Finite Group Databasebeta

← #30638810

Subgroups — #30638810

Order 16256.

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

normal subgroups only

This group has 316 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#22549840C212yesmaximal
2#22549645C214yesmaximal, complemented
3C22×C4⋊C4×C127C211yesmaximal, characteristic
4C2×#14523232C2214yes—
5C22⋊(C127⋊C8)—22no—
6C22⋊(C127⋊C8)—22no—
7C23×C4×C127C412yes—
8C23×C4×C127C2211yescharacteristic
9C2×C4⋊C4×C127—22no—
10C2×C4⋊C4×C127—22no—
11C2×C4⋊C4×C127C412yes—
12C2×C4⋊C4×C127C2212yes—
13#14523232—28no—
14C2×C127⋊C8—24no—
15C24×C127C2×C411yescharacteristic
16C22×C4×C127—24no—
17C22×C4×C127—22no—
18C22×C4×C127—22no—
19C22×C4×C127—44no—
20C22×C4×C127C2×C411yescharacteristic
21C22×C4×C127C2×C414yes—
22C22×C4×C127D412yes—
23C22×C4×C127D412yes—
24C22×C4×C127C2311yescharacteristic
25C4⋊C4×C127—24no—
26C4⋊C4×C127—24no—
27C127⋊C8—44no—
28C23×C127—22no—
29C23×C127—22no—
30C23×C127—22no—
31C23×C127C22⋊C411yescharacteristic
32C23×C127C22⋊C411yescharacteristic
33C23×C127C22×C411yescharacteristic
34C2×C4×C127—22no—
35C2×C4×C127—24no—
36C2×C4×C127—24no—
37C2×C4×C127—24no—
38C2×C4×C127—44no—
39C2×C4×C127—44no—
40C2×C4×C127—44no—
41C2×C4×C127C22⋊C412yes—
42C2×C4×C127C2×D412yes—
43C2×C4×C127#4114yes—
44C22×C127—28no—
45C22×C127—21no—
46C22×C127—21no—
47C22×C127—21no—
48C22×C127—21no—
49C22×C127—21no—
50C22×C127—21no—
51C22×C127#10012yes—
52C22×C127#10112yes—
53C22×C127C2×C22⋊C411yescharacteristic
54C22×C127#11712yes—
55C4×C127—22no—
56C4×C127—44no—
57C4×C127#12314yes—
58C2×C127—22no—
59C2×C127—22no—
60C2×C127—22no—
61C2×C127#38111yescharacteristic
62C2×C127C2×#10011yescharacteristic
63C2×C127C2×#10111yescharacteristic
64#1821—1271nomaximal, Sylow
65C127#182111yescharacteristic, complemented, Sylow
66C22×C4⋊C4D12711yescharacteristic
67#407—1272no—
68#343—1274no—
69C23×C4Dic12712yes—
70C23×C4C2×D12711yescharacteristic
71C2×M16—1274no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4Dic12712yes—
75C2×C4⋊C4C2×D12712yes—
76C22⋊C8—2542no—
77C22⋊C8—2542no—
78C24C2×Dic12711yescharacteristic
79C22×C4—24no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—44no—
83C22×C4C2×Dic12711yescharacteristic
84C22×C4C2×Dic12714yes—
85C22×C4C22⋊D12712yes—
86C22×C4C22⋊D12712yes—
87C22×C4C22×D12711yescharacteristic
88M16—2548no—
89C2×C8—2544no—
90C4⋊C4—24no—
91C4⋊C4—24no—
92C23—22no—
93C23—22no—
94C23—22no—
95C23C22⋊Dic12711yescharacteristic
96C23C22⋊Dic12711yescharacteristic
97C23C22×Dic12711yescharacteristic
98C2×C4—22no—
99C2×C4—24no—
100C2×C4—24no—
101C2×C4—24no—
102C2×C4—44no—
103C2×C4—44no—
104C2×C4—44no—
105C2×C4C22⋊Dic12712yes—
106C2×C4C2×C22⋊D12712yes—
107C2×C4#1452324814yes—
108C8—5084no—
109C22—28no—
110C22—21no—
111C22—21no—
112C22—21no—
113C22—21no—
114C22—21no—
115C22—21no—
116C22#1686112612yes—
117C22#1686112812yes—
118C22C2×C22⋊Dic12711yescharacteristic
119C22#1686116812yes—
120C4—22no—
121C4—44no—
122C4#1686119214yes—
123C2—22no—
124C2—22no—
125C2—22no—
126C2#2254973911yescharacteristic
127C2C2×#1686112611yescharacteristic
128C2C2×#1686112811yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
FittingC22×C4⋊C4×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC24×C127C2×C4
cosocleC22×C4×C127C23
hypercenterC22×C4⋊C4D127
hypocenterC127#1821
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22549645C2semidirect
65. C127#1821semidirect

Maximal subgroups

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

subgroupindex
#225498402
#225496452
C22×C4⋊C4×C1272
#1821127