Finite Group Databasebeta

← #27156997

Subgroups — #27156997

Order 12416.

300 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 64 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#20261755C212yesmaximal, complemented
2#20261579C214yesmaximal, complemented
3C2×#15690772C211yesmaximal, characteristic
4#15690877—22no—
5C22⋊#11715897C2214yes—
6#15690865—22no—
7C22×D388C411yescharacteristic
8C22×C4×D97C412yes—
9#15690772—22no—
10#15690772—22no—
11C2×#13095488C2212yes—
12C2×C4×Dic97C2211yescharacteristic
13C2×C4⋊C4×C97C411yescharacteristic
14C2×#11715897—24no—
15C23×D97C2×C412yes—
16C2×D388—24no—
17C2×D388—22no—
18C22×Dic97C2×C411yescharacteristic
19C22×Dic97C2311yescharacteristic
20C2×C4×D97—22no—
21C2×C4×D97—22no—
22C2×C4×D97—44no—
23#13095488—24no—
24C4×Dic97D412yes—
25C4×Dic97D412yes—
26C22×C4×C97C2×C412yes—
27C22×C4×C97C2×C411yescharacteristic
28C4⋊C4×C97—22no—
29C23×C97C22×C411yescharacteristic
30C22×D97—22no—
31C22×D97—22no—
32C22×D97—42no—
33C22×D97—42no—
34C22×D97M1614yes—
35C2×C4×C97—22no—
36C2×C4×C97—22no—
37C2×C4×C97—42no—
38C2×C4×C97C22⋊C412yes—
39C2×Dic97—22no—
40C2×Dic97—41no—
41C2×Dic97C22⋊C411yescharacteristic
42C2×Dic97C22⋊C411yescharacteristic
43C2×Dic97C2×D412yes—
44D388—28no—
45C4×D97—44no—
46C4×D97—44no—
47#11715897—44no—
48C22×C97—21no—
49C22×C97—21no—
50C22×C97C2×C22⋊C411yescharacteristic
51C22×C97C2×M1612yes—
52C2×D97—24no—
53C2×D97—44no—
54C2×D97—42no—
55C2×D97—42no—
56C2×D97—44no—
57C4×C97—22no—
58C4×C97—42no—
59Dic97—22no—
60Dic97—21no—
61Dic97#10012yes—
62C2×C97—22no—
63C2×C97#40711yescharacteristic
64C2×C97#41011yescharacteristic
65C2×C97C2×#10011yescharacteristic
66D97—44no—
67#1466—971nomaximal, Sylow
68C97#146611yescharacteristic, complemented, Sylow
69C2×C4×D4—971no—
70#432—972no—
71#323—974no—
72C22×D4—971no—
73C23×C4—972no—
74C4×D4—1942no—
75C4×D4—1942no—
76C2×C4⋊C4#8760411yescharacteristic
77C2×C22⋊C4—972no—
78C2×C42—971no—
79C4⋊C8—1942no—
80C22⋊C8—974no—
81#97—1942no—
82C24—972no—
83C2×D4—1944no—
84C2×D4—1942no—
85C22×C4—971no—
86C22×C4—971no—
87C22×C4—3884no—
88C22×C4—1942no—
89C22×C4—1942no—
90C22×C4C2×#8760412yes—
91C22×C4C2×#8760411yescharacteristic
92C2×C8—1944no—
93C4⋊C4—22no—
94C22⋊C4—1944no—
95C42—972no—
96C42—972no—
97C23—974no—
98C23—3882no—
99C23—3882no—
100C23—1942no—
101C23—1942no—
102C23C22×#8760411yescharacteristic
103D4—1948no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—42no—
107C2×C4—971no—
108C2×C4—972no—
109C2×C4—971no—
110C2×C4—3884no—
111C2×C4—3884no—
112C2×C4—3881no—
113C2×C4—1942no—
114C2×C4C22⋊#8760412yes—
115C8—3884no—
116C22—21no—
117C22—21no—
118C22—3884no—
119C22—3884no—
120C22—3882no—
121C22—3882no—
122C22—1944no—
123C22C2×C22⋊#8760411yescharacteristic
124C22D97⋊M1612yes—
125C4—22no—
126C4—42no—
127C4—972no—
128C4—1942no—
129C4—1941no—
130C2—22no—
131C2—3884no—
132C2#2026170711yescharacteristic
133C2#2026171611yescharacteristic
134C2D97⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D388C4
cosocleC22×Dic97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1466
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20261755C2semidirect
2. #20261579C2semidirect
68. C97#1466semidirect

Maximal subgroups

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

subgroupindex
#202617552
#202615792
C2×#156907722
#146697