Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
954 conjugacy classes of subgroups; only the 282 normal subgroups are shown.
This group has 954 conjugacy classes of subgroups — too many to draw. Only its 282 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 | #33313 | C2 | 1 | 2 | yes | maximal |
| 2 | #33242 | C2 | 1 | 4 | yes | maximal |
| 3 | C2×#1723 | C2 | 1 | 8 | yes | maximal |
| 4 | C4×#322 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C2×#500 | C22 | 1 | 4 | yes | — |
| 6 | C2×#500 | C22 | 1 | 4 | yes | — |
| 7 | C2×#498 | C22 | 1 | 4 | yes | — |
| 8 | C2×#494 | C22 | 1 | 4 | yes | — |
| 9 | C2×#487 | C22 | 1 | 8 | yes | — |
| 10 | #2862 | C22 | 1 | 4 | yes | — |
| 11 | #1969 | — | 2 | 16 | no | — |
| 12 | #1846 | — | 2 | 8 | no | — |
| 13 | C42×C8 | C22 | 1 | 1 | yes | characteristic |
| 14 | C42×C8 | C22 | 1 | 2 | yes | — |
| 15 | #1723 | — | 2 | 32 | no | — |
| 16 | C2×#322 | C22 | 1 | 4 | yes | — |
| 17 | C22×Q16 | — | 2 | 8 | no | — |
| 18 | C2×C4⋊Q8 | C23 | 1 | 8 | yes | — |
| 19 | #500 | — | 2 | 16 | no | — |
| 20 | #500 | — | 2 | 16 | no | — |
| 21 | #498 | — | 2 | 16 | no | — |
| 22 | #494 | — | 2 | 16 | no | — |
| 23 | #487 | — | 2 | 16 | no | — |
| 24 | #487 | — | 2 | 16 | no | — |
| 25 | C2×#107 | — | 2 | 8 | no | — |
| 26 | C2×#107 | — | 2 | 8 | no | — |
| 27 | C2×#106 | — | 2 | 8 | no | — |
| 28 | C2×#103 | — | 2 | 16 | no | — |
| 29 | C2×C4×C8 | D4 | 1 | 8 | yes | — |
| 30 | C2×C4×C8 | D4 | 1 | 4 | yes | — |
| 31 | C2×C4×C8 | C23 | 1 | 4 | yes | — |
| 32 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 33 | #391 | — | 2 | 16 | no | — |
| 34 | #382 | — | 2 | 8 | no | — |
| 35 | C43 | C23 | 1 | 1 | yes | characteristic |
| 36 | #322 | D4 | 1 | 16 | yes | — |
| 37 | C22×Q8 | — | 2 | 8 | no | — |
| 38 | C2×Q16 | — | 4 | 32 | no | — |
| 39 | C2×Q16 | — | 4 | 16 | no | — |
| 40 | C22×C8 | C2×D4 | 1 | 8 | yes | — |
| 41 | C22×C8 | C2×D4 | 1 | 4 | yes | — |
| 42 | C4⋊Q8 | — | 2 | 8 | no | — |
| 43 | C4⋊Q8 | — | 2 | 8 | no | — |
| 44 | C4⋊Q8 | — | 2 | 8 | no | — |
| 45 | C4⋊Q8 | — | 2 | 8 | no | — |
| 46 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 47 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 48 | C2×C42 | C2×D4 | 1 | 4 | yes | — |
| 49 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 50 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 51 | #107 | — | 4 | 16 | no | — |
| 52 | #107 | — | 4 | 16 | no | — |
| 53 | #106 | — | 4 | 16 | no | — |
| 54 | #103 | — | 4 | 32 | no | — |
| 55 | C4×C8 | — | 4 | 8 | no | — |
| 56 | C4×C8 | Q16 | 1 | 16 | yes | complemented |
| 57 | C4×C8 | C2×D4 | 1 | 16 | yes | — |
| 58 | C4×C8 | C2×D4 | 1 | 8 | yes | — |
| 59 | #95 | — | 4 | 16 | no | — |
| 60 | C2×Q8 | — | 4 | 8 | no | — |
| 61 | C2×Q8 | — | 4 | 8 | no | — |
| 62 | C2×Q8 | — | 4 | 32 | no | — |
| 63 | C22×C4 | — | 4 | 8 | no | — |
| 64 | C22×C4 | #127 | 1 | 4 | yes | — |
| 65 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 66 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 67 | Q16 | — | 8 | 32 | no | — |
| 68 | C2×C8 | — | 2 | 16 | no | — |
| 69 | C2×C8 | #127 | 1 | 16 | yes | — |
| 70 | C2×C8 | #127 | 1 | 8 | yes | — |
| 71 | C2×C8 | C2×Q16 | 1 | 16 | yes | — |
| 72 | C4⋊C4 | — | 4 | 8 | no | — |
| 73 | C4⋊C4 | — | 4 | 8 | no | — |
| 74 | C4⋊C4 | — | 4 | 8 | no | — |
| 75 | C4⋊C4 | — | 4 | 8 | no | — |
| 76 | C42 | — | 2 | 8 | no | — |
| 77 | C42 | C2×Q16 | 1 | 8 | yes | — |
| 78 | C42 | C22×D4 | 1 | 4 | yes | — |
| 79 | C23 | C2×#127 | 1 | 1 | yes | characteristic |
| 80 | Q8 | — | 8 | 16 | no | — |
| 81 | Q8 | — | 8 | 16 | no | — |
| 82 | C2×C4 | — | 2 | 8 | no | — |
| 83 | C2×C4 | — | 8 | 8 | no | — |
| 84 | C2×C4 | — | 8 | 16 | no | — |
| 85 | C2×C4 | #494 | 1 | 8 | yes | — |
| 86 | C2×C4 | C2×#127 | 1 | 2 | yes | — |
| 87 | C2×C4 | C2×#127 | 1 | 4 | yes | — |
| 88 | C2×C4 | C22×Q16 | 1 | 4 | yes | — |
| 89 | C2×C4 | #575 | 1 | 8 | yes | — |
| 90 | C8 | — | 2 | 16 | no | — |
| 91 | C8 | #494 | 1 | 16 | yes | — |
| 92 | C22 | #2867 | 1 | 1 | yes | characteristic |
| 93 | C22 | C2×#494 | 1 | 2 | yes | — |
| 94 | C22 | #3150 | 1 | 4 | yes | — |
| 95 | C4 | — | 2 | 8 | no | — |
| 96 | C4 | — | 16 | 8 | no | — |
| 97 | C4 | C2×#494 | 1 | 4 | yes | — |
| 98 | C4 | #3153 | 1 | 8 | yes | — |
| 99 | C2 | #19618 | 1 | 4 | yes | — |
| 100 | C2 | #33302 | 1 | 1 | yes | characteristic |
| 101 | C2 | #33306 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 56. C4×C8 | Q16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33313 | 2 |
| #33242 | 2 |
| C2×#1723 | 2 |
| C4×#322 | 2 |