 19 | C8×D8 | C22 | 1 | 1 | yes | characteristic |
| 20 | #1370 | C22 | 1 | 1 | yes | characteristic, complemented |
| 21 | #1370 | C22 | 1 | 1 | yes | characteristic, complemented |
| 22 | #1364 | — | 2 | 1 | no | — |
| 23 | #1362 | — | 2 | 1 | no | — |
| 24 | C8×C16 | C22 | 1 | 1 | yes | characteristic, complemented |
| 25 | C22×D8 | — | 4 | 1 | no | — |
| 26 | C22×D8 | — | 4 | 1 | no | — |
| 27 | D42 | — | 2 | 2 | no | — |
| 28 | C2×SD32 | — | 4 | 1 | no | — |
| 29 | #500 | — | 2 | 1 | no | — |
| 30 | #500 | — | 2 | 1 | no | — |
| 31 | #494 | — | 2 | 1 | no | — |
| 32 | #493 | — | 2 | 1 | no | — |
| 33 | #493 | — | 2 | 1 | no | — |
| 34 | #493 | D4 | 1 | 2 | yes | — |
| 35 | #466 | — | 4 | 1 | no | — |
| 36 | #466 | — | 4 | 1 | no | — |
| 37 | #459 | — | 2 | 2 | no | — |
| 38 | #459 | — | 2 | 2 | no | — |
| 39 | #447 | — | 4 | 2 | no | — |
| 40 | #447 | — | 4 | 2 | no | — |
| 41 | C4×D8 | D4 | 1 | 1 | yes | characteristic |
| 42 | C4×D8 | D4 | 1 | 1 | yes | characteristic |
| 43 | C8×D4 | — | 2 | 1 | no | — |
| 44 | C8×D4 | — | 2 | 1 | no | — |
| 45 | #366 | — | 4 | 1 | no | — |
| 46 | C4⋊C16 | — | 2 | 1 | no | — |
| 47 | C4⋊C16 | — | 2 | 1 | no | — |
| 48 | #358 | — | 4 | 1 | no | — |
| 49 | #357 | — | 4 | 4 | no | — |
| 50 | #357 | — | 4 | 1 | no | — |
| 51 | C4×C16 | — | 2 | 1 | no | — |
| 52 | #335 | D4 | 1 | 1 | yes | characteristic |
| 53 | #335 | D4 | 1 | 1 | yes | characteristic |
| 54 | #331 | — | 2 | 2 | no | — |
| 55 | D4⋊C8 | — | 2 | 1 | no | — |
| 56 | D4⋊C8 | — | 2 | 1 | no | — |
| 57 | C82 | C23 | 1 | 1 | yes | characteristic |
| 58 | C22×D4 | — | 4 | 2 | no | — |
| 59 | C22×D4 | — | 4 | 2 | no | — |
| 60 | C2×Q16 | — | 4 | 1 | no | — |
| 61 | C2×Q16 | — | 4 | 1 | no | — |
| 62 | C2×D8 | — | 4 | 2 | no | — |
| 63 | C2×D8 | — | 4 | 2 | no | — |
| 64 | C2×D8 | — | 4 | 2 | no | — |
| 65 | C2×D8 | — | 4 | 1 | no | — |
| 66 | C2×D8 | — | 4 | 1 | no | — |
| 67 | C2×D8 | — | 4 | 1 | no | — |
| 68 | C2×D8 | — | 4 | 1 | no | — |
| 69 | C2×D8 | — | 8 | 2 | no | — |
| 70 | C2×D8 | — | 8 | 2 | no | — |
| 71 | C2×D8 | — | 8 | 2 | no | — |
| 72 | C2×D8 | — | 8 | 2 | no | — |
| 73 | C2×D8 | D8 | 1 | 1 | yes | characteristic |
| 74 | C2×D8 | D8 | 1 | 1 | yes | characteristic |
| 75 | C22×C8 | — | 4 | 1 | no | — |
| 76 | C22×C8 | — | 4 | 1 | no | — |
| 77 | C4⋊Q8 | — | 4 | 1 | no | — |
| 78 | #127 | — | 2 | 2 | no | — |
| 79 | #121 | — | 4 | 2 | no | — |
| 80 | #121 | — | 4 | 2 | no | — |
| 81 | #120 | — | 8 | 2 | no | — |
| 82 | C4×D4 | — | 2 | 1 | no | — |
| 83 | C4×D4 | — | 2 | 1 | no | — |
| 84 | SD32 | — | 8 | 2 | no | — |
| 85 | C2×C16 | — | 2 | 2 | no | — |
| 86 | C2×C16 | — | 4 | 1 | no | — |
| 87 | C2×C16 | — | 4 | 1 | no | — |
| 88 | #107 | — | 4 | 2 | no | — |
| 89 | #107 | — | 4 | 1 | no | — |
| 90 | #107 | — | 4 | 1 | no | — |
| 91 | #107 | D8 | 1 | 1 | yes | characteristic |
