Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1135 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 1135 conjugacy classes of subgroups — too many to draw. Only its 222 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 | C24×C16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#2217 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #32931 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #32930 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #13060 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #13059 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #12835 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C24×C8 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×SD32 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | C23×C16 | — | 2 | 12 | no | — |
| 11 | C23×C16 | C22 | 1 | 3 | yes | — |
| 12 | C23×C16 | C22 | 1 | 3 | yes | — |
| 13 | C2×#467 | C22 | 1 | 3 | yes | — |
| 14 | C2×#466 | C22 | 1 | 3 | yes | — |
| 15 | #2848 | — | 2 | 1 | no | — |
| 16 | #2847 | — | 2 | 1 | no | — |
| 17 | #2217 | — | 2 | 12 | no | — |
| 18 | C2×#367 | C22 | 1 | 3 | yes | complemented |
| 19 | C2×#358 | C22 | 1 | 3 | yes | complemented |
| 20 | C2×#357 | C22 | 1 | 3 | yes | complemented |
| 21 | #1893 | — | 2 | 3 | no | — |
| 22 | #1892 | — | 2 | 3 | no | — |
| 23 | #1808 | C22 | 1 | 3 | yes | — |
| 24 | #1808 | C22 | 1 | 3 | yes | — |
| 25 | #1380 | C22 | 1 | 6 | yes | complemented |
| 26 | C24×C4 | D4 | 1 | 1 | yes | characteristic |
| 27 | C22×Q16 | C23 | 1 | 1 | yes | characteristic |
| 28 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 29 | C23×C8 | — | 2 | 6 | no | — |
| 30 | C23×C8 | — | 2 | 6 | no | — |
| 31 | C23×C8 | D4 | 1 | 3 | yes | — |
| 32 | C23×C8 | C23 | 1 | 3 | yes | — |
| 33 | C2×C22⋊Q8 | — | 2 | 3 | no | — |
| 34 | C2×#121 | — | 2 | 3 | no | — |
| 35 | C2×#120 | — | 4 | 1 | no | — |
| 36 | C2×SD32 | — | 2 | 6 | no | — |
| 37 | C22×C16 | — | 2 | 32 | no | — |
| 38 | C22×C16 | — | 2 | 6 | no | — |
| 39 | C22×C16 | — | 2 | 12 | no | — |
| 40 | C22×C16 | — | 2 | 12 | no | — |
| 41 | C22×C16 | D4 | 1 | 12 | yes | complemented |
| 42 | C22×C16 | C23 | 1 | 3 | yes | — |
| 43 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 44 | #467 | — | 2 | 6 | no | — |
| 45 | #467 | — | 2 | 6 | no | — |
| 46 | #466 | — | 2 | 6 | no | — |
| 47 | #466 | — | 2 | 6 | no | — |
| 48 | C2×#107 | C23 | 1 | 3 | yes | — |
| 49 | C2×#107 | C23 | 1 | 3 | yes | — |
| 50 | C2×#103 | — | 2 | 3 | no | — |
| 51 | C2×#102 | — | 2 | 3 | no | — |
| 52 | #386 | — | 4 | 3 | no | — |
| 53 | #385 | — | 4 | 3 | no | — |
| 54 | #380 | — | 2 | 3 | no | — |
| 55 | #380 | — | 2 | 3 | no | — |
| 56 | #379 | — | 4 | 1 | no | — |
| 57 | #367 | — | 2 | 6 | no | — |
| 58 | #358 | — | 2 | 6 | no | — |
| 59 | #357 | — | 2 | 6 | no | — |
| 60 | #340 | — | 2 | 3 | no | — |
| 61 | #340 | — | 2 | 3 | no | — |
| 62 | C25 | D8 | 1 | 1 | yes | characteristic |
| 63 | C22×Q8 | — | 2 | 1 | no | — |
| 64 | C22×D4 | — | 2 | 1 | no | — |
| 65 | C22×D4 | — | 4 | 3 | no | — |
| 66 | C23×C4 | — | 2 | 6 | no | — |
| 67 | C23×C4 | — | 2 | 6 | no | — |
| 68 | C23×C4 | D8 | 1 | 3 | yes | — |
| 69 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 70 | C2×Q16 | — | 2 | 3 | no | — |
