Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
23774 conjugacy classes of subgroups; only the 4258 normal subgroups are shown.
This group has 23774 conjugacy classes of subgroups — too many to draw. Only its 4258 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 | C25×Q8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C23×#123 | C2 | 1 | 3 | yes | maximal |
| 3 | C2×#2614 | C2 | 1 | 48 | yes | maximal, direct product factor |
| 4 | C22×#395 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C22×#393 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | C22×#381 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C24×Q8 | — | 2 | 48 | no | — |
| 8 | C24×Q8 | C22 | 1 | 4 | yes | — |
| 9 | C24×Q8 | C22 | 1 | 24 | yes | complemented |
| 10 | C24×D4 | C22 | 1 | 1 | yes | characteristic |
| 11 | C25×C4 | C22 | 1 | 3 | yes | — |
| 12 | C22×#123 | — | 2 | 48 | no | — |
| 13 | C22×#123 | C22 | 1 | 72 | yes | — |
| 14 | C23×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 15 | C23×C22⋊C4 | C22 | 1 | 3 | yes | — |
| 16 | C23×C22⋊C4 | C22 | 1 | 6 | yes | — |
| 17 | #2614 | C22 | 1 | 256 | yes | direct product factor |
| 18 | C2×#395 | C22 | 1 | 96 | yes | complemented |
| 19 | C2×#393 | C22 | 1 | 96 | yes | complemented |
| 20 | C2×#381 | C22 | 1 | 72 | yes | complemented |
| 21 | C22×#95 | C22 | 1 | 12 | yes | — |
| 22 | C26 | C23 | 1 | 1 | yes | characteristic |
| 23 | C26 | C23 | 1 | 1 | yes | characteristic |
| 24 | C23×Q8 | — | 2 | 512 | no | — |
| 25 | C23×Q8 | — | 2 | 96 | no | — |
| 26 | C23×Q8 | — | 2 | 576 | no | — |
| 27 | C23×Q8 | C23 | 1 | 48 | yes | — |
| 28 | C23×Q8 | C23 | 1 | 64 | yes | complemented |
| 29 | C23×D4 | — | 2 | 24 | no | — |
| 30 | C23×D4 | C23 | 1 | 12 | yes | — |
| 31 | C24×C4 | — | 2 | 48 | no | — |
| 32 | C24×C4 | D4 | 1 | 48 | yes | — |
| 33 | C24×C4 | C23 | 1 | 1 | yes | characteristic |
| 34 | C24×C4 | C23 | 1 | 6 | yes | — |
| 35 | C24×C4 | C23 | 1 | 36 | yes | — |
| 36 | C24×C4 | C23 | 1 | 6 | yes | — |
| 37 | C2×#123 | — | 2 | 576 | no | — |
| 38 | C2×#123 | C23 | 1 | 192 | yes | — |
| 39 | C22×C4⋊C4 | — | 2 | 48 | no | — |
| 40 | C22×C4⋊C4 | C23 | 1 | 72 | yes | — |
| 41 | C22×C22⋊C4 | — | 2 | 24 | no | — |
| 42 | C22×C22⋊C4 | — | 2 | 48 | no | — |
| 43 | C22×C22⋊C4 | C23 | 1 | 72 | yes | — |
| 44 | C22×C22⋊C4 | C23 | 1 | 36 | yes | — |
| 45 | #395 | C23 | 1 | 256 | yes | complemented |
| 46 | #393 | C23 | 1 | 256 | yes | complemented |
| 47 | #381 | C23 | 1 | 192 | yes | complemented |
| 48 | C2×#95 | C23 | 1 | 144 | yes | — |
| 49 | C25 | — | 2 | 16 | no | — |
| 50 | C25 | — | 2 | 12 | no | — |
