Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1124 conjugacy classes of subgroups; only the 214 normal subgroups are shown.
This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 214 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 | #21203 | C2 | 1 | 3 | yes | maximal |
| 2 | #20824 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #10683 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10677 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #10673 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10671 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #7852 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×#127 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#418 | — | 2 | 6 | no | — |
| 10 | C2×C4×M16 | C22 | 1 | 3 | yes | — |
| 11 | #2285 | C22 | 1 | 3 | yes | — |
| 12 | #1961 | — | 2 | 6 | no | — |
| 13 | #1957 | C22 | 1 | 3 | yes | — |
| 14 | #1957 | C22 | 1 | 3 | yes | — |
| 15 | #1957 | C22 | 1 | 3 | yes | — |
| 16 | #1957 | C22 | 1 | 3 | yes | — |
| 17 | #1845 | C22 | 1 | 3 | yes | complemented |
| 18 | #1844 | C22 | 1 | 3 | yes | complemented |
| 19 | #1834 | C22 | 1 | 6 | yes | complemented |
| 20 | #1809 | — | 2 | 6 | no | — |
| 21 | #1776 | — | 2 | 6 | no | — |
| 22 | #1273 | C22 | 1 | 3 | yes | complemented |
| 23 | #1273 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | C23×D4 | — | 2 | 3 | no | — |
| 25 | C22×M16 | D4 | 1 | 6 | yes | — |
| 26 | C2×#127 | — | 2 | 3 | no | — |
| 27 | C2×#127 | — | 2 | 6 | no | — |
| 28 | C2×#127 | Q8 | 1 | 4 | yes | — |
| 29 | C2×#127 | C23 | 1 | 1 | yes | characteristic |
| 30 | C2×#127 | C23 | 1 | 1 | yes | characteristic |
| 31 | C2×#117 | — | 2 | 3 | no | — |
| 32 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 33 | C2×#107 | — | 2 | 6 | no | — |
| 34 | C2×#106 | — | 2 | 6 | no | — |
| 35 | #418 | — | 4 | 24 | no | — |
| 36 | C2×#102 | — | 2 | 6 | no | — |
| 37 | C2×#102 | — | 2 | 6 | no | — |
| 38 | C2×#102 | — | 2 | 6 | no | — |
| 39 | C2×#102 | — | 2 | 6 | no | — |
| 40 | C4×M16 | — | 2 | 12 | no | — |
| 41 | C2×#97 | D4 | 1 | 6 | yes | — |
| 42 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 43 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 44 | #384 | C23 | 1 | 3 | yes | — |
| 45 | #384 | C23 | 1 | 3 | yes | — |
| 46 | #382 | — | 2 | 3 | no | — |
| 47 | #380 | — | 2 | 6 | no | — |
| 48 | C4×C4⋊C4 | — | 2 | 6 | no | — |
| 49 | #340 | — | 2 | 12 | no | — |
| 50 | C25 | — | 4 | 1 | no | — |
| 51 | C22×D4 | — | 2 | 12 | no | — |
| 52 | C22×D4 | — | 2 | 3 | no | — |
| 53 | C22×D4 | — | 2 | 3 | no | — |
| 54 | C22×D4 | — | 4 | 12 | no | — |
| 55 | C22×D4 | — | 4 | 12 | no | — |
| 56 | C22×D4 | — | 4 | 6 | no | — |
| 57 | C22×D4 | — | 4 | 3 | no | — |
| 58 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 59 | C2×M16 | — | 2 | 12 | no | — |
| 60 | C2×M16 | — | 2 | 12 | no | — |
