Finite Group Databasebeta

← #30638654

Subgroups — #30638654

Order 16256.

564 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16861128C214yesmaximal, complemented
2#22549840C212yesmaximal
3C23×Q8×C127C211yesmaximal, characteristic
4C2×#14523232C2214yes—
5#16861128—28no—
6C22⋊(C127⋊C8)—24no—
7C22×Q8×C127—28no—
8C22×Q8×C127—24no—
9C22×Q8×C127C412yes—
10C22×Q8×C127C2212yes—
11C23×C4×C127C412yes—
12C23×C4×C127C2211yescharacteristic
13#14523232—28no—
14C2×C127⋊C8—24no—
15C24×C127C2×C411yescharacteristic
16C2×Q8×C127—232no—
17C2×Q8×C127—28no—
18C2×Q8×C127—24no—
19C2×Q8×C127—24no—
20C2×Q8×C127—24no—
21C2×Q8×C127D418yes—
22C22×C4×C127—24no—
23C22×C4×C127—28no—
24C22×C4×C127—24no—
25C22×C4×C127C2×C414yes—
26C22×C4×C127C2×C411yescharacteristic
27C22×C4×C127D414yes—
28C22×C4×C127C2311yescharacteristic
29C127⋊C8—44no—
30C23×C127—24no—
31C23×C127—22no—
32C23×C127C22⋊C412yes—
33C23×C127C22×C411yescharacteristic
34Q8×C127—216no—
35Q8×C127—216no—
36C2×C4×C127—216no—
37C2×C4×C127—24no—
38C2×C4×C127—22no—
39C2×C4×C127—24no—
40C2×C4×C127—28no—
41C2×C4×C127C22⋊C418yes—
42C2×C4×C127C22⋊C412yes—
43C2×C4×C127C2×D414yes—
44C2×C4×C127C2×D412yes—
45C22×C127—28no—
46C22×C127—22no—
47C22×C127—22no—
48C22×C127—22no—
49C22×C127#10114yescomplemented
50C22×C127C2×C22⋊C412yes—
51C22×C127C2×C22⋊C411yescharacteristic
52C4×C127—28no—
53C4×C127—22no—
54C4×C127#12014yes—
55C2×C127—24no—
56C2×C127—22no—
57C2×C127#37911yescharacteristic
58C2×C127C2×#10112yes—
59#1784—1271nomaximal, Sylow
60C127#178411yescharacteristic, complemented, Sylow
61C23×Q8D12711yescharacteristic
62C2×#101—1274no—
63#407—1272no—
64C22×Q8—28no—
65C22×Q8—24no—
66C22×Q8Dic12712yes—
67C22×Q8C2×D12712yes—
68C23×C4Dic12712yes—
69C23×C4C2×D12711yescharacteristic
70C2×M16—1274no—
71#101—2548no—
72C22⋊C8—2544no—
73C24C2×Dic12711yescharacteristic
74C2×Q8—232no—
75C2×Q8—28no—
76C2×Q8—24no—
77C2×Q8—24no—
78C2×Q8—24no—
79C2×Q8C22⋊D12718yes—
80C22×C4—24no—
81C22×C4—28no—
82C22×C4—24no—
83C22×C4C2×Dic12714yes—
84C22×C4C2×Dic12711yescharacteristic
85C22×C4C22⋊D12714yes—
86C22×C4C22×D12711yescharacteristic
87M16—2548no—
88C2×C8—2544no—
89C23—24no—
90C23—22no—
91C23C22⋊Dic12712yes—
92C23C22×Dic12711yescharacteristic
93Q8—216no—
94Q8—216no—
95C2×C4—216no—
96C2×C4—24no—
97C2×C4—22no—
98C2×C4—24no—
99C2×C4—28no—
100C2×C4C22⋊Dic12718yes—
101C2×C4C22⋊Dic12712yes—
102C2×C4C2×C22⋊D12714yes—
103C2×C4C2×C22⋊D12712yes—
104C8—5084no—
105C22—28no—
106C22—22no—
107C22—22no—
108C22—22no—
109C22#1686112814yescomplemented
110C22C2×C22⋊Dic12712yes—
111C22C2×C22⋊Dic12711yescharacteristic
112C4—28no—
113C4—22no—
114C4#1686117814yes—
115C2—24no—
116C2—22no—
117C2#2254973211yescharacteristic
118C2C2×#1686112812yes—
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
FittingC23×Q8×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC24×C127C2×C4
cosocleC22×C4×C127C23
hypercenterC23×Q8D127
hypocenterC127#1784
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16861128C2semidirect
49. C22×C127#101semidirect
60. C127#1784semidirect
109. C22#16861128semidirect

Maximal subgroups

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

subgroupindex
C2×#168611282
#225498402
C23×Q8×C1272
#1784127