Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
634 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 634 conjugacy classes of subgroups — too many to draw. Only its 106 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 | C22×#508 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#1420 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C2×#1419 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #13553 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #12837 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×#135 | — | 2 | 1 | no | — |
| 7 | C2×#508 | — | 2 | 4 | no | — |
| 8 | C2×#508 | — | 2 | 4 | no | — |
| 9 | C2×#508 | — | 2 | 2 | no | — |
| 10 | C2×#508 | C4 | 1 | 4 | yes | — |
| 11 | C22×Q32 | C22 | 1 | 1 | yes | characteristic |
| 12 | C22×SD32 | — | 2 | 1 | no | — |
| 13 | C22×D16 | C22 | 1 | 1 | yes | characteristic |
| 14 | C23×C16 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×M64 | C22 | 1 | 2 | yes | — |
| 16 | C2×#368 | C22 | 1 | 2 | yes | — |
| 17 | #1810 | — | 2 | 1 | no | — |
| 18 | #1420 | — | 2 | 4 | no | — |
| 19 | #1419 | — | 2 | 4 | no | — |
| 20 | C22⋊C32 | — | 2 | 2 | no | — |
| 21 | #1381 | — | 2 | 1 | no | — |
| 22 | C22×#41 | — | 4 | 1 | no | — |
| 23 | C2×#135 | — | 2 | 8 | no | — |
| 24 | C2×#135 | — | 2 | 4 | no | — |
| 25 | C2×#135 | — | 4 | 4 | no | — |
| 26 | C2×#135 | — | 4 | 2 | no | — |
| 27 | C22×Q16 | — | 2 | 1 | no | — |
| 28 | C22×SD16 | — | 4 | 1 | no | — |
| 29 | C22×D8 | — | 2 | 1 | no | — |
| 30 | C23×C8 | D4 | 1 | 1 | yes | characteristic |
| 31 | #508 | — | 2 | 16 | no | — |
| 32 | #508 | — | 2 | 8 | no | — |
| 33 | #508 | — | 2 | 8 | no | — |
| 34 | C2×Q32 | — | 2 | 2 | no | — |
| 35 | C2×Q32 | — | 2 | 1 | no | — |
| 36 | C2×Q32 | C2×C4 | 1 | 2 | yes | — |
| 37 | C2×Q32 | D4 | 1 | 4 | yes | — |
| 38 | C2×SD32 | — | 2 | 8 | no | — |
| 39 | C2×SD32 | — | 2 | 4 | no | — |
| 40 | C2×D16 | — | 2 | 2 | no | — |
| 41 | C2×D16 | — | 2 | 1 | no | — |
| 42 | C2×D16 | C2×C4 | 1 | 2 | yes | — |
| 43 | C2×D16 | D4 | 1 | 4 | yes | — |
| 44 | C22×C16 | — | 2 | 2 | no | — |
| 45 | C22×C16 | — | 2 | 2 | no | — |
| 46 | C22×C16 | C2×C4 | 1 | 2 | yes | — |
| 47 | C22×C16 | D4 | 1 | 1 | yes | characteristic |
| 48 | C22×C16 | D4 | 1 | 2 | yes | — |
| 49 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 50 | C2×#108 | — | 2 | 2 | no | — |
| 51 | #407 | — | 4 | 1 | no | — |
| 52 | M64 | — | 2 | 2 | no | — |
| 53 | M64 | — | 2 | 2 | no | — |
| 54 | C2×C32 | — | 2 | 2 | no | — |
| 55 | #368 | — | 2 | 2 | no | — |
| 56 | #368 | — | 2 | 2 | no | — |
| 57 | #341 | — | 4 | 1 | no | — |
| 58 | C2×#41 | — | 4 | 8 | no | — |
| 59 | C2×#41 | — | 4 | 4 | no | — |
| 60 | C2×#41 | — | 8 | 4 | no | — |
| 61 | C2×#41 | — | 8 | 2 | no | — |
| 62 | C22×Q8 | — | 4 | 1 | no | — |
| 63 | C22×D4 | — | 4 | 1 | no | — |
| 64 | C22×D4 | — | 8 | 1 | no | — |
| 65 | C23×C4 | — | 8 | 1 | no | — |
| 66 | C23×C4 | D8 | 1 | 1 | yes | characteristic |
