Finite Group Databasebeta

← #27157217

Subgroups — #27157217

Order 12416.

248 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 58 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, complemented
2#20261591C214yesmaximal, complemented
3C2×#15690799C211yesmaximal, characteristic
4#15690877—22no—
5C22⋊#11715897—22no—
6#15690865C2214yescomplemented
7C22×D388C411yescharacteristic
8#15690799—22no—
9#15690799C412yes—
10#15690799C2212yes—
11C2×#13095488—22no—
12C2×C4×Dic97C2211yescharacteristic
13C22×Q8×C97C411yescharacteristic
14C2×#11715897—24no—
15C23×D97—21no—
16C2×D388—21no—
17C2×D388—42no—
18C2×D388C2×C412yes—
19C22×Dic97D412yes—
20#13095488—24no—
21#13095488—24no—
22C4×Dic97C2×C411yescharacteristic
23C4×Dic97D412yes—
24C4×Dic97C2311yescharacteristic
25C2×Q8×C97—21no—
26C2×Q8×C97—42no—
27C2×Q8×C97C2×C412yes—
28C22×C4×C97—21no—
29C22×C4×C97C2×C411yescharacteristic
30C23×C97C22⋊C411yescharacteristic
31C22×D97—22no—
32C22×D97—44no—
33C22×D97—42no—
34Q8×C97—42no—
35Q8×C97—42no—
36C2×C4×C97—22no—
37C2×C4×C97—22no—
38C2×C4×C97—42no—
39C2×C4×C97C22⋊C411yescharacteristic
40C2×C4×C97C22×C411yescharacteristic
41C2×Dic97—42no—
42C2×Dic97C22⋊C412yes—
43C2×Dic97C2×D412yes—
44D388—42no—
45D388—42no—
46#11715897—44no—
47C22×C97—22no—
48C22×C97#10012yes—
49C22×C97C2×C22⋊C411yescharacteristic
50C2×D97—42no—
51C2×D97—44no—
52C2×D97—84no—
53C4×C97—21no—
54C4×C97—21no—
55C4×C97—42no—
56Dic97—22no—
57Dic97—22no—
58C2×C97—22no—
59C2×C97C2×#10011yescharacteristic
60C2×C97#42112yes—
61D97—82no—
62#1533—971nomaximal, Sylow
63C97#153311yescharacteristic, complemented, Sylow
64C2×#124—971no—
65#432—972no—
66#329—974no—
67C22×Q8#8760411yescharacteristic
68C22×D4—971no—
69#124—972no—
70#124—972no—
71#124—1942no—
72C2×C22⋊C4—1942no—
73C2×C42—971no—
74C4⋊C8—1942no—
75C22⋊C8—1942no—
76#97—974no—
77C24—1941no—
78C2×Q8—21no—
79C2×Q8—42no—
80C2×Q8C2×#8760412yes—
81C2×D4—972no—
82C2×D4—3882no—
83C2×D4—1941no—
84C22×C4—21no—
85C22×C4—972no—
86C22×C4C2×#8760411yescharacteristic
87C2×C8—1944no—
88C22⋊C4—1944no—
89C22⋊C4—1944no—
90C42—972no—
91C42—971no—
92C42—971no—
93C23—3884no—
94C23—3882no—
95C23—1942no—
96C23C22⋊#8760411yescharacteristic
97Q8—42no—
98Q8—42no—
99D4—3882no—
100D4—3882no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—42no—
104C2×C4—972no—
105C2×C4—972no—
106C2×C4—3882no—
107C2×C4C22⋊#8760411yescharacteristic
108C2×C4C22×#8760411yescharacteristic
109C8—3884no—
110C22—22no—
111C22—7764no—
112C22—3882no—
113C22—3884no—
114C22#1569087012yes—
115C22C2×C22⋊#8760411yescharacteristic
116C4—21no—
117C4—21no—
118C4—42no—
119C4—1942no—
120C4—1942no—
121C2—22no—
122C2—7762no—
123C2C2×#1569087011yescharacteristic
124C2#2026173412yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC22×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D388C4
cosocleC4×Dic97C23
hypercenterC22×Q8#87604
hypocenterC97#1533
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. #20261755C2semidirect
2. #20261591C2semidirect
6. #15690865C22semidirect
63. C97#1533semidirect

Maximal subgroups

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

subgroupindex
#202617552
#202615912
C2×#156907992
#153397