| 92 | #107 | D8 | 1 | 1 | yes | characteristic |
| 93 | C4⋊C8 | — | 2 | 1 | no | — |
| 94 | C4⋊C8 | — | 2 | 1 | no | — |
| 95 | #102 | — | 2 | 1 | no | — |
| 96 | #102 | — | 2 | 1 | no | — |
| 97 | #102 | — | 4 | 2 | no | — |
| 98 | #102 | — | 4 | 2 | no | — |
| 99 | C22⋊C8 | — | 4 | 1 | no | — |
| 100 | C22⋊C8 | — | 4 | 1 | no | — |
| 101 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 102 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 103 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 104 | C24 | — | 8 | 2 | no | — |
| 105 | C2×Q8 | — | 8 | 1 | no | — |
| 106 | C2×D4 | — | 2 | 1 | no | — |
| 107 | C2×D4 | — | 2 | 1 | no | — |
| 108 | C2×D4 | — | 4 | 2 | no | — |
| 109 | C2×D4 | — | 4 | 2 | no | — |
| 110 | C2×D4 | — | 4 | 2 | no | — |
| 111 | C2×D4 | — | 8 | 2 | no | — |
| 112 | C2×D4 | — | 8 | 2 | no | — |
| 113 | C2×D4 | — | 8 | 4 | no | — |
| 114 | C2×D4 | — | 8 | 4 | no | — |
| 115 | C2×D4 | — | 16 | 2 | no | — |
| 116 | C2×D4 | — | 16 | 2 | no | — |
| 117 | C22×C4 | — | 4 | 1 | no | — |
| 118 | C22×C4 | — | 4 | 1 | no | — |
| 119 | Q16 | — | 8 | 1 | no | — |
| 120 | Q16 | — | 8 | 1 | no | — |
| 121 | Q16 | — | 16 | 1 | no | — |
| 122 | D8 | — | 2 | 1 | no | — |
| 123 | D8 | — | 2 | 1 | no | — |
| 124 | D8 | — | 8 | 4 | no | — |
| 125 | D8 | — | 8 | 4 | no | — |
| 126 | D8 | — | 8 | 4 | no | — |
| 127 | D8 | — | 8 | 1 | no | — |
| 128 | D8 | — | 8 | 1 | no | — |
| 129 | D8 | — | 8 | 1 | no | — |
| 130 | D8 | — | 8 | 1 | no | — |
| 131 | D8 | — | 16 | 1 | no | — |
| 132 | D8 | — | 16 | 1 | no | — |
| 133 | D8 | D16 | 1 | 2 | yes | — |
| 134 | D8 | SD32 | 1 | 2 | yes | complemented |
| 135 | C2×C8 | — | 2 | 1 | no | — |
| 136 | C2×C8 | — | 4 | 1 | no | — |
| 137 | C2×C8 | — | 4 | 1 | no | — |
| 138 | C2×C8 | — | 8 | 1 | no | — |
| 139 | C2×C8 | — | 8 | 1 | no | — |
| 140 | C2×C8 | — | 8 | 1 | no | — |
| 141 | C2×C8 | — | 8 | 1 | no | — |
| 142 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 143 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 144 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 145 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 146 | C4⋊C4 | — | 2 | 1 | no | — |
| 147 | C4⋊C4 | — | 2 | 1 | no | — |
| 148 | C4⋊C4 | — | 8 | 1 | no | — |
| 149 | C4⋊C4 | — | 8 | 1 | no | — |
| 150 | C22⋊C4 | — | 4 | 1 | no | — |
| 151 | C22⋊C4 | — | 4 | 1 | no | — |
| 152 | C22⋊C4 | — | 8 | 2 | no | — |
| 153 | C42 | #120 | 1 | 1 | yes | characteristic |
| 154 | C16 | — | 4 | 2 | no | — |
| 155 | C16 | — | 8 | 1 | no | — |
| 156 | C16 | — | 8 | 1 | no | — |
| 157 | C23 | — | 4 | 1 | no | — |
| 158 | C23 | — | 4 | 1 | no | — |
| 159 | C23 | — | 8 | 2 | no | — |
| 160 | C23 | — | 16 | 4 | no | — |
| 161 | C23 | — | 16 | 4 | no | — |
| 162 | C23 | — | 32 | 2 | no | — |
| 163 | Q8 | — | 16 | 1 | no | — |
| 164 | Q8 | — | 16 | 1 | no | — |