| 51 | C25 | — | 2 | 16 | no | — |
| 52 | C25 | C2×D4 | 1 | 24 | yes | — |
| 53 | C25 | C24 | 1 | 6 | yes | — |
| 54 | C25 | C24 | 1 | 6 | yes | — |
| 55 | C25 | C24 | 1 | 1 | yes | characteristic |
| 56 | C22×Q8 | — | 2 | 512 | no | — |
| 57 | C22×Q8 | — | 2 | 3072 | no | — |
| 58 | C22×Q8 | — | 2 | 576 | no | — |
| 59 | C22×Q8 | — | 2 | 768 | no | — |
| 60 | C22×Q8 | C24 | 1 | 64 | yes | — |
| 61 | C22×D4 | — | 2 | 144 | no | — |
| 62 | C22×D4 | — | 4 | 64 | no | — |
| 63 | C22×D4 | C24 | 1 | 16 | yes | — |
| 64 | C23×C4 | — | 2 | 384 | no | — |
| 65 | C23×C4 | — | 2 | 288 | no | — |
| 66 | C23×C4 | — | 2 | 48 | no | — |
| 67 | C23×C4 | — | 2 | 24 | no | — |
| 68 | C23×C4 | — | 2 | 12 | no | — |
| 69 | C23×C4 | — | 2 | 48 | no | — |
| 70 | C23×C4 | C2×D4 | 1 | 48 | yes | — |
| 71 | C23×C4 | C2×D4 | 1 | 288 | yes | — |
| 72 | C23×C4 | C24 | 1 | 36 | yes | — |
| 73 | C23×C4 | C24 | 1 | 6 | yes | — |
| 74 | C23×C4 | C24 | 1 | 36 | yes | — |
| 75 | C23×C4 | C24 | 1 | 48 | yes | — |
| 76 | #123 | — | 2 | 768 | no | — |
| 77 | C2×C4⋊C4 | — | 2 | 288 | no | — |
| 78 | C2×C4⋊C4 | C24 | 1 | 96 | yes | — |
| 79 | C2×C22⋊C4 | — | 2 | 144 | no | — |
| 80 | C2×C22⋊C4 | — | 2 | 288 | no | — |
| 81 | C2×C22⋊C4 | C24 | 1 | 48 | yes | — |
| 82 | C2×C22⋊C4 | C24 | 1 | 96 | yes | — |
| 83 | #95 | C24 | 1 | 192 | yes | — |
| 84 | C24 | — | 2 | 192 | no | — |
| 85 | C24 | — | 2 | 48 | no | — |
| 86 | C24 | — | 2 | 36 | no | — |
| 87 | C24 | — | 2 | 48 | no | — |
| 88 | C24 | — | 4 | 16 | no | — |
| 89 | C24 | — | 4 | 96 | no | — |
| 90 | C24 | #120 | 1 | 64 | yes | — |
| 91 | C24 | C22×D4 | 1 | 72 | yes | — |
| 92 | C24 | C22×D4 | 1 | 12 | yes | — |
| 93 | C24 | 2−1+4 | 1 | 16 | yes | — |
| 94 | C24 | C25 | 1 | 4 | yes | — |
| 95 | C24 | C25 | 1 | 4 | yes | — |
| 96 | C24 | C25 | 1 | 3 | yes | — |
| 97 | C2×Q8 | — | 2 | 1536 | no | — |
| 98 | C2×Q8 | — | 2 | 2048 | no | — |
| 99 | C2×Q8 | — | 2 | 384 | no | — |
| 100 | C2×D4 | — | 2 | 96 | no | — |
| 101 | C2×D4 | — | 4 | 192 | no | — |
| 102 | C22×C4 | — | 2 | 1152 | no | — |
| 103 | C22×C4 | — | 2 | 192 | no | — |
| 104 | C22×C4 | — | 2 | 192 | no | — |
| 105 | C22×C4 | — | 2 | 144 | no | — |
| 106 | C22×C4 | — | 2 | 72 | no | — |
| 107 | C22×C4 | — | 2 | 36 | no | — |
| 108 | C22×C4 | — | 2 | 144 | no | — |
| 109 | C22×C4 | — | 4 | 16 | no | — |
| 110 | C22×C4 | — | 4 | 96 | no | — |
| 111 | C22×C4 | C22×D4 | 1 | 144 | yes | — |
| 112 | C22×C4 | C22×D4 | 1 | 192 | yes | — |
| 113 | C22×C4 | C25 | 1 | 24 | yes | — |
| 114 | C22×C4 | C25 | 1 | 4 | yes | — |
| 115 | C22×C4 | C25 | 1 | 24 | yes | — |
| 116 | C4⋊C4 | — | 2 | 192 | no | — |
