Finite Group Databasebeta

← #27158134

Subgroups — #27158134

Order 12416.

172 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 48 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
1D8×#87604C212yesmaximal, complemented
2#20261662C211yesmaximal, characteristic, complemented
3#20261647C212yesmaximal, complemented
4#20261622C211yesmaximal, characteristic, complemented
5D16×D97C211yesmaximal, characteristic
6D4×#87604—22no—
7#15690883C2211yescharacteristic
8#15690879C2212yes—
9#15690872—22no—
10C8×#87604C2211yescharacteristic
11D8×D97C2212yes—
12D1552C411yescharacteristic, complemented
13#15690751C412yescomplemented
14C16×D97C2211yescharacteristic
15D16×C97C411yescharacteristic, complemented
16C22×#87604—42no—
17#13095519D411yescharacteristic
18C4⋊#87604—22no—
19C22⋊#87604—42no—
20C4×#87604D411yescharacteristic
21#13095514—21no—
22D4×D97—22no—
23D776C2×C412yescomplemented
24#13095496—22no—
25C8×D97C2311yescharacteristic
26#13095486C2×C411yescharacteristic, complemented
27D8×C97C2×C412yescomplemented
28C16×C97C2×C411yescharacteristic, complemented
29C22×D97—42no—
30C2×#87604—42no—
31C2×#87604—84no—
32C2×#87604D812yes—
33D4×C97—22no—
34C22⋊D97—42no—
35D388—22no—
36C4×D97C2×D411yescharacteristic
37#11715897—21no—
38C8×C97C22×C411yescharacteristic
39#11715895#4111yescharacteristic
40C22×C97—42no—
41C2×D97—42no—
42C2×D97—82no—
43C2×D97—82no—
44C2×D97C2×D811yescharacteristic
45#87604—22no—
46#87604—82no—
47C4×C97C4×D411yescharacteristic
48Dic97#13511yescharacteristic
49C2×C97—82no—
50C2×C97C4×D811yescharacteristic
51D97—82no—
52D97#50912yes—
53#2177—971nomaximal, Sylow
54C97#217711yescharacteristic, complemented, Sylow
55C2×D16—971no—
56C4×D8—972no—
57#367—971no—
58#357—972no—
59#346—971no—
60C2×D8—972no—
61C4×D4—1942no—
62D16—972no—
63D16—971no—
64D16#8760411yescharacteristic, complemented
65C2×C16—971no—
66C2×C16—971no—
67#107—972no—
68#102—1942no—
69C4×C8—971no—
70C2×D4—1942no—
71C22×C4—3882no—
72D8—972no—
73D8—1942no—
74D8C2×#8760412yescomplemented
75C2×C8—971no—
76C2×C8—971no—
77C4⋊C4—1942no—
78C22⋊C4—3882no—
79C42—971no—
80C16—971no—
81C16—1941no—
82C16C2×#8760411yescharacteristic, complemented
83C23—3882no—
84D4—22no—
85D4—3882no—
86D4—1942no—
87C2×C4—7764no—
88C2×C4—971no—
89C2×C4—972no—
90C2×C4—3882no—
91C8—971no—
92C8—1941no—
93C8C22×#8760411yescharacteristic
94C22—42no—
95C22—7762no—
96C22—7762no—
97C22—971no—
98C22—3882no—
99C4—7762no—
100C4—971no—
101C4—1942no—
102C4D4×#8760411yescharacteristic
103C2—82no—
104C2—7762no—
105C2—972no—
106C2D8×#8760411yescharacteristic
subgroupG/H
centerC2D8×#87604
commutatorC8×C97C22×C4
FrattiniC8C22×#87604
FittingD16×C97C4
socleC2×C97C4×D8
prime radicalD16×D97C2
cosocleC8×D97C23
hypercenterD16#87604
hypocenterC97#2177
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ D16

Complements

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

NH ≅ G/Ntype
1. D8×#87604C2semidirect
2. #20261662C2semidirect
3. #20261647C2semidirect
4. #20261622C2semidirect
12. D1552C4semidirect
13. #15690751C4semidirect
15. D16×C97C4semidirect
23. D776C2×C4semidirect
26. #13095486C2×C4semidirect
27. D8×C97C2×C4semidirect
28. C16×C97C2×C4semidirect
54. C97#2177semidirect
64. D16#87604semidirect
74. D8C2×#87604semidirect
82. C16C2×#87604semidirect

Maximal subgroups

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

subgroupindex
D8×#876042
#202616622
#202616472
#202616222
D16×D972
#217797