Finite Group Databasebeta

← #67640

Subgroups — #67640

Order 384.

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

normal subgroups only

This group has 170 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×#390C211yesmaximal, characteristic
2#4528C212yesmaximal, complemented
3#1318—31nomaximal, Sylow
4C22×C3×D4C411yescharacteristic
5C2×C3×C4⋊C4C2211yescharacteristic
6C2×C3×C22⋊C4—22no—
7C2×C3×C42C411yescharacteristic
8#830—22no—
9#390D311yescharacteristic
10#324—32no—
11C24×C3—21no—
12C2×C3×D4—21no—
13C2×C3×D4—41no—
14C2×C3×D4—41no—
15C2×C3×D4C812yescomplemented
16C22×C3×C4—21no—
17C22×C3×C4C2×C411yescharacteristic
18C22×C3×C4D412yes—
19C3×C4⋊C4—41no—
20C3×C22⋊C4—44no—
21C3×C42—41no—
22C2×C3⋊C8—42no—
23C22×D4Dic311yescharacteristic
24C2×C4⋊C4C2×D311yescharacteristic
25C2×C22⋊C4—22no—
26C2×C42Dic311yescharacteristic
27C22⋊C8—62no—
28C23×C3—22no—
29C23×C3—42no—
30C23×C3—42no—
31C23×C3—41no—
32C23×C3—41no—
33C23×C3C22⋊C411yescharacteristic
34C3×D4—42no—
35C3×D4—42no—
36C2×C3×C4—21no—
37C2×C3×C4—21no—
38C2×C3×C4—22no—
39C2×C3×C4—22no—
40C2×C3×C4—41no—
41C2×C3×C4—41no—
42C2×C3×C4—42no—
43C2×C3×C4C2×C811yescharacteristic
44C2×C3×C4M1611yescharacteristic
45C3⋊C8—82no—
46C24—21no—
47C2×D4—21no—
48C2×D4—41no—
49C2×D4—41no—
50C2×D4C3⋊C812yescomplemented
51C22×C4—21no—
52C22×C4C2×Dic311yescharacteristic
53C22×C4#6712yes—
54C2×C8—122no—
55C4⋊C4—41no—
56C22⋊C4—44no—
57C42—41no—
58C22×C3—21no—
59C22×C3—21no—
60C22×C3—44no—
61C22×C3—42no—
62C22×C3—82no—
63C22×C3—81no—
64C22×C3—81no—
65C22×C3C22⋊C811yescharacteristic
66C22×C3#9911yescharacteristic
67C22×C3#10011yescharacteristic
68C3×C4—21no—
69C3×C4—21no—
70C3×C4—42no—
71C3×C4—42no—
72C23—22no—
73C23—42no—
74C23—42no—
75C23—41no—
76C23—41no—
77C23#21711yescharacteristic
78D4—42no—
79D4—42no—
80C2×C4—21no—
81C2×C4—21no—
82C2×C4—22no—
83C2×C4—22no—
84C2×C4—41no—
85C2×C4—41no—
86C2×C4—42no—
87C2×C4C2×C3⋊C811yescharacteristic
88C2×C4#20811yescharacteristic
89C8—242no—
90C2×C3—21no—
91C2×C3—21no—
92C2×C3—82no—
93C2×C3#32311yescharacteristic
94C2×C3#35311yescharacteristic
95C2×C3#35511yescharacteristic
96C22—21no—
97C22—21no—
98C22—44no—
99C22—42no—
100C22—82no—
101C22—81no—
102C22—81no—
103C22#83011yescharacteristic
104C22#83311yescharacteristic
105C22#83411yescharacteristic
106C4—21no—
107C4—21no—
108C4—42no—
109C4—42no—
110C3#131811yescharacteristic, complemented, Sylow
111C2—21no—
112C2—21no—
113C2—82no—
114C2#452511yescharacteristic
115C2#454211yescharacteristic
116C2#454511yescharacteristic
subgroupG/H
centerC22#834
commutatorC2×C3×C4C2×C8
FrattiniC2×C4⋊C4C2×D3
FittingC3×#390C2
socleC22×C3#99
prime radicalC22×C3×D4C4
cosocleC2×C3×C4⋊C4C22
hypercenter#390D3
hypocenterC3#1318
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #4528C2semidirect
15. C2×C3×D4C8semidirect
50. C2×D4C3⋊C8semidirect
110. C3#1318semidirect

Maximal subgroups

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

subgroupindex
C3×#3902
#45282
#13183