Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2068 conjugacy classes of subgroups; only the 638 normal subgroups are shown.
This group has 2068 conjugacy classes of subgroups — too many to draw. Only its 638 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 | #60673 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | #19604 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #19603 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #19600 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | #19596 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | C22×C82 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #3444 | C22 | 1 | 4 | yes | — |
| 8 | #3170 | C22 | 1 | 12 | yes | — |
| 9 | #3169 | C22 | 1 | 12 | yes | — |
| 10 | #3168 | — | 2 | 24 | no | — |
| 11 | #3167 | C22 | 1 | 24 | yes | — |
| 12 | #3160 | C22 | 1 | 6 | yes | — |
| 13 | #3147 | C22 | 1 | 6 | yes | — |
| 14 | #3131 | C22 | 1 | 12 | yes | — |
| 15 | #3050 | — | 2 | 12 | no | — |
| 16 | C4×#135 | — | 2 | 12 | no | — |
| 17 | C22×C4×C8 | C22 | 1 | 3 | yes | — |
| 18 | #1713 | C22 | 1 | 3 | yes | complemented |
| 19 | #1712 | C22 | 1 | 1 | yes | characteristic, complemented |
| 20 | #1711 | C22 | 1 | 8 | yes | complemented |
| 21 | #1710 | C22 | 1 | 6 | yes | complemented |
| 22 | #1709 | C22 | 1 | 6 | yes | complemented |
| 23 | #1708 | C22 | 1 | 12 | yes | complemented |
| 24 | #1707 | C22 | 1 | 12 | yes | complemented |
| 25 | #1706 | C22 | 1 | 12 | yes | complemented |
| 26 | #1705 | C22 | 1 | 4 | yes | complemented |
| 27 | C2×C82 | C22 | 1 | 3 | yes | — |
| 28 | C2×C82 | C22 | 1 | 6 | yes | — |
| 29 | C2×C82 | C22 | 1 | 3 | yes | — |
| 30 | C2×#135 | — | 2 | 12 | no | — |
| 31 | C23×C8 | D4 | 1 | 6 | yes | — |
| 32 | #533 | C23 | 1 | 12 | yes | — |
| 33 | #532 | C23 | 1 | 12 | yes | — |
| 34 | #529 | — | 2 | 24 | no | — |
| 35 | #525 | — | 2 | 12 | no | — |
| 36 | C4×#41 | — | 2 | 12 | no | — |
| 37 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 38 | #500 | C23 | 1 | 12 | yes | — |
| 39 | #499 | — | 2 | 12 | no | — |
| 40 | #498 | C23 | 1 | 12 | yes | — |
| 41 | #495 | — | 2 | 12 | no | — |
| 42 | #494 | C23 | 1 | 6 | yes | — |
| 43 | #493 | C23 | 1 | 6 | yes | — |
| 44 | #492 | C23 | 1 | 12 | yes | — |
| 45 | #488 | — | 2 | 24 | no | — |
| 46 | #487 | C23 | 1 | 24 | yes | — |
| 47 | #486 | C23 | 1 | 24 | yes | — |
| 48 | #467 | — | 2 | 24 | no | — |
| 49 | #466 | — | 2 | 24 | no | — |
| 50 | #465 | — | 2 | 48 | no | — |
| 51 | C4×Q16 | — | 2 | 24 | no | — |
| 52 | C4×SD16 | — | 2 | 48 | no | — |
| 53 | C4×D8 | — | 2 | 24 | no | — |
| 54 | #427 | — | 2 | 12 | no | — |
| 55 | #416 | — | 2 | 24 | no | — |
| 56 | C2×C4×C8 | D4 | 1 | 12 | yes | — |
