Finite Group Databasebeta

← #27157923

Subgroups — #27157923

Order 12416.

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

normal subgroups only

This group has 244 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#20261739C211yesmaximal, characteristic
2#20261671C211yesmaximal, characteristic
3#20261605C214yesmaximal
4#20261291C211yesmaximal, characteristic
5C2×#13095521C2212yes—
6C2×C4×#87604C2212yes—
7C4⋊#11715897—22no—
8(C2×#87604)⋊C4—22no—
9#15690809C412yescomplemented
10#15690802C411yescharacteristic, complemented
11#15690788—22no—
12#15690775—22no—
13#15690768C2212yes—
14C42×D97C2211yescharacteristic
15#125×C97C411yescharacteristic, complemented
16C22×#87604—22no—
17#13095521—24no—
18C2×#11715897—22no—
19C4×#87604—22no—
20C4×#87604Q814yescomplemented
21C2×Dic194—22no—
22C2×C4×D97D412yes—
23C2×C4×D97C2311yescharacteristic
24C4⋊Dic97C2×C412yes—
25#13095490—22no—
26#13095490—22no—
27#13095488—22no—
28C4×Dic97—22no—
29C4×Dic97C2×C411yescharacteristic
30C4×Dic97D412yes—
31C4⋊C4×C97—22no—
32C4⋊C4×C97C2×C412yes—
33C42×C97C2×C411yescharacteristic
34C22×D97C2×D411yescharacteristic
35C2×#87604—24no—
36C2×#87604—44no—
37C2×C4×C97—22no—
38C2×C4×C97C22⋊C412yes—
39C2×C4×C97C22×C411yescharacteristic
40C2×Dic97—22no—
41C2×Dic97C22⋊C412yes—
42C2×Dic97C2×D411yescharacteristic
43C4×D97—42no—
44C4×D97C2×Q812yes—
45C4×D97#4112yes—
46Dic194—44no—
47#11715897—42no—
48C22×C97C2×C22⋊C411yescharacteristic
49C2×D97—22no—
50C2×D97C22⋊Q812yes—
51#87604—44no—
52C4×C97—22no—
53C4×C97—42no—
54C4×C97C4×Q812yes—
55Dic97—22no—
56Dic97—42no—
57Dic97#12411yescharacteristic
58Dic97C4⋊Q811yescharacteristic
59C2×C97#39111yescharacteristic
60C2×C97#42112yes—
61D97—22no—
62#1995—971nomaximal, Sylow
63C97#199511yescharacteristic, complemented, Sylow
64#533—971no—
65#423—971no—
66#376—971no—
67#339—974no—
68C2×M16—972no—
69C4⋊Q8—972no—
70#125—971no—
71#125#8760411yescharacteristic, complemented
72C22⋊Q8—1942no—
73C4×Q8—1942no—
74#117—972no—
75C2×C42—971no—
76C2×C42—972no—
77C4⋊C8—1942no—
78#95—1942no—
79C2×Q8—1942no—
80C22×C4—971no—
81C22×C4—972no—
82C22×C4—1942no—
83M16—1944no—
84C2×C8—1942no—
85C4⋊C4—22no—
86C4⋊C4—972no—
87C4⋊C4—1942no—
88C4⋊C4—1942no—
89C4⋊C4C2×#8760412yes—
90C22⋊C4—1942no—
91C42—972no—
92C42—971no—
93C42—974no—
94C42—1942no—
95C42—1942no—
96C42C2×#8760411yescharacteristic
97C23—971no—
98Q8—3884no—
99C2×C4—22no—
100C2×C4—971no—
101C2×C4—972no—
102C2×C4—972no—
103C2×C4—972no—
104C2×C4—3882no—
105C2×C4—3884no—
106C2×C4—1944no—
107C2×C4—1942no—
108C2×C4C22⋊#8760412yes—
109C2×C4C22×#8760411yescharacteristic
110C8—3882no—
111C22—972no—
112C22—1942no—
113C22C2×C22⋊#8760411yescharacteristic
114C4—22no—
115C4—42no—
116C4—971no—
117C4—971no—
118C4—3884no—
119C4—3882no—
120C4—1942no—
121C4Q8×#8760412yes—
122C2—1942no—
123C2(C2×#87604)⋊Q811yescharacteristic
124C2#2026173212yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#125×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C2×D4
cosocleC2×C4×D97C23
hypercenter#125#87604
hypocenterC97#1995
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
9. #15690809C4semidirect
10. #15690802C4semidirect
15. #125×C97C4semidirect
20. C4×#87604Q8semidirect
63. C97#1995semidirect
71. #125#87604semidirect

Maximal subgroups

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

subgroupindex
#202617392
#202616712
#202616052
#202612912
#199597