Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
357 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 30 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 | D82 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #13673 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #12310 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #7529 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #7515 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #3292 | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | D4×D8 | — | 2 | 2 | no | — |
| 8 | #2213 | — | 2 | 1 | no | — |
| 9 | #2196 | — | 2 | 2 | no | — |
| 10 | #1712 | C22 | 1 | 1 | yes | characteristic |
| 11 | #1667 | — | 2 | 2 | no | — |
| 12 | #1619 | — | 2 | 1 | no | — |
| 13 | #1619 | C22 | 1 | 2 | yes | — |
| 14 | C8×D8 | — | 2 | 1 | no | — |
| 15 | #1408 | C22 | 1 | 2 | yes | complemented |
| 16 | #1402 | — | 2 | 2 | no | — |
| 17 | #1357 | C22 | 1 | 1 | yes | characteristic, complemented |
| 18 | #1335 | — | 2 | 1 | no | — |
| 19 | #576 | — | 4 | 1 | no | — |
| 20 | C22×D8 | — | 4 | 2 | no | — |
| 21 | D42 | — | 2 | 1 | no | — |
| 22 | D42 | D4 | 1 | 2 | yes | complemented |
| 23 | #509 | — | 4 | 1 | no | — |
| 24 | #493 | — | 2 | 2 | no | — |
| 25 | #493 | D4 | 1 | 1 | yes | characteristic, complemented |
| 26 | #493 | C23 | 1 | 1 | yes | characteristic |
| 27 | #472 | — | 4 | 1 | no | — |
| 28 | #466 | — | 4 | 2 | no | — |
| 29 | #459 | — | 2 | 2 | no | — |
| 30 | #459 | — | 2 | 2 | no | — |
| 31 | B(4,𝔽2) | — | 4 | 2 | no | — |
| 32 | #453 | — | 2 | 2 | no | — |
| 33 | #447 | — | 4 | 2 | no | — |
| 34 | #447 | — | 4 | 2 | no | — |
| 35 | #443 | — | 2 | 1 | no | — |
| 36 | C4×D8 | — | 2 | 1 | no | — |
| 37 | C8×D4 | — | 2 | 2 | no | — |
| 38 | #364 | — | 2 | 1 | no | — |
| 39 | #361 | — | 4 | 1 | no | — |
| 40 | #353 | — | 2 | 2 | no | — |
| 41 | AΣL(1,𝔽16) | — | 4 | 2 | no | — |
| 42 | #335 | — | 2 | 1 | no | — |
| 43 | #331 | — | 2 | 1 | no | — |
| 44 | #331 | D4 | 1 | 2 | yes | complemented |
| 45 | D4⋊C8 | — | 2 | 2 | no | — |
| 46 | C82 | D4 | 1 | 1 | yes | characteristic, complemented |
| 47 | H(5,𝔽2) | — | 2 | 2 | no | — |
| 48 | C22×D4 | — | 4 | 2 | no | — |
| 49 | C22×D4 | — | 4 | 2 | no | — |
| 50 | AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 51 | AGL(1,ℤ/8) | — | 8 | 2 | no | — |
| 52 | #135 | — | 2 | 1 | no | — |
| 53 | #135 | — | 8 | 1 | no | — |
| 54 | C2×D8 | — | 2 | 1 | no | — |
| 55 | C2×D8 | — | 4 | 1 | no | — |
| 56 | C2×D8 | — | 4 | 1 | no | — |
| 57 | C2×D8 | — | 4 | 2 | no | — |
| 58 | C2×D8 | — | 4 | 2 | no | — |
| 59 | C2×D8 | — | 4 | 2 | no | — |
| 60 | C2×D8 | — | 4 | 2 | no | — |
| 61 | C2×D8 | — | 8 | 8 | no | — |
| 62 | C2×D8 | — | 8 | 1 | no | — |
| 63 | #131 | — | 4 | 1 | no | — |
| 64 | C22×C8 | — | 4 | 2 | no | — |
