Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
557 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 557 conjugacy classes of subgroups — too many to draw. Only its 134 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 | #19332 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #17981 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #10530 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #8826 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #8640 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #3253 | C22 | 1 | 4 | yes | — |
| 7 | #3169 | C22 | 1 | 2 | yes | — |
| 8 | #2931 | C22 | 1 | 4 | yes | — |
| 9 | #1848 | C22 | 1 | 2 | yes | — |
| 10 | #1697 | — | 2 | 8 | no | — |
| 11 | #1675 | — | 2 | 4 | no | — |
| 12 | #1667 | — | 2 | 4 | no | — |
| 13 | #1586 | — | 2 | 8 | no | — |
| 14 | #1567 | C22 | 1 | 4 | yes | — |
| 15 | #1566 | C22 | 1 | 2 | yes | — |
| 16 | #1560 | C22 | 1 | 8 | yes | — |
| 17 | #1492 | C22 | 1 | 4 | yes | — |
| 18 | C2×#322 | C22 | 1 | 1 | yes | characteristic |
| 19 | #1309 | C22 | 1 | 4 | yes | — |
| 20 | C2×#131 | — | 2 | 4 | no | — |
| 21 | #532 | C23 | 1 | 4 | yes | — |
| 22 | #500 | D4 | 1 | 2 | yes | — |
| 23 | #493 | D4 | 1 | 2 | yes | — |
| 24 | #486 | D4 | 1 | 4 | yes | — |
| 25 | #476 | — | 2 | 8 | no | — |
| 26 | #469 | — | 4 | 4 | no | — |
| 27 | #466 | — | 2 | 4 | no | — |
| 28 | #459 | — | 2 | 8 | no | — |
| 29 | C4×D8 | — | 2 | 4 | no | — |
| 30 | #436 | — | 2 | 4 | no | — |
| 31 | #436 | — | 2 | 4 | no | — |
| 32 | C8×D4 | — | 2 | 4 | no | — |
| 33 | C8×D4 | — | 2 | 4 | no | — |
| 34 | C2×#108 | — | 4 | 2 | no | — |
| 35 | #423 | C23 | 1 | 4 | yes | — |
| 36 | #423 | C23 | 1 | 4 | yes | — |
| 37 | #408 | — | 2 | 4 | no | — |
| 38 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 39 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 40 | #335 | — | 2 | 4 | no | — |
| 41 | #335 | — | 2 | 4 | no | — |
| 42 | #334 | — | 2 | 4 | no | — |
| 43 | #333 | — | 2 | 8 | no | — |
| 44 | #331 | — | 2 | 8 | no | — |
| 45 | D4⋊C8 | — | 2 | 8 | no | — |
| 46 | #322 | D4 | 1 | 4 | yes | — |
| 47 | C2×#41 | — | 2 | 4 | no | — |
| 48 | C2×D8 | — | 2 | 4 | no | — |
| 49 | C2×D8 | — | 4 | 4 | no | — |
| 50 | #131 | — | 8 | 8 | no | — |
| 51 | C2×M16 | — | 2 | 4 | no | — |
| 52 | C2×M16 | — | 2 | 4 | no | — |
| 53 | C2×M16 | — | 4 | 4 | no | — |
| 54 | C22×C8 | — | 2 | 1 | no | — |
| 55 | C22×C8 | — | 4 | 4 | no | — |
| 56 | C22×C8 | #41 | 1 | 4 | yes | — |
| 57 | C4⋊Q8 | C2×D4 | 1 | 4 | yes | — |
| 58 | #127 | C2×D4 | 1 | 4 | yes | — |
| 59 | #124 | — | 2 | 4 | no | — |
| 60 | #121 | — | 2 | 4 | no | — |
| 61 | #121 | — | 2 | 4 | no | — |
| 62 | C4×D4 | — | 2 | 4 | no | — |
| 63 | C4×D4 | — | 2 | 4 | no | — |
| 64 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 65 | #108 | — | 8 | 4 | no | — |
| 66 | #107 | — | 2 | 4 | no | — |
| 67 | #106 | — | 4 | 4 | no | — |
| 68 | C4⋊C8 | — | 2 | 4 | no | — |
| 69 | C4⋊C8 | — | 2 | 4 | no | — |
| 70 | C4⋊C8 | — | 2 | 8 | no | — |
| 71 | #102 | — | 2 | 8 | no | — |
| 72 | #102 | — | 4 | 8 | no | — |
| 73 | C22⋊C8 | — | 4 | 4 | no | — |
| 74 | C22⋊C8 | — | 4 | 4 | no | — |
| 75 | C4×C8 | — | 2 | 1 | no | — |
