Finite Group Databasebeta

← #67650

Subgroups — #67650

Order 384.

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

normal subgroups only

This group has 124 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×#398C211yesmaximal, characteristic
2#4530C211yesmaximal, characteristic
3#4528C211yesmaximal, characteristic
4#1346—31nomaximal, Sylow
5C2×C3×C4⋊C4—21no—
6C2×C3×C4⋊C4C411yescharacteristic
7C2×C3×C4⋊C4C2211yescharacteristic
8C2×#208—21no—
9C3×#95—21no—
10C3×#95C411yescharacteristic
11#830—21no—
12#398D311yescharacteristic
13#343—31no—
14#324—31no—
15C22×C3×C4—21no—
16C22×C3×C4—21no—
17C22×C3×C4C2×C411yescharacteristic
18C22×C3×C4D411yescharacteristic
19C22×C3×C4D411yescharacteristic
20C3×C4⋊C4—41no—
21C3×C4⋊C4—41no—
22C3×C4⋊C4—41no—
23C3×C4⋊C4—41no—
24#208—22no—
25#208—41no—
26C2×C3⋊C8—41no—
27C2×C3⋊C8—41no—
28C2×M16—61no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4Dic311yescharacteristic
31C2×C4⋊C4C2×D311yescharacteristic
32C22⋊C8—61no—
33#95—21no—
34#95Dic311yescharacteristic
35C23×C3C22⋊C411yescharacteristic
36C2×C3×C4—21no—
37C2×C3×C4—21no—
38C2×C3×C4—22no—
39C2×C3×C4—21no—
40C2×C3×C4—21no—
41C2×C3×C4—41no—
42C2×C3×C4—41no—
43C2×C3×C4—41no—
44C2×C3×C4—41no—
45C2×C3×C4—41no—
46C2×C3×C4—41no—
47C2×C3×C4—41no—
48C2×C3×C4D811yescharacteristic
49C2×C3×C4SD1611yescharacteristic
50C3⋊C8—42no—
51C3⋊C8—81no—
52C22×C4—21no—
53C22×C4—21no—
54C22×C4C2×Dic311yescharacteristic
55C22×C4#6711yescharacteristic
56C22×C4#6711yescharacteristic
57M16—62no—
58M16—121no—
59C2×C8—121no—
60C2×C8—121no—
61C4⋊C4—41no—
62C4⋊C4—41no—
63C4⋊C4—41no—
64C4⋊C4—41no—
65C22×C3—21no—
66C22×C3—21no—
67C22×C3#9911yescharacteristic
68C22×C3#10211yescharacteristic
69C22×C3#10411yescharacteristic
70C3×C4—21no—
71C3×C4—21no—
72C3×C4—42no—
73C3×C4—41no—
74C3×C4—41no—
75C3×C4—81no—
76C23#21711yescharacteristic
77C2×C4—21no—
78C2×C4—21no—
79C2×C4—22no—
80C2×C4—21no—
81C2×C4—21no—
82C2×C4—41no—
83C2×C4—41no—
84C2×C4—41no—
85C2×C4—41no—
86C2×C4—41no—
87C2×C4—41no—
88C2×C4—41no—
89C2×C4#21311yescharacteristic
90C2×C4#21411yescharacteristic
91C8—122no—
92C8—241no—
93C2×C3—21no—
94C2×C3—21no—
95C2×C3#32711yescharacteristic
96C2×C3#35211yescharacteristic
97C2×C3#35611yescharacteristic
98C22—21no—
99C22—21no—
100C22#83211yescharacteristic
101C22#83411yescharacteristic
102C22#83711yescharacteristic
103C4—21no—
104C4—21no—
105C4—42no—
106C4—41no—
107C4—41no—
108C4—81no—
109C3#134611yescharacteristic, complemented, Sylow
110C2—21no—
111C2—21no—
112C2#453711yescharacteristic
113C2#453911yescharacteristic
114C2#454811yescharacteristic
subgroupG/H
centerC22#834
commutatorC22×C3×C4C2×C4
FrattiniC2×C4⋊C4C2×D3
FittingC3×#398C2
socleC22×C3#99
prime radicalC23×C3C22⋊C4
cosocleC2×C3×C4⋊C4C22
hypercenter#398D3
hypocenterC3#1346
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #398

Complements

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

NH ≅ G/Ntype
109. C3#1346semidirect

Maximal subgroups

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

subgroupindex
C3×#3982
#45302
#45282
#13463