Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
56 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #455 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #355 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #329 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | H(5,𝔽2) | C4 | 1 | 1 | yes | characteristic |
| 5 | #137 | — | 2 | 1 | no | — |
| 6 | #124 | C22 | 1 | 1 | yes | characteristic |
| 7 | #104 | — | 2 | 1 | no | — |
| 8 | #101 | — | 2 | 1 | no | — |
| 9 | #100 | C4 | 1 | 1 | yes | characteristic |
| 10 | #97 | — | 2 | 1 | no | — |
| 11 | #41 | — | 2 | 1 | no | — |
| 12 | #41 | — | 4 | 1 | no | — |
| 13 | C2×Q8 | D4 | 1 | 1 | yes | characteristic |
| 14 | C2×D4 | — | 4 | 1 | no | — |
| 15 | C2×D4 | — | 4 | 1 | no | — |
| 16 | C2×D4 | C2×C4 | 1 | 1 | yes | characteristic |
| 17 | Q16 | — | 4 | 1 | no | — |
| 18 | SD16 | — | 4 | 1 | no | — |
| 19 | M16 | — | 2 | 1 | no | — |
| 20 | M16 | — | 4 | 1 | no | — |
| 21 | C2×C8 | — | 4 | 1 | no | — |
| 22 | C22⋊C4 | — | 4 | 1 | no | — |
| 23 | C42 | D4 | 1 | 1 | yes | characteristic |
| 24 | C23 | — | 2 | 1 | no | — |
| 25 | C23 | — | 4 | 1 | no | — |
| 26 | Q8 | — | 2 | 1 | no | — |
| 27 | Q8 | — | 4 | 1 | no | — |
| 28 | D4 | — | 2 | 1 | no | — |
| 29 | D4 | — | 4 | 1 | no | — |
| 30 | D4 | — | 4 | 2 | no | — |
| 31 | D4 | — | 4 | 1 | no | — |
| 32 | C2×C4 | — | 2 | 1 | no | — |
| 33 | C2×C4 | — | 2 | 1 | no | — |
| 34 | C2×C4 | — | 4 | 1 | no | — |
| 35 | C2×C4 | — | 4 | 1 | no | — |
| 36 | C2×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 37 | C8 | — | 4 | 1 | no | — |
| 38 | C8 | — | 4 | 2 | no | — |
| 39 | C8 | — | 8 | 1 | no | — |
| 40 | C22 | — | 4 | 1 | no | — |
| 41 | C22 | — | 4 | 1 | no | — |
| 42 | C22 | — | 8 | 1 | no | — |
| 43 | C22 | — | 16 | 1 | no | — |
| 44 | C22 | #99 | 1 | 1 | yes | characteristic |
| 45 | C4 | — | 2 | 1 | no | — |
| 46 | C4 | — | 4 | 1 | no | — |
| 47 | C4 | — | 4 | 1 | no | — |
| 48 | C4 | — | 4 | 1 | no | — |
| 49 | C2 | — | 2 | 1 | no | — |
| 50 | C2 | — | 8 | 1 | no | — |
| 51 | C2 | — | 8 | 1 | no | — |
| 52 | C2 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C2×C4 |
| Frattini, cosocle | #124 | C22 |
| prime radical | H(5,𝔽2) | C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #355 | C2 | semidirect |
| 3. #329 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #455 | 2 |
| #355 | 2 |
| #329 | 2 |