Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
154 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 26 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 | #9862 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #7373 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #3147 | C4 | 1 | 1 | yes | characteristic |
| 4 | #1848 | C22 | 1 | 1 | yes | characteristic |
| 5 | #1492 | — | 2 | 2 | no | — |
| 6 | C2×#322 | C4 | 1 | 1 | yes | characteristic |
| 7 | #1327 | — | 2 | 2 | no | — |
| 8 | C2×#135 | — | 4 | 1 | no | — |
| 9 | #532 | — | 2 | 1 | no | — |
| 10 | #495 | — | 4 | 1 | no | — |
| 11 | #494 | C8 | 1 | 1 | yes | characteristic |
| 12 | #493 | C8 | 1 | 1 | yes | characteristic |
| 13 | #492 | — | 2 | 1 | no | — |
| 14 | C2×#108 | — | 4 | 1 | no | — |
| 15 | #423 | D4 | 1 | 2 | yes | — |
| 16 | C2×C4×C8 | — | 2 | 1 | no | — |
| 17 | C2×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 18 | #349 | — | 4 | 2 | no | — |
| 19 | D4⋊C8 | — | 4 | 4 | no | — |
| 20 | #322 | — | 4 | 1 | no | — |
| 21 | C2×#41 | — | 4 | 1 | no | — |
| 22 | #135 | — | 8 | 2 | no | — |
| 23 | #135 | — | 8 | 2 | no | — |
| 24 | C2×Q16 | — | 4 | 1 | no | — |
| 25 | C2×SD16 | — | 4 | 2 | no | — |
| 26 | C2×D8 | — | 4 | 1 | no | — |
| 27 | C2×M16 | — | 2 | 2 | no | — |
| 28 | C22×C8 | — | 2 | 1 | no | — |
| 29 | C22×C8 | — | 4 | 1 | no | — |
| 30 | C4⋊Q8 | — | 2 | 1 | no | — |
| 31 | #127 | — | 2 | 1 | no | — |
| 32 | #124 | — | 4 | 1 | no | — |
| 33 | #121 | — | 4 | 2 | no | — |
| 34 | C4×D4 | — | 4 | 2 | no | — |
| 35 | C2×C42 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 36 | M32 | — | 8 | 2 | no | — |
| 37 | #108 | — | 8 | 2 | no | — |
| 38 | C4⋊C8 | — | 4 | 2 | no | — |
| 39 | C4×C8 | — | 2 | 2 | no | — |
| 40 | C4×C8 | — | 2 | 1 | no | — |
| 41 | C4×C8 | — | 4 | 1 | no | — |
| 42 | C4×C8 | C2×C8 | 1 | 1 | yes | characteristic |
| 43 | C4×C8 | M16 | 1 | 1 | yes | characteristic |
| 44 | #41 | — | 8 | 2 | no | — |
| 45 | #41 | — | 8 | 2 | no | — |
| 46 | C2×Q8 | — | 4 | 1 | no | — |
| 47 | C2×D4 | — | 4 | 2 | no | — |
| 48 | C2×D4 | — | 4 | 1 | no | — |
| 49 | C2×D4 | — | 8 | 1 | no | — |
| 50 | C22×C4 | — | 2 | 1 | no | — |
| 51 | C22×C4 | — | 8 | 1 | no | — |
| 52 | C22×C4 | #99 | 1 | 1 | yes | characteristic |
| 53 | Q16 | — | 8 | 2 | no | — |
| 54 | SD16 | — | 8 | 4 | no | — |
| 55 | D8 | — | 8 | 2 | no | — |
| 56 | M16 | — | 8 | 2 | no | — |
| 57 | C2×C8 | — | 2 | 2 | no | — |
| 58 | C2×C8 | — | 2 | 2 | no | — |
| 59 | C2×C8 | — | 2 | 1 | no | — |
| 60 | C2×C8 | — | 2 | 1 | no | — |
| 61 | C2×C8 | — | 4 | 2 | no | — |
| 62 | C2×C8 | — | 4 | 2 | no | — |
| 63 | C2×C8 | — | 8 | 2 | no | — |
| 64 | C4⋊C4 | — | 4 | 2 | no | — |
| 65 | C22⋊C4 | — | 8 | 2 | no | — |
| 66 | C42 | — | 2 | 1 | no | — |
| 67 | C42 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 68 | C42 | #100 | 1 | 1 | yes | characteristic |
| 69 | C16 | — | 16 | 2 | no | — |
| 70 | C23 | — | 8 | 1 | no | — |
| 71 | C23 | #355 | 1 | 1 | yes | characteristic |
| 72 | Q8 | — | 8 | 2 | no | — |
| 73 | D4 | — | 8 | 2 | no | — |
| 74 | D4 | — | 8 | 4 | no | — |
| 75 | D4 | — | 16 | 2 | no | — |
| 76 | C2×C4 | — | 2 | 1 | no | — |
| 77 | C2×C4 | — | 2 | 1 | no | — |
| 78 | C2×C4 | — | 2 | 1 | no | — |
| 79 | C2×C4 | — | 2 | 1 | no | — |
| 80 | C2×C4 | — | 4 | 2 | no | — |
| 81 | C2×C4 | — | 8 | 1 | no | — |
| 82 | C2×C4 | — | 16 | 1 | no | — |
| 83 | C2×C4 | — | 16 | 1 | no | — |
| 84 | C2×C4 | #323 | 1 | 1 | yes | characteristic |
| 85 | C2×C4 | #353 | 1 | 1 | yes | characteristic |
| 86 | C8 | — | 2 | 2 | no | — |
| 87 | C8 | — | 2 | 2 | no | — |
| 88 | C8 | — | 8 | 2 | no | — |
| 89 | C8 | — | 8 | 2 | no | — |
| 90 | C22 | — | 2 | 1 | no | — |
| 91 | C22 | — | 4 | 1 | no | — |
| 92 | C22 | — | 16 | 2 | no | — |
| 93 | C22 | — | 16 | 1 | no | — |
| 94 | C22 | #1318 | 1 | 1 | yes | characteristic |
| 95 | C4 | — | 2 | 2 | no | — |
| 96 | C4 | — | 2 | 1 | no | — |
| 97 | C4 | — | 4 | 1 | no | — |
| 98 | C4 | — | 16 | 1 | no | — |
| 99 | C4 | #1408 | 1 | 1 | yes | characteristic |
| 100 | C4 | #1410 | 1 | 1 | yes | characteristic |
| 101 | C2 | — | 2 | 1 | no | — |
| 102 | C2 | — | 4 | 1 | no | — |
| 103 | C2 | — | 32 | 1 | no | — |
| 104 | C2 | #7354 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1408 |
| commutator | C4×C8 | C2×C8 |
| Frattini, cosocle | #1848 | C22 |
| socle | C2 | #7354 |
| prime radical | #3147 | C4 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1848 ◁ G | |
| LECS | G ▷ #1848 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #7373 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #9862 | 2 |
| #7373 | 2 |