Finite Group Databasebeta

← #30638838

Subgroups — #30638838

Order 16256.

296 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C2×#16861124C212yesmaximal
2#22549642C214yesmaximal, complemented
3#518×C127C211yesmaximal, characteristic
4C2×C22⋊Dic127C2214yes—
5#16861124—24no—
6C22×D4×C127C2211yescharacteristic
7C4×D4×C127—24no—
8#117×C127C412yescomplemented
9C2×C22⋊C4×C127C412yescomplemented
10C2×C22⋊C4×C127C2212yes—
11C22×Dic127—24no—
12C22⋊Dic127—24no—
13C22⋊Dic127—24no—
14C24×C127C2×C411yescharacteristic
15C24×C127C2311yescharacteristic
16C2×D4×C127—24no—
17C2×D4×C127D414yes—
18C22×C4×C127—22no—
19C22×C4×C127—22no—
20C22×C4×C127C2×C411yescharacteristic
21C4⋊C4×C127—22no—
22C22⋊C4×C127—22no—
23C22⋊C4×C127—22no—
24C22⋊C4×C127C2×C414yes—
25C42×C127—22no—
26C2×Dic127—44no—
27C2×Dic127—84no—
28C23×C127—22no—
29C23×C127—81no—
30C23×C127—81no—
31C23×C127C22⋊C412yes—
32C23×C127C22×C411yescharacteristic
33C23×C127C2×D412yes—
34C23×C127#4114yes—
35D4×C127—28no—
36C2×C4×C127—22no—
37C2×C4×C127—24no—
38C2×C4×C127—82no—
39C2×C4×C127—82no—
40C2×C4×C127C22⋊C412yes—
41C22×C127—22no—
42C22×C127—42no—
43C22×C127—41no—
44C22×C127—44no—
45C22×C127—42no—
46C22×C127—82no—
47C22×C127C2×C22⋊C411yescharacteristic
48C22×C127#11712yes—
49C22×C127#12314yes—
50C4×C127—22no—
51C4×C127—44no—
52Dic127—84no—
53C2×C127—24no—
54C2×C127—21no—
55C2×C127—22no—
56C2×C127—42no—
57C2×C127#38111yescharacteristic
58#1828—1271nomaximal, Sylow
59C127#182811yescharacteristic, complemented, Sylow
60#518D12711yescharacteristic
61C2×#99—1272no—
62#342—1274no—
63C22×D4C2×D12711yescharacteristic
64C4×D4—24no—
65#117Dic12712yescomplemented
66C2×C22⋊C4—1274no—
67C2×C22⋊C4Dic12712yescomplemented
68C2×C22⋊C4C2×D12712yes—
69#99—2544no—
70C24C2×Dic12711yescharacteristic
71C24C22×D12711yescharacteristic
72C2×D4—24no—
73C2×D4C22⋊D12714yes—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—2544no—
77C22×C4C2×Dic12711yescharacteristic
78C4⋊C4—22no—
79C22⋊C4—22no—
80C22⋊C4—22no—
81C22⋊C4—2544no—
82C22⋊C4—2544no—
83C22⋊C4C2×Dic12714yes—
84C42—22no—
85C23—22no—
86C23—81no—
87C23—81no—
88C23C22⋊Dic12712yes—
89C23C22×Dic12711yescharacteristic
90C23C2×C22⋊D12712yes—
91C23#1452324814yes—
92D4—28no—
93C2×C4—22no—
94C2×C4—24no—
95C2×C4—82no—
96C2×C4—82no—
97C2×C4—10164no—
98C2×C4—5084no—
99C2×C4C22⋊Dic12712yes—
100C22—22no—
101C22—42no—
102C22—44no—
103C22—41no—
104C22—42no—
105C22—82no—
106C22C2×C22⋊Dic12711yescharacteristic
107C22#1686116812yes—
108C22#1686119214yes—
109C4—22no—
110C4—44no—
111C4—10164no—
112C2—21no—
113C2—24no—
114C2—22no—
115C2—42no—
116C2#2254973911yescharacteristic
subgroupG/H
centerC2#22549739
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting#518×C127C2
socleC2×C127#381
prime radicalC22×D4×C127C22
cosocleC24×C127C23
hypercenter#518D127
hypocenterC127#1828
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #22549642C2semidirect
8. #117×C127C4semidirect
9. C2×C22⋊C4×C127C4semidirect
59. C127#1828semidirect
65. #117Dic127semidirect
67. C2×C22⋊C4Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611242
#225496422
#518×C1272
#1828127