Finite Group Databasebeta

← #67681

Subgroups — #67681

Order 384.

128 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 37 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
1C3×#407C211yesmaximal, characteristic
2#4553C212yesmaximal, complemented
3#1320—31nomaximal, Sylow
4C23×C3×C4C411yescharacteristic
5C2×C3×M16C411yescharacteristic
6C2×C3×M16C2211yescharacteristic
7C3×C22⋊C8—22no—
8#812—22no—
9#407D311yescharacteristic
10#349—32no—
11C24×C3C811yescharacteristic
12C22×C3×C4—21no—
13C22×C3×C4—21no—
14C22×C3×C4—41no—
15C22×C3×C4—41no—
16C22×C3×C4C811yescharacteristic
17C22×C3×C4C2×C411yescharacteristic
18C3×M16—41no—
19C3×M16—41no—
20C2×C3×C8—21no—
21C2×C3×C8D412yes—
22C3⋊C16—42no—
23C23×C4Dic311yescharacteristic
24C2×M16Dic311yescharacteristic
25C2×M16C2×D311yescharacteristic
26M32—62no—
27C22⋊C8—22no—
28C23×C3—21no—
29C23×C3—41no—
30C23×C3—42no—
31C23×C3C2×C811yescharacteristic
32C2×C3×C4—21no—
33C2×C3×C4—21no—
34C2×C3×C4—21no—
35C2×C3×C4—42no—
36C2×C3×C4—41no—
37C2×C3×C4—41no—
38C2×C3×C4—41no—
39C2×C3×C4C22⋊C411yescharacteristic
40C2×C3×C4M1611yescharacteristic
41C3×C8—22no—
42C3×C8—41no—
43C24C3⋊C811yescharacteristic
44C22×C4—21no—
45C22×C4—21no—
46C22×C4—41no—
47C22×C4—41no—
48C22×C4C3⋊C811yescharacteristic
49C22×C4C2×Dic311yescharacteristic
50M16—41no—
51M16—41no—
52C2×C8—21no—
53C2×C8#6712yes—
54C16—122no—
55C22×C3—22no—
56C22×C3—21no—
57C22×C3—41no—
58C22×C3—41no—
59C22×C3—42no—
60C22×C3—42no—
61C22×C3—41no—
62C22×C3C22⋊C811yescharacteristic
63C3×C4—21no—
64C3×C4—41no—
65C3×C4#9911yescharacteristic
66C3×C4#10011yescharacteristic
67C23—21no—
68C23—41no—
69C23—42no—
70C23C2×C3⋊C811yescharacteristic
71C2×C4—21no—
72C2×C4—21no—
73C2×C4—21no—
74C2×C4—42no—
75C2×C4—41no—
76C2×C4—41no—
77C2×C4—41no—
78C2×C4#20811yescharacteristic
79C2×C4#21711yescharacteristic
80C8—22no—
81C8—41no—
82C2×C3—21no—
83C2×C3—42no—
84C2×C3—41no—
85C2×C3#32311yescharacteristic
86C22—22no—
87C22—21no—
88C22—41no—
89C22—41no—
90C22—42no—
91C22—42no—
92C22—41no—
93C22#83011yescharacteristic
94C4—21no—
95C4—41no—
96C4#83311yescharacteristic
97C4#83411yescharacteristic
98C3#132011yescharacteristic, complemented, Sylow
99C2—21no—
100C2—42no—
101C2—41no—
102C2#452511yescharacteristic
subgroupG/H
centerC4#834
commutatorC23×C3C2×C8
FrattiniC2×M16C2×D3
FittingC3×#407C2
socleC2×C3#323
prime radicalC24×C3C8
cosocleC2×C3×M16C22
hypercenter#407D3
hypocenterC3#1320
LCSG ▷ C23×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #4553C2semidirect
98. C3#1320semidirect

Maximal subgroups

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

subgroupindex
C3×#4072
#45532
#13203