Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
375 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 375 conjugacy classes of subgroups — too many to draw. Only its 76 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 | #33815 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #19256 | C2 | 1 | 2 | yes | maximal |
| 3 | #13189 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11672 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #7241 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #3170 | C22 | 1 | 2 | yes | — |
| 7 | #3077 | — | 2 | 2 | no | — |
| 8 | C2×C8×D4 | C22 | 1 | 1 | yes | characteristic |
| 9 | #2227 | — | 2 | 2 | no | — |
| 10 | #2216 | — | 2 | 2 | no | — |
| 11 | #2209 | — | 2 | 2 | no | — |
| 12 | C2×C4⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C22⋊C16 | — | 2 | 2 | no | — |
| 14 | C2×C4×C16 | C22 | 1 | 1 | yes | characteristic |
| 15 | #1683 | — | 2 | 4 | no | — |
| 16 | #1675 | — | 2 | 2 | no | — |
| 17 | #1669 | — | 2 | 2 | no | — |
| 18 | #1567 | C22 | 1 | 2 | yes | — |
| 19 | #1550 | — | 2 | 2 | no | — |
| 20 | #1332 | — | 2 | 4 | no | — |
| 21 | C23×C8 | — | 2 | 1 | no | — |
| 22 | #533 | — | 2 | 2 | no | — |
| 23 | C2×C4×D4 | D4 | 1 | 1 | yes | characteristic |
| 24 | C2×Q32 | — | 2 | 2 | no | — |
| 25 | C22×C16 | — | 2 | 1 | no | — |
| 26 | C22×C16 | D4 | 1 | 2 | yes | — |
| 27 | #500 | D4 | 1 | 2 | yes | — |
| 28 | #494 | D4 | 1 | 2 | yes | — |
| 29 | #487 | — | 2 | 2 | no | — |
| 30 | #478 | — | 4 | 4 | no | — |
| 31 | #476 | — | 2 | 4 | no | — |
| 32 | #474 | — | 2 | 4 | no | — |
| 33 | #467 | — | 2 | 2 | no | — |
| 34 | #467 | — | 2 | 2 | no | — |
| 35 | C4×Q16 | — | 2 | 2 | no | — |
| 36 | C4×Q16 | — | 2 | 2 | no | — |
| 37 | C8×D4 | — | 2 | 2 | no | — |
| 38 | C8×D4 | — | 2 | 2 | no | — |
| 39 | #428 | — | 4 | 2 | no | — |
| 40 | #423 | — | 2 | 2 | no | — |
| 41 | C2×C4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 42 | #418 | — | 4 | 2 | no | — |
| 43 | C2×C22⋊C8 | — | 2 | 1 | no | — |
| 44 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 45 | #366 | — | 2 | 2 | no | — |
| 46 | C4⋊C16 | — | 2 | 2 | no | — |
| 47 | #358 | — | 2 | 4 | no | — |
| 48 | C22⋊C16 | — | 4 | 4 | no | — |
| 49 | C4×C16 | — | 2 | 2 | no | — |
| 50 | #335 | — | 2 | 2 | no | — |
| 51 | #335 | D4 | 1 | 4 | yes | — |
| 52 | #333 | — | 2 | 4 | no | — |
| 53 | #332 | — | 2 | 4 | no | — |
| 54 | C22×D4 | D8 | 1 | 1 | yes | characteristic |
| 55 | C23×C4 | — | 2 | 1 | no | — |
| 56 | C2×Q16 | — | 2 | 2 | no | — |
| 57 | C2×Q16 | — | 2 | 2 | no | — |
| 58 | C2×M16 | — | 4 | 2 | no | — |
| 59 | C22×C8 | — | 2 | 1 | no | — |
| 60 | C22×C8 | — | 4 | 2 | no | — |
| 61 | C22×C8 | — | 4 | 2 | no | — |
| 62 | C22×C8 | — | 4 | 1 | no | — |
| 63 | C22×C8 | — | 4 | 1 | no | — |
| 64 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 65 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 66 | C4⋊Q8 | — | 2 | 2 | no | — |
