Finite Group Databasebeta

← #30640747

Subgroups — #30640747

Order 16256.

248 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550048C213yesmaximal, complemented
2#22549673C213yesmaximal, complemented
3H(3,𝔽4)×C127C211yesmaximal, characteristic, complemented
4#16861257C2211yescharacteristic, complemented
5#16861235—23no—
6#16861137—23no—
7#16861128C2213yescomplemented
8#124×C127—23no—
9#124×C127C2213yes—
10#120×C127—23no—
11#14523249—41no—
12#14523248—23no—
13Q8×D127—41no—
14#14523236—43no—
15#14523233—43no—
16#14523232—23no—
17C24×C127—21no—
18C2×Q8×C127—21no—
19C2×Q8×C127C2311yescharacteristic
20C2×D4×C127—23no—
21C2×D4×C127D413yes—
22C22⋊C4×C127—23no—
23C22⋊C4×C127—26no—
24C42×C127D413yes—
25Dic254—23no—
26D508—23no—
27C22⋊D127—41no—
28C4×D127—43no—
29C127⋊C8—43no—
30C23×C127—23no—
31C23×C127—82no—
32C23×C127—81no—
33Q8×C127—41no—
34Q8×C127—81no—
35D4×C127—43no—
36D4×C127—83no—
37C2×C4×C127—23no—
38C2×C4×C127C2×D413yes—
39C22×C127—46no—
40C22×C127—43no—
41C22×C127—84no—
42C22×C127#12011yescharacteristic
43C2×D127—41no—
44C4×C127—23no—
45C4×C127—43no—
46Dic127—41no—
47C2×C127—21no—
48C2×C127—83no—
49C2×C127B(4,𝔽2)11yescharacteristic
50D127—81no—
51#2202—1271nomaximal, Sylow
52C127#220211yescharacteristic, complemented, Sylow
53H(3,𝔽4)D12711yescharacteristic, complemented
54#454—1273no—
55#355—1273no—
562−1+4—1271no—
57AGL(1,ℤ/8)—2543no—
58#124—23no—
59#124C2×D12713yes—
60#120—23no—
61#104—2543no—
62#101—1273no—
63C24—21no—
64#41—5081no—
65#41—2543no—
66C2×Q8—21no—
67C2×Q8—5081no—
68C2×Q8C22×D12711yescharacteristic
69C2×D4—23no—
70C2×D4C22⋊D12713yes—
71SD16—5083no—
72D8—5083no—
73M16—2543no—
74C22⋊C4—23no—
75C22⋊C4—26no—
76C42C22⋊D12713yes—
77C23—23no—
78C23—82no—
79C23—81no—
80Q8—41no—
81Q8—81no—
82Q8—2543no—
83D4—43no—
84D4—83no—
85D4—5081no—
86D4—2543no—
87C2×C4—23no—
88C2×C4—5083no—
89C2×C4C2×C22⋊D12713yes—
90C8—5083no—
91C22—46no—
92C22—43no—
93C22—84no—
94C22—5081no—
95C22#1686117811yescharacteristic
96C4—23no—
97C4—43no—
98C4—5081no—
99C2—21no—
100C2—83no—
101C2—10161no—
102C2#2255006311yescharacteristic
subgroupG/H
centerC2#22550063
commutator, cosocleC2×Q8×C127C23
FrattiniC2×Q8C22×D127
FittingH(3,𝔽4)×C127C2
socleC2×C127B(4,𝔽2)
hypercenterH(3,𝔽4)D127
hypocenterC127#2202
LCSG ▷ C2×Q8×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ H(3,𝔽4)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22550048C2semidirect
2. #22549673C2semidirect
3. H(3,𝔽4)×C127C2semidirect
4. #16861257C22semidirect
7. #16861128C22semidirect
52. C127#2202semidirect
53. H(3,𝔽4)D127semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225500482
#225496732
H(3,𝔽4)×C1272
#2202127