| 61 | C2×M16 | — | 4 | 6 | no | — |
| 62 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 63 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 64 | #127 | — | 2 | 12 | no | — |
| 65 | #127 | — | 2 | 3 | no | — |
| 66 | #127 | — | 2 | 3 | no | — |
| 67 | #127 | — | 4 | 6 | no | — |
| 68 | #127 | C2×Q8 | 1 | 2 | yes | — |
| 69 | #127 | C2×Q8 | 1 | 2 | yes | — |
| 70 | #117 | — | 4 | 12 | no | — |
| 71 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 72 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 73 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 74 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 75 | C2×C22⋊C4 | — | 4 | 3 | no | — |
| 76 | C2×C42 | — | 4 | 3 | no | — |
| 77 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 78 | C2×C42 | C2×Q8 | 1 | 2 | yes | — |
| 79 | C2×C42 | #41 | 1 | 6 | yes | — |
| 80 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 81 | #107 | — | 4 | 12 | no | — |
| 82 | #106 | — | 4 | 12 | no | — |
| 83 | #102 | — | 4 | 12 | no | — |
| 84 | #102 | — | 4 | 12 | no | — |
| 85 | #102 | — | 4 | 12 | no | — |
| 86 | #102 | — | 4 | 12 | no | — |
| 87 | #97 | — | 2 | 12 | no | — |
| 88 | C4×C8 | — | 2 | 6 | no | — |
| 89 | C4×C8 | — | 2 | 6 | no | — |
| 90 | #95 | — | 4 | 6 | no | — |
| 91 | C24 | — | 4 | 1 | no | — |
| 92 | C24 | — | 4 | 1 | no | — |
| 93 | C24 | — | 4 | 4 | no | — |
| 94 | C24 | — | 8 | 6 | no | — |
| 95 | C24 | — | 8 | 6 | no | — |
| 96 | C24 | #127 | 1 | 1 | yes | characteristic |
| 97 | C2×D4 | — | 2 | 6 | no | — |
| 98 | C2×D4 | — | 2 | 6 | no | — |
| 99 | C2×D4 | — | 4 | 48 | no | — |
| 100 | C2×D4 | — | 4 | 3 | no | — |
| 101 | C2×D4 | — | 4 | 3 | no | — |
| 102 | C2×D4 | — | 4 | 6 | no | — |
| 103 | C2×D4 | — | 4 | 6 | no | — |
| 104 | C2×D4 | — | 4 | 6 | no | — |
| 105 | C2×D4 | — | 4 | 6 | no | — |
| 106 | C2×D4 | — | 4 | 12 | no | — |
| 107 | C2×D4 | — | 8 | 6 | no | — |
| 108 | C2×D4 | — | 8 | 6 | no | — |
| 109 | C2×D4 | — | 8 | 24 | no | — |
| 110 | C22×C4 | — | 2 | 6 | no | — |
| 111 | C22×C4 | — | 2 | 6 | no | — |
| 112 | C22×C4 | — | 4 | 3 | no | — |
| 113 | C22×C4 | — | 4 | 3 | no | — |
| 114 | C22×C4 | C22⋊Q8 | 1 | 6 | yes | — |
| 115 | C22×C4 | #123 | 1 | 6 | yes | — |
| 116 | C22×C4 | #127 | 1 | 3 | yes | — |
| 117 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 118 | M16 | — | 4 | 24 | no | — |
| 119 | C2×C8 | — | 2 | 6 | no | — |
| 120 | C2×C8 | — | 2 | 6 | no | — |
| 121 | C2×C8 | — | 2 | 6 | no | — |
| 122 | C2×C8 | — | 2 | 6 | no | — |
| 123 | C2×C8 | — | 2 | 6 | no | — |
| 124 | C2×C8 | AGL(1,ℤ/8) | 1 | 12 | yes | — |
| 125 | C4⋊C4 | — | 4 | 6 | no | — |
| 126 | C4⋊C4 | — | 4 | 6 | no | — |
| 127 | C4⋊C4 | — | 4 | 12 | no | — |
| 128 | C4⋊C4 | — | 4 | 12 | no | — |
| 129 | C22⋊C4 | — | 8 | 6 | no | — |
| 130 | C42 | — | 8 | 6 | no | — |
| 131 | C42 | C22⋊Q8 | 1 | 6 | yes | — |
