Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
230 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 42 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 | #13200 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #12623 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #8239 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #2929 | C22 | 1 | 1 | yes | characteristic |
| 5 | #2153 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | #2120 | — | 2 | 1 | no | — |
| 7 | #2117 | — | 2 | 1 | no | — |
| 8 | #2112 | — | 2 | 1 | no | — |
| 9 | #2110 | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | #1757 | — | 2 | 1 | no | — |
| 11 | #1752 | — | 2 | 1 | no | — |
| 12 | #1501 | — | 2 | 1 | no | — |
| 13 | #1392 | C4 | 1 | 1 | yes | characteristic, complemented |
| 14 | #1391 | C4 | 1 | 1 | yes | characteristic, complemented |
| 15 | #1298 | C4 | 1 | 1 | yes | characteristic, complemented |
| 16 | #1297 | C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | C2×#131 | D4 | 1 | 1 | yes | characteristic |
| 18 | C23×C8 | D4 | 1 | 1 | yes | characteristic |
| 19 | #525 | D4 | 1 | 1 | yes | characteristic |
| 20 | C2×M32 | — | 2 | 1 | no | — |
| 21 | C2×M32 | — | 2 | 1 | no | — |
| 22 | #435 | — | 4 | 1 | no | — |
| 23 | C8×D4 | — | 4 | 1 | no | — |
| 24 | #424 | Q8 | 1 | 1 | yes | characteristic |
| 25 | #413 | — | 2 | 1 | no | — |
| 26 | #410 | — | 2 | 1 | no | — |
| 27 | #408 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | #407 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | C2×C22⋊C8 | — | 2 | 1 | no | — |
| 30 | #405 | — | 4 | 1 | no | — |
| 31 | #405 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C2×#97 | — | 4 | 1 | no | — |
| 33 | C4×C22⋊C4 | — | 2 | 1 | no | — |
| 34 | #364 | — | 4 | 1 | no | — |
| 35 | #352 | — | 2 | 2 | no | — |
| 36 | AΣL(1,𝔽16) | — | 2 | 2 | no | — |
| 37 | #350 | — | 4 | 2 | no | — |
| 38 | C22⋊C16 | — | 4 | 2 | no | — |
| 39 | #347 | — | 4 | 1 | no | — |
| 40 | #344 | — | 4 | 1 | no | — |
| 41 | #336 | — | 4 | 1 | no | — |
| 42 | #328 | — | 2 | 2 | no | — |
| 43 | #327 | — | 2 | 2 | no | — |
| 44 | C2×#41 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 45 | C23×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 46 | #131 | — | 4 | 1 | no | — |
| 47 | #131 | — | 4 | 1 | no | — |
| 48 | C2×M16 | — | 2 | 1 | no | — |
| 49 | C2×M16 | — | 4 | 1 | no | — |
| 50 | C2×M16 | C42 | 1 | 1 | yes | characteristic |
| 51 | C2×M16 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 52 | C22×C8 | — | 2 | 1 | no | — |
| 53 | C22×C8 | — | 2 | 1 | no | — |
| 54 | C22×C8 | — | 2 | 1 | no | — |
| 55 | C22×C8 | — | 4 | 1 | no | — |
| 56 | C22×C8 | — | 4 | 1 | no | — |
| 57 | C22×C8 | — | 4 | 1 | no | — |
| 58 | C22×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 59 | C22×C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 60 | #123 | — | 2 | 1 | no | — |