| 67 | #135 | — | 2 | 16 | no | — |
| 68 | #135 | — | 4 | 8 | no | — |
| 69 | #135 | — | 4 | 16 | no | — |
| 70 | C2×Q16 | — | 2 | 2 | no | — |
| 71 | C2×Q16 | — | 2 | 4 | no | — |
| 72 | C2×Q16 | — | 4 | 1 | no | — |
| 73 | C2×Q16 | — | 4 | 2 | no | — |
| 74 | C2×SD16 | — | 4 | 8 | no | — |
| 75 | C2×SD16 | — | 4 | 4 | no | — |
| 76 | C2×D8 | — | 2 | 2 | no | — |
| 77 | C2×D8 | — | 2 | 4 | no | — |
| 78 | C2×D8 | — | 4 | 1 | no | — |
| 79 | C2×D8 | — | 4 | 2 | no | — |
| 80 | C2×M16 | — | 4 | 2 | no | — |
| 81 | C22×C8 | — | 2 | 2 | no | — |
| 82 | C22×C8 | — | 2 | 2 | no | — |
| 83 | C22×C8 | C22⋊C4 | 1 | 2 | yes | — |
| 84 | C22×C8 | D8 | 1 | 1 | yes | characteristic |
| 85 | C22×C8 | SD16 | 1 | 2 | yes | — |
| 86 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | Q32 | — | 2 | 2 | no | — |
| 88 | Q32 | — | 2 | 4 | no | — |
| 89 | Q32 | C22⋊C4 | 1 | 4 | yes | — |
| 90 | SD32 | — | 2 | 16 | no | — |
| 91 | D16 | — | 2 | 2 | no | — |
| 92 | D16 | — | 2 | 4 | no | — |
| 93 | D16 | C22⋊C4 | 1 | 4 | yes | — |
| 94 | C2×C16 | — | 2 | 2 | no | — |
| 95 | C2×C16 | — | 2 | 2 | no | — |
| 96 | C2×C16 | — | 2 | 2 | no | — |
| 97 | C2×C16 | — | 2 | 4 | no | — |
| 98 | C2×C16 | C22⋊C4 | 1 | 2 | yes | — |
| 99 | C2×C16 | C22×C4 | 1 | 1 | yes | characteristic |
| 100 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 101 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 102 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 103 | #108 | — | 4 | 2 | no | — |
| 104 | #108 | — | 4 | 2 | no | — |
| 105 | C22⋊C8 | — | 8 | 2 | no | — |
| 106 | C32 | — | 4 | 2 | no | — |
| 107 | C24 | — | 8 | 1 | no | — |
| 108 | C24 | SD32 | 1 | 1 | yes | characteristic |
| 109 | #41 | — | 4 | 16 | no | — |
| 110 | #41 | — | 8 | 8 | no | — |
| 111 | #41 | — | 8 | 16 | no | — |
| 112 | C2×Q8 | — | 4 | 2 | no | — |
| 113 | C2×Q8 | — | 4 | 4 | no | — |
| 114 | C2×Q8 | — | 8 | 1 | no | — |
| 115 | C2×Q8 | — | 8 | 2 | no | — |
| 116 | C2×D4 | — | 4 | 2 | no | — |
| 117 | C2×D4 | — | 4 | 4 | no | — |
| 118 | C2×D4 | — | 8 | 8 | no | — |
| 119 | C2×D4 | — | 8 | 4 | no | — |
| 120 | C2×D4 | — | 8 | 1 | no | — |
| 121 | C2×D4 | — | 8 | 2 | no | — |
| 122 | C22×C4 | — | 2 | 2 | no | — |
| 123 | C22×C4 | — | 2 | 2 | no | — |
| 124 | C22×C4 | — | 8 | 8 | no | — |
| 125 | C22×C4 | — | 8 | 1 | no | — |
| 126 | C22×C4 | — | 8 | 4 | no | — |
| 127 | C22×C4 | #102 | 1 | 2 | yes | — |
| 128 | C22×C4 | D16 | 1 | 2 | yes | — |
| 129 | C22×C4 | SD32 | 1 | 1 | yes | characteristic |
| 130 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 131 | Q16 | — | 2 | 4 | no | — |
| 132 | Q16 | — | 4 | 2 | no | — |
| 133 | Q16 | — | 4 | 4 | no | — |
| 134 | SD16 | — | 4 | 16 | no | — |
| 135 | D8 | — | 2 | 4 | no | — |
| 136 | D8 | — | 4 | 2 | no | — |
| 137 | D8 | — | 4 | 4 | no | — |
| 138 | M16 | — | 8 | 2 | no | — |
| 139 | M16 | — | 8 | 2 | no | — |
| 140 | C2×C8 | — | 2 | 2 | no | — |