| 67 | C4⋊Q8 | — | 2 | 2 | no | — |
| 68 | #125 | — | 4 | 2 | no | — |
| 69 | C22⋊Q8 | — | 4 | 2 | no | — |
| 70 | C22⋊Q8 | — | 4 | 2 | no | — |
| 71 | C4×Q8 | — | 4 | 2 | no | — |
| 72 | C4×Q8 | — | 4 | 2 | no | — |
| 73 | C4×D4 | — | 2 | 2 | no | — |
| 74 | C4×D4 | — | 2 | 2 | no | — |
| 75 | #117 | — | 4 | 2 | no | — |
| 76 | C2×C4⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 77 | C2×C22⋊C4 | — | 2 | 1 | no | — |
| 78 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 79 | Q32 | — | 8 | 2 | no | — |
| 80 | C2×C16 | — | 2 | 2 | no | — |
| 81 | C2×C16 | — | 2 | 2 | no | — |
| 82 | C2×C16 | — | 4 | 1 | no | — |
| 83 | C2×C16 | — | 4 | 1 | no | — |
| 84 | C2×C16 | — | 4 | 2 | no | — |
| 85 | #107 | — | 2 | 2 | no | — |
| 86 | #107 | — | 2 | 2 | no | — |
| 87 | #107 | — | 4 | 4 | no | — |
| 88 | #106 | — | 4 | 4 | no | — |
| 89 | C4⋊C8 | — | 2 | 2 | no | — |
| 90 | C4⋊C8 | — | 2 | 4 | no | — |
| 91 | C4⋊C8 | — | 4 | 2 | no | — |
| 92 | #103 | — | 4 | 2 | no | — |
| 93 | #103 | — | 4 | 2 | no | — |
| 94 | #102 | — | 4 | 4 | no | — |
| 95 | #102 | — | 4 | 4 | no | — |
| 96 | C22⋊C8 | — | 4 | 2 | no | — |
| 97 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 98 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 99 | C24 | — | 2 | 1 | no | — |
| 100 | C2×Q8 | — | 4 | 2 | no | — |
| 101 | C2×Q8 | — | 4 | 2 | no | — |
| 102 | C2×D4 | — | 2 | 2 | no | — |
| 103 | C2×D4 | — | 2 | 1 | no | — |
| 104 | C2×D4 | — | 2 | 2 | no | — |
| 105 | C2×D4 | Q32 | 1 | 2 | yes | complemented |
| 106 | C22×C4 | — | 2 | 1 | no | — |
| 107 | C22×C4 | — | 4 | 2 | no | — |
| 108 | C22×C4 | — | 4 | 2 | no | — |
| 109 | C22×C4 | — | 4 | 1 | no | — |
| 110 | C22×C4 | — | 4 | 1 | no | — |
| 111 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 112 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 113 | C22×C4 | #135 | 1 | 1 | yes | characteristic |
| 114 | Q16 | — | 4 | 4 | no | — |
| 115 | Q16 | — | 4 | 2 | no | — |
| 116 | Q16 | — | 4 | 2 | no | — |
| 117 | M16 | — | 8 | 2 | no | — |
| 118 | M16 | — | 8 | 2 | no | — |
| 119 | C2×C8 | — | 2 | 1 | no | — |
| 120 | C2×C8 | — | 2 | 1 | no | — |
| 121 | C2×C8 | — | 4 | 1 | no | — |
| 122 | C2×C8 | — | 4 | 2 | no | — |
| 123 | C2×C8 | — | 4 | 1 | no | — |
| 124 | C2×C8 | — | 4 | 4 | no | — |
| 125 | C2×C8 | — | 8 | 2 | no | — |
| 126 | C2×C8 | — | 8 | 1 | no | — |
| 127 | C2×C8 | — | 8 | 1 | no | — |
| 128 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 129 | C2×C8 | #121 | 1 | 2 | yes | — |
| 130 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 131 | C4⋊C4 | — | 2 | 2 | no | — |
| 132 | C4⋊C4 | — | 4 | 2 | no | — |
| 133 | C4⋊C4 | — | 4 | 2 | no | — |
| 134 | C4⋊C4 | — | 4 | 2 | no | — |
| 135 | C4⋊C4 | — | 4 | 2 | no | — |
| 136 | C4⋊C4 | — | 8 | 2 | no | — |
| 137 | C4⋊C4 | — | 8 | 2 | no | — |
| 138 | C22⋊C4 | — | 4 | 2 | no | — |
| 139 | C22⋊C4 | — | 8 | 2 | no | — |