| 57 | C2×C4×C8 | D4 | 1 | 6 | yes | — |
| 58 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 59 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 60 | C2×C4×C8 | C23 | 1 | 6 | yes | — |
| 61 | C2×C4×C8 | C23 | 1 | 6 | yes | — |
| 62 | C82 | C23 | 1 | 1 | yes | characteristic |
| 63 | C82 | C23 | 1 | 6 | yes | — |
| 64 | C82 | C23 | 1 | 3 | yes | — |
| 65 | C82 | C23 | 1 | 3 | yes | — |
| 66 | C82 | C23 | 1 | 2 | yes | — |
| 67 | C82 | C23 | 1 | 1 | yes | characteristic |
| 68 | C2×#41 | — | 2 | 12 | no | — |
| 69 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 70 | #135 | — | 4 | 48 | no | — |
| 71 | C2×Q16 | — | 2 | 12 | no | — |
| 72 | C2×SD16 | — | 2 | 24 | no | — |
| 73 | C2×D8 | — | 2 | 12 | no | — |
| 74 | C22×C8 | — | 2 | 24 | no | — |
| 75 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 76 | C22×C8 | C2×D4 | 1 | 12 | yes | — |
| 77 | C22×C8 | C2×D4 | 1 | 12 | yes | — |
| 78 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 79 | C4⋊Q8 | C24 | 1 | 12 | yes | — |
| 80 | #127 | C24 | 1 | 4 | yes | — |
| 81 | #126 | — | 4 | 8 | no | — |
| 82 | #125 | — | 2 | 12 | no | — |
| 83 | #124 | — | 2 | 12 | no | — |
| 84 | #123 | — | 4 | 12 | no | — |
| 85 | C22⋊Q8 | — | 2 | 24 | no | — |
| 86 | #121 | — | 2 | 24 | no | — |
| 87 | #120 | — | 4 | 4 | no | — |
| 88 | C4×Q8 | — | 2 | 24 | no | — |
| 89 | C4×D4 | — | 2 | 24 | no | — |
| 90 | C4×D4 | — | 4 | 24 | no | — |
| 91 | #117 | — | 2 | 12 | no | — |
| 92 | #117 | — | 4 | 12 | no | — |
| 93 | C2×C42 | — | 4 | 4 | no | — |
| 94 | C2×C42 | C2×D4 | 1 | 6 | yes | — |
| 95 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 96 | C2×C42 | C24 | 1 | 3 | yes | — |
| 97 | #107 | — | 2 | 24 | no | — |
| 98 | #106 | — | 2 | 24 | no | — |
| 99 | #103 | — | 2 | 48 | no | — |
| 100 | #102 | — | 2 | 48 | no | — |
| 101 | C4×C8 | C2×D4 | 1 | 12 | yes | — |
| 102 | C4×C8 | C2×D4 | 1 | 12 | yes | — |
| 103 | C4×C8 | C2×D4 | 1 | 6 | yes | — |
| 104 | C4×C8 | C2×D4 | 1 | 6 | yes | — |
| 105 | C4×C8 | C24 | 1 | 6 | yes | — |
| 106 | C4×C8 | C24 | 1 | 3 | yes | — |
| 107 | C4×C8 | C24 | 1 | 3 | yes | — |
| 108 | C24 | #127 | 1 | 1 | yes | characteristic |
| 109 | #41 | — | 4 | 48 | no | — |
| 110 | C2×Q8 | — | 2 | 12 | no | — |
| 111 | C2×D4 | — | 2 | 12 | no | — |
| 112 | C2×D4 | — | 4 | 12 | no | — |
| 113 | C22×C4 | — | 2 | 12 | no | — |
| 114 | C22×C4 | — | 4 | 12 | no | — |
| 115 | C22×C4 | #127 | 1 | 6 | yes | — |
| 116 | C22×C4 | #127 | 1 | 3 | yes | — |
| 117 | C22×C4 | C22×D4 | 1 | 6 | yes | — |
| 118 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 119 | Q16 | — | 4 | 24 | no | — |
| 120 | SD16 | — | 4 | 48 | no | — |
| 121 | D8 | — | 4 | 24 | no | — |
| 122 | C2×C8 | — | 2 | 12 | no | — |
