Finite Group Databasebeta

← #30636660

Subgroups — #30636660

Order 16256.

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

normal subgroups only

This group has 150 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#22549600C212yesmaximal
2#381×C127C211yesmaximal, characteristic
3C22⋊(C127⋊C8)—22no—
4C23×C4×C127C411yescharacteristic
5C2×C22⋊C4×C127C411yescharacteristic
6C2×C22⋊C4×C127C2211yescharacteristic
7#95×C127—22no—
8C2×C127⋊C8—42no—
9C24×C127C2×C411yescharacteristic
10C22×C4×C127—21no—
11C22×C4×C127—21no—
12C22×C4×C127—21no—
13C22×C4×C127—41no—
14C22×C4×C127—41no—
15C22×C4×C127C812yescomplemented
16C22×C4×C127D412yes—
17C22⋊C4×C127—41no—
18C22⋊C4×C127—41no—
19C127⋊C8—82no—
20C23×C127—21no—
21C23×C127—21no—
22C23×C127—42no—
23C23×C127C22⋊C411yescharacteristic
24C23×C127C2×C811yescharacteristic
25C23×C127M1611yescharacteristic
26C2×C4×C127—22no—
27C2×C4×C127—22no—
28C2×C4×C127—42no—
29C2×C4×C127—41no—
30C2×C4×C127—41no—
31C2×C4×C127—42no—
32C2×C4×C127—42no—
33C2×C4×C127—42no—
34C2×C4×C127—41no—
35C22×C127—22no—
36C22×C127—21no—
37C22×C127—21no—
38C22×C127—41no—
39C22×C127—41no—
40C22×C127—42no—
41C22×C127—42no—
42C22×C127C22⋊C811yescharacteristic
43C22×C127#9911yescharacteristic
44C22×C127#10011yescharacteristic
45C4×C127—42no—
46C4×C127—42no—
47C4×C127—81no—
48C2×C127—21no—
49C2×C127—21no—
50C2×C127—41no—
51C2×C127—41no—
52C2×C127#32311yescharacteristic
53C2×C127#35212yes—
54#1317—1271nomaximal, Sylow
55C127#131711yescharacteristic, complemented, Sylow
56#381D12711yescharacteristic
57#323—1272no—
58C23×C4Dic12711yescharacteristic
59C2×C22⋊C4Dic12711yescharacteristic
60C2×C22⋊C4C2×D12711yescharacteristic
61C22⋊C8—2542no—
62#95—22no—
63C24C2×Dic12711yescharacteristic
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4C127⋊C812yescomplemented
70C22×C4C22⋊D12712yes—
71C2×C8—5082no—
72C22⋊C4—41no—
73C22⋊C4—41no—
74C23—21no—
75C23—21no—
76C23—42no—
77C23C22⋊Dic12711yescharacteristic
78C23C2×C127⋊C811yescharacteristic
79C23#1452323211yescharacteristic
80C2×C4—22no—
81C2×C4—22no—
82C2×C4—42no—
83C2×C4—41no—
84C2×C4—41no—
85C2×C4—42no—
86C2×C4—42no—
87C2×C4—42no—
88C2×C4—41no—
89C8—10162no—
90C22—22no—
91C22—21no—
92C22—21no—
93C22—41no—
94C22—41no—
95C22—42no—
96C22—42no—
97C22C22⋊(C127⋊C8)11yescharacteristic
98C22#1686112411yescharacteristic
99C22#1686112611yescharacteristic
100C4—42no—
101C4—42no—
102C4—81no—
103C2—21no—
104C2—21no—
105C2—41no—
106C2—41no—
107C2#2254960011yescharacteristic
108C2#2254966612yes—
subgroupG/H
centerC22#16861124
commutatorC23×C127C2×C8
FrattiniC2×C22⋊C4C2×D127
Fitting#381×C127C2
socleC22×C127#99
prime radicalC24×C127C2×C4
cosocleC2×C22⋊C4×C127C22
hypercenter#381D127
hypocenterC127#1317
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #381

Complements

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

NH ≅ G/Ntype
15. C22×C4×C127C8semidirect
55. C127#1317semidirect
69. C22×C4C127⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225496002
#381×C1272
#1317127