| 76 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 77 | C4×C8 | C2×D4 | 1 | 4 | yes | — |
| 78 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 79 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 80 | #41 | — | 8 | 4 | no | — |
| 81 | #41 | — | 8 | 4 | no | — |
| 82 | C2×Q8 | — | 2 | 4 | no | — |
| 83 | C2×D4 | — | 2 | 4 | no | — |
| 84 | C2×D4 | — | 2 | 4 | no | — |
| 85 | C2×D4 | — | 2 | 4 | no | — |
| 86 | C2×D4 | — | 4 | 4 | no | — |
| 87 | C22×C4 | — | 4 | 4 | no | — |
| 88 | C22×C4 | C2×#41 | 1 | 2 | yes | — |
| 89 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 90 | D8 | — | 8 | 4 | no | — |
| 91 | D8 | — | 16 | 4 | no | — |
| 92 | M16 | — | 8 | 8 | no | — |
| 93 | M16 | — | 8 | 4 | no | — |
| 94 | M16 | — | 8 | 4 | no | — |
| 95 | C2×C8 | — | 2 | 2 | no | — |
| 96 | C2×C8 | — | 2 | 2 | no | — |
| 97 | C2×C8 | — | 2 | 2 | no | — |
| 98 | C2×C8 | — | 4 | 4 | no | — |
| 99 | C2×C8 | — | 4 | 4 | no | — |
| 100 | C2×C8 | — | 4 | 4 | no | — |
| 101 | C2×C8 | — | 8 | 8 | no | — |
| 102 | C2×C8 | #121 | 1 | 4 | yes | — |
| 103 | C2×C8 | #121 | 1 | 4 | yes | — |
| 104 | C4⋊C4 | — | 2 | 4 | no | — |
| 105 | C4⋊C4 | — | 2 | 4 | no | — |
| 106 | C22⋊C4 | — | 4 | 4 | no | — |
| 107 | C22⋊C4 | — | 4 | 4 | no | — |
| 108 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 109 | C42 | C22×D4 | 1 | 2 | yes | — |
| 110 | C42 | 2−1+4 | 1 | 1 | yes | characteristic |
| 111 | C23 | — | 4 | 4 | no | — |
| 112 | C23 | #555 | 1 | 1 | yes | characteristic |
| 113 | Q8 | — | 8 | 4 | no | — |
| 114 | D4 | — | 4 | 4 | no | — |
| 115 | D4 | — | 4 | 4 | no | — |
| 116 | D4 | — | 4 | 8 | no | — |
| 117 | D4 | — | 8 | 4 | no | — |
| 118 | D4 | — | 8 | 8 | no | — |
| 119 | C2×C4 | — | 2 | 1 | no | — |
| 120 | C2×C4 | — | 2 | 1 | no | — |
| 121 | C2×C4 | — | 2 | 4 | no | — |
| 122 | C2×C4 | — | 4 | 4 | no | — |
| 123 | C2×C4 | — | 8 | 4 | no | — |
| 124 | C2×C4 | — | 8 | 4 | no | — |
| 125 | C2×C4 | C2×#121 | 1 | 2 | yes | — |
| 126 | C2×C4 | #537 | 1 | 1 | yes | characteristic |
| 127 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 128 | C2×C4 | #547 | 1 | 2 | yes | — |
| 129 | C8 | — | 2 | 4 | no | — |
| 130 | C8 | — | 2 | 4 | no | — |
| 131 | C8 | — | 2 | 4 | no | — |
| 132 | C8 | — | 8 | 4 | no | — |
| 133 | C8 | — | 8 | 4 | no | — |
| 134 | C22 | — | 2 | 2 | no | — |
| 135 | C22 | — | 2 | 1 | no | — |
| 136 | C22 | — | 8 | 4 | no | — |
| 137 | C22 | — | 8 | 4 | no | — |
| 138 | C22 | — | 8 | 4 | no | — |
| 139 | C22 | #2537 | 1 | 1 | yes | characteristic |
| 140 | C4 | — | 2 | 2 | no | — |
| 141 | C4 | — | 2 | 1 | no | — |
| 142 | C4 | — | 8 | 4 | no | — |
| 143 | C4 | #3068 | 1 | 4 | yes | — |
| 144 | C4 | #3292 | 1 | 1 | yes | characteristic |
| 145 | C4 | #3293 | 1 | 1 | yes | characteristic |
| 146 | C2 | — | 4 | 1 | no | — |
| 147 | C2 | — | 16 | 4 | no | — |
| 148 | C2 | #28243 | 1 | 2 | yes | — |
| 149 | C2 | #28247 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2537 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #19332 | C2 | semidirect |
| 2. #17981 | C2 | semidirect |
| 3. #10530 | C2 | semidirect |
| 4. #8826 | C2 | semidirect |
| 5. #8640 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #19332 | 2 |
| #17981 | 2 |
| #10530 | 2 |
| #8826 | 2 |
| #8640 | 2 |