Finite Group Databasebeta

← #27158020

Subgroups — #27158020

Order 12416.

238 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 54 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#20261714C212yesmaximal
2#20261642C214yesmaximal, complemented
3#20261311C211yesmaximal, characteristic
4#15690889—22no—
5#15690869C2214yes—
6#15690868—22no—
7C2×#13095511C2211yescharacteristic
8C22×Dic194C411yescharacteristic, complemented
9#15690810C412yescomplemented
10#15690798C2212yes—
11#15690794—22no—
12#15690790—22no—
13#117×C97C411yescharacteristic, complemented
14C4⋊#87604—42no—
15C22⋊#87604—24no—
16C4×#87604—42no—
17#13095511—42no—
18C2×Dic194—21no—
19C2×Dic194—42no—
20C2×Dic194C2×C412yes—
21C2×Dic194C2×C411yescharacteristic
22C2×D388C2311yescharacteristic
23C2×C22⋊D97D412yes—
24C22×Dic97—21no—
25C2×C4×D97D412yes—
26C4⋊Dic97—24no—
27#13095488—24no—
28C22×C4×C97C2×C411yescharacteristic
29C4⋊C4×C97—21no—
30C22⋊C4×C97—21no—
31C42×C97C2×C412yes—
32C23×C97C22⋊C411yescharacteristic
33C22×D97C2×D412yes—
34C2×#87604—44no—
35C2×C4×C97—22no—
36C2×C4×C97—22no—
37C2×C4×C97C22⋊C411yescharacteristic
38C2×C4×C97C22×C411yescharacteristic
39C22⋊D97—42no—
40C2×Dic97—22no—
41C2×Dic97—42no—
42C2×Dic97C22⋊C412yes—
43D388—41no—
44C4×D97—22no—
45C4×D97—22no—
46Dic194—41no—
47Dic194—42no—
48Dic194—42no—
49C22×C97—21no—
50C22×C97—41no—
51C22×C97C2×C22⋊C411yescharacteristic
52C2×D97—22no—
53C2×D97—42no—
54#87604—84no—
55C4×C97—21no—
56C4×C97—42no—
57C4×C97#9912yes—
58Dic97—22no—
59Dic97—42no—
60C2×C97—21no—
61C2×C97—41no—
62C2×C97C2×#9911yescharacteristic
63D97—42no—
64#2129—971nomaximal, Sylow
65C97#212911yescharacteristic, complemented, Sylow
66#536—971no—
67#410—972no—
68#354—974no—
69C2×#41—971no—
70C22×Q8—971no—
71C4⋊Q8—972no—
72#124—972no—
73#123—1942no—
74C22⋊Q8—1942no—
75#117—1942no—
76#117#8760411yescharacteristic, complemented
77#99—974no—
78#99—1942no—
79#41—3882no—
80C2×Q8—971no—
81C2×Q8—972no—
82C2×Q8—3882no—
83C2×Q8—1941no—
84C2×D4—972no—
85C2×D4—971no—
86C22×C4—972no—
87C22×C4—1941no—
88C22×C4C2×#8760411yescharacteristic
89C4⋊C4—21no—
90C4⋊C4—3882no—
91C4⋊C4—1944no—
92C22⋊C4—21no—
93C22⋊C4—1944no—
94C22⋊C4—1944no—
95C42—3882no—
96C42C2×#8760412yes—
97C23—972no—
98C23C22⋊#8760411yescharacteristic
99Q8—3882no—
100Q8—3881no—
101Q8—3882no—
102D4—3881no—
103D4—3882no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—972no—
107C2×C4—3884no—
108C2×C4—3882no—
109C2×C4—1942no—
110C2×C4—1942no—
111C2×C4—1942no—
112C2×C4C22⋊#8760411yescharacteristic
113C2×C4C22×#8760411yescharacteristic
114C22—21no—
115C22—41no—
116C22—3882no—
117C22—1942no—
118C22C2×C22⋊#8760411yescharacteristic
119C4—21no—
120C4—42no—
121C4—7764no—
122C4—3882no—
123C4—1942no—
124C4#1569086912yes—
125C2—21no—
126C2—41no—
127C2—3882no—
128C2C2×#1569086911yescharacteristic
subgroupG/H
centerC2C2×#15690869
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#117×C97C4
socleC2×C97C2×#99
prime radicalC2×#13095511C22
cosocleC2×D388C23
hypercenter#117#87604
hypocenterC97#2129
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #20261642C2semidirect
8. C22×Dic194C4semidirect
9. #15690810C4semidirect
13. #117×C97C4semidirect
65. C97#2129semidirect
76. #117#87604semidirect

Maximal subgroups

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

subgroupindex
#202617142
#202616422
#202613112
#212997