Finite Group Databasebeta

← #30638651

Subgroups — #30638651

Order 16256.

820 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16861126C214yesmaximal, complemented
2#22549840C212yesmaximal, complemented
3C23×D4×C127C211yesmaximal, characteristic
4C2×#14523232C2214yescomplemented
5#16861126—28no—
6C22⋊(C127⋊C8)—24no—
7C25×C127C412yes—
8C22×D4×C127—28no—
9C22×D4×C127—24no—
10C22×D4×C127C412yes—
11C22×D4×C127C2212yes—
12C23×C4×C127C2211yescharacteristic
13#14523232—28no—
14C2×C127⋊C8—24no—
15C24×C127—28no—
16C24×C127—24no—
17C24×C127—48no—
18C24×C127C2×C411yescharacteristic
19C24×C127C2×C414yes—
20C2×D4×C127—232no—
21C2×D4×C127—28no—
22C2×D4×C127—24no—
23C2×D4×C127—24no—
24C2×D4×C127—24no—
25C2×D4×C127D418yes—
26C22×C4×C127—24no—
27C22×C4×C127C2×C411yescharacteristic
28C22×C4×C127D414yes—
29C22×C4×C127C2311yescharacteristic
30C127⋊C8—44no—
31C23×C127—24no—
32C23×C127—22no—
33C23×C127—216no—
34C23×C127—24no—
35C23×C127—24no—
36C23×C127—48no—
37C23×C127—432no—
38C23×C127—48no—
39C23×C127—48no—
40C23×C127C22⋊C412yes—
41C23×C127C22⋊C418yes—
42C23×C127C22×C411yescharacteristic
43D4×C127—216no—
44D4×C127—216no—
45C2×C4×C127—22no—
46C2×C4×C127—28no—
47C2×C4×C127C22⋊C412yes—
48C2×C4×C127C2×D414yes—
49C2×C4×C127C2×D412yes—
50C22×C127—28no—
51C22×C127—22no—
52C22×C127—22no—
53C22×C127—22no—
54C22×C127—28no—
55C22×C127—48no—
56C22×C127—432no—
57C22×C127—48no—
58C22×C127—48no—
59C22×C127#10014yescomplemented
60C22×C127C2×C22⋊C412yes—
61C22×C127C2×C22⋊C411yescharacteristic
62C4×C127—22no—
63C4×C127#12014yes—
64C2×C127—24no—
65C2×C127—22no—
66C2×C127—48no—
67C2×C127#37911yescharacteristic
68C2×C127C2×#10012yes—
69#1783—1271nomaximal, Sylow
70C127#178311yescharacteristic, complemented, Sylow
71C23×D4D12711yescharacteristic
72C2×#100—1274no—
73#407—1272no—
74C25Dic12712yes—
75C22×D4—28no—
76C22×D4—24no—
77C22×D4Dic12712yes—
78C22×D4C2×D12712yes—
79C23×C4C2×D12711yescharacteristic
80C2×M16—1274no—
81#100—2548no—
82C22⋊C8—2544no—
83C24—28no—
84C24—24no—
85C24—48no—
86C24C2×Dic12711yescharacteristic
87C24C2×Dic12714yes—
88C2×D4—232no—
89C2×D4—28no—
90C2×D4—24no—
91C2×D4—24no—
92C2×D4—24no—
93C2×D4C22⋊D12718yes—
94C22×C4—24no—
95C22×C4C2×Dic12711yescharacteristic
96C22×C4C22⋊D12714yes—
97C22×C4C22×D12711yescharacteristic
98M16—2548no—
99C2×C8—2544no—
100C23—24no—
101C23—22no—
102C23—216no—
103C23—24no—
104C23—24no—
105C23—48no—
106C23—432no—
107C23—48no—
108C23—48no—
109C23C22⋊Dic12712yes—
110C23C22⋊Dic12718yes—
111C23C22×Dic12711yescharacteristic
112D4—216no—
113D4—216no—
114C2×C4—22no—
115C2×C4—28no—
116C2×C4C22⋊Dic12712yes—
117C2×C4C2×C22⋊D12714yes—
118C2×C4C2×C22⋊D12712yes—
119C8—5084no—
120C22—28no—
121C22—22no—
122C22—22no—
123C22—22no—
124C22—28no—
125C22—48no—
126C22—432no—
127C22—48no—
128C22—48no—
129C22#1686112614yescomplemented
130C22C2×C22⋊Dic12712yes—
131C22C2×C22⋊Dic12711yescharacteristic
132C4—22no—
133C4#1686117814yes—
134C2—24no—
135C2—22no—
136C2—48no—
137C2#2254973211yescharacteristic
138C2C2×#1686112612yes—
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radicalC23×D4×C127C2
socleC22×C127C2×C22⋊C4
cosocleC22×C4×C127C23
hypercenterC23×D4D127
hypocenterC127#1783
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16861126C2semidirect
2. #22549840C2semidirect
4. C2×#14523232C22semidirect
59. C22×C127#100semidirect
70. C127#1783semidirect
129. C22#16861126semidirect

Maximal subgroups

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

subgroupindex
C2×#168611262
#225498402
C23×D4×C1272
#1783127