| 123 | C2×C8 | — | 2 | 24 | no | — |
| 124 | C2×C8 | — | 2 | 24 | no | — |
| 125 | C2×C8 | #135 | 1 | 24 | yes | complemented |
| 126 | C2×C8 | C22×D4 | 1 | 12 | yes | — |
| 127 | C2×C8 | C22×D4 | 1 | 6 | yes | — |
| 128 | C2×C8 | C22×D4 | 1 | 6 | yes | — |
| 129 | C4⋊C4 | — | 2 | 24 | no | — |
| 130 | C4⋊C4 | — | 4 | 24 | no | — |
| 131 | C22⋊C4 | — | 4 | 12 | no | — |
| 132 | C22⋊C4 | — | 4 | 24 | no | — |
| 133 | C42 | — | 4 | 12 | no | — |
| 134 | C42 | #127 | 1 | 3 | yes | — |
| 135 | C42 | #127 | 1 | 2 | yes | — |
| 136 | C42 | #127 | 1 | 1 | yes | characteristic |
| 137 | C42 | C22×D4 | 1 | 6 | yes | — |
| 138 | C42 | C22×D4 | 1 | 3 | yes | — |
| 139 | C42 | C25 | 1 | 1 | yes | characteristic |
| 140 | C23 | — | 2 | 6 | no | — |
| 141 | C23 | — | 4 | 4 | no | — |
| 142 | C23 | C2×#127 | 1 | 3 | yes | — |
| 143 | Q8 | — | 4 | 24 | no | — |
| 144 | D4 | — | 4 | 24 | no | — |
| 145 | D4 | — | 8 | 24 | no | — |
| 146 | C2×C4 | — | 2 | 6 | no | — |
| 147 | C2×C4 | — | 2 | 12 | no | — |
| 148 | C2×C4 | — | 2 | 12 | no | — |
| 149 | C2×C4 | — | 4 | 12 | no | — |
| 150 | C2×C4 | — | 8 | 24 | no | — |
| 151 | C2×C4 | C2×#127 | 1 | 6 | yes | — |
| 152 | C2×C4 | C2×#127 | 1 | 3 | yes | — |
| 153 | C2×C4 | C2×#135 | 1 | 12 | yes | — |
| 154 | C2×C4 | C23×D4 | 1 | 3 | yes | — |
| 155 | C8 | — | 2 | 12 | no | — |
| 156 | C8 | C2×#135 | 1 | 12 | yes | — |
| 157 | C8 | C2×#135 | 1 | 12 | yes | — |
| 158 | C22 | — | 2 | 8 | no | — |
| 159 | C22 | — | 2 | 6 | no | — |
| 160 | C22 | — | 8 | 12 | no | — |
| 161 | C22 | #3147 | 1 | 6 | yes | — |
| 162 | C22 | C22×#127 | 1 | 1 | yes | characteristic |
| 163 | C4 | — | 2 | 6 | no | — |
| 164 | C4 | — | 8 | 12 | no | — |
| 165 | C4 | #3147 | 1 | 6 | yes | — |
| 166 | C4 | C22×#135 | 1 | 6 | yes | — |
| 167 | C2 | — | 2 | 6 | no | — |
| 168 | C2 | — | 16 | 4 | no | — |
| 169 | C2 | C2×#3147 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C42 | #127 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×#127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #60673 | C2 | semidirect |
| 2. #19604 | C2 | semidirect |
| 3. #19603 | C2 | semidirect |
| 4. #19600 | C2 | semidirect |
| 5. #19596 | C2 | semidirect |
| 6. C22×C82 | C2 | semidirect |
| 18. #1713 | C22 | semidirect |
| 19. #1712 | C22 | semidirect |
| 20. #1711 | C22 | semidirect |
| 21. #1710 | C22 | semidirect |
| 22. #1709 | C22 | semidirect |
| 23. #1708 | C22 | semidirect |
| 24. #1707 | C22 | semidirect |
| 25. #1706 | C22 | semidirect |
| 26. #1705 | C22 | semidirect |
| 125. C2×C8 | #135 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60673 | 2 |
| #19604 | 2 |
| #19603 | 2 |
| #19600 | 2 |
| #19596 | 2 |
| C22×C82 | 2 |