Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
506 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 506 conjugacy classes of subgroups — too many to draw. Only its 142 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 | #33475 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#2177 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #28295 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #13518 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #13439 | C2 | 1 | 2 | yes | maximal |
| 6 | #13144 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #12892 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #12818 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C22×D16 | C22 | 1 | 1 | yes | characteristic |
| 10 | #3077 | — | 2 | 2 | no | — |
| 11 | #2960 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C4×D8 | C22 | 1 | 2 | yes | — |
| 13 | #2183 | — | 2 | 4 | no | — |
| 14 | #2177 | — | 2 | 4 | no | — |
| 15 | C2×#367 | C22 | 1 | 1 | yes | characteristic |
| 16 | #2151 | C22 | 1 | 2 | yes | — |
| 17 | #2150 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#361 | C22 | 1 | 4 | yes | — |
| 19 | C2×#357 | C22 | 1 | 2 | yes | — |
| 20 | #2108 | C22 | 1 | 2 | yes | — |
| 21 | C2×#346 | C22 | 1 | 1 | yes | characteristic |
| 22 | #2063 | — | 2 | 4 | no | — |
| 23 | #1978 | C22 | 1 | 4 | yes | — |
| 24 | #1952 | C22 | 1 | 4 | yes | — |
| 25 | #1848 | C22 | 1 | 2 | yes | — |
| 26 | #1800 | — | 2 | 2 | no | — |
| 27 | #1387 | C22 | 1 | 4 | yes | — |
| 28 | #1381 | C22 | 1 | 2 | yes | — |
| 29 | #1381 | C22 | 1 | 2 | yes | — |
| 30 | C22×D8 | C23 | 1 | 2 | yes | — |
| 31 | #533 | D4 | 1 | 1 | yes | characteristic |
| 32 | C2×C4×D4 | — | 2 | 2 | no | — |
| 33 | C2×D16 | — | 2 | 4 | no | — |
| 34 | C2×D16 | — | 2 | 2 | no | — |
| 35 | C2×M32 | C23 | 1 | 2 | yes | — |
| 36 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 37 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 38 | #478 | — | 4 | 4 | no | — |
| 39 | #476 | — | 4 | 2 | no | — |
| 40 | #474 | — | 4 | 2 | no | — |
| 41 | #446 | — | 2 | 2 | no | — |
| 42 | C8×Q8 | — | 2 | 2 | no | — |
| 43 | C4×D8 | — | 2 | 2 | no | — |
| 44 | C4×D8 | — | 2 | 2 | no | — |
| 45 | C4×D8 | — | 2 | 2 | no | — |
| 46 | C4×D8 | — | 2 | 2 | no | — |
| 47 | C2×#108 | — | 2 | 2 | no | — |
| 48 | C2×#108 | C23 | 1 | 4 | yes | — |
| 49 | #428 | — | 2 | 4 | no | — |
| 50 | C2×#107 | C23 | 1 | 2 | yes | — |
| 51 | #424 | D4 | 1 | 2 | yes | — |
| 52 | #423 | — | 2 | 2 | no | — |
| 53 | #423 | D4 | 1 | 1 | yes | characteristic |
| 54 | #418 | — | 2 | 4 | no | — |
| 55 | C2×#102 | — | 2 | 2 | no | — |
| 56 | C2×#100 | — | 2 | 4 | no | — |
| 57 | #405 | C23 | 1 | 2 | yes | — |
| 58 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 59 | #367 | — | 2 | 2 | no | — |
| 60 | C4⋊C16 | — | 2 | 4 | no | — |
| 61 | C4⋊C16 | — | 2 | 2 | no | — |
| 62 | #361 | — | 2 | 8 | no | — |
| 63 | #357 | — | 2 | 4 | no | — |
| 64 | #346 | — | 2 | 2 | no | — |
| 65 | #346 | D4 | 1 | 4 | yes | — |
| 66 | C4×C16 | — | 2 | 2 | no | — |
