Finite Group Databasebeta

← #30637071

Subgroups — #30637071

Order 16256.

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

normal subgroups only

This group has 300 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
1#22549933C212yesmaximal, complemented
2#22549600C214yesmaximal, complemented
3C2×C4×D4×C127C211yesmaximal, characteristic
4C4⋊(C127⋊C8)—22no—
5C22⋊(C127⋊C8)C2214yes—
6#16861120—22no—
7C22×D4×C127C411yescharacteristic
8C23×C4×C127C412yes—
9C4×D4×C127—22no—
10C4×D4×C127—22no—
11C2×C4⋊C4×C127C411yescharacteristic
12C2×C22⋊C4×C127C2212yes—
13C2×C42×C127C2211yescharacteristic
14C2×C127⋊C8—24no—
15C24×C127C2×C412yes—
16C2×D4×C127—24no—
17C2×D4×C127—22no—
18C22×C4×C127—22no—
19C22×C4×C127—22no—
20C22×C4×C127—44no—
21C22×C4×C127C2×C411yescharacteristic
22C22×C4×C127C2×C411yescharacteristic
23C22×C4×C127C2×C412yes—
24C22×C4×C127C2311yescharacteristic
25C4⋊C4×C127—22no—
26C22⋊C4×C127—24no—
27C42×C127D412yes—
28C42×C127D412yes—
29C127⋊C8—44no—
30C23×C127—22no—
31C23×C127—22no—
32C23×C127—42no—
33C23×C127—42no—
34C23×C127M1614yes—
35C23×C127C22×C411yescharacteristic
36D4×C127—28no—
37C2×C4×C127—22no—
38C2×C4×C127—22no—
39C2×C4×C127—22no—
40C2×C4×C127—41no—
41C2×C4×C127—44no—
42C2×C4×C127—44no—
43C2×C4×C127—42no—
44C2×C4×C127C22⋊C412yes—
45C2×C4×C127C22⋊C411yescharacteristic
46C2×C4×C127C22⋊C411yescharacteristic
47C2×C4×C127C2×D412yes—
48C22×C127—21no—
49C22×C127—21no—
50C22×C127—24no—
51C22×C127—44no—
52C22×C127—44no—
53C22×C127—42no—
54C22×C127—42no—
55C22×C127C2×C22⋊C411yescharacteristic
56C22×C127C2×M1612yes—
57C4×C127—22no—
58C4×C127—22no—
59C4×C127—21no—
60C4×C127—42no—
61C4×C127#10012yes—
62C2×C127—22no—
63C2×C127—44no—
64C2×C127#40711yescharacteristic
65C2×C127#41011yescharacteristic
66C2×C127C2×#10011yescharacteristic
67#1466—1271nomaximal, Sylow
68C127#146611yescharacteristic, complemented, Sylow
69C2×C4×D4D12711yescharacteristic
70#432—1272no—
71#323—1274no—
72C22×D4Dic12711yescharacteristic
73C23×C4Dic12712yes—
74C4×D4—22no—
75C4×D4—22no—
76C2×C4⋊C4Dic12711yescharacteristic
77C2×C22⋊C4C2×D12712yes—
78C2×C42C2×D12711yescharacteristic
79C4⋊C8—2542no—
80C22⋊C8—1274no—
81#97—2542no—
82C24C2×Dic12712yes—
83C2×D4—24no—
84C2×D4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—44no—
88C22×C4C2×Dic12711yescharacteristic
89C22×C4C2×Dic12711yescharacteristic
90C22×C4C2×Dic12712yes—
91C22×C4C22×D12711yescharacteristic
92C2×C8—2544no—
93C4⋊C4—22no—
94C22⋊C4—24no—
95C42C22⋊D12712yes—
96C42C22⋊D12712yes—
97C23—22no—
98C23—22no—
99C23—42no—
100C23—42no—
101C23#1452323214yes—
102C23C22×Dic12711yescharacteristic
103D4—28no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—41no—
108C2×C4—44no—
109C2×C4—44no—
110C2×C4—42no—
111C2×C4C22⋊Dic12712yes—
112C2×C4C22⋊Dic12711yescharacteristic
113C2×C4C22⋊Dic12711yescharacteristic
114C2×C4C2×C22⋊D12712yes—
115C8—5084no—
116C22—21no—
117C22—21no—
118C22—24no—
119C22—44no—
120C22—44no—
121C22—42no—
122C22—42no—
123C22C2×C22⋊Dic12711yescharacteristic
124C22C2×#1452323212yes—
125C4—22no—
126C4—22no—
127C4—21no—
128C4—42no—
129C4#1686112612yes—
130C2—22no—
131C2—44no—
132C2#2254984011yescharacteristic
133C2#2254985011yescharacteristic
134C2C2×#1686112611yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×C4×D4×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D4×C127C4
cosocleC22×C4×C127C23
hypercenterC2×C4×D4D127
hypocenterC127#1466
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #22549933C2semidirect
2. #22549600C2semidirect
68. C127#1466semidirect

Maximal subgroups

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

subgroupindex
#225499332
#225496002
C2×C4×D4×C1272
#1466127