Finite Group Databasebeta

← #67643

Subgroups — #67643

Order 384.

134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 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×#391C211yesmaximal, characteristic
2#4528C212yesmaximal
3#1319—31nomaximal, Sylow
4C22×C3×Q8C411yescharacteristic
5C2×C3×C4⋊C4C2211yescharacteristic
6C2×C3×C42C411yescharacteristic
7C3×#95—22no—
8#830—22no—
9#391D311yescharacteristic
10#324—32no—
11C2×C3×Q8—21no—
12C2×C3×Q8—41no—
13C2×C3×Q8—41no—
14C2×C3×Q8C812yescomplemented
15C22×C3×C4—21no—
16C22×C3×C4—21no—
17C22×C3×C4C2×C411yescharacteristic
18C22×C3×C4D412yes—
19C3×C4⋊C4—41no—
20C3×C42—41no—
21C2×C3⋊C8—42no—
22C22×Q8Dic311yescharacteristic
23C2×C4⋊C4C2×D311yescharacteristic
24C2×C42Dic311yescharacteristic
25C22⋊C8—62no—
26#95—22no—
27C23×C3C22⋊C411yescharacteristic
28C3×Q8—42no—
29C3×Q8—42no—
30C2×C3×C4—21no—
31C2×C3×C4—21no—
32C2×C3×C4—22no—
33C2×C3×C4—22no—
34C2×C3×C4—22no—
35C2×C3×C4—41no—
36C2×C3×C4—41no—
37C2×C3×C4—41no—
38C2×C3×C4—41no—
39C2×C3×C4—42no—
40C2×C3×C4C2×C811yescharacteristic
41C2×C3×C4M1611yescharacteristic
42C3⋊C8—82no—
43C2×Q8—21no—
44C2×Q8—41no—
45C2×Q8—41no—
46C2×Q8C3⋊C812yescomplemented
47C22×C4—21no—
48C22×C4—21no—
49C22×C4C2×Dic311yescharacteristic
50C22×C4#6712yes—
51C2×C8—122no—
52C4⋊C4—41no—
53C42—41no—
54C22×C3—21no—
55C22×C3—21no—
56C22×C3C22⋊C811yescharacteristic
57C22×C3#9911yescharacteristic
58C22×C3#10011yescharacteristic
59C3×C4—21no—
60C3×C4—21no—
61C3×C4—42no—
62C3×C4—42no—
63C3×C4—42no—
64C23#21711yescharacteristic
65Q8—42no—
66Q8—42no—
67C2×C4—21no—
68C2×C4—21no—
69C2×C4—22no—
70C2×C4—22no—
71C2×C4—22no—
72C2×C4—41no—
73C2×C4—41no—
74C2×C4—41no—
75C2×C4—41no—
76C2×C4—42no—
77C2×C4C2×C3⋊C811yescharacteristic
78C2×C4#20811yescharacteristic
79C8—242no—
80C2×C3—21no—
81C2×C3—21no—
82C2×C3#32311yescharacteristic
83C2×C3#35411yescharacteristic
84C2×C3#35611yescharacteristic
85C22—21no—
86C22—21no—
87C22#83011yescharacteristic
88C22#83311yescharacteristic
89C22#83411yescharacteristic
90C4—21no—
91C4—21no—
92C4—42no—
93C4—42no—
94C4—42no—
95C3#131911yescharacteristic, complemented, Sylow
96C2—21no—
97C2—21no—
98C2#452511yescharacteristic
99C2#454111yescharacteristic
100C2#454811yescharacteristic
subgroupG/H
centerC22#834
commutatorC2×C3×C4C2×C8
FrattiniC2×C4⋊C4C2×D3
FittingC3×#391C2
socleC22×C3#99
prime radicalC23×C3C22⋊C4
cosocleC2×C3×C4⋊C4C22
hypercenter#391D3
hypocenterC3#1319
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #391

Complements

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

NH ≅ G/Ntype
14. C2×C3×Q8C8semidirect
46. C2×Q8C3⋊C8semidirect
95. C3#1319semidirect

Maximal subgroups

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

subgroupindex
C3×#3912
#45282
#13193