Finite Group Databasebeta

← #27156984

Subgroups — #27156984

Order 12416.

312 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 74 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#20261705C212yesmaximal
2#20261580C214yesmaximal, complemented
3C22×C4⋊Dic97C211yesmaximal, characteristic
4C2×#13095520—22no—
5C22⋊#11715897—22no—
6C22⋊#11715897C2214yes—
7C23×Dic97C411yescharacteristic
8C23×Dic97C2211yescharacteristic
9C2×C4⋊Dic97—22no—
10C2×C4⋊Dic97—22no—
11C2×C4⋊Dic97C412yes—
12C2×C4⋊Dic97C2212yes—
13C23×C4×C97C411yescharacteristic
14#13095520—44no—
15C2×#11715897—24no—
16C22×Dic97—21no—
17C22×Dic97—21no—
18C22×Dic97—42no—
19C22×Dic97—42no—
20C22×Dic97C2×C412yes—
21C22×Dic97C2×C411yescharacteristic
22C22×Dic97D412yes—
23C22×Dic97D412yes—
24C22×Dic97C2311yescharacteristic
25C4⋊Dic97—24no—
26C4⋊Dic97—24no—
27C24×C97C2×C411yescharacteristic
28C22×C4×C97—21no—
29C22×C4×C97—21no—
30C22×C4×C97—24no—
31C22×C4×C97C2×C412yes—
32C23×C97—22no—
33C23×C97—22no—
34C23×C97—22no—
35C23×C97C22⋊C411yescharacteristic
36C23×C97C22⋊C411yescharacteristic
37C23×C97C22×C411yescharacteristic
38C2×C4×C97—24no—
39C2×C4×C97—24no—
40C2×C4×C97—22no—
41C2×C4×C97—22no—
42C2×C4×C97M1614yescomplemented
43C2×Dic97—24no—
44C2×Dic97—22no—
45C2×Dic97—41no—
46C2×Dic97—42no—
47C2×Dic97—41no—
48C2×Dic97—42no—
49C2×Dic97—42no—
50C2×Dic97—42no—
51C2×Dic97C22⋊C412yes—
52C2×Dic97C2×D412yes—
53#11715897—44no—
54C22×C97—28no—
55C22×C97—21no—
56C22×C97—21no—
57C22×C97—21no—
58C22×C97—21no—
59C22×C97—21no—
60C22×C97—21no—
61C22×C97#9912yes—
62C22×C97#10112yes—
63C22×C97C2×C22⋊C411yescharacteristic
64C22×C97C2×M1612yes—
65C4×C97—24no—
66Dic97—22no—
67Dic97—22no—
68Dic97—42no—
69C2×C97—22no—
70C2×C97—22no—
71C2×C97—22no—
72C2×C97#40711yescharacteristic
73C2×C97C2×#9911yescharacteristic
74C2×C97C2×#10111yescharacteristic
75#1463—971nomaximal, Sylow
76C97#146311yescharacteristic, complemented, Sylow
77C22×C4⋊C4—971no—
78#407—972no—
79#324—974no—
80C23×C4—971no—
81C23×C4—971no—
82C23×C4#8760411yescharacteristic
83C2×M16—1942no—
84C2×C4⋊C4—972no—
85C2×C4⋊C4—972no—
86C2×C4⋊C4—1942no—
87C2×C4⋊C4—1942no—
88C22⋊C8—974no—
89C22⋊C8—1942no—
90C24C2×#8760411yescharacteristic
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—24no—
94C22×C4—972no—
95C22×C4—971no—
96C22×C4—971no—
97C22×C4—972no—
98C22×C4—972no—
99C22×C4—3882no—
100C22×C4—3882no—
101C22×C4—1941no—
102C22×C4—1941no—
103C22×C4C2×#8760412yes—
104M16—3884no—
105C2×C8—1944no—
106C4⋊C4—1944no—
107C4⋊C4—1944no—
108C23—22no—
109C23—22no—
110C23—22no—
111C23C22⋊#8760411yescharacteristic
112C23C22⋊#8760411yescharacteristic
113C23C22×#8760411yescharacteristic
114C2×C4—24no—
115C2×C4—24no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—972no—
119C2×C4—972no—
120C2×C4—3882no—
121C2×C4—3882no—
122C2×C4—3882no—
123C2×C4—3881no—
124C2×C4—3882no—
125C2×C4—3881no—
126C2×C4—1942no—
127C2×C4—1944no—
128C2×C4#1309552014yescomplemented
129C8—3884no—
130C22—28no—
131C22—21no—
132C22—21no—
133C22—21no—
134C22—21no—
135C22—21no—
136C22—21no—
137C22#1569086912yes—
138C22#1569087112yes—
139C22C2×C22⋊#8760411yescharacteristic
140C22C2×#1309552012yes—
141C4—24no—
142C4—3882no—
143C4—1942no—
144C4—1942no—
145C2—22no—
146C2—22no—
147C2—22no—
148C2#2026170511yescharacteristic
149C2C2×#1569086911yescharacteristic
150C2C2×#1569087111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC23×C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC24×C97C2×C4
cosocleC22×Dic97C23
hypercenterC23×C4#87604
hypocenterC97#1463
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. #20261580C2semidirect
42. C2×C4×C97M16semidirect
76. C97#1463semidirect
128. C2×C4#13095520semidirect

Maximal subgroups

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

subgroupindex
#202617052
#202615802
C22×C4⋊Dic972
#146397