Finite Group Databasebeta

← #27157636

Subgroups — #27157636

Order 12416.

316 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 70 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#20261707C212yesmaximal
2#20261615C214yesmaximal, complemented
3C2×C4⋊C4×D97C211yesmaximal, characteristic
4C2×#13095520C2214yes—
5#15690867—22no—
6#15690867—22no—
7C22×C4×D97C412yes—
8C22×C4×D97C2211yescharacteristic
9C2×C4⋊Dic97C411yescharacteristic
10C2×#13095490C2212yes—
11C4⋊C4×D97—22no—
12C4⋊C4×D97—22no—
13C2×C4⋊C4×C97C411yescharacteristic
14#13095520—28no—
15C2×#11715897—24no—
16C23×D97C2×C411yescharacteristic
17C22×Dic97C2×C412yes—
18C22×Dic97C2311yescharacteristic
19C2×C4×D97—24no—
20C2×C4×D97—22no—
21C2×C4×D97—22no—
22C2×C4×D97—44no—
23C2×C4×D97D412yes—
24C2×C4×D97D412yes—
25C4⋊Dic97—22no—
26#13095490—24no—
27C22×C4×C97C2×C412yes—
28C22×C4×C97C2×C411yescharacteristic
29C4⋊C4×C97—22no—
30C23×C97C22×C411yescharacteristic
31C22×D97—22no—
32C22×D97—22no—
33C22×D97—22no—
34C22×D97C22⋊C411yescharacteristic
35C22×D97C22⋊C411yescharacteristic
36C2×C4×C97—22no—
37C2×C4×C97—22no—
38C2×C4×C97—42no—
39C2×C4×C97C22⋊C412yes—
40C2×Dic97—22no—
41C2×Dic97—42no—
42C2×Dic97C2×D412yes—
43C2×Dic97#4114yes—
44C4×D97—24no—
45C4×D97—24no—
46C4×D97—44no—
47C4×D97—44no—
48#11715897—44no—
49C22×C97—21no—
50C22×C97—21no—
51C22×C97C2×C22⋊C411yescharacteristic
52C22×C97#11712yes—
53C2×D97—28no—
54C2×D97—21no—
55C2×D97—21no—
56C2×D97—21no—
57C2×D97—21no—
58C2×D97#10012yes—
59C2×D97#10112yes—
60C4×C97—22no—
61C4×C97—42no—
62Dic97—42no—
63Dic97#12314yes—
64C2×C97—22no—
65C2×C97#38111yescharacteristic
66C2×C97C2×#10011yescharacteristic
67C2×C97C2×#10111yescharacteristic
68D97—22no—
69D97—22no—
70#1821—971nomaximal, Sylow
71C97#182111yescharacteristic, complemented, Sylow
72C22×C4⋊C4—971no—
73#407—972no—
74#343—974no—
75C23×C4—971no—
76C23×C4—972no—
77C2×M16—974no—
78C2×C4⋊C4—972no—
79C2×C4⋊C4—971no—
80C2×C4⋊C4—1942no—
81C2×C4⋊C4—1942no—
82C2×C4⋊C4#8760411yescharacteristic
83C22⋊C8—1942no—
84C22⋊C8—1942no—
85C24—971no—
86C22×C4—971no—
87C22×C4—972no—
88C22×C4—972no—
89C22×C4—972no—
90C22×C4—3884no—
91C22×C4—1944no—
92C22×C4—1942no—
93C22×C4—1942no—
94C22×C4C2×#8760412yes—
95C22×C4C2×#8760411yescharacteristic
96M16—1948no—
97C2×C8—1944no—
98C4⋊C4—22no—
99C4⋊C4—1944no—
100C4⋊C4—1942no—
101C23—971no—
102C23—971no—
103C23—1942no—
104C23—1942no—
105C23—1942no—
106C23C22×#8760411yescharacteristic
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—42no—
110C2×C4—972no—
111C2×C4—974no—
112C2×C4—3884no—
113C2×C4—3884no—
114C2×C4—3882no—
115C2×C4—1944no—
116C2×C4—1944no—
117C2×C4—1942no—
118C2×C4C22⋊#8760412yes—
119C8—3884no—
120C22—21no—
121C22—21no—
122C22—972no—
123C22—972no—
124C22—1948no—
125C22—1941no—
126C22—1941no—
127C22—1941no—
128C22—1941no—
129C22C2×C22⋊#8760411yescharacteristic
130C22#1569088912yes—
131C4—22no—
132C4—42no—
133C4—974no—
134C4—3882no—
135C2—22no—
136C2—1942no—
137C2—1942no—
138C2#2026167811yescharacteristic
139C2D97⋊#10011yescharacteristic
140C2D97⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×D97C2×C4
cosocleC22×Dic97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1821
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
2. #20261615C2semidirect
71. C97#1821semidirect

Maximal subgroups

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

subgroupindex
#202617072
#202616152
C2×C4⋊C4×D972
#182197