| 117 | C22⋊C4 | — | 2 | 96 | no | — |
| 118 | C22⋊C4 | — | 2 | 192 | no | — |
| 119 | C23 | — | 2 | 16 | no | — |
| 120 | C23 | — | 2 | 256 | no | — |
| 121 | C23 | — | 2 | 288 | no | — |
| 122 | C23 | — | 2 | 12 | no | — |
| 123 | C23 | — | 2 | 16 | no | — |
| 124 | C23 | — | 4 | 24 | no | — |
| 125 | C23 | — | 4 | 144 | no | — |
| 126 | C23 | — | 8 | 64 | no | — |
| 127 | C23 | C2×#120 | 1 | 96 | yes | — |
| 128 | C23 | C2×#120 | 1 | 16 | yes | — |
| 129 | C23 | #536 | 1 | 96 | yes | — |
| 130 | C23 | C23×D4 | 1 | 24 | yes | — |
| 131 | C23 | C23×D4 | 1 | 18 | yes | — |
| 132 | C23 | C2×2−1+4 | 1 | 24 | yes | — |
| 133 | C23 | C26 | 1 | 1 | yes | characteristic |
| 134 | Q8 | — | 2 | 512 | no | — |
| 135 | D4 | — | 4 | 64 | no | — |
| 136 | C2×C4 | — | 2 | 48 | no | — |
| 137 | C2×C4 | — | 2 | 384 | no | — |
| 138 | C2×C4 | — | 2 | 288 | no | — |
| 139 | C2×C4 | — | 2 | 24 | no | — |
| 140 | C2×C4 | — | 2 | 12 | no | — |
| 141 | C2×C4 | — | 2 | 48 | no | — |
| 142 | C2×C4 | — | 4 | 24 | no | — |
| 143 | C2×C4 | — | 4 | 144 | no | — |
| 144 | C2×C4 | C23×D4 | 1 | 48 | yes | — |
| 145 | C22 | — | 2 | 192 | no | — |
| 146 | C22 | — | 2 | 48 | no | — |
| 147 | C22 | — | 4 | 4 | no | — |
| 148 | C22 | — | 4 | 24 | no | — |
| 149 | C22 | — | 8 | 48 | no | — |
| 150 | C22 | #2614 | 1 | 64 | yes | direct product factor |
| 151 | C22 | C22×#120 | 1 | 16 | yes | — |
| 152 | C22 | C22×#120 | 1 | 12 | yes | — |
| 153 | C22 | C2×#536 | 1 | 72 | yes | — |
| 154 | C22 | C24×D4 | 1 | 3 | yes | — |
| 155 | C22 | C22×2−1+4 | 1 | 4 | yes | — |
| 156 | C4 | — | 2 | 48 | no | — |
| 157 | C4 | — | 4 | 4 | no | — |
| 158 | C4 | — | 4 | 24 | no | — |
| 159 | C2 | — | 2 | 16 | no | — |
| 160 | C2 | — | 8 | 4 | no | — |
| 161 | C2 | C2×#2614 | 1 | 24 | yes | direct product factor |
| 162 | C2 | C23×#120 | 1 | 1 | yes | characteristic |
| 163 | C2 | C22×#536 | 1 | 6 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C25×Q8 | C2 | semidirect |
| 3. C2×#2614 | C2 | direct (N × H) |
| 4. C22×#395 | C2 | semidirect |
| 5. C22×#393 | C2 | semidirect |
| 6. C22×#381 | C2 | semidirect |
| 9. C24×Q8 | C22 | semidirect |
| 17. #2614 | C22 | direct (N × H) |
| 18. C2×#395 | C22 | semidirect |
| 19. C2×#393 | C22 | semidirect |
| 20. C2×#381 | C22 | semidirect |
| 28. C23×Q8 | C23 | semidirect |
| 45. #395 | C23 | semidirect |
| 46. #393 | C23 | semidirect |
| 47. #381 | C23 | semidirect |
| 150. C22 | #2614 | direct (N × H) |
| 161. C2 | C2×#2614 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×Q8 | 2 |
| C23×#123 | 2 |
| C2×#2614 | 2 |
| C22×#395 | 2 |
| C22×#393 | 2 |
| C22×#381 | 2 |