Finite Group Databasebeta

← #67648

Subgroups — #67648

Order 384.

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

normal subgroups only

This group has 140 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×#395C211yesmaximal, characteristic
2#4530C211yesmaximal, characteristic
3#4528C211yesmaximal, characteristic
4#1354—31nomaximal, Sylow
5C22×C3×Q8C411yescharacteristic
6C2×C3×C4⋊C4—21no—
7C2×C3×C4⋊C4C2211yescharacteristic
8C2×#208—21no—
9C3×#95—21no—
10C3×#95C411yescharacteristic
11#830—21no—
12#395D311yescharacteristic
13#343—31no—
14#324—31no—
15C2×C3×Q8—42no—
16C2×C3×Q8—41no—
17C22×C3×C4—21no—
18C22×C3×C4—21no—
19C22×C3×C4C2×C411yescharacteristic
20C22×C3×C4D411yescharacteristic
21C22×C3×C4D411yescharacteristic
22C3×C4⋊C4—41no—
23C3×C4⋊C4—42no—
24#208—22no—
25#208—41no—
26C2×C3⋊C8—41no—
27C2×C3⋊C8—41no—
28C22×Q8Dic311yescharacteristic
29C2×M16—61no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4C2×D311yescharacteristic
32C22⋊C8—61no—
33#95—21no—
34#95Dic311yescharacteristic
35C23×C3C22⋊C411yescharacteristic
36C3×Q8—42no—
37C3×Q8—42no—
38C2×C3×C4—21no—
39C2×C3×C4—21no—
40C2×C3×C4—22no—
41C2×C3×C4—22no—
42C2×C3×C4—21no—
43C2×C3×C4—21no—
44C2×C3×C4—41no—
45C2×C3×C4—41no—
46C2×C3×C4—41no—
47C2×C3×C4—41no—
48C2×C3×C4—41no—
49C2×C3×C4—41no—
50C2×C3×C4SD1611yescharacteristic
51C2×C3×C4Q1611yescharacteristic
52C3⋊C8—42no—
53C3⋊C8—81no—
54C2×Q8—42no—
55C2×Q8—41no—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4C2×Dic311yescharacteristic
59C22×C4#6711yescharacteristic
60C22×C4#6711yescharacteristic
61M16—62no—
62M16—121no—
63C2×C8—121no—
64C2×C8—121no—
65C4⋊C4—41no—
66C4⋊C4—42no—
67C22×C3—21no—
68C22×C3—21no—
69C22×C3#9911yescharacteristic
70C22×C3#10311yescharacteristic
71C22×C3#10411yescharacteristic
72C3×C4—21no—
73C3×C4—21no—
74C3×C4—42no—
75C3×C4—41no—
76C3×C4—42no—
77C3×C4—41no—
78C23#21711yescharacteristic
79Q8—42no—
80Q8—42no—
81C2×C4—21no—
82C2×C4—21no—
83C2×C4—22no—
84C2×C4—22no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—41no—
88C2×C4—41no—
89C2×C4—41no—
90C2×C4—41no—
91C2×C4—41no—
92C2×C4—41no—
93C2×C4#21511yescharacteristic
94C2×C4#21611yescharacteristic
95C8—122no—
96C8—241no—
97C2×C3—21no—
98C2×C3—21no—
99C2×C3#32811yescharacteristic
100C2×C3AΣL(1,𝔽16)11yescharacteristic
101C2×C3#35611yescharacteristic
102C22—21no—
103C22—21no—
104C22#83411yescharacteristic
105C22#83511yescharacteristic
106C22#83711yescharacteristic
107C4—21no—
108C4—21no—
109C4—42no—
110C4—41no—
111C4—42no—
112C4—41no—
113C3#135411yescharacteristic, complemented, Sylow
114C2—21no—
115C2—21no—
116C2#453611yescharacteristic
117C2#453811yescharacteristic
118C2#454811yescharacteristic
subgroupG/H
centerC22#834
commutatorC22×C3×C4C2×C4
FrattiniC2×C4⋊C4C2×D3
FittingC3×#395C2
socleC22×C3#99
prime radicalC23×C3C22⋊C4
cosocleC2×C3×C4⋊C4C22
hypercenter#395D3
hypocenterC3#1354
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #395

Complements

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

NH ≅ G/Ntype
113. C3#1354semidirect

Maximal subgroups

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

subgroupindex
C3×#3952
#45302
#45282
#13543