Finite Group Databasebeta

← #27157000

Subgroups — #27157000

Order 12416.

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

normal subgroups only

This group has 236 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
2#20261581C214yesmaximal
3C2×#15690775C211yesmaximal, characteristic
4#15690877—22no—
5C22⋊#11715897C2214yes—
6#15690865—22no—
7C22×Dic194C411yescharacteristic
8#15690775—22no—
9#15690775—22no—
10C2×#13095490C2212yes—
11C2×C4×Dic97C412yes—
12C2×C4×Dic97C2211yescharacteristic
13C2×C4⋊C4×C97C411yescharacteristic
14C2×#11715897—24no—
15C2×Dic194—22no—
16C2×Dic194—24no—
17C22×Dic97C2×C411yescharacteristic
18C22×Dic97C2×C412yes—
19C22×Dic97C2311yescharacteristic
20#13095490—24no—
21C4×Dic97—22no—
22C4×Dic97—22no—
23C4×Dic97D412yes—
24C4×Dic97D412yes—
25C22×C4×C97C2×C412yes—
26C22×C4×C97C2×C411yescharacteristic
27C4⋊C4×C97—22no—
28C23×C97C22×C411yescharacteristic
29C2×C4×C97—22no—
30C2×C4×C97—22no—
31C2×C4×C97—42no—
32C2×C4×C97C22⋊C412yes—
33C2×Dic97—22no—
34C2×Dic97—22no—
35C2×Dic97—22no—
36C2×Dic97—41no—
37C2×Dic97C22⋊C411yescharacteristic
38C2×Dic97C22⋊C411yescharacteristic
39C2×Dic97M1614yes—
40C2×Dic97C2×D412yes—
41Dic194—28no—
42#11715897—44no—
43C22×C97—21no—
44C22×C97—21no—
45C22×C97C2×C22⋊C411yescharacteristic
46C22×C97C2×M1612yes—
47C4×C97—22no—
48C4×C97—42no—
49Dic97—24no—
50Dic97—22no—
51Dic97—21no—
52Dic97#10112yes—
53C2×C97—22no—
54C2×C97#40711yescharacteristic
55C2×C97#41011yescharacteristic
56C2×C97C2×#10111yescharacteristic
57#1467—971nomaximal, Sylow
58C97#146711yescharacteristic, complemented, Sylow
59C2×C4×Q8—971no—
60#432—972no—
61#324—974no—
62C22×Q8—971no—
63C4×Q8—1942no—
64C4×Q8—1942no—
65C2×C4⋊C4—972no—
66C2×C4⋊C4#8760411yescharacteristic
67C2×C42—971no—
68C2×C42—972no—
69C4⋊C8—1942no—
70C22⋊C8—974no—
71#97—1942no—
72C2×Q8—1944no—
73C2×Q8—1942no—
74C22×C4—971no—
75C22×C4—971no—
76C22×C4—972no—
77C22×C4C2×#8760412yes—
78C22×C4C2×#8760411yescharacteristic
79C2×C8—1944no—
80C4⋊C4—22no—
81C4⋊C4—1944no—
82C42—972no—
83C42—972no—
84C42—1942no—
85C42—1942no—
86C23C22×#8760411yescharacteristic
87Q8—1948no—
88C2×C4—22no—
89C2×C4—22no—
90C2×C4—42no—
91C2×C4—971no—
92C2×C4—971no—
93C2×C4—972no—
94C2×C4—974no—
95C2×C4—3881no—
96C2×C4—1942no—
97C2×C4—1942no—
98C2×C4—1942no—
99C2×C4C22⋊#8760412yes—
100C8—3884no—
101C22—21no—
102C22—21no—
103C22C2×C22⋊#8760411yescharacteristic
104C22D97⋊M1612yes—
105C4—22no—
106C4—42no—
107C4—972no—
108C4—1944no—
109C4—1942no—
110C4—1941no—
111C2—22no—
112C2#2026170711yescharacteristic
113C2#2026171511yescharacteristic
114C2D97⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutator, prime radicalC23×C97C22×C4
FrattiniC23C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
cosocleC22×Dic97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1467
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
58. C97#1467semidirect

Maximal subgroups

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

subgroupindex
#202617552
#202615812
C2×#156907752
#146797