| 61 | C22⋊Q8 | — | 2 | 1 | no | — |
| 62 | #121 | — | 2 | 1 | no | — |
| 63 | #120 | — | 2 | 1 | no | — |
| 64 | C4×D4 | — | 4 | 1 | no | — |
| 65 | #117 | — | 4 | 1 | no | — |
| 66 | #117 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 67 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 68 | C2×C42 | — | 4 | 1 | no | — |
| 69 | M32 | — | 4 | 1 | no | — |
| 70 | M32 | — | 4 | 1 | no | — |
| 71 | M32 | — | 4 | 1 | no | — |
| 72 | M32 | — | 4 | 1 | no | — |
| 73 | C2×C16 | — | 2 | 2 | no | — |
| 74 | C2×C16 | — | 4 | 1 | no | — |
| 75 | C4⋊C8 | — | 4 | 1 | no | — |
| 76 | #101 | — | 4 | 1 | no | — |
| 77 | #100 | — | 2 | 2 | no | — |
| 78 | #99 | — | 2 | 2 | no | — |
| 79 | #99 | — | 4 | 1 | no | — |
| 80 | C22⋊C8 | — | 2 | 2 | no | — |
| 81 | C22⋊C8 | — | 4 | 1 | no | — |
| 82 | C22⋊C8 | — | 4 | 1 | no | — |
| 83 | C22⋊C8 | — | 4 | 1 | no | — |
| 84 | #97 | — | 2 | 1 | no | — |
| 85 | #97 | — | 8 | 1 | no | — |
| 86 | #97 | — | 8 | 1 | no | — |
| 87 | #97 | — | 8 | 1 | no | — |
| 88 | C4×C8 | — | 2 | 1 | no | — |
| 89 | C4×C8 | — | 8 | 1 | no | — |
| 90 | #95 | — | 2 | 2 | no | — |
| 91 | C24 | #104 | 1 | 1 | yes | characteristic |
| 92 | #41 | — | 4 | 1 | no | — |
| 93 | #41 | — | 4 | 1 | no | — |
| 94 | C2×Q8 | #104 | 1 | 1 | yes | characteristic |
| 95 | C2×D4 | — | 2 | 1 | no | — |
| 96 | C2×D4 | — | 4 | 1 | no | — |
| 97 | C2×D4 | #104 | 1 | 1 | yes | characteristic |
| 98 | C22×C4 | — | 2 | 1 | no | — |
| 99 | C22×C4 | — | 2 | 1 | no | — |
| 100 | C22×C4 | — | 2 | 1 | no | — |
| 101 | C22×C4 | — | 4 | 1 | no | — |
| 102 | C22×C4 | — | 4 | 1 | no | — |
| 103 | C22×C4 | — | 4 | 2 | no | — |
| 104 | C22×C4 | #95 | 1 | 1 | yes | characteristic |
| 105 | C22×C4 | #104 | 1 | 1 | yes | characteristic |
| 106 | M16 | — | 4 | 1 | no | — |
| 107 | M16 | — | 4 | 1 | no | — |
| 108 | M16 | — | 4 | 2 | no | — |
| 109 | M16 | — | 8 | 1 | no | — |
| 110 | M16 | — | 8 | 1 | no | — |
| 111 | C2×C8 | — | 2 | 1 | no | — |
| 112 | C2×C8 | — | 2 | 1 | no | — |
| 113 | C2×C8 | — | 2 | 1 | no | — |
| 114 | C2×C8 | — | 2 | 1 | no | — |
| 115 | C2×C8 | — | 2 | 1 | no | — |
| 116 | C2×C8 | — | 4 | 1 | no | — |
| 117 | C2×C8 | — | 4 | 1 | no | — |
| 118 | C2×C8 | — | 4 | 1 | no | — |
| 119 | C2×C8 | — | 4 | 1 | no | — |
| 120 | C2×C8 | — | 4 | 1 | no | — |
| 121 | C2×C8 | — | 4 | 2 | no | — |
| 122 | C2×C8 | — | 8 | 1 | no | — |
| 123 | C2×C8 | — | 8 | 1 | no | — |
| 124 | C2×C8 | — | 8 | 1 | no | — |
| 125 | C2×C8 | — | 8 | 1 | no | — |
| 126 | C2×C8 | #99 | 1 | 1 | yes | characteristic |
| 127 | C2×C8 | #99 | 1 | 1 | yes | characteristic |
| 128 | C4⋊C4 | — | 2 | 1 | no | — |
| 129 | C4⋊C4 | — | 4 | 1 | no | — |
| 130 | C4⋊C4 | — | 8 | 1 | no | — |
| 131 | C22⋊C4 | — | 2 | 1 | no | — |
| 132 | C22⋊C4 | — | 4 | 1 | no | — |
| 133 | C22⋊C4 | — | 4 | 1 | no | — |
| 134 | C22⋊C4 | — | 4 | 2 | no | — |
| 135 | C22⋊C4 | — | 8 | 2 | no | — |
| 136 | C42 | — | 2 | 1 | no | — |