| 71 | C2×Q16 | — | 2 | 3 | no | — |
| 72 | C2×D8 | — | 2 | 3 | no | — |
| 73 | C2×D8 | — | 2 | 3 | no | — |
| 74 | C22×C8 | — | 2 | 24 | no | — |
| 75 | C22×C8 | — | 2 | 8 | no | — |
| 76 | C22×C8 | — | 2 | 6 | no | — |
| 77 | C22×C8 | — | 2 | 12 | no | — |
| 78 | C22×C8 | — | 2 | 6 | no | — |
| 79 | C22×C8 | — | 2 | 6 | no | — |
| 80 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 81 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 82 | C22×C8 | #41 | 1 | 6 | yes | — |
| 83 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 84 | C22⋊Q8 | — | 4 | 6 | no | — |
| 85 | C22⋊Q8 | — | 4 | 6 | no | — |
| 86 | #121 | — | 4 | 6 | no | — |
| 87 | #121 | — | 4 | 6 | no | — |
| 88 | #120 | — | 8 | 4 | no | — |
| 89 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 90 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 91 | C2×C4⋊C4 | — | 4 | 3 | no | — |
| 92 | C2×C22⋊C4 | — | 4 | 3 | no | — |
| 93 | C2×C22⋊C4 | — | 4 | 3 | no | — |
| 94 | C2×C22⋊C4 | — | 4 | 3 | no | — |
| 95 | C2×C22⋊C4 | — | 4 | 3 | no | — |
| 96 | SD32 | — | 4 | 4 | no | — |
| 97 | C2×C16 | — | 2 | 24 | no | — |
| 98 | C2×C16 | — | 2 | 6 | no | — |
| 99 | C2×C16 | — | 2 | 24 | no | — |
| 100 | C2×C16 | C2×D4 | 1 | 6 | yes | — |
| 101 | C2×C16 | C2×D4 | 1 | 6 | yes | — |
| 102 | #107 | — | 2 | 3 | no | — |
| 103 | #107 | — | 2 | 3 | no | — |
| 104 | #107 | — | 2 | 3 | no | — |
| 105 | #107 | — | 2 | 3 | no | — |
| 106 | #103 | — | 4 | 6 | no | — |
| 107 | #102 | — | 4 | 6 | no | — |
| 108 | #95 | — | 4 | 3 | no | — |
| 109 | #95 | — | 4 | 3 | no | — |
| 110 | C24 | — | 2 | 6 | no | — |
| 111 | C24 | — | 2 | 6 | no | — |
| 112 | C24 | — | 4 | 1 | no | — |
| 113 | C24 | SD32 | 1 | 4 | yes | complemented |
| 114 | C24 | C2×D8 | 1 | 3 | yes | — |
| 115 | C2×Q8 | — | 4 | 3 | no | — |
| 116 | C2×Q8 | — | 4 | 3 | no | — |
| 117 | C2×D4 | — | 4 | 3 | no | — |
| 118 | C2×D4 | — | 4 | 3 | no | — |
| 119 | C2×D4 | — | 8 | 6 | no | — |
| 120 | C2×D4 | — | 8 | 6 | no | — |
| 121 | C2×D4 | — | 8 | 6 | no | — |
| 122 | C22×C4 | — | 2 | 8 | no | — |
| 123 | C22×C4 | — | 2 | 24 | no | — |
| 124 | C22×C4 | — | 2 | 6 | no | — |
| 125 | C22×C4 | — | 2 | 6 | no | — |
| 126 | C22×C4 | — | 2 | 6 | no | — |
| 127 | C22×C4 | — | 2 | 12 | no | — |
| 128 | C22×C4 | — | 4 | 3 | no | — |
| 129 | C22×C4 | — | 4 | 3 | no | — |
| 130 | C22×C4 | — | 4 | 1 | no | — |
| 131 | C22×C4 | #121 | 1 | 6 | yes | — |
| 132 | C22×C4 | C2×D8 | 1 | 3 | yes | — |
| 133 | C22×C4 | #135 | 1 | 6 | yes | — |
| 134 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 135 | Q16 | — | 4 | 1 | no | — |
| 136 | Q16 | — | 4 | 3 | no | — |
| 137 | D8 | — | 4 | 1 | no | — |
| 138 | D8 | — | 4 | 3 | no | — |
| 139 | C2×C8 | — | 2 | 24 | no | — |
| 140 | C2×C8 | — | 2 | 6 | no | — |
| 141 | C2×C8 | — | 2 | 6 | no | — |
| 142 | C2×C8 | — | 2 | 6 | no | — |
| 143 | C2×C8 | — | 2 | 12 | no | — |
| 144 | C2×C8 | #121 | 1 | 6 | yes | — |
| 145 | C2×C8 | C22×D4 | 1 | 3 | yes | — |
| 146 | C2×C8 | C2×#41 | 1 | 3 | yes | — |
| 147 | C4⋊C4 | — | 4 | 3 | no | — |
| 148 | C4⋊C4 | — | 4 | 3 | no | — |
| 149 | C4⋊C4 | — | 4 | 3 | no | — |
| 150 | C4⋊C4 | — | 4 | 3 | no | — |