| 140 | C42 | — | 8 | 2 | no | — |
| 141 | C42 | #120 | 1 | 2 | yes | — |
| 142 | C42 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 143 | C16 | — | 4 | 2 | no | — |
| 144 | C16 | — | 4 | 2 | no | — |
| 145 | C23 | — | 2 | 2 | no | — |
| 146 | C23 | — | 4 | 2 | no | — |
| 147 | C23 | — | 4 | 2 | no | — |
| 148 | C23 | — | 4 | 1 | no | — |
| 149 | C23 | — | 4 | 1 | no | — |
| 150 | C23 | #466 | 1 | 1 | yes | characteristic |
| 151 | Q8 | — | 8 | 2 | no | — |
| 152 | Q8 | — | 8 | 2 | no | — |
| 153 | Q8 | — | 16 | 2 | no | — |
| 154 | D4 | — | 2 | 4 | no | — |
| 155 | D4 | — | 4 | 2 | no | — |
| 156 | C2×C4 | — | 2 | 1 | no | — |
| 157 | C2×C4 | — | 2 | 1 | no | — |
| 158 | C2×C4 | — | 4 | 1 | no | — |
| 159 | C2×C4 | — | 4 | 2 | no | — |
| 160 | C2×C4 | — | 4 | 1 | no | — |
| 161 | C2×C4 | — | 8 | 2 | no | — |
| 162 | C2×C4 | — | 8 | 1 | no | — |
| 163 | C2×C4 | — | 8 | 1 | no | — |
| 164 | C2×C4 | — | 8 | 2 | no | — |
| 165 | C2×C4 | — | 8 | 2 | no | — |
| 166 | C2×C4 | #392 | 1 | 1 | yes | characteristic |
| 167 | C2×C4 | #447 | 1 | 2 | yes | — |
| 168 | C2×C4 | #449 | 1 | 2 | yes | — |
| 169 | C2×C4 | #469 | 1 | 1 | yes | characteristic |
| 170 | C2×C4 | C2×Q32 | 1 | 1 | yes | characteristic |
| 171 | C2×C4 | #508 | 1 | 1 | yes | characteristic |
| 172 | C2×C4 | #510 | 1 | 2 | yes | — |
| 173 | C8 | — | 2 | 1 | no | — |
| 174 | C8 | — | 2 | 2 | no | — |
| 175 | C8 | — | 8 | 4 | no | — |
| 176 | C8 | — | 8 | 1 | no | — |
| 177 | C8 | #453 | 1 | 1 | yes | characteristic |
| 178 | C8 | #454 | 1 | 1 | yes | characteristic |
| 179 | C22 | — | 2 | 1 | no | — |
| 180 | C22 | — | 2 | 1 | no | — |
| 181 | C22 | — | 4 | 4 | no | — |
| 182 | C22 | — | 4 | 2 | no | — |
| 183 | C22 | — | 8 | 2 | no | — |
| 184 | C22 | — | 8 | 1 | no | — |
| 185 | C22 | — | 8 | 1 | no | — |
| 186 | C22 | #1999 | 1 | 1 | yes | characteristic |
| 187 | C22 | #2216 | 1 | 1 | yes | characteristic |
| 188 | C22 | #2219 | 1 | 1 | yes | characteristic |
| 189 | C4 | — | 2 | 2 | no | — |
| 190 | C4 | — | 2 | 1 | no | — |
| 191 | C4 | — | 8 | 1 | no | — |
| 192 | C4 | — | 16 | 2 | no | — |
| 193 | C4 | — | 16 | 2 | no | — |
| 194 | C4 | #2006 | 1 | 1 | yes | characteristic |
| 195 | C4 | #2017 | 1 | 1 | yes | characteristic |
| 196 | C4 | #2188 | 1 | 2 | yes | — |
| 197 | C4 | #2189 | 1 | 2 | yes | — |
| 198 | C2 | — | 2 | 1 | no | — |
| 199 | C2 | — | 2 | 1 | no | — |
| 200 | C2 | — | 8 | 2 | no | — |
| 201 | C2 | #10254 | 1 | 1 | yes | characteristic |
| 202 | C2 | #13306 | 1 | 1 | yes | characteristic |
| 203 | C2 | #13312 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D4 | D8 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33815 | C2 | semidirect |
| 4. #11672 | C2 | semidirect |
| 5. #7241 | C2 | semidirect |
| 105. C2×D4 | Q32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33815 | 2 |
| #19256 | 2 |
| #13189 | 2 |
| #11672 | 2 |
| #7241 | 2 |