Finite Group Databasebeta

← #30636639

Subgroups — #30636639

Order 16256.

192 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 57 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#22549840C211yesmaximal, characteristic
2#22549739C211yesmaximal, characteristic
3#407×C127C211yesmaximal, characteristic
4C2×#14523232—21no—
5C2×C22⋊Dic127C412yes—
6C22⋊(C127⋊C8)—21no—
7C22⋊(C127⋊C8)C412yescomplemented
8C127⋊#95—21no—
9C127⋊#95—21no—
10C23×C4×C127C2211yescharacteristic
11C2×M16×C127—21no—
12C22⋊C8×C127—21no—
13C22⋊C8×C127C412yescomplemented
14C22×Dic127—22no—
15#14523232—42no—
16C2×C127⋊C8—22no—
17C22⋊Dic127—42no—
18C24×C127C2×C411yescharacteristic
19C22×C4×C127—42no—
20C22×C4×C127C2×C411yescharacteristic
21C22×C4×C127C2×C411yescharacteristic
22C22×C4×C127D411yescharacteristic
23C22×C4×C127D411yescharacteristic
24C22×C4×C127D411yescharacteristic
25C22×C4×C127Q811yescharacteristic
26M16×C127—42no—
27C2×C8×C127—22no—
28C2×Dic127—42no—
29C2×Dic127—42no—
30C2×Dic127—42no—
31C127⋊C8—42no—
32C23×C127—22no—
33C23×C127—41no—
34C23×C127—41no—
35C23×C127C4211yescharacteristic
36C23×C127C22⋊C411yescharacteristic
37C23×C127C4⋊C411yescharacteristic
38C2×C4×C127—22no—
39C2×C4×C127—22no—
40C2×C4×C127—41no—
41C2×C4×C127—41no—
42C2×C4×C127—41no—
43C2×C4×C127—41no—
44C2×C4×C127C22⋊C411yescharacteristic
45C2×C4×C127C22⋊C411yescharacteristic
46C2×C4×C127C4⋊C411yescharacteristic
47C2×C4×C127C4⋊C411yescharacteristic
48C8×C127—42no—
49C22×C127—21no—
50C22×C127—21no—
51C22×C127—21no—
52C22×C127—21no—
53C22×C127—21no—
54C22×C127—21no—
55C22×C127—21no—
56C22×C127—21no—
57C22×C127—44no—
58C22×C127#9511yescharacteristic
59C22×C127#9912yes—
60C4×C127—21no—
61C4×C127—21no—
62C4×C127—21no—
63C4×C127—21no—
64Dic127—82no—
65C2×C127—22no—
66C2×C127—41no—
67C2×C127—41no—
68C2×C127#34211yescharacteristic
69C2×C127#34411yescharacteristic
70C2×C127#34411yescharacteristic
71#1306—1271nomaximal, Sylow
72C127#130611yescharacteristic, complemented, Sylow
73#407—1271no—
74#407D12711yescharacteristic
75#381—1271no—
76C23×C4C2×D12711yescharacteristic
77C2×M16—21no—
78C2×M16—2541no—
79C2×C22⋊C4—1272no—
80C22⋊C8—21no—
81C22⋊C8—1272no—
82C22⋊C8—2541no—
83C22⋊C8Dic12712yescomplemented
84#95—2541no—
85#95—2541no—
86C24C4×D12711yescharacteristic
87C22×C4—42no—
88C22×C4—2542no—
89C22×C4C4×D12711yescharacteristic
90C22×C4C2×Dic12711yescharacteristic
91C22×C4C22⋊D12711yescharacteristic
92C22×C4C22⋊D12711yescharacteristic
93C22×C4D50811yescharacteristic
94C22×C4Dic25411yescharacteristic
95M16—42no—
96M16—5082no—
97C2×C8—22no—
98C2×C8—2542no—
99C22⋊C4—5082no—
100C23—22no—
101C23—41no—
102C23—41no—
103C23C4×Dic12711yescharacteristic
104C23(C2×D127)⋊C411yescharacteristic
105C23Dic127⋊C411yescharacteristic
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—5082no—
113C2×C4—5082no—
114C2×C4—5082no—
115C2×C4(C2×D127)⋊C411yescharacteristic
116C2×C4C22⋊Dic12711yescharacteristic
117C2×C4Dic127⋊C411yescharacteristic
118C2×C4C4⋊Dic12711yescharacteristic
119C8—42no—
120C8—5082no—
121C22—21no—
122C22—21no—
123C22—21no—
124C22—21no—
125C22—21no—
126C22—21no—
127C22—21no—
128C22—21no—
129C22—44no—
130C22C127⋊#9511yescharacteristic
131C22#1686112312yes—
132C4—21no—
133C4—21no—
134C4—21no—
135C4—21no—
136C4—10162no—
137C2—22no—
138C2—41no—
139C2—41no—
140C2#2254964211yescharacteristic
141C2#2254964711yescharacteristic
142C2#2254964811yescharacteristic
subgroupG/H
centerC22C127⋊#95
commutatorC23×C127C42
FrattiniC23×C4C2×D127
Fitting#407×C127C2
socleC22×C127#95
prime radicalC24×C127C2×C4
cosocleC23×C4×C127C22
hypercenter#407D127
hypocenterC127#1306
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
7. C22⋊(C127⋊C8)C4semidirect
13. C22⋊C8×C127C4semidirect
72. C127#1306semidirect
83. C22⋊C8Dic127semidirect

Maximal subgroups

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

subgroupindex
#225498402
#225497392
#407×C1272
#1306127