| 67 | #343 | — | 2 | 4 | no | — |
| 68 | C22×D4 | — | 2 | 2 | no | — |
| 69 | C23×C4 | — | 4 | 2 | no | — |
| 70 | C2×D8 | — | 2 | 2 | no | — |
| 71 | C2×D8 | — | 2 | 2 | no | — |
| 72 | C2×D8 | — | 2 | 2 | no | — |
| 73 | C2×D8 | — | 2 | 2 | no | — |
| 74 | C2×D8 | — | 2 | 2 | no | — |
| 75 | C2×D8 | #41 | 1 | 4 | yes | — |
| 76 | C2×M16 | — | 2 | 4 | no | — |
| 77 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 78 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 79 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 80 | C4⋊Q8 | D8 | 1 | 2 | yes | — |
| 81 | #125 | D8 | 1 | 2 | yes | — |
| 82 | C22⋊Q8 | — | 2 | 2 | no | — |
| 83 | C4×Q8 | — | 2 | 2 | no | — |
| 84 | C4×D4 | — | 4 | 2 | no | — |
| 85 | C4×D4 | — | 4 | 2 | no | — |
| 86 | C4×D4 | — | 4 | 2 | no | — |
| 87 | C4×D4 | — | 4 | 2 | no | — |
| 88 | #117 | C2×D4 | 1 | 2 | yes | — |
| 89 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 90 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 91 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | D16 | — | 2 | 8 | no | — |
| 93 | M32 | — | 2 | 4 | no | — |
| 94 | C2×C16 | — | 2 | 2 | no | — |
| 95 | C2×C16 | — | 2 | 2 | no | — |
| 96 | C2×C16 | — | 2 | 2 | no | — |
| 97 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 98 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 99 | #108 | — | 2 | 4 | no | — |
| 100 | #108 | — | 2 | 4 | no | — |
| 101 | #108 | — | 8 | 2 | no | — |
| 102 | #107 | — | 2 | 2 | no | — |
| 103 | #107 | — | 2 | 2 | no | — |
| 104 | #107 | — | 4 | 4 | no | — |
| 105 | #106 | — | 4 | 4 | no | — |
| 106 | C4⋊C8 | — | 2 | 4 | no | — |
| 107 | C4⋊C8 | — | 2 | 2 | no | — |
| 108 | C4⋊C8 | — | 4 | 2 | no | — |
| 109 | C4⋊C8 | — | 4 | 2 | no | — |
| 110 | #102 | — | 4 | 2 | no | — |
| 111 | #102 | — | 4 | 2 | no | — |
| 112 | #102 | — | 4 | 4 | no | — |
| 113 | #102 | — | 4 | 4 | no | — |
| 114 | #100 | — | 8 | 4 | no | — |
| 115 | #97 | — | 2 | 2 | no | — |
| 116 | C4×C8 | — | 2 | 2 | no | — |
| 117 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 118 | C4×C8 | #41 | 1 | 2 | yes | — |
| 119 | C24 | — | 4 | 2 | no | — |
| 120 | C2×Q8 | — | 2 | 2 | no | — |
| 121 | C2×D4 | — | 2 | 4 | no | — |
| 122 | C2×D4 | — | 4 | 2 | no | — |
| 123 | C2×D4 | — | 4 | 2 | no | — |
| 124 | C2×D4 | — | 4 | 2 | no | — |
| 125 | C2×D4 | — | 4 | 2 | no | — |
| 126 | C2×D4 | — | 4 | 2 | no | — |
| 127 | C22×C4 | — | 4 | 2 | no | — |
| 128 | C22×C4 | — | 8 | 2 | no | — |
| 129 | C22×C4 | — | 8 | 2 | no | — |
| 130 | C22×C4 | — | 16 | 4 | no | — |
| 131 | C22×C4 | #135 | 1 | 2 | yes | — |
| 132 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 133 | D8 | — | 2 | 4 | no | — |
| 134 | D8 | — | 2 | 4 | no | — |
| 135 | D8 | — | 4 | 2 | no | — |
| 136 | D8 | — | 4 | 2 | no | — |
| 137 | M16 | — | 2 | 4 | no | — |
| 138 | M16 | — | 4 | 4 | no | — |
| 139 | M16 | — | 4 | 4 | no | — |
| 140 | C2×C8 | — | 2 | 2 | no | — |
| 141 | C2×C8 | — | 2 | 2 | no | — |
| 142 | C2×C8 | — | 4 | 4 | no | — |
| 143 | C2×C8 | #121 | 1 | 2 | yes | — |