| 151 | C4⋊C4 | — | 8 | 6 | no | — |
| 152 | C22⋊C4 | — | 8 | 6 | no | — |
| 153 | C22⋊C4 | — | 8 | 6 | no | — |
| 154 | C22⋊C4 | — | 8 | 6 | no | — |
| 155 | C22⋊C4 | — | 8 | 6 | no | — |
| 156 | C16 | — | 2 | 6 | no | — |
| 157 | C16 | #120 | 1 | 4 | yes | complemented |
| 158 | C23 | — | 2 | 8 | no | — |
| 159 | C23 | — | 2 | 24 | no | — |
| 160 | C23 | — | 2 | 6 | no | — |
| 161 | C23 | — | 2 | 24 | no | — |
| 162 | C23 | — | 2 | 6 | no | — |
| 163 | C23 | — | 8 | 1 | no | — |
| 164 | C23 | — | 8 | 3 | no | — |
| 165 | C23 | — | 8 | 3 | no | — |
| 166 | C23 | #466 | 1 | 6 | yes | — |
| 167 | C23 | C2×SD32 | 1 | 6 | yes | — |
| 168 | C23 | #508 | 1 | 6 | yes | — |
| 169 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 170 | Q8 | — | 8 | 1 | no | — |
| 171 | Q8 | — | 8 | 3 | no | — |
| 172 | D4 | — | 8 | 1 | no | — |
| 173 | D4 | — | 8 | 3 | no | — |
| 174 | D4 | — | 16 | 12 | no | — |
| 175 | C2×C4 | — | 2 | 24 | no | — |
| 176 | C2×C4 | — | 2 | 6 | no | — |
| 177 | C2×C4 | — | 2 | 6 | no | — |
| 178 | C2×C4 | — | 2 | 6 | no | — |
| 179 | C2×C4 | — | 2 | 12 | no | — |
| 180 | C2×C4 | — | 8 | 3 | no | — |
| 181 | C2×C4 | — | 8 | 6 | no | — |
| 182 | C2×C4 | — | 8 | 3 | no | — |
| 183 | C2×C4 | — | 8 | 3 | no | — |
| 184 | C2×C4 | — | 8 | 3 | no | — |
| 185 | C2×C4 | — | 8 | 3 | no | — |
| 186 | C2×C4 | #466 | 1 | 6 | yes | — |
| 187 | C2×C4 | C2×#121 | 1 | 3 | yes | — |
| 188 | C2×C4 | C2×#135 | 1 | 3 | yes | — |
| 189 | C8 | — | 2 | 6 | no | — |
| 190 | C8 | C2×#120 | 1 | 1 | yes | characteristic |
| 191 | C8 | #525 | 1 | 3 | yes | — |
| 192 | C22 | — | 2 | 32 | no | — |
| 193 | C22 | — | 2 | 12 | no | — |
| 194 | C22 | — | 2 | 6 | no | — |
| 195 | C22 | — | 2 | 12 | no | — |
| 196 | C22 | — | 2 | 6 | no | — |
| 197 | C22 | — | 16 | 3 | no | — |
| 198 | C22 | — | 16 | 3 | no | — |
| 199 | C22 | — | 16 | 1 | no | — |
| 200 | C22 | #2217 | 1 | 12 | yes | complemented |
| 201 | C22 | C2×#466 | 1 | 3 | yes | — |
| 202 | C22 | C22×SD32 | 1 | 1 | yes | characteristic |
| 203 | C22 | C2×#508 | 1 | 3 | yes | — |
| 204 | C4 | — | 2 | 6 | no | — |
| 205 | C4 | — | 16 | 3 | no | — |
| 206 | C4 | — | 16 | 3 | no | — |
| 207 | C4 | — | 16 | 1 | no | — |
| 208 | C4 | #2847 | 1 | 1 | yes | characteristic |
| 209 | C4 | #3050 | 1 | 3 | yes | — |
| 210 | C2 | — | 2 | 12 | no | — |
| 211 | C2 | — | 32 | 1 | no | — |
| 212 | C2 | #32930 | 1 | 1 | yes | characteristic |
| 213 | C2 | C2×#2217 | 1 | 3 | yes | — |
| 214 | C2 | #33600 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #32930 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C16 | C2 | semidirect |
| 2. C2×#2217 | C2 | semidirect |
| 3. #32931 | C2 | semidirect |
| 5. #13060 | C2 | semidirect |
| 6. #13059 | C2 | semidirect |
| 7. #12835 | C2 | semidirect |
| 9. C22×SD32 | C22 | semidirect |
| 18. C2×#367 | C22 | semidirect |
| 19. C2×#358 | C22 | semidirect |
| 20. C2×#357 | C22 | semidirect |
| 25. #1380 | C22 | semidirect |
| 41. C22×C16 | D4 | semidirect |
| 113. C24 | SD32 | semidirect |
| 157. C16 | #120 | semidirect |
| 200. C22 | #2217 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C16 | 2 |
| C2×#2217 | 2 |
| #32931 | 2 |
| #32930 | 2 |
| #13060 | 2 |
| #13059 | 2 |
| #12835 | 2 |