| 65 | #127 | — | 2 | 1 | no | — |
| 66 | #127 | C2×D4 | 1 | 2 | yes | — |
| 67 | #121 | — | 4 | 2 | no | — |
| 68 | #121 | — | 4 | 2 | no | — |
| 69 | #120 | — | 4 | 2 | no | — |
| 70 | #120 | — | 8 | 1 | no | — |
| 71 | #120 | — | 8 | 2 | no | — |
| 72 | C4×D4 | — | 2 | 2 | no | — |
| 73 | SD32 | — | 8 | 1 | no | — |
| 74 | D16 | — | 8 | 1 | no | — |
| 75 | M32 | — | 4 | 1 | no | — |
| 76 | #108 | — | 2 | 1 | no | — |
| 77 | #107 | — | 2 | 1 | no | — |
| 78 | C4⋊C8 | — | 2 | 2 | no | — |
| 79 | #104 | — | 4 | 1 | no | — |
| 80 | #102 | — | 2 | 2 | no | — |
| 81 | #102 | — | 4 | 2 | no | — |
| 82 | #102 | — | 4 | 2 | no | — |
| 83 | #100 | — | 2 | 2 | no | — |
| 84 | #99 | — | 4 | 2 | no | — |
| 85 | #99 | — | 8 | 2 | no | — |
| 86 | C22⋊C8 | — | 4 | 2 | no | — |
| 87 | C4×C8 | — | 2 | 1 | no | — |
| 88 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | C24 | — | 4 | 2 | no | — |
| 90 | C24 | — | 8 | 1 | no | — |
| 91 | #41 | — | 4 | 1 | no | — |
| 92 | #41 | — | 8 | 2 | no | — |
| 93 | C2×D4 | — | 2 | 2 | no | — |
| 94 | C2×D4 | — | 2 | 2 | no | — |
| 95 | C2×D4 | — | 4 | 2 | no | — |
| 96 | C2×D4 | — | 4 | 2 | no | — |
| 97 | C2×D4 | — | 4 | 1 | no | — |
| 98 | C2×D4 | — | 8 | 2 | no | — |
| 99 | C2×D4 | — | 8 | 2 | no | — |
| 100 | C2×D4 | — | 8 | 4 | no | — |
| 101 | C2×D4 | — | 8 | 4 | no | — |
| 102 | C2×D4 | — | 8 | 2 | no | — |
| 103 | C2×D4 | — | 8 | 2 | no | — |
| 104 | C2×D4 | — | 16 | 2 | no | — |
| 105 | C2×D4 | — | 16 | 2 | no | — |
| 106 | C2×D4 | — | 16 | 2 | no | — |
| 107 | C22×C4 | — | 4 | 2 | no | — |
| 108 | Q16 | — | 2 | 1 | no | — |
| 109 | SD16 | — | 4 | 1 | no | — |
| 110 | SD16 | — | 8 | 2 | no | — |
| 111 | D8 | — | 2 | 4 | no | — |
| 112 | D8 | — | 2 | 1 | no | — |
| 113 | D8 | — | 8 | 4 | no | — |
| 114 | D8 | — | 8 | 1 | no | — |
| 115 | D8 | — | 8 | 1 | no | — |
| 116 | D8 | — | 8 | 1 | no | — |
| 117 | D8 | — | 8 | 1 | no | — |
| 118 | D8 | — | 8 | 4 | no | — |
| 119 | D8 | — | 8 | 4 | no | — |
| 120 | D8 | — | 8 | 2 | no | — |
| 121 | D8 | — | 16 | 2 | no | — |
| 122 | M16 | — | 4 | 1 | no | — |
| 123 | M16 | — | 8 | 1 | no | — |
| 124 | C2×C8 | — | 2 | 1 | no | — |
| 125 | C2×C8 | — | 2 | 1 | no | — |
| 126 | C2×C8 | — | 2 | 1 | no | — |
| 127 | C2×C8 | — | 4 | 2 | no | — |
| 128 | C2×C8 | — | 8 | 4 | no | — |
| 129 | C2×C8 | — | 8 | 1 | no | — |
| 130 | C4⋊C4 | — | 2 | 2 | no | — |
| 131 | C22⋊C4 | — | 4 | 2 | no | — |
| 132 | C22⋊C4 | — | 4 | 2 | no | — |
| 133 | C22⋊C4 | — | 8 | 1 | no | — |
| 134 | C22⋊C4 | — | 8 | 2 | no | — |
| 135 | C22⋊C4 | — | 16 | 2 | no | — |
| 136 | C42 | #120 | 1 | 1 | yes | characteristic |
| 137 | C16 | — | 8 | 1 | no | — |
| 138 | C23 | — | 4 | 2 | no | — |
| 139 | C23 | — | 4 | 2 | no | — |
| 140 | C23 | — | 8 | 2 | no | — |
| 141 | C23 | — | 8 | 1 | no | — |
