Finite Group Databasebeta

← #27140852

Subgroups — #27140852

Order 12416.

314 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 65 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#20261107C211yesmaximal, characteristic, complemented
2#20261025C212yesmaximal, complemented
3#20260479C211yesmaximal, characteristic, complemented
4D4⋊#11715895C212yesmaximal, complemented
5D42×C97C211yesmaximal, characteristic
6C2×#13095499—22no—
7#15690810C2211yescharacteristic, complemented
8#15690741—22no—
9C4⋊#11715895C2212yescomplemented
10#15690731—22no—
11C4×#11715895C2211yescharacteristic, complemented
12C22×D4×C97—22no—
13#127×C97C2211yescharacteristic
14#121×C97—22no—
15#120×C97—22no—
16C4×D4×C97C2212yes—
17C2×Dic194—21no—
18#13095499—44no—
19C2×#11715895—21no—
20C2×#11715895—22no—
21C4⋊Dic97—22no—
22C24×C97—22no—
23C2×D4×C97—24no—
24C2×D4×C97—22no—
25C2×D4×C97—44no—
26C2×D4×C97—44no—
27C2×D4×C97D412yes—
28C2×D4×C97D412yes—
29C22×C4×C97—22no—
30C4⋊C4×C97D412yes—
31C22⋊C4×C97—22no—
32C22⋊C4×C97—22no—
33C42×C97C2311yescharacteristic
34C23×C97—22no—
35C23×C97—22no—
36C23×C97—48no—
37C23×C97—82no—
38D4×C97—22no—
39D4×C97—42no—
40D4×C97—48no—
41D4×C97—42no—
42D4×C97—42no—
43D4×C97SD1614yescomplemented
44C2×C4×C97—22no—
45C2×C4×C97—42no—
46C2×C4×C97—42no—
47C2×C4×C97C2×D411yescharacteristic
48C2×C4×C97C2×D412yes—
49C2×Dic97—41no—
50Dic194—42no—
51#11715895—22no—
52#11715895—42no—
53C22×C97—24no—
54C22×C97—42no—
55C22×C97—42no—
56C22×C97—44no—
57C22×C97—42no—
58C22×C97—88no—
59C22×C97#12011yescharacteristic
60C4×C97—21no—
61C4×C97—42no—
62C4×C97C2×SD1612yes—
63C4×C97AGL(1,ℤ/8)12yes—
64Dic97—81no—
65C2×C97—44no—
66C2×C97—82no—
67C2×C97#45012yes—
68C2×C97#45311yescharacteristic
69#1629—971nomaximal, Sylow
70C97#162911yescharacteristic, complemented, Sylow
71D42D9711yescharacteristic
72#492—971no—
73#476—972no—
74#331—971no—
75D4⋊C8—972no—
76C22×D4—22no—
77C2×SD16—1942no—
78C4⋊Q8—971no—
79#127C2×D9711yescharacteristic
80#121—22no—
81#120—22no—
82C4×D4C2×D9712yes—
83#106—1942no—
84C4⋊C8—972no—
85#102—1942no—
86C4×C8—971no—
87C24—22no—
88C2×Q8—1941no—
89C2×D4—24no—
90C2×D4—22no—
91C2×D4—44no—
92C2×D4—44no—
93C2×D4C22⋊D9712yes—
94C2×D4C22⋊D9712yes—
95C22×C4—22no—
96SD16—3884no—
97C2×C8—1942no—
98C2×C8—1941no—
99C4⋊C4—1942no—
100C4⋊C4C22⋊D9712yes—
101C22⋊C4—22no—
102C22⋊C4—22no—
103C42C22×D9711yescharacteristic
104C23—22no—
105C23—22no—
106C23—48no—
107C23—82no—
108Q8—3882no—
109D4—22no—
110D4—42no—
111D4—48no—
112D4—42no—
113D4—42no—
114D4#1309549914yescomplemented
115C2×C4—22no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—3881no—
119C2×C4C2×C22⋊D9711yescharacteristic
120C2×C4C2×C22⋊D9712yes—
121C8—3882no—
122C8—1942no—
123C22—24no—
124C22—42no—
125C22—42no—
126C22—44no—
127C22—42no—
128C22—88no—
129C22#1569077911yescharacteristic
130C4—21no—
131C4—42no—
132C4—7761no—
133C4C2×#1309549912yes—
134C4#1569083612yes—
135C2—44no—
136C2—82no—
137C2#2026088612yes—
138C2#2026090311yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting, prime radicalD42×C97C2
socleC22×C97#120
hypercenterD42D97
hypocenterC97#1629
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #20261107C2semidirect
2. #20261025C2semidirect
3. #20260479C2semidirect
4. D4⋊#11715895C2semidirect
7. #15690810C22semidirect
9. C4⋊#11715895C22semidirect
11. C4×#11715895C22semidirect
43. D4×C97SD16semidirect
70. C97#1629semidirect
114. D4#13095499semidirect

Maximal subgroups

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

subgroupindex
#202611072
#202610252
#202604792
D4⋊#117158952
D42×C972
#162997