Finite Group Databasebeta

← #30636664

Subgroups — #30636664

Order 16256.

134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 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#22549602C212yesmaximal
2#391×C127C211yesmaximal, characteristic
3C22⋊(C127⋊C8)—22no—
4C22×Q8×C127C411yescharacteristic
5C2×C4⋊C4×C127C2211yescharacteristic
6C2×C42×C127C411yescharacteristic
7#95×C127—22no—
8C2×C127⋊C8—42no—
9C2×Q8×C127—21no—
10C2×Q8×C127—41no—
11C2×Q8×C127—41no—
12C2×Q8×C127C812yescomplemented
13C22×C4×C127—21no—
14C22×C4×C127—21no—
15C22×C4×C127C2×C411yescharacteristic
16C22×C4×C127D412yes—
17C4⋊C4×C127—41no—
18C42×C127—41no—
19C127⋊C8—82no—
20C23×C127C22⋊C411yescharacteristic
21Q8×C127—42no—
22Q8×C127—42no—
23C2×C4×C127—21no—
24C2×C4×C127—21no—
25C2×C4×C127—22no—
26C2×C4×C127—22no—
27C2×C4×C127—22no—
28C2×C4×C127—41no—
29C2×C4×C127—41no—
30C2×C4×C127—41no—
31C2×C4×C127—41no—
32C2×C4×C127—42no—
33C2×C4×C127C2×C811yescharacteristic
34C2×C4×C127M1611yescharacteristic
35C22×C127—21no—
36C22×C127—21no—
37C22×C127C22⋊C811yescharacteristic
38C22×C127#9911yescharacteristic
39C22×C127#10011yescharacteristic
40C4×C127—21no—
41C4×C127—21no—
42C4×C127—42no—
43C4×C127—42no—
44C4×C127—42no—
45C2×C127—21no—
46C2×C127—21no—
47C2×C127#32311yescharacteristic
48C2×C127#35411yescharacteristic
49C2×C127#35611yescharacteristic
50#1319—1271nomaximal, Sylow
51C127#131911yescharacteristic, complemented, Sylow
52#391D12711yescharacteristic
53#324—1272no—
54C22×Q8Dic12711yescharacteristic
55C2×C4⋊C4C2×D12711yescharacteristic
56C2×C42Dic12711yescharacteristic
57C22⋊C8—2542no—
58#95—22no—
59C2×Q8—21no—
60C2×Q8—41no—
61C2×Q8—41no—
62C2×Q8C127⋊C812yescomplemented
63C22×C4—21no—
64C22×C4—21no—
65C22×C4C2×Dic12711yescharacteristic
66C22×C4C22⋊D12712yes—
67C2×C8—5082no—
68C4⋊C4—41no—
69C42—41no—
70C23C22⋊Dic12711yescharacteristic
71Q8—42no—
72Q8—42no—
73C2×C4—21no—
74C2×C4—21no—
75C2×C4—22no—
76C2×C4—22no—
77C2×C4—22no—
78C2×C4—41no—
79C2×C4—41no—
80C2×C4—41no—
81C2×C4—41no—
82C2×C4—42no—
83C2×C4C2×C127⋊C811yescharacteristic
84C2×C4#1452323211yescharacteristic
85C8—10162no—
86C22—21no—
87C22—21no—
88C22C22⋊(C127⋊C8)11yescharacteristic
89C22#1686112411yescharacteristic
90C22#1686112611yescharacteristic
91C4—21no—
92C4—21no—
93C4—42no—
94C4—42no—
95C4—42no—
96C2—21no—
97C2—21no—
98C2#2254960011yescharacteristic
99C2#2254967111yescharacteristic
100C2#2254967511yescharacteristic
subgroupG/H
centerC22#16861124
commutatorC2×C4×C127C2×C8
FrattiniC2×C4⋊C4C2×D127
Fitting#391×C127C2
socleC22×C127#99
prime radicalC23×C127C22⋊C4
cosocleC2×C4⋊C4×C127C22
hypercenter#391D127
hypocenterC127#1319
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #391

Complements

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

NH ≅ G/Ntype
12. C2×Q8×C127C8semidirect
51. C127#1319semidirect
62. C2×Q8C127⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225496022
#391×C1272
#1319127