Finite Group Databasebeta

← #30638805

Subgroups — #30638805

Order 16256.

300 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 85 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×#16861168C211yesmaximal, characteristic
2#22549739C211yesmaximal, characteristic
3#22549642C214yesmaximal
4#381×C127C211yesmaximal, characteristic
5#16861168—22no—
6#16861168—22no—
7C2×C4⋊Dic127C412yescomplemented
8C2×C22⋊Dic127C2212yes—
9C2×C22⋊Dic127C2212yes—
10C2×C4×Dic127C412yescomplemented
11C127⋊#95—22no—
12C23×C4×C127C2211yescharacteristic
13C2×C22⋊C4×C127C2212yes—
14#95×C127—22no—
15C22×Dic127—22no—
16C22×Dic127C2×C414yes—
17C4⋊Dic127—24no—
18C22⋊Dic127—22no—
19C22⋊Dic127—22no—
20C22⋊Dic127—24no—
21C4×Dic127—24no—
22C24×C127C2311yescharacteristic
23C22×C4×C127—24no—
24C22×C4×C127—22no—
25C22×C4×C127C2×C411yescharacteristic
26C22×C4×C127C2×C411yescharacteristic
27C22×C4×C127D412yes—
28C22×C4×C127D412yes—
29C22⋊C4×C127—24no—
30C2×Dic127—24no—
31C2×Dic127—44no—
32C2×Dic127—44no—
33C2×Dic127—42no—
34C23×C127—42no—
35C23×C127C22×C411yescharacteristic
36C23×C127C2×D411yescharacteristic
37C23×C127C2×D411yescharacteristic
38C23×C127#4114yes—
39C2×C4×C127—22no—
40C2×C4×C127—22no—
41C2×C4×C127—24no—
42C2×C4×C127—24no—
43C2×C4×C127—44no—
44C2×C4×C127—42no—
45C2×C4×C127C22⋊C412yes—
46C2×C4×C127C22⋊C412yes—
47C22×C127—22no—
48C22×C127—22no—
49C22×C127—22no—
50C22×C127—44no—
51C22×C127C2×C22⋊C411yescharacteristic
52C22×C127#11712yes—
53C22×C127#12312yes—
54C22×C127#12312yes—
55C4×C127—22no—
56C4×C127—22no—
57C4×C127—82no—
58Dic127—44no—
59Dic127—82no—
60C2×C127—22no—
61C2×C127—22no—
62C2×C127—22no—
63C2×C127#38111yescharacteristic
64C2×C127#41012yes—
65#1820—1271nomaximal, Sylow
66C127#182011yescharacteristic, complemented, Sylow
67C2×#117—1271no—
68#381—1271no—
69#381D12711yescharacteristic
70#342—1274no—
71C23×C4C2×D12711yescharacteristic
72#117—2542no—
73#117—2542no—
74C2×C4⋊C4—1272no—
75C2×C22⋊C4—1272no—
76C2×C22⋊C4—1272no—
77C2×C22⋊C4C2×D12712yes—
78C2×C42—1272no—
79#95—22no—
80#95—2542no—
81C24C22×D12711yescharacteristic
82C22×C4—24no—
83C22×C4—22no—
84C22×C4—1274no—
85C22×C4—2542no—
86C22×C4C4×D12711yescharacteristic
87C22×C4C4×D12711yescharacteristic
88C22×C4C22⋊D12712yes—
89C22×C4D50812yes—
90C4⋊C4—2544no—
91C22⋊C4—24no—
92C22⋊C4—2544no—
93C22⋊C4—2542no—
94C22⋊C4—2542no—
95C42—2544no—
96C23—42no—
97C23C2×C4×D12711yescharacteristic
98C23C2×C22⋊D12711yescharacteristic
99C23C2×D50811yescharacteristic
100C23#1452324714yes—
101C2×C4—24no—
102C2×C4—24no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—44no—
106C2×C4—42no—
107C2×C4—5084no—
108C2×C4—5082no—
109C2×C4—5084no—
110C2×C4—2544no—
111C2×C4(C2×D127)⋊C412yes—
112C2×C4(C2×D127)⋊C412yes—
113C22—22no—
114C22—22no—
115C22—22no—
116C22—44no—
117C22C2×(C2×D127)⋊C411yescharacteristic
118C22#1686116512yes—
119C22#1686119012yes—
120C22#1686119312yes—
121C4—22no—
122C4—22no—
123C4—82no—
124C4—10162no—
125C4—5084no—
126C2—22no—
127C2—22no—
128C2—22no—
129C2#2254973711yescharacteristic
130C2#2254984912yes—
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting#381×C127C2
socleC22×C127C2×C22⋊C4
prime radical, cosocleC24×C127C23
hypercenter#381D127
hypocenterC127#1820
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
7. C2×C4⋊Dic127C4semidirect
10. C2×C4×Dic127C4semidirect
66. C127#1820semidirect

Maximal subgroups

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

subgroupindex
C2×#168611682
#225497392
#225496422
#381×C1272
#1820127