Finite Group Databasebeta

← #30637444

Subgroups — #30637444

Order 16256.

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

normal subgroups only

This group has 332 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
1#22549933C212yesmaximal, complemented
2#22549620C214yesmaximal, complemented
3C2×#127×C127C211yesmaximal, characteristic
4C4⋊(C127⋊C8)C2214yescomplemented
5C22⋊(C127⋊C8)—22no—
6#16861120—22no—
7C22×D4×C127—22no—
8C22×D4×C127C412yes—
9#127×C127—22no—
10#127×C127C412yes—
11#127×C127C2212yes—
12C2×C42×C127C2211yescharacteristic
13C2×C127⋊C8—24no—
14C24×C127—22no—
15C2×D4×C127—22no—
16C2×D4×C127—24no—
17C2×D4×C127—24no—
18C2×D4×C127—44no—
19C2×D4×C127—48no—
20C2×D4×C127C2×C414yes—
21C22×C4×C127C2×C411yescharacteristic
22C22×C4×C127D412yes—
23C42×C127C2×C411yescharacteristic
24C42×C127D412yes—
25C42×C127C2311yescharacteristic
26C127⋊C8—44no—
27C23×C127—24no—
28C23×C127—48no—
29C23×C127—44no—
30C23×C127C22⋊C411yescharacteristic
31D4×C127—44no—
32D4×C127—44no—
33D4×C127—48no—
34D4×C127—48no—
35C2×C4×C127—22no—
36C2×C4×C127—24no—
37C2×C4×C127C22⋊C412yes—
38C2×C4×C127C22⋊C411yescharacteristic
39C2×C4×C127C22×C411yescharacteristic
40C2×C4×C127C2×D412yes—
41C22×C127—22no—
42C22×C127—44no—
43C22×C127—48no—
44C22×C127—88no—
45C22×C127#10012yes—
46C22×C127C2×C22⋊C411yescharacteristic
47C4×C127—21no—
48C4×C127—21no—
49C4×C127—22no—
50C4×C127AGL(1,ℤ/8)14yes—
51C2×C127—22no—
52C2×C127—84no—
53C2×C127C2×#10011yescharacteristic
54C2×C127#41812yes—
55#1555—1271nomaximal, Sylow
56C127#155511yescharacteristic, complemented, Sylow
57C2×#127D12711yescharacteristic
58#432—1272no—
59#331—1274no—
60C22×D4—22no—
61C22×D4Dic12712yes—
62#127—22no—
63#127Dic12712yes—
64#127C2×D12712yes—
65C2×C42C2×D12711yescharacteristic
66C4⋊C8—1274no—
67C22⋊C8—2542no—
68#97—2542no—
69C24—22no—
70C2×D4—22no—
71C2×D4—24no—
72C2×D4—24no—
73C2×D4—44no—
74C2×D4—48no—
75C2×D4C2×Dic12714yes—
76C22×C4C2×Dic12711yescharacteristic
77C22×C4C22⋊D12712yes—
78C2×C8—2544no—
79C42C2×Dic12711yescharacteristic
80C42C22⋊D12712yes—
81C42C22×D12711yescharacteristic
82C23—24no—
83C23—48no—
84C23—44no—
85C23C22⋊Dic12711yescharacteristic
86D4—44no—
87D4—44no—
88D4—48no—
89D4—48no—
90C2×C4—22no—
91C2×C4—24no—
92C2×C4C22⋊Dic12712yes—
93C2×C4C22⋊Dic12711yescharacteristic
94C2×C4C22×Dic12711yescharacteristic
95C2×C4C2×C22⋊D12712yes—
96C8—5084no—
97C22—22no—
98C22—44no—
99C22—48no—
100C22—88no—
101C22#1686112612yes—
102C22C2×C22⋊Dic12711yescharacteristic
103C4—21no—
104C4—21no—
105C4—22no—
106C4#1686123514yes—
107C2—22no—
108C2—84no—
109C2C2×#1686112611yescharacteristic
110C2#2254988112yes—
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting, prime radicalC2×#127×C127C2
socleC22×C127C2×C22⋊C4
cosocleC42×C127C23
hypercenterC2×#127D127
hypocenterC127#1555
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #22549933C2semidirect
2. #22549620C2semidirect
4. C4⋊(C127⋊C8)C22semidirect
56. C127#1555semidirect

Maximal subgroups

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

subgroupindex
#225499332
#225496202
C2×#127×C1272
#1555127