| 132 | C42 | #123 | 1 | 6 | yes | — |
| 133 | C42 | C22×Q8 | 1 | 1 | yes | characteristic |
| 134 | C42 | C2×#41 | 1 | 3 | yes | — |
| 135 | C23 | — | 2 | 3 | no | — |
| 136 | C23 | — | 2 | 3 | no | — |
| 137 | C23 | — | 4 | 2 | no | — |
| 138 | C23 | — | 4 | 2 | no | — |
| 139 | C23 | — | 8 | 24 | no | — |
| 140 | C23 | — | 8 | 3 | no | — |
| 141 | C23 | — | 8 | 3 | no | — |
| 142 | C23 | — | 8 | 3 | no | — |
| 143 | C23 | — | 8 | 3 | no | — |
| 144 | C23 | — | 16 | 4 | no | — |
| 145 | C23 | — | 16 | 12 | no | — |
| 146 | C23 | #399 | 1 | 2 | yes | — |
| 147 | C23 | C2×#127 | 1 | 1 | yes | characteristic |
| 148 | D4 | — | 4 | 12 | no | — |
| 149 | D4 | — | 4 | 12 | no | — |
| 150 | D4 | — | 8 | 24 | no | — |
| 151 | D4 | — | 8 | 6 | no | — |
| 152 | D4 | — | 8 | 6 | no | — |
| 153 | C2×C4 | — | 2 | 3 | no | — |
| 154 | C2×C4 | — | 2 | 3 | no | — |
| 155 | C2×C4 | — | 2 | 12 | no | — |
| 156 | C2×C4 | — | 2 | 12 | no | — |
| 157 | C2×C4 | — | 8 | 3 | no | — |
| 158 | C2×C4 | — | 8 | 3 | no | — |
| 159 | C2×C4 | — | 8 | 6 | no | — |
| 160 | C2×C4 | — | 8 | 6 | no | — |
| 161 | C2×C4 | #399 | 1 | 6 | yes | — |
| 162 | C2×C4 | C2×C22⋊Q8 | 1 | 3 | yes | — |
| 163 | C2×C4 | C2×#123 | 1 | 3 | yes | — |
| 164 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 6 | yes | — |
| 165 | C2×C4 | #575 | 1 | 6 | yes | — |
| 166 | C8 | — | 2 | 12 | no | — |
| 167 | C8 | — | 4 | 6 | no | — |
| 168 | C22 | — | 2 | 6 | no | — |
| 169 | C22 | — | 4 | 1 | no | — |
| 170 | C22 | — | 4 | 3 | no | — |
| 171 | C22 | — | 8 | 6 | no | — |
| 172 | C22 | — | 8 | 6 | no | — |
| 173 | C22 | — | 16 | 16 | no | — |
| 174 | C22 | — | 16 | 1 | no | — |
| 175 | C22 | — | 16 | 1 | no | — |
| 176 | C22 | — | 16 | 3 | no | — |
| 177 | C22 | — | 16 | 3 | no | — |
| 178 | C22 | C2×#399 | 1 | 1 | yes | characteristic |
| 179 | C22 | #3151 | 1 | 3 | yes | — |
| 180 | C22 | #3153 | 1 | 3 | yes | — |
| 181 | C4 | — | 2 | 6 | no | — |
| 182 | C4 | — | 16 | 3 | no | — |
| 183 | C4 | — | 16 | 3 | no | — |
| 184 | C4 | #3077 | 1 | 6 | yes | — |
| 185 | C4 | #3094 | 1 | 6 | yes | — |
| 186 | C2 | — | 4 | 2 | no | — |
| 187 | C2 | — | 16 | 2 | no | — |
| 188 | C2 | — | 16 | 2 | no | — |
| 189 | C2 | #19745 | 1 | 1 | yes | characteristic |
| 190 | C2 | #19749 | 1 | 3 | yes | — |
| 191 | C2 | #23731 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20824 | C2 | semidirect |
| 3. #10683 | C2 | semidirect |
| 4. #10677 | C2 | semidirect |
| 5. #10673 | C2 | semidirect |
| 6. #10671 | C2 | semidirect |
| 7. #7852 | C2 | semidirect |
| 17. #1845 | C22 | semidirect |
| 18. #1844 | C22 | semidirect |
| 19. #1834 | C22 | semidirect |
| 22. #1273 | C22 | semidirect |
| 23. #1273 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21203 | 2 |
| #20824 | 2 |
| #10683 | 2 |
| #10677 | 2 |
| #10673 | 2 |
| #10671 | 2 |
| #7852 | 2 |