| 165 | D4 | — | 2 | 2 | no | — |
| 166 | D4 | — | 2 | 2 | no | — |
| 167 | D4 | — | 4 | 1 | no | — |
| 168 | D4 | — | 4 | 1 | no | — |
| 169 | D4 | — | 8 | 2 | no | — |
| 170 | D4 | — | 8 | 2 | no | — |
| 171 | D4 | — | 8 | 2 | no | — |
| 172 | D4 | — | 8 | 2 | no | — |
| 173 | D4 | — | 16 | 2 | no | — |
| 174 | D4 | — | 16 | 4 | no | — |
| 175 | D4 | — | 16 | 4 | no | — |
| 176 | C2×C4 | — | 4 | 1 | no | — |
| 177 | C2×C4 | — | 4 | 1 | no | — |
| 178 | C2×C4 | — | 8 | 1 | no | — |
| 179 | C2×C4 | — | 8 | 1 | no | — |
| 180 | C2×C4 | — | 8 | 1 | no | — |
| 181 | C2×C4 | — | 8 | 1 | no | — |
| 182 | C2×C4 | — | 16 | 1 | no | — |
| 183 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 184 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 185 | C2×C4 | #453 | 1 | 1 | yes | characteristic |
| 186 | C8 | — | 2 | 2 | no | — |
| 187 | C8 | — | 2 | 1 | no | — |
| 188 | C8 | — | 2 | 1 | no | — |
| 189 | C8 | — | 8 | 1 | no | — |
| 190 | C8 | — | 8 | 1 | no | — |
| 191 | C8 | C2×D16 | 1 | 1 | yes | characteristic |
| 192 | C8 | C2×SD32 | 1 | 1 | yes | characteristic |
| 193 | C8 | #509 | 1 | 1 | yes | characteristic |
| 194 | C8 | #509 | 1 | 1 | yes | characteristic |
| 195 | C22 | — | 4 | 2 | no | — |
| 196 | C22 | — | 4 | 2 | no | — |
| 197 | C22 | — | 8 | 1 | no | — |
| 198 | C22 | — | 8 | 1 | no | — |
| 199 | C22 | — | 8 | 1 | no | — |
| 200 | C22 | — | 8 | 1 | no | — |
| 201 | C22 | — | 16 | 2 | no | — |
| 202 | C22 | — | 16 | 2 | no | — |
| 203 | C22 | — | 16 | 2 | no | — |
| 204 | C22 | — | 32 | 8 | no | — |
| 205 | C22 | #1619 | 1 | 1 | yes | characteristic |
| 206 | C4 | — | 2 | 1 | no | — |
| 207 | C4 | — | 8 | 1 | no | — |
| 208 | C4 | — | 8 | 1 | no | — |
| 209 | C4 | — | 32 | 1 | no | — |
| 210 | C4 | #2184 | 1 | 1 | yes | characteristic |
| 211 | C4 | #2187 | 1 | 1 | yes | characteristic |
| 212 | C4 | #2192 | 1 | 1 | yes | characteristic |
| 213 | C4 | #2194 | 1 | 1 | yes | characteristic |
| 214 | C2 | — | 8 | 2 | no | — |
| 215 | C2 | — | 8 | 2 | no | — |
| 216 | C2 | — | 32 | 2 | no | — |
| 217 | C2 | #12312 | 1 | 1 | yes | characteristic |
| 218 | C2 | #12316 | 1 | 1 | yes | characteristic |
| 219 | C2 | #12320 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator, Frattini, cosocle | C82 | C23 |
| prime radical | D82 | C2 |
| LCS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12509 | C2 | semidirect |
| 3. #12445 | C2 | semidirect |
| 4. #12426 | C2 | semidirect |
| 5. #7096 | C2 | semidirect |
| 6. #7068 | C2 | semidirect |
| 7. #7068 | C2 | semidirect |
| 13. #1713 | C22 | semidirect |
| 20. #1370 | C22 | semidirect |
| 21. #1370 | C22 | semidirect |
| 24. C8×C16 | C22 | semidirect |
| 134. D8 | SD32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D82 | 2 |
| #12509 | 2 |
| #12445 | 2 |
| #12426 | 2 |
| #7096 | 2 |
| #7068 | 2 |