| 137 | C42 | — | 8 | 1 | no | — |
| 138 | C42 | — | 8 | 1 | no | — |
| 139 | C42 | — | 8 | 1 | no | — |
| 140 | C16 | — | 4 | 2 | no | — |
| 141 | C16 | — | 8 | 1 | no | — |
| 142 | C23 | — | 2 | 1 | no | — |
| 143 | C23 | — | 2 | 1 | no | — |
| 144 | C23 | — | 4 | 2 | no | — |
| 145 | C23 | — | 4 | 1 | no | — |
| 146 | C23 | #339 | 1 | 1 | yes | characteristic |
| 147 | Q8 | — | 4 | 1 | no | — |
| 148 | D4 | — | 4 | 1 | no | — |
| 149 | D4 | — | 8 | 1 | no | — |
| 150 | D4 | — | 8 | 2 | no | — |
| 151 | C2×C4 | — | 2 | 1 | no | — |
| 152 | C2×C4 | — | 2 | 1 | no | — |
| 153 | C2×C4 | — | 2 | 1 | no | — |
| 154 | C2×C4 | — | 2 | 1 | no | — |
| 155 | C2×C4 | — | 4 | 1 | no | — |
| 156 | C2×C4 | — | 4 | 1 | no | — |
| 157 | C2×C4 | — | 4 | 1 | no | — |
| 158 | C2×C4 | — | 4 | 1 | no | — |
| 159 | C2×C4 | — | 4 | 1 | no | — |
| 160 | C2×C4 | — | 4 | 1 | no | — |
| 161 | C2×C4 | — | 8 | 1 | no | — |
| 162 | C2×C4 | — | 8 | 2 | no | — |
| 163 | C2×C4 | — | 8 | 2 | no | — |
| 164 | C2×C4 | — | 8 | 2 | no | — |
| 165 | C2×C4 | — | 8 | 2 | no | — |
| 166 | C2×C4 | #339 | 1 | 1 | yes | characteristic |
| 167 | C2×C4 | #342 | 1 | 1 | yes | characteristic |
| 168 | C8 | — | 2 | 1 | no | — |
| 169 | C8 | — | 4 | 1 | no | — |
| 170 | C8 | — | 4 | 1 | no | — |
| 171 | C8 | — | 4 | 1 | no | — |
| 172 | C8 | — | 8 | 2 | no | — |
| 173 | C8 | — | 8 | 2 | no | — |
| 174 | C8 | #353 | 1 | 1 | yes | characteristic |
| 175 | C8 | #354 | 1 | 1 | yes | characteristic |
| 176 | C22 | — | 2 | 2 | no | — |
| 177 | C22 | — | 2 | 1 | no | — |
| 178 | C22 | — | 4 | 1 | no | — |
| 179 | C22 | — | 4 | 1 | no | — |
| 180 | C22 | — | 4 | 2 | no | — |
| 181 | C22 | — | 4 | 1 | no | — |
| 182 | C22 | — | 4 | 1 | no | — |
| 183 | C22 | — | 8 | 1 | no | — |
| 184 | C22 | — | 8 | 1 | no | — |
| 185 | C22 | #1297 | 1 | 1 | yes | characteristic |
| 186 | C4 | — | 2 | 1 | no | — |
| 187 | C4 | — | 4 | 1 | no | — |
| 188 | C4 | — | 4 | 1 | no | — |
| 189 | C4 | — | 8 | 1 | no | — |
| 190 | C4 | — | 16 | 2 | no | — |
| 191 | C4 | — | 16 | 2 | no | — |
| 192 | C4 | #1391 | 1 | 1 | yes | characteristic |
| 193 | C4 | #1393 | 1 | 1 | yes | characteristic |
| 194 | C2 | — | 2 | 1 | no | — |
| 195 | C2 | — | 4 | 2 | no | — |
| 196 | C2 | — | 4 | 1 | no | — |
| 197 | C2 | — | 8 | 1 | no | — |
| 198 | C2 | #7283 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1393 |
| commutator | C2×M16 | C42 |
| Frattini, cosocle | #2929 | C22 |
| socle | C2 | #7283 |
| prime radical | #525 | D4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ #2929 ◁ G | |
| LECS | G ▷ #2929 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. #2153 | C4 | semidirect |
| 9. #2110 | C4 | semidirect |
| 13. #1392 | C4 | semidirect |
| 14. #1391 | C4 | semidirect |
| 15. #1298 | C4 | semidirect |
| 16. #1297 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13200 | 2 |
| #12623 | 2 |
| #8239 | 2 |