Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1054 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 1054 conjugacy classes of subgroups — too many to draw. Only its 250 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 | C2×#2862 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20772 | C2 | 1 | 2 | yes | maximal |
| 3 | #11382 | C2 | 1 | 4 | yes | maximal |
| 4 | #11355 | C2 | 1 | 8 | yes | maximal |
| 5 | C22×C4⋊Q8 | C22 | 1 | 2 | yes | — |
| 6 | #2862 | — | 2 | 8 | no | — |
| 7 | C2×#391 | C4 | 1 | 8 | yes | — |
| 8 | C2×#382 | C22 | 1 | 4 | yes | — |
| 9 | C2×C43 | C22 | 1 | 1 | yes | characteristic |
| 10 | #1839 | — | 2 | 8 | no | — |
| 11 | C2×#343 | C22 | 1 | 16 | yes | — |
| 12 | #1728 | C22 | 1 | 8 | yes | — |
| 13 | #1728 | C22 | 1 | 4 | yes | — |
| 14 | C23×Q8 | C2×C4 | 1 | 4 | yes | — |
| 15 | C22×M16 | C23 | 1 | 8 | yes | — |
| 16 | C2×C4⋊Q8 | — | 2 | 8 | no | — |
| 17 | C2×C4⋊Q8 | — | 4 | 8 | no | — |
| 18 | C22×C4⋊C4 | C2×C4 | 1 | 4 | yes | — |
| 19 | C22×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 20 | C22×C42 | C2×C4 | 1 | 4 | yes | — |
| 21 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 22 | C22×C42 | C23 | 1 | 2 | yes | — |
| 23 | #391 | — | 2 | 16 | no | — |
| 24 | #391 | — | 2 | 16 | no | — |
| 25 | #382 | — | 2 | 16 | no | — |
| 26 | C2×#95 | C2×C4 | 1 | 16 | yes | — |
| 27 | C43 | D4 | 1 | 8 | yes | — |
| 28 | #343 | — | 4 | 16 | no | — |
| 29 | #336 | — | 2 | 8 | no | — |
| 30 | #336 | — | 2 | 16 | no | — |
| 31 | C22×Q8 | — | 2 | 8 | no | — |
| 32 | C22×Q8 | — | 4 | 8 | no | — |
| 33 | C22×Q8 | — | 4 | 16 | no | — |
| 34 | C23×C4 | C22×C4 | 1 | 2 | yes | — |
| 35 | C23×C4 | C22×C4 | 1 | 8 | yes | — |
| 36 | C23×C4 | C22×C4 | 1 | 4 | yes | — |
| 37 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 38 | C2×M16 | — | 2 | 16 | no | — |
| 39 | C2×M16 | — | 2 | 32 | no | — |
| 40 | C22×C8 | — | 2 | 8 | no | — |
| 41 | C4⋊Q8 | — | 4 | 32 | no | — |
| 42 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 43 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 44 | C2×C4⋊C4 | — | 2 | 16 | no | — |
| 45 | C2×C4⋊C4 | — | 4 | 8 | no | — |
| 46 | C2×C42 | — | 2 | 4 | no | — |
| 47 | C2×C42 | — | 2 | 8 | no | — |
| 48 | C2×C42 | — | 4 | 8 | no | — |
| 49 | C2×C42 | C22⋊C4 | 1 | 8 | yes | — |
| 50 | C2×C42 | C22⋊C4 | 1 | 8 | yes | — |
| 51 | C2×C42 | C2×D4 | 1 | 4 | yes | — |
| 52 | C2×C42 | C2×D4 | 1 | 8 | yes | — |
| 53 | #95 | — | 2 | 32 | no | — |
| 54 | C24 | C23×C4 | 1 | 1 | yes | characteristic |
| 55 | C2×Q8 | — | 4 | 16 | no | — |
| 56 | C2×Q8 | — | 4 | 16 | no | — |
| 57 | C2×Q8 | — | 4 | 16 | no | — |
| 58 | C2×Q8 | — | 8 | 32 | no | — |
| 59 | C22×C4 | — | 2 | 4 | no | — |
| 60 | C22×C4 | — | 2 | 8 | no | — |
| 61 | C22×C4 | — | 2 | 8 | no | — |
| 62 | C22×C4 | — | 2 | 8 | no | — |
| 63 | C22×C4 | — | 2 | 16 | no | — |
| 64 | C22×C4 | — | 4 | 8 | no | — |
| 65 | C22×C4 | — | 4 | 16 | no | — |
| 66 | C22×C4 | C2×C22⋊C4 | 1 | 8 | yes | — |
| 67 | C22×C4 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 68 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 69 | C22×C4 | H(5,𝔽2) | 1 | 4 | yes | — |
| 70 | C22×C4 | 2−1+4 | 1 | 8 | yes | — |
| 71 | M16 | — | 4 | 32 | no | — |
| 72 | C2×C8 | — | 4 | 8 | no | — |
| 73 | C2×C8 | — | 4 | 16 | no | — |
| 74 | C4⋊C4 | — | 4 | 16 | no | — |
| 75 | C4⋊C4 | — | 4 | 16 | no | — |
| 76 | C42 | — | 2 | 8 | no | — |
| 77 | C42 | — | 2 | 16 | no | — |
| 78 | C42 | — | 2 | 16 | no | — |
| 79 | C42 | — | 4 | 8 | no | — |
| 80 | C23 | — | 2 | 4 | no | — |
| 81 | C23 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 82 | C23 | #518 | 1 | 2 | yes | — |
| 83 | C23 | #519 | 1 | 4 | yes | — |
| 84 | Q8 | — | 8 | 32 | no | — |
| 85 | C2×C4 | — | 2 | 16 | no | — |
| 86 | C2×C4 | — | 2 | 4 | no | — |
| 87 | C2×C4 | — | 2 | 4 | no | — |
| 88 | C2×C4 | — | 2 | 8 | no | — |
| 89 | C2×C4 | — | 2 | 16 | no | — |
| 90 | C2×C4 | — | 4 | 16 | no | — |
| 91 | C2×C4 | — | 4 | 8 | no | — |
| 92 | C2×C4 | — | 4 | 16 | no | — |
| 93 | C2×C4 | — | 8 | 16 | no | — |
| 94 | C2×C4 | #413 | 1 | 8 | yes | — |
| 95 | C2×C4 | #535 | 1 | 4 | yes | — |
| 96 | C2×C4 | #536 | 1 | 8 | yes | — |
| 97 | C2×C4 | #563 | 1 | 16 | yes | — |
| 98 | C8 | — | 8 | 8 | no | — |
| 99 | C22 | — | 2 | 4 | no | — |
| 100 | C22 | — | 2 | 8 | no | — |
| 101 | C22 | #2309 | 1 | 1 | yes | characteristic |
| 102 | C22 | #2310 | 1 | 2 | yes | — |
| 103 | C22 | #2381 | 1 | 4 | yes | — |
| 104 | C22 | C2×#413 | 1 | 4 | yes | — |
| 105 | C4 | — | 2 | 8 | no | — |
| 106 | C4 | — | 4 | 8 | no | — |
| 107 | C4 | — | 8 | 8 | no | — |
| 108 | C4 | #2821 | 1 | 8 | yes | — |
| 109 | C2 | — | 2 | 4 | no | — |
| 110 | C2 | #14663 | 1 | 1 | yes | characteristic |
| 111 | C2 | #21798 | 1 | 2 | yes | — |
| 112 | C2 | #21831 | 1 | 4 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator, prime radical | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2862 | 2 |
| #20772 | 2 |
| #11382 | 2 |
| #11355 | 2 |