Finite Group Databasebeta

← #27157634

Subgroups — #27157634

Order 12416.

300 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 66 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#20261678C212yesmaximal
2#20261613C214yesmaximal
3C2×#15690768C211yesmaximal, characteristic
4C2×C22⋊#87604C2214yes—
5(C2×#87604)⋊C4—24no—
6C22×C4×D97C2211yescharacteristic
7#15690768—24no—
8C2×C4⋊Dic97C411yescharacteristic, complemented
9C2×#13095488C2212yes—
10C2×C4×Dic97C412yescomplemented
11C2×C4⋊C4×C97C411yescharacteristic, complemented
12C22×#87604—24no—
13C22⋊#87604—28no—
14C23×D97C2311yescharacteristic
15C22×Dic97C2×C411yescharacteristic
16C22×Dic97C2×C412yes—
17C2×C4×D97—24no—
18C2×C4×D97D414yes—
19C4⋊Dic97—22no—
20#13095488—24no—
21C4×Dic97—24no—
22C22×C4×C97C2×C412yes—
23C22×C4×C97C2×C411yescharacteristic
24C4⋊C4×C97—22no—
25C23×C97C22×C411yescharacteristic
26C22×D97—41no—
27C22×D97—41no—
28C22×D97C2×D412yes—
29C22×D97#4114yes—
30C2×#87604—44no—
31C2×#87604—44no—
32C2×#87604—44no—
33C2×C4×C97—22no—
34C2×C4×C97—22no—
35C2×C4×C97—42no—
36C2×C4×C97C22⋊C412yes—
37C2×Dic97—22no—
38C2×Dic97—22no—
39C2×Dic97—42no—
40C2×Dic97C22⋊C412yes—
41C4×D97—28no—
42C22×C97—21no—
43C22×C97—21no—
44C22×C97C2×C22⋊C411yescharacteristic
45C22×C97#11712yes—
46C2×D97—22no—
47C2×D97—22no—
48C2×D97—44no—
49C2×D97#12314yes—
50#87604—84no—
51C4×C97—22no—
52C4×C97—42no—
53Dic97—22no—
54Dic97—42no—
55C2×C97—22no—
56C2×C97#38111yescharacteristic
57C2×C97#41011yescharacteristic
58C2×C97#41011yescharacteristic
59D97—24no—
60#1820—971nomaximal, Sylow
61C97#182011yescharacteristic, complemented, Sylow
62C2×#117—971no—
63#381—972no—
64#342—974no—
65C23×C4—971no—
66#117—1944no—
67C2×C4⋊C4—971no—
68C2×C4⋊C4#8760411yescharacteristic, complemented
69C2×C22⋊C4—972no—
70C2×C22⋊C4—974no—
71C2×C42—972no—
72#95—1944no—
73C24—971no—
74C22×C4—971no—
75C22×C4—974no—
76C22×C4—972no—
77C22×C4—1944no—
78C22×C4—1944no—
79C22×C4C2×#8760412yes—
80C22×C4C2×#8760411yescharacteristic
81C4⋊C4—22no—
82C4⋊C4—1942no—
83C22⋊C4—1944no—
84C22⋊C4—1948no—
85C42—1944no—
86C23—974no—
87C23—972no—
88C23—3881no—
89C23—3881no—
90C23C22×#8760411yescharacteristic
91C2×C4—22no—
92C2×C4—22no—
93C2×C4—42no—
94C2×C4—972no—
95C2×C4—3884no—
96C2×C4—3884no—
97C2×C4—3884no—
98C2×C4—3882no—
99C2×C4—1948no—
100C2×C4—1942no—
101C2×C4—1942no—
102C2×C4C22⋊#8760412yes—
103C22—21no—
104C22—21no—
105C22—974no—
106C22—3884no—
107C22—1942no—
108C22—1942no—
109C22C2×C22⋊#8760411yescharacteristic
110C22#1569088912yes—
111C4—22no—
112C4—42no—
113C4—7764no—
114C4—3882no—
115C4—1942no—
116C2—22no—
117C2—1944no—
118C2#2026167811yescharacteristic
119C2#2026171511yescharacteristic
120C2#2026171611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radical, cosocleC23×D97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1820
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
8. C2×C4⋊Dic97C4semidirect
10. C2×C4×Dic97C4semidirect
11. C2×C4⋊C4×C97C4semidirect
61. C97#1820semidirect
68. C2×C4⋊C4#87604semidirect

Maximal subgroups

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

subgroupindex
#202616782
#202616132
C2×#156907682
#182097