Finite Group Databasebeta

← #30636638

Subgroups — #30636638

Order 16256.

192 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 57 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#22549840C212yesmaximal
2#381×C127C211yesmaximal, characteristic
3C2×#14523232—22no—
4C22⋊(C127⋊C8)—22no—
5C22⋊(C127⋊C8)C414yescomplemented
6C23×C4×C127C2211yescharacteristic
7C2×C22⋊C4×C127C412yes—
8#95×C127—22no—
9#14523232—44no—
10C2×C127⋊C8—24no—
11C24×C127C2×C411yescharacteristic
12C22×C4×C127—22no—
13C22×C4×C127—42no—
14C22×C4×C127C2×C412yes—
15C22×C4×C127D412yes—
16C22×C4×C127D411yescharacteristic
17C22×C4×C127Q811yescharacteristic
18C22⋊C4×C127—42no—
19C127⋊C8—44no—
20C23×C127—22no—
21C23×C127—42no—
22C23×C127C4211yescharacteristic
23C23×C127C22⋊C411yescharacteristic
24C23×C127C4⋊C411yescharacteristic
25C2×C4×C127—24no—
26C2×C4×C127—42no—
27C2×C4×C127—41no—
28C2×C4×C127—41no—
29C2×C4×C127—44no—
30C2×C4×C127—42no—
31C2×C4×C127C22⋊C412yes—
32C2×C4×C127C4⋊C412yes—
33C22×C127—22no—
34C22×C127—21no—
35C22×C127—21no—
36C22×C127—22no—
37C22×C127—22no—
38C22×C127—44no—
39C22×C127#9511yescharacteristic
40C22×C127#9912yes—
41C4×C127—22no—
42C4×C127—22no—
43C4×C127—82no—
44C2×C127—22no—
45C2×C127—41no—
46C2×C127—41no—
47C2×C127#34211yescharacteristic
48C2×C127#34412yes—
49#1306—1271nomaximal, Sylow
50C127#130611yescharacteristic, complemented, Sylow
51#407—1272no—
52#381D12711yescharacteristic
53C23×C4C2×D12711yescharacteristic
54C2×M16—2542no—
55C2×C22⋊C4Dic12712yes—
56C22⋊C8—1274no—
57C22⋊C8—2542no—
58#95—22no—
59C24C2×Dic12711yescharacteristic
60C22×C4—22no—
61C22×C4—42no—
62C22×C4C4×D12712yes—
63C22×C4C22⋊D12712yes—
64C22×C4D50811yescharacteristic
65C22×C4Dic25411yescharacteristic
66M16—5084no—
67C2×C8—2544no—
68C22⋊C4—42no—
69C23—22no—
70C23—42no—
71C23C4×Dic12711yescharacteristic
72C23C22⋊Dic12711yescharacteristic
73C23C4⋊Dic12711yescharacteristic
74C2×C4—24no—
75C2×C4—42no—
76C2×C4—41no—
77C2×C4—41no—
78C2×C4—44no—
79C2×C4—42no—
80C2×C4(C2×D127)⋊C412yes—
81C2×C4Dic127⋊C412yes—
82C8—5084no—
83C22—22no—
84C22—21no—
85C22—21no—
86C22—22no—
87C22—22no—
88C22—44no—
89C22C127⋊#9511yescharacteristic
90C22#1686112412yes—
91C4—22no—
92C4—22no—
93C4—82no—
94C2—22no—
95C2—41no—
96C2—41no—
97C2#2254964311yescharacteristic
98C2#2254964612yes—
subgroupG/H
centerC22C127⋊#95
commutatorC23×C127C42
FrattiniC23×C4C2×D127
Fitting#381×C127C2
socleC22×C127#95
prime radicalC24×C127C2×C4
cosocleC23×C4×C127C22
hypercenter#381D127
hypocenterC127#1306
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
5. C22⋊(C127⋊C8)C4semidirect
50. C127#1306semidirect

Maximal subgroups

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

subgroupindex
#225498402
#381×C1272
#1306127