| 144 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 145 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 146 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 147 | C2×C8 | 2−1+4 | 1 | 2 | yes | — |
| 148 | C4⋊C4 | — | 2 | 4 | no | — |
| 149 | C4⋊C4 | — | 2 | 2 | no | — |
| 150 | C4⋊C4 | — | 4 | 2 | no | — |
| 151 | C4⋊C4 | — | 4 | 2 | no | — |
| 152 | C4⋊C4 | C2×D8 | 1 | 4 | yes | — |
| 153 | C22⋊C4 | — | 2 | 2 | no | — |
| 154 | C22⋊C4 | — | 8 | 2 | no | — |
| 155 | C22⋊C4 | — | 8 | 2 | no | — |
| 156 | C42 | — | 2 | 2 | no | — |
| 157 | C42 | #121 | 1 | 2 | yes | — |
| 158 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 159 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 160 | C16 | — | 2 | 2 | no | — |
| 161 | C16 | — | 2 | 2 | no | — |
| 162 | C16 | — | 4 | 2 | no | — |
| 163 | C23 | — | 4 | 4 | no | — |
| 164 | C23 | — | 8 | 2 | no | — |
| 165 | C23 | — | 8 | 2 | no | — |
| 166 | C23 | — | 16 | 2 | no | — |
| 167 | C23 | — | 16 | 2 | no | — |
| 168 | C23 | C2×#135 | 1 | 1 | yes | characteristic |
| 169 | Q8 | — | 8 | 2 | no | — |
| 170 | D4 | — | 4 | 4 | no | — |
| 171 | D4 | — | 4 | 4 | no | — |
| 172 | D4 | — | 8 | 2 | no | — |
| 173 | D4 | — | 8 | 2 | no | — |
| 174 | C2×C4 | — | 2 | 4 | no | — |
| 175 | C2×C4 | — | 4 | 2 | no | — |
| 176 | C2×C4 | — | 8 | 2 | no | — |
| 177 | C2×C4 | — | 16 | 4 | no | — |
| 178 | C2×C4 | — | 16 | 4 | no | — |
| 179 | C2×C4 | — | 16 | 2 | no | — |
| 180 | C2×C4 | #466 | 1 | 2 | yes | — |
| 181 | C2×C4 | #466 | 1 | 2 | yes | — |
| 182 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 183 | C2×C4 | #548 | 1 | 2 | yes | — |
| 184 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 185 | C2×C4 | #577 | 1 | 2 | yes | — |
| 186 | C8 | — | 2 | 2 | no | — |
| 187 | C8 | — | 4 | 2 | no | — |
| 188 | C8 | — | 8 | 4 | no | — |
| 189 | C8 | #525 | 1 | 1 | yes | characteristic |
| 190 | C8 | #536 | 1 | 1 | yes | characteristic |
| 191 | C8 | #552 | 1 | 2 | yes | — |
| 192 | C22 | — | 2 | 2 | no | — |
| 193 | C22 | — | 8 | 4 | no | — |
| 194 | C22 | — | 16 | 4 | no | — |
| 195 | C22 | — | 16 | 4 | no | — |
| 196 | C22 | — | 16 | 2 | no | — |
| 197 | C22 | — | 16 | 2 | no | — |
| 198 | C22 | C2×#466 | 1 | 1 | yes | characteristic |
| 199 | C22 | #3360 | 1 | 2 | yes | — |
| 200 | C4 | — | 2 | 2 | no | — |
| 201 | C4 | — | 2 | 2 | no | — |
| 202 | C4 | — | 4 | 4 | no | — |
| 203 | C4 | — | 16 | 2 | no | — |
| 204 | C4 | #2536 | 1 | 1 | yes | characteristic |
| 205 | C4 | #3060 | 1 | 1 | yes | characteristic |
| 206 | C4 | #3358 | 1 | 2 | yes | — |
| 207 | C2 | — | 2 | 2 | no | — |
| 208 | C2 | — | 16 | 4 | no | — |
| 209 | C2 | #28185 | 1 | 1 | yes | characteristic |
| 210 | C2 | #33618 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33475 | C2 | semidirect |
| 3. #28295 | C2 | semidirect |
| 4. #13518 | C2 | semidirect |
| 6. #13144 | C2 | semidirect |
| 7. #12892 | C2 | semidirect |
| 8. #12818 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33475 | 2 |
| C2×#2177 | 2 |
| #28295 | 2 |
| #13518 | 2 |
| #13439 | 2 |
| #13144 | 2 |
| #12892 | 2 |
| #12818 | 2 |