Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
747 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 747 conjugacy classes of subgroups — too many to draw. Only its 122 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 | #18686 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | #10530 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #9856 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #3549 | C22 | 1 | 2 | yes | complemented |
| 5 | #3253 | C22 | 1 | 6 | yes | — |
| 6 | #3147 | C22 | 1 | 3 | yes | — |
| 7 | #1848 | C22 | 1 | 3 | yes | complemented |
| 8 | #1634 | — | 2 | 6 | no | — |
| 9 | #1629 | — | 2 | 6 | no | — |
| 10 | #1624 | — | 2 | 6 | no | — |
| 11 | #1619 | — | 2 | 6 | no | — |
| 12 | #1560 | C22 | 1 | 6 | yes | complemented |
| 13 | #1492 | C22 | 1 | 12 | yes | complemented |
| 14 | C2×C82 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | #1309 | C22 | 1 | 2 | yes | complemented |
| 16 | C2×#135 | — | 4 | 3 | no | — |
| 17 | H(3,𝔽4) | — | 4 | 2 | no | — |
| 18 | #555 | — | 2 | 6 | no | — |
| 19 | D42 | — | 2 | 6 | no | — |
| 20 | #535 | — | 4 | 6 | no | — |
| 21 | #532 | C23 | 1 | 6 | yes | — |
| 22 | #495 | — | 4 | 3 | no | — |
| 23 | #494 | D4 | 1 | 3 | yes | — |
| 24 | #493 | D4 | 1 | 3 | yes | — |
| 25 | #492 | D4 | 1 | 6 | yes | — |
| 26 | #476 | — | 2 | 12 | no | — |
| 27 | #469 | — | 4 | 6 | no | — |
| 28 | #459 | — | 2 | 12 | no | — |
| 29 | C2×#108 | — | 4 | 3 | no | — |
| 30 | #423 | C23 | 1 | 6 | yes | — |
| 31 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 32 | #333 | — | 2 | 6 | no | — |
| 33 | #331 | — | 2 | 6 | no | — |
| 34 | D4⋊C8 | — | 2 | 12 | no | — |
| 35 | D4⋊C8 | — | 2 | 12 | no | — |
| 36 | C82 | — | 4 | 1 | no | — |
| 37 | C2×#41 | — | 2 | 6 | no | — |
| 38 | C22×D4 | — | 4 | 6 | no | — |
| 39 | #135 | — | 8 | 6 | no | — |
| 40 | #135 | — | 8 | 6 | no | — |
| 41 | C2×Q16 | — | 4 | 3 | no | — |
| 42 | C2×SD16 | — | 4 | 6 | no | — |
| 43 | C2×D8 | — | 4 | 3 | no | — |
| 44 | C2×D8 | — | 4 | 6 | no | — |
| 45 | C2×M16 | — | 2 | 6 | no | — |
| 46 | C22×C8 | — | 2 | 3 | no | — |
| 47 | C4⋊Q8 | C2×D4 | 1 | 6 | yes | — |
| 48 | #127 | — | 4 | 4 | no | — |
| 49 | #127 | C2×D4 | 1 | 6 | yes | — |
| 50 | #124 | — | 2 | 6 | no | — |
| 51 | #124 | — | 4 | 12 | no | — |
| 52 | #121 | — | 2 | 12 | no | — |
| 53 | #121 | — | 4 | 12 | no | — |
| 54 | #120 | — | 4 | 12 | no | — |
| 55 | #120 | — | 4 | 4 | no | — |
| 56 | C4×D4 | — | 2 | 12 | no | — |
| 57 | #117 | — | 4 | 6 | no | — |
| 58 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 59 | #108 | — | 8 | 6 | no | — |
| 60 | #106 | — | 4 | 6 | no | — |
| 61 | C4⋊C8 | — | 2 | 12 | no | — |
| 62 | #102 | — | 4 | 12 | no | — |
| 63 | C4×C8 | — | 2 | 3 | no | — |
