Finite Group Databasebeta

← #27156639

Subgroups — #27156639

Order 12416.

192 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 44 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#20260596C211yesmaximal, characteristic
3C2×#13095520—22no—
4C22⋊#11715897—22no—
5C22⋊#11715897C414yescomplemented
6C23×Dic97C2211yescharacteristic
7C2×C22⋊Dic97C411yescharacteristic
8#15690717—22no—
9C2×C22⋊C4×C97C411yescharacteristic
10#13095520—42no—
11#13095520—42no—
12C2×#11715897—24no—
13C22×Dic97—21no—
14C22×Dic97—41no—
15C22×Dic97—41no—
16C22×Dic97C2×C412yes—
17C22×Dic97D412yes—
18C22×Dic97D411yescharacteristic
19C22×Dic97Q811yescharacteristic
20C22⋊Dic97—41no—
21C24×C97C2×C411yescharacteristic
22C22×C4×C97—21no—
23C22⋊C4×C97—41no—
24C23×C97—21no—
25C23×C97—21no—
26C23×C97—42no—
27C23×C97C4211yescharacteristic
28C23×C97C22⋊C411yescharacteristic
29C23×C97C4⋊C411yescharacteristic
30C2×C4×C97—42no—
31C2×C4×C97—41no—
32C2×Dic97—22no—
33C2×Dic97—22no—
34C2×Dic97—42no—
35C2×Dic97—41no—
36C2×Dic97—41no—
37C2×Dic97—42no—
38C2×Dic97—41no—
39C2×Dic97C22⋊C412yes—
40C2×Dic97C4⋊C412yes—
41#11715897—44no—
42C22×C97—22no—
43C22×C97—21no—
44C22×C97—21no—
45C22×C97—22no—
46C22×C97—22no—
47C22×C97—42no—
48C22×C97—42no—
49C22×C97#9511yescharacteristic
50C22×C97#9911yescharacteristic
51C22×C97#9911yescharacteristic
52C4×C97—81no—
53Dic97—22no—
54Dic97—22no—
55Dic97—81no—
56C2×C97—21no—
57C2×C97—21no—
58C2×C97—41no—
59C2×C97—41no—
60C2×C97#34211yescharacteristic
61C2×C97#34412yes—
62#1306—971nomaximal, Sylow
63C97#130611yescharacteristic, complemented, Sylow
64#407—972no—
65#381—971no—
66C23×C4—971no—
67C2×M16—1942no—
68C2×C22⋊C4—971no—
69C2×C22⋊C4#8760411yescharacteristic
70C22⋊C8—974no—
71C22⋊C8—1942no—
72#95—1942no—
73C24C2×#8760411yescharacteristic
74C22×C4—21no—
75C22×C4—972no—
76C22×C4—971no—
77C22×C4—971no—
78C22×C4—972no—
79C22×C4—3881no—
80C22×C4—3881no—
81C22×C4—1941no—
82M16—3882no—
83M16—3882no—
84C2×C8—1944no—
85C22⋊C4—41no—
86C22⋊C4—3881no—
87C23—21no—
88C23—21no—
89C23—42no—
90C23C4×#8760411yescharacteristic
91C23C22⋊#8760411yescharacteristic
92C23C4⋊#8760411yescharacteristic
93C2×C4—42no—
94C2×C4—41no—
95C2×C4—972no—
96C2×C4—972no—
97C2×C4—3882no—
98C2×C4—3881no—
99C2×C4—3881no—
100C2×C4—3882no—
101C2×C4—3881no—
102C2×C4—1942no—
103C2×C4—1942no—
104C8—3884no—
105C22—22no—
106C22—21no—
107C22—21no—
108C22—22no—
109C22—22no—
110C22—42no—
111C22—42no—
112C22(C2×#87604)⋊C411yescharacteristic
113C22#1569086811yescharacteristic
114C22#1569086911yescharacteristic
115C4—81no—
116C4—7761no—
117C4—1942no—
118C4—1942no—
119C2—21no—
120C2—21no—
121C2—41no—
122C2—41no—
123C2#2026161311yescharacteristic
124C2#2026161612yes—
subgroupG/H
centerC22(C2×#87604)⋊C4
commutatorC23×C97C42
FrattiniC24C2×#87604
FittingC2×C22⋊C4×C97C4
socleC22×C97#95
prime radicalC24×C97C2×C4
cosocleC23×Dic97C22
hypercenterC2×C22⋊C4#87604
hypocenterC97#1306
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
5. C22⋊#11715897C4semidirect
63. C97#1306semidirect

Maximal subgroups

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

subgroupindex
#202617052
#202605962
#130697