Finite Group Databasebeta

← #30638804

Subgroups — #30638804

Order 16256.

504 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 101 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#22549732C212yesmaximal
2#22549643C214yesmaximal, complemented
3C22×C22⋊C4×C127C211yesmaximal, characteristic
4C2×C22⋊Dic127—24no—
5C2×C22⋊Dic127C2214yes—
6C25×C127C2211yescharacteristic
7C23×C4×C127C412yescomplemented
8C2×C22⋊C4×C127—24no—
9C2×C22⋊C4×C127C412yescomplemented
10C2×C22⋊C4×C127C2212yes—
11C22×Dic127—24no—
12C22⋊Dic127—28no—
13C22⋊Dic127—48no—
14C24×C127—24no—
15C24×C127—44no—
16C24×C127C2×C411yescharacteristic
17C24×C127C2×C411yescharacteristic
18C24×C127D414yes—
19C24×C127C2311yescharacteristic
20C22×C4×C127—22no—
21C22×C4×C127—22no—
22C22×C4×C127—44no—
23C22×C4×C127C2×C414yes—
24C22⋊C4×C127—24no—
25C22⋊C4×C127—24no—
26C2×Dic127—48no—
27C2×Dic127—44no—
28C23×C127—24no—
29C23×C127—22no—
30C23×C127—28no—
31C23×C127—22no—
32C23×C127—28no—
33C23×C127—42no—
34C23×C127—44no—
35C23×C127—42no—
36C23×C127—416no—
37C23×C127C22⋊C412yes—
38C23×C127C22⋊C412yes—
39C23×C127C22×C411yescharacteristic
40C23×C127C2×D412yes—
41C23×C127#4114yes—
42C2×C4×C127—24no—
43C2×C4×C127—44no—
44C2×C4×C127—44no—
45C2×C4×C127—44no—
46C22×C127—28no—
47C22×C127—22no—
48C22×C127—22no—
49C22×C127—22no—
50C22×C127—24no—
51C22×C127—22no—
52C22×C127—24no—
53C22×C127—44no—
54C22×C127—416no—
55C22×C127—44no—
56C22×C127#9914yes—
57C22×C127C2×C22⋊C411yescharacteristic
58C22×C127#11712yes—
59C22×C127#12314yes—
60C4×C127—44no—
61Dic127—84no—
62C2×C127—24no—
63C2×C127—24no—
64C2×C127—22no—
65C2×C127—42no—
66C2×C127#38111yescharacteristic
67C2×C127C2×#9912yes—
68#1819—1271nomaximal, Sylow
69C127#181911yescharacteristic, complemented, Sylow
70C22×C22⋊C4D12711yescharacteristic
71#379—1272no—
72#342—1274no—
73C25C2×D12711yescharacteristic
74C23×C4Dic12712yescomplemented
75C2×C22⋊C4—24no—
76C2×C22⋊C4—1274no—
77C2×C22⋊C4—2544no—
78C2×C22⋊C4Dic12712yescomplemented
79C2×C22⋊C4C2×D12712yes—
80C24—24no—
81C24—44no—
82C24C2×Dic12711yescharacteristic
83C24C2×Dic12711yescharacteristic
84C24C22⋊D12714yes—
85C24C22×D12711yescharacteristic
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—44no—
89C22×C4—2544no—
90C22×C4C2×Dic12714yes—
91C22⋊C4—24no—
92C22⋊C4—24no—
93C22⋊C4—5088no—
94C22⋊C4—2548no—
95C23—24no—
96C23—22no—
97C23—28no—
98C23—28no—
99C23—22no—
100C23—42no—
101C23—44no—
102C23—42no—
103C23—416no—
104C23C22⋊Dic12712yes—
105C23C22⋊Dic12712yes—
106C23C22×Dic12711yescharacteristic
107C23C2×C22⋊D12712yes—
108C23#1452324814yes—
109C2×C4—24no—
110C2×C4—44no—
111C2×C4—44no—
112C2×C4—44no—
113C2×C4—5088no—
114C2×C4—5084no—
115C22—28no—
116C22—22no—
117C22—22no—
118C22—24no—
119C22—22no—
120C22—24no—
121C22—22no—
122C22—44no—
123C22—416no—
124C22—44no—
125C22#1686112414yes—
126C22C2×C22⋊Dic12711yescharacteristic
127C22#1686116812yes—
128C22#1686119214yes—
129C4—44no—
130C4—10164no—
131C2—24no—
132C2—24no—
133C2—22no—
134C2—42no—
135C2#2254973911yescharacteristic
136C2C2×#1686112412yes—
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC23×C127C22×C4
FrattiniC24C22×D127
FittingC22×C22⋊C4×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC25×C127C22
cosocleC24×C127C23
hypercenterC22×C22⋊C4D127
hypocenterC127#1819
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #22549643C2semidirect
7. C23×C4×C127C4semidirect
9. C2×C22⋊C4×C127C4semidirect
69. C127#1819semidirect
74. C23×C4Dic127semidirect
78. C2×C22⋊C4Dic127semidirect

Maximal subgroups

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

subgroupindex
#225497322
#225496432
C22×C22⋊C4×C1272
#1819127