Finite Group Databasebeta

← #30638648

Subgroups — #30638648

Order 16256.

612 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16861124C214yesmaximal, complemented
2#22549739C212yesmaximal, complemented
3C22×#41×C127C211yesmaximal, characteristic
4C2×C22⋊Dic127C2214yes—
5#16861124—28no—
6C127⋊#95—24no—
7C2×#41×C127—28no—
8C2×#41×C127—24no—
9C22×Q8×C127C411yescharacteristic, complemented
10C22×D4×C127C411yescharacteristic, complemented
11C22×D4×C127C2212yes—
12C23×C4×C127C412yescomplemented
13C23×C4×C127C2211yescharacteristic
14C22×Dic127—24no—
15C22⋊Dic127—28no—
16C24×C127C2×C412yes—
17C24×C127C2311yescharacteristic
18#41×C127—232no—
19C2×Q8×C127—24no—
20C2×Q8×C127—22no—
21C2×D4×C127—24no—
22C2×D4×C127—24no—
23C2×D4×C127—24no—
24C2×D4×C127—22no—
25C2×D4×C127D418yes—
26C22×C4×C127—24no—
27C22×C4×C127—24no—
28C22×C4×C127—24no—
29C22×C4×C127—24no—
30C22×C4×C127C2×C411yescharacteristic
31C22×C4×C127C2×C412yes—
32C22×C4×C127C2×C411yescharacteristic
33C22×C4×C127D414yes—
34C2×Dic127—48no—
35C2×Dic127—44no—
36C23×C127—22no—
37C23×C127—22no—
38C23×C127—42no—
39C23×C127—44no—
40C23×C127C22⋊C414yes—
41C23×C127C22×C411yescharacteristic
42C23×C127C2×D412yes—
43C23×C127C2×D414yes—
44Q8×C127—28no—
45D4×C127—28no—
46D4×C127—216no—
47C2×C4×C127—216no—
48C2×C4×C127—22no—
49C2×C4×C127—22no—
50C2×C4×C127—22no—
51C2×C4×C127—22no—
52C2×C4×C127—28no—
53C2×C4×C127C22⋊C412yes—
54C2×C4×C127C22⋊C414yes—
55C2×C4×C127C22⋊C412yes—
56C22×C127—22no—
57C22×C127—24no—
58C22×C127—24no—
59C22×C127—44no—
60C22×C127—44no—
61C22×C127—44no—
62C22×C127—44no—
63C22×C127C2×C22⋊C412yes—
64C22×C127C2×C22⋊C411yescharacteristic
65C22×C127#12014yes—
66C4×C127—22no—
67C4×C127—24no—
68C4×C127—22no—
69Dic127—84no—
70C2×C127—24no—
71C2×C127—22no—
72C2×C127—44no—
73C2×C127#37911yescharacteristic
74C2×C127#41012yes—
75#1782—1271nomaximal, Sylow
76C127#178211yescharacteristic, complemented, Sylow
77C22×#41D12711yescharacteristic
78C2×#99—1274no—
79#381—1272no—
80C2×#41—28no—
81C2×#41—24no—
82C22×Q8Dic12711yescharacteristic, complemented
83C22×D4Dic12711yescharacteristic, complemented
84C22×D4C2×D12712yes—
85C23×C4Dic12712yescomplemented
86C23×C4C2×D12711yescharacteristic
87C2×C22⋊C4—1274no—
88#99—2548no—
89#95—2544no—
90C24C2×Dic12712yes—
91C24C22×D12711yescharacteristic
92#41—232no—
93C2×Q8—24no—
94C2×Q8—22no—
95C2×D4—24no—
96C2×D4—24no—
97C2×D4—24no—
98C2×D4—22no—
99C2×D4C22⋊D12718yes—
100C22×C4—24no—
101C22×C4—24no—
102C22×C4—24no—
103C22×C4—24no—
104C22×C4—2544no—
105C22×C4C2×Dic12711yescharacteristic
106C22×C4C2×Dic12712yes—
107C22×C4C2×Dic12711yescharacteristic
108C22×C4C22⋊D12714yes—
109C22⋊C4—2548no—
110C23—22no—
111C23—22no—
112C23—42no—
113C23—44no—
114C23C22⋊Dic12714yes—
115C23C22×Dic12711yescharacteristic
116C23C2×C22⋊D12712yes—
117C23C2×C22⋊D12714yes—
118Q8—28no—
119D4—28no—
120D4—216no—
121C2×C4—216no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—28no—
127C2×C4—5088no—
128C2×C4—5084no—
129C2×C4C22⋊Dic12714yes—
130C2×C4C22⋊Dic12712yes—
131C2×C4C22⋊Dic12712yes—
132C22—24no—
133C22—22no—
134C22—24no—
135C22—44no—
136C22—44no—
137C22—44no—
138C22—44no—
139C22C2×C22⋊Dic12712yes—
140C22C2×C22⋊Dic12711yescharacteristic
141C22#1686117814yes—
142C4—22no—
143C4—24no—
144C4—22no—
145C4—10164no—
146C2—24no—
147C2—22no—
148C2—44no—
149C2#2254973211yescharacteristic
150C2#2254985012yes—
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting, prime radicalC22×#41×C127C2
socleC22×C127C2×C22⋊C4
cosocleC24×C127C23
hypercenterC22×#41D127
hypocenterC127#1782
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16861124C2semidirect
2. #22549739C2semidirect
9. C22×Q8×C127C4semidirect
10. C22×D4×C127C4semidirect
12. C23×C4×C127C4semidirect
76. C127#1782semidirect
82. C22×Q8Dic127semidirect
83. C22×D4Dic127semidirect
85. C23×C4Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611242
#225497392
C22×#41×C1272
#1782127