Finite Group Databasebeta

← #30638807

Subgroups — #30638807

Order 16256.

300 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 93 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#22549739C212yesmaximal
2#22549643C214yesmaximal
3C2×#117×C127C211yesmaximal, characteristic
4C2×C22⋊Dic127C2214yes—
5C127⋊#95—24no—
6C23×C4×C127C2211yescharacteristic
7#117×C127—24no—
8C2×C4⋊C4×C127C412yescomplemented
9C2×C22⋊C4×C127C2212yes—
10C2×C42×C127C412yescomplemented
11C22×Dic127—24no—
12C22⋊Dic127—28no—
13C24×C127C2311yescharacteristic
14C22×C4×C127—24no—
15C22×C4×C127C2×C411yescharacteristic
16C22×C4×C127C2×C411yescharacteristic
17C22×C4×C127C2×C414yes—
18C22×C4×C127D414yes—
19C4⋊C4×C127—24no—
20C22⋊C4×C127—24no—
21C42×C127—24no—
22C2×Dic127—48no—
23C2×Dic127—44no—
24C23×C127—42no—
25C23×C127C22×C411yescharacteristic
26C23×C127C2×D412yes—
27C23×C127#4114yes—
28C2×C4×C127—22no—
29C2×C4×C127—22no—
30C2×C4×C127—28no—
31C2×C4×C127—24no—
32C2×C4×C127—44no—
33C2×C4×C127C22⋊C412yes—
34C2×C4×C127C22⋊C412yes—
35C22×C127—22no—
36C22×C127—24no—
37C22×C127—44no—
38C22×C127C2×C22⋊C411yescharacteristic
39C22×C127#11712yes—
40C22×C127#12314yes—
41C4×C127—22no—
42C4×C127—22no—
43C4×C127—44no—
44Dic127—84no—
45C2×C127—24no—
46C2×C127—22no—
47C2×C127#38111yescharacteristic
48C2×C127#41012yes—
49#1820—1271nomaximal, Sylow
50C127#182011yescharacteristic, complemented, Sylow
51C2×#117D12711yescharacteristic
52#381—1272no—
53#342—1274no—
54C23×C4C2×D12711yescharacteristic
55#117—24no—
56C2×C4⋊C4Dic12712yescomplemented
57C2×C22⋊C4—1274no—
58C2×C22⋊C4C2×D12712yes—
59C2×C42Dic12712yescomplemented
60#95—2544no—
61C24C22×D12711yescharacteristic
62C22×C4—24no—
63C22×C4—2544no—
64C22×C4C2×Dic12711yescharacteristic
65C22×C4C2×Dic12711yescharacteristic
66C22×C4C2×Dic12714yes—
67C22×C4C22⋊D12714yes—
68C4⋊C4—24no—
69C22⋊C4—24no—
70C22⋊C4—2548no—
71C42—24no—
72C23—42no—
73C23C22×Dic12711yescharacteristic
74C23C2×C22⋊D12712yes—
75C23#1452324814yes—
76C2×C4—28no—
77C2×C4—22no—
78C2×C4—22no—
79C2×C4—24no—
80C2×C4—44no—
81C2×C4—5088no—
82C2×C4—5084no—
83C2×C4C22⋊Dic12712yes—
84C2×C4C22⋊Dic12712yes—
85C22—22no—
86C22—24no—
87C22—44no—
88C22C2×C22⋊Dic12711yescharacteristic
89C22#1686116812yes—
90C22#1686119214yes—
91C4—22no—
92C4—22no—
93C4—44no—
94C4—10164no—
95C2—24no—
96C2—22no—
97C2#2254973911yescharacteristic
98C2#2254985012yes—
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC23×C127C22×C4
FrattiniC24C22×D127
FittingC2×#117×C127C2
socleC22×C127C2×C22⋊C4
prime radical, cosocleC24×C127C23
hypercenterC2×#117D127
hypocenterC127#1820
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
8. C2×C4⋊C4×C127C4semidirect
10. C2×C42×C127C4semidirect
50. C127#1820semidirect
56. C2×C4⋊C4Dic127semidirect
59. C2×C42Dic127semidirect

Maximal subgroups

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

subgroupindex
#225497392
#225496432
C2×#117×C1272
#1820127