| 141 | C2×C8 | — | 2 | 2 | no | — |
| 142 | C2×C8 | — | 2 | 2 | no | — |
| 143 | C2×C8 | — | 2 | 4 | no | — |
| 144 | C2×C8 | — | 8 | 2 | no | — |
| 145 | C2×C8 | #102 | 1 | 2 | yes | — |
| 146 | C2×C8 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 147 | C2×C8 | #120 | 1 | 2 | yes | — |
| 148 | C2×C8 | C2×SD16 | 1 | 1 | yes | characteristic |
| 149 | C2×C8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 150 | C16 | — | 2 | 2 | no | — |
| 151 | C16 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 152 | C16 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 153 | C16 | #120 | 1 | 2 | yes | — |
| 154 | C23 | — | 2 | 2 | no | — |
| 155 | C23 | — | 4 | 2 | no | — |
| 156 | C23 | — | 8 | 2 | no | — |
| 157 | C23 | — | 8 | 4 | no | — |
| 158 | C23 | — | 16 | 4 | no | — |
| 159 | C23 | #357 | 1 | 2 | yes | — |
| 160 | C23 | C2×SD32 | 1 | 1 | yes | characteristic |
| 161 | Q8 | — | 4 | 4 | no | — |
| 162 | Q8 | — | 8 | 2 | no | — |
| 163 | Q8 | — | 8 | 4 | no | — |
| 164 | D4 | — | 4 | 4 | no | — |
| 165 | D4 | — | 8 | 16 | no | — |
| 166 | D4 | — | 8 | 2 | no | — |
| 167 | D4 | — | 8 | 4 | no | — |
| 168 | C2×C4 | — | 2 | 2 | no | — |
| 169 | C2×C4 | — | 2 | 2 | no | — |
| 170 | C2×C4 | — | 2 | 4 | no | — |
| 171 | C2×C4 | — | 2 | 2 | no | — |
| 172 | C2×C4 | — | 8 | 16 | no | — |
| 173 | C2×C4 | — | 8 | 2 | no | — |
| 174 | C2×C4 | — | 8 | 4 | no | — |
| 175 | C2×C4 | #357 | 1 | 2 | yes | — |
| 176 | C2×C4 | C2×#102 | 1 | 1 | yes | characteristic |
| 177 | C2×C4 | #447 | 1 | 2 | yes | — |
| 178 | C2×C4 | C2×D16 | 1 | 1 | yes | characteristic |
| 179 | C2×C4 | #509 | 1 | 2 | yes | — |
| 180 | C8 | — | 2 | 2 | no | — |
| 181 | C8 | — | 16 | 2 | no | — |
| 182 | C8 | #379 | 1 | 1 | yes | characteristic |
| 183 | C8 | #418 | 1 | 1 | yes | characteristic |
| 184 | C8 | #450 | 1 | 2 | yes | — |
| 185 | C22 | — | 2 | 2 | no | — |
| 186 | C22 | — | 2 | 4 | no | — |
| 187 | C22 | — | 2 | 2 | no | — |
| 188 | C22 | — | 4 | 4 | no | — |
| 189 | C22 | — | 8 | 4 | no | — |
| 190 | C22 | — | 16 | 8 | no | — |
| 191 | C22 | — | 16 | 4 | no | — |
| 192 | C22 | C2×#357 | 1 | 1 | yes | characteristic |
| 193 | C22 | #2187 | 1 | 2 | yes | — |
| 194 | C4 | — | 2 | 2 | no | — |
| 195 | C4 | — | 8 | 4 | no | — |
| 196 | C4 | #1786 | 1 | 1 | yes | characteristic |
| 197 | C4 | #2140 | 1 | 1 | yes | characteristic |
| 198 | C4 | #2184 | 1 | 2 | yes | — |
| 199 | C2 | — | 2 | 2 | no | — |
| 200 | C2 | — | 4 | 2 | no | — |
| 201 | C2 | — | 16 | 4 | no | — |
| 202 | C2 | #12731 | 1 | 1 | yes | characteristic |
| 203 | C2 | #13625 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C22×#508 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#1420 | C2 | semidirect |
| 3. C2×#1419 | C2 | semidirect |
| 4. #13553 | C2 | semidirect |
| 5. #12837 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#508 | 2 |
| C2×#1420 | 2 |
| C2×#1419 | 2 |
| #13553 | 2 |
| #12837 | 2 |