Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
604 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 604 conjugacy classes of subgroups — too many to draw. Only its 128 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 | C4×#564 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #9872 | C2 | 1 | 3 | yes | maximal |
| 3 | #7568 | C2 | 1 | 3 | yes | maximal |
| 4 | C2×#564 | C4 | 1 | 1 | yes | characteristic |
| 5 | #2382 | — | 2 | 6 | no | — |
| 6 | #2382 | C4 | 1 | 3 | yes | — |
| 7 | C4×#125 | — | 2 | 6 | no | — |
| 8 | C4×#125 | C22 | 1 | 3 | yes | — |
| 9 | #1842 | C22 | 1 | 1 | yes | characteristic |
| 10 | #1744 | C22 | 1 | 3 | yes | — |
| 11 | #1279 | — | 2 | 6 | no | — |
| 12 | #1271 | — | 2 | 6 | no | — |
| 13 | #564 | — | 2 | 6 | no | — |
| 14 | #564 | C8 | 1 | 4 | yes | complemented |
| 15 | C2×#125 | — | 2 | 6 | no | — |
| 16 | C2×#125 | C2×C4 | 1 | 3 | yes | — |
| 17 | C2×C4⋊C8 | — | 2 | 3 | no | — |
| 18 | C2×C4×C8 | — | 2 | 3 | no | — |
| 19 | #389 | — | 2 | 6 | no | — |
| 20 | #389 | — | 2 | 6 | no | — |
| 21 | #389 | D4 | 1 | 6 | yes | — |
| 22 | #387 | — | 2 | 6 | no | — |
| 23 | #387 | — | 2 | 6 | no | — |
| 24 | #387 | — | 2 | 6 | no | — |
| 25 | #387 | — | 2 | 6 | no | — |
| 26 | #387 | — | 2 | 6 | no | — |
| 27 | #382 | — | 2 | 6 | no | — |
| 28 | #382 | C2×C4 | 1 | 3 | yes | — |
| 29 | C4×C4⋊C4 | — | 2 | 6 | no | — |
| 30 | C4×C4⋊C4 | — | 2 | 6 | no | — |
| 31 | C4×C4⋊C4 | D4 | 1 | 6 | yes | — |
| 32 | C43 | — | 2 | 2 | no | — |
| 33 | C43 | C23 | 1 | 1 | yes | characteristic |
| 34 | #336 | — | 2 | 6 | no | — |
| 35 | C22×C8 | — | 4 | 3 | no | — |
| 36 | C22×C8 | — | 4 | 1 | no | — |
| 37 | #125 | — | 2 | 24 | no | — |
| 38 | #125 | — | 2 | 24 | no | — |
| 39 | #125 | — | 2 | 6 | no | — |
| 40 | #125 | C2×C8 | 1 | 6 | yes | — |
| 41 | #125 | M16 | 1 | 6 | yes | — |
| 42 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 43 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 44 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 45 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 46 | C2×C4⋊C4 | C22⋊C4 | 1 | 6 | yes | — |
| 47 | C2×C4⋊C4 | C22⋊C4 | 1 | 6 | yes | — |
| 48 | C2×C42 | — | 2 | 2 | no | — |
| 49 | C2×C42 | — | 2 | 6 | no | — |
| 50 | C2×C42 | — | 2 | 6 | no | — |
| 51 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 52 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 53 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 54 | C4⋊C8 | — | 4 | 6 | no | — |
| 55 | C4×C8 | — | 2 | 6 | no | — |
| 56 | C4×C8 | — | 4 | 3 | no | — |
| 57 | #95 | — | 2 | 6 | no | — |
| 58 | #95 | — | 2 | 6 | no | — |
| 59 | #95 | — | 2 | 6 | no | — |
| 60 | #95 | — | 2 | 6 | no | — |
| 61 | #95 | — | 2 | 6 | no | — |
| 62 | C22×C4 | — | 2 | 6 | no | — |
| 63 | C22×C4 | — | 2 | 6 | no | — |
| 64 | C22×C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 65 | C22×C4 | #120 | 1 | 3 | yes | — |
| 66 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 67 | C2×C8 | — | 4 | 6 | no | — |
| 68 | C2×C8 | — | 8 | 6 | no | — |
| 69 | C2×C8 | — | 8 | 3 | no | — |
| 70 | C4⋊C4 | — | 2 | 24 | no | — |
| 71 | C4⋊C4 | — | 2 | 12 | no | — |
| 72 | C4⋊C4 | — | 2 | 12 | no | — |
| 73 | C4⋊C4 | — | 2 | 6 | no | — |
| 74 | C4⋊C4 | — | 4 | 6 | no | — |
| 75 | C4⋊C4 | — | 8 | 6 | no | — |
| 76 | C4⋊C4 | — | 8 | 6 | no | — |
| 77 | C4⋊C4 | C22⋊C8 | 1 | 12 | yes | — |
| 78 | C42 | — | 2 | 8 | no | — |
| 79 | C42 | — | 2 | 3 | no | — |
| 80 | C42 | — | 2 | 3 | no | — |
| 81 | C42 | — | 4 | 3 | no | — |
| 82 | C42 | — | 4 | 6 | no | — |
| 83 | C42 | — | 4 | 6 | no | — |
| 84 | C42 | — | 8 | 6 | no | — |
| 85 | C42 | C22×C8 | 1 | 1 | yes | characteristic |
| 86 | C42 | C2×M16 | 1 | 3 | yes | — |
| 87 | C23 | #379 | 1 | 1 | yes | characteristic |
| 88 | C2×C4 | — | 2 | 3 | no | — |
| 89 | C2×C4 | — | 2 | 3 | no | — |
| 90 | C2×C4 | — | 2 | 3 | no | — |
| 91 | C2×C4 | — | 2 | 6 | no | — |
| 92 | C2×C4 | — | 2 | 12 | no | — |
| 93 | C2×C4 | — | 4 | 6 | no | — |
| 94 | C2×C4 | — | 4 | 6 | no | — |
| 95 | C2×C4 | — | 4 | 6 | no | — |
| 96 | C2×C4 | — | 4 | 6 | no | — |
| 97 | C2×C4 | — | 4 | 6 | no | — |
| 98 | C2×C4 | C2×C22⋊C8 | 1 | 3 | yes | — |
| 99 | C2×C4 | #407 | 1 | 3 | yes | — |
| 100 | C2×C4 | #454 | 1 | 3 | yes | — |
| 101 | C2×C4 | #456 | 1 | 3 | yes | — |
| 102 | C8 | — | 8 | 6 | no | — |
| 103 | C8 | — | 16 | 1 | no | — |
| 104 | C22 | #1779 | 1 | 1 | yes | characteristic |
| 105 | C22 | #1793 | 1 | 3 | yes | — |
| 106 | C22 | #1794 | 1 | 3 | yes | — |
| 107 | C4 | — | 2 | 3 | no | — |
| 108 | C4 | — | 2 | 3 | no | — |
| 109 | C4 | — | 4 | 3 | no | — |
| 110 | C4 | — | 4 | 12 | no | — |
| 111 | C4 | — | 8 | 6 | no | — |
| 112 | C4 | #1658 | 1 | 1 | yes | characteristic |
| 113 | C4 | #1663 | 1 | 3 | yes | — |
| 114 | C2 | #8319 | 1 | 3 | yes | — |
| 115 | C2 | #8505 | 1 | 3 | yes | — |
| 116 | C2 | #8509 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C43 | C23 |
| socle, prime radical | C23 | #379 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 14. #564 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#564 | 2 |
| #9872 | 2 |
| #7568 | 2 |