| 64 | C4×C8 | C2×D4 | 1 | 3 | yes | — |
| 65 | C4×C8 | C2×D4 | 1 | 3 | yes | — |
| 66 | C24 | — | 4 | 4 | no | — |
| 67 | #41 | — | 8 | 6 | no | — |
| 68 | #41 | — | 8 | 6 | no | — |
| 69 | C2×Q8 | — | 2 | 6 | no | — |
| 70 | C2×D4 | — | 2 | 6 | no | — |
| 71 | C2×D4 | — | 2 | 12 | no | — |
| 72 | C2×D4 | — | 4 | 12 | no | — |
| 73 | C2×D4 | — | 4 | 6 | no | — |
| 74 | C2×D4 | — | 16 | 6 | no | — |
| 75 | C2×D4 | — | 16 | 6 | no | — |
| 76 | C22×C4 | — | 4 | 6 | no | — |
| 77 | C22×C4 | H(5,𝔽2) | 1 | 3 | yes | — |
| 78 | Q16 | — | 8 | 6 | no | — |
| 79 | SD16 | — | 8 | 12 | no | — |
| 80 | D8 | — | 8 | 6 | no | — |
| 81 | D8 | — | 16 | 6 | no | — |
| 82 | M16 | — | 8 | 6 | no | — |
| 83 | C2×C8 | — | 2 | 6 | no | — |
| 84 | C2×C8 | — | 2 | 6 | no | — |
| 85 | C2×C8 | — | 2 | 3 | no | — |
| 86 | C2×C8 | — | 2 | 3 | no | — |
| 87 | C2×C8 | — | 4 | 6 | no | — |
| 88 | C4⋊C4 | — | 2 | 12 | no | — |
| 89 | C22⋊C4 | — | 4 | 12 | no | — |
| 90 | C22⋊C4 | — | 4 | 12 | no | — |
| 91 | C42 | — | 4 | 4 | no | — |
| 92 | C42 | C22×D4 | 1 | 3 | yes | — |
| 93 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 94 | C23 | — | 4 | 6 | no | — |
| 95 | C23 | — | 16 | 12 | no | — |
| 96 | C23 | H(3,𝔽4) | 1 | 1 | yes | characteristic |
| 97 | Q8 | — | 8 | 6 | no | — |
| 98 | D4 | — | 4 | 12 | no | — |
| 99 | D4 | — | 4 | 12 | no | — |
| 100 | D4 | — | 8 | 6 | no | — |
| 101 | D4 | — | 16 | 6 | no | — |
| 102 | D4 | — | 16 | 12 | no | — |
| 103 | C2×C4 | — | 2 | 3 | no | — |
| 104 | C2×C4 | — | 2 | 3 | no | — |
| 105 | C2×C4 | — | 4 | 6 | no | — |
| 106 | C2×C4 | — | 8 | 6 | no | — |
| 107 | C2×C4 | — | 8 | 6 | no | — |
| 108 | C2×C4 | #534 | 1 | 3 | yes | — |
| 109 | C2×C4 | D42 | 1 | 3 | yes | — |
| 110 | C8 | — | 2 | 6 | no | — |
| 111 | C8 | — | 2 | 6 | no | — |
| 112 | C8 | — | 8 | 6 | no | — |
| 113 | C22 | — | 2 | 3 | no | — |
| 114 | C22 | — | 8 | 6 | no | — |
| 115 | C22 | — | 8 | 12 | no | — |
| 116 | C22 | — | 32 | 8 | no | — |
| 117 | C22 | #2679 | 1 | 1 | yes | characteristic |
| 118 | C4 | — | 2 | 3 | no | — |
| 119 | C4 | — | 8 | 6 | no | — |
| 120 | C4 | #3291 | 1 | 3 | yes | — |
| 121 | C4 | #3292 | 1 | 3 | yes | — |
| 122 | C2 | — | 4 | 1 | no | — |
| 123 | C2 | — | 16 | 6 | no | — |
| 124 | C2 | #31070 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2679 |
| 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. #18686 | C2 | semidirect |
| 2. #10530 | C2 | semidirect |
| 3. #9856 | C2 | semidirect |
| 4. #3549 | C22 | semidirect |
| 7. #1848 | C22 | semidirect |
| 12. #1560 | C22 | semidirect |
| 13. #1492 | C22 | semidirect |
| 14. C2×C82 | C22 | semidirect |
| 15. #1309 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #18686 | 2 |
| #10530 | 2 |
| #9856 | 2 |