| 142 | C23 | — | 16 | 4 | no | — |
| 143 | C23 | — | 16 | 4 | no | — |
| 144 | C23 | — | 16 | 2 | no | — |
| 145 | C23 | — | 32 | 1 | no | — |
| 146 | Q8 | — | 4 | 1 | no | — |
| 147 | D4 | — | 2 | 4 | no | — |
| 148 | D4 | — | 4 | 2 | no | — |
| 149 | D4 | — | 4 | 1 | no | — |
| 150 | D4 | — | 8 | 2 | no | — |
| 151 | D4 | — | 8 | 2 | no | — |
| 152 | D4 | — | 8 | 2 | no | — |
| 153 | D4 | — | 8 | 2 | no | — |
| 154 | D4 | — | 8 | 2 | no | — |
| 155 | D4 | — | 8 | 1 | no | — |
| 156 | D4 | — | 16 | 1 | no | — |
| 157 | D4 | — | 16 | 4 | no | — |
| 158 | D4 | — | 16 | 4 | no | — |
| 159 | D4 | — | 16 | 2 | no | — |
| 160 | D4 | — | 32 | 4 | no | — |
| 161 | C2×C4 | — | 2 | 1 | no | — |
| 162 | C2×C4 | — | 4 | 2 | no | — |
| 163 | C2×C4 | — | 8 | 2 | no | — |
| 164 | C2×C4 | — | 8 | 1 | no | — |
| 165 | C2×C4 | — | 8 | 2 | no | — |
| 166 | C2×C4 | — | 8 | 2 | no | — |
| 167 | C2×C4 | — | 16 | 2 | no | — |
| 168 | C2×C4 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 169 | C8 | — | 2 | 1 | no | — |
| 170 | C8 | — | 2 | 2 | no | — |
| 171 | C8 | — | 2 | 1 | no | — |
| 172 | C8 | — | 4 | 1 | no | — |
| 173 | C8 | — | 8 | 2 | no | — |
| 174 | C8 | — | 8 | 1 | no | — |
| 175 | C22 | — | 4 | 4 | no | — |
| 176 | C22 | — | 8 | 2 | no | — |
| 177 | C22 | — | 8 | 2 | no | — |
| 178 | C22 | — | 8 | 2 | no | — |
| 179 | C22 | — | 8 | 1 | no | — |
| 180 | C22 | — | 16 | 2 | no | — |
| 181 | C22 | — | 16 | 2 | no | — |
| 182 | C22 | — | 16 | 4 | no | — |
| 183 | C22 | — | 16 | 1 | no | — |
| 184 | C22 | — | 32 | 2 | no | — |
| 185 | C22 | — | 32 | 4 | no | — |
| 186 | C22 | — | 32 | 2 | no | — |
| 187 | C22 | #2196 | 1 | 1 | yes | characteristic |
| 188 | C4 | — | 2 | 1 | no | — |
| 189 | C4 | — | 2 | 1 | no | — |
| 190 | C4 | — | 2 | 1 | no | — |
| 191 | C4 | — | 8 | 2 | no | — |
| 192 | C4 | — | 8 | 1 | no | — |
| 193 | C4 | — | 32 | 2 | no | — |
| 194 | C2 | — | 2 | 1 | no | — |
| 195 | C2 | — | 8 | 4 | no | — |
| 196 | C2 | — | 16 | 2 | no | — |
| 197 | C2 | — | 16 | 1 | no | — |
| 198 | C2 | — | 32 | 1 | no | — |
| 199 | C2 | #13673 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13673 |
| commutator, Frattini, cosocle | #493 | C23 |
| LCS | G ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #493 ◁ G | |
| LECS | G ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D82 | C2 | semidirect |
| 2. #13673 | C2 | semidirect |
| 3. #12310 | C2 | semidirect |
| 4. #7529 | C2 | semidirect |
| 5. #7515 | C2 | semidirect |
| 6. #3292 | C22 | semidirect |
| 15. #1408 | C22 | semidirect |
| 17. #1357 | C22 | semidirect |
| 22. D42 | D4 | semidirect |
| 25. #493 | D4 | semidirect |
| 44. #331 | D4 | semidirect |
| 46. C82 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D82 | 2 |
| #13673 | 2 |
| #12310 | 2 |
| #7529 | 2 |
| #7515 | 2 |