Finite Group Databasebeta

← #67646

Subgroups — #67646

Order 384.

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

normal subgroups only

This group has 162 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×#393C211yesmaximal, characteristic
2#4526C211yesmaximal, characteristic
3#4525C211yesmaximal, characteristic
4#1351—31nomaximal, Sylow
5C22×C3×Q8C411yescharacteristic, complemented
6C2×C3×C22⋊C4—21no—
7C2×C3×C22⋊C4C2211yescharacteristic
8C2×#217—21no—
9C3×#95—21no—
10C3×#95C411yescharacteristic, complemented
11#830—21no—
12#393D311yescharacteristic
13#342—31no—
14#323—31no—
15C24×C3D411yescharacteristic
16C2×C3×Q8—42no—
17C2×C3×Q8—41no—
18C22×C3×C4—21no—
19C22×C3×C4—21no—
20C22×C3×C4C2×C411yescharacteristic
21C22×C3×C4D411yescharacteristic
22C22×Dic3—41no—
23C3×C22⋊C4—41no—
24C3×C22⋊C4—42no—
25#217—22no—
26#217—41no—
27C2×C3⋊C8—41no—
28C22×Q8Dic311yescharacteristic, complemented
29C2×C22⋊C4—21no—
30C2×C22⋊C4—61no—
31C2×C22⋊C4C2×D311yescharacteristic
32C22⋊C8—61no—
33#95—21no—
34#95Dic311yescharacteristic, complemented
35C23×C3—21no—
36C23×C3—21no—
37C23×C3—41no—
38C23×C3—41no—
39C23×C3C22⋊C411yescharacteristic
40C23×C3SD1611yescharacteristic
41C23×C3Q1611yescharacteristic
42C3×Q8—42no—
43C3×Q8—42no—
44C2×C3×C4—22no—
45C2×C3×C4—21no—
46C2×C3×C4—21no—
47C2×C3×C4—41no—
48C2×C3×C4—41no—
49C2×C3×C4—41no—
50C2×C3×C4—41no—
51C2×C3×C4—41no—
52C2×C3×C4—41no—
53C2×C3×C4—41no—
54C2×Dic3—42no—
55C2×Dic3—81no—
56C2×Dic3—81no—
57C3⋊C8—81no—
58C24#6711yescharacteristic
59C2×Q8—42no—
60C2×Q8—41no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—121no—
64C22×C4C2×Dic311yescharacteristic
65C22×C4#6711yescharacteristic
66C2×C8—121no—
67C22⋊C4—41no—
68C22⋊C4—42no—
69C22⋊C4—62no—
70C22⋊C4—121no—
71C22×C3—21no—
72C22×C3—21no—
73C22×C3—21no—
74C22×C3—21no—
75C22×C3—41no—
76C22×C3—41no—
77C22×C3—41no—
78C22×C3—41no—
79C22×C3—81no—
80C22×C3#9911yescharacteristic
81C22×C3#10311yescharacteristic
82C22×C3#10411yescharacteristic
83C3×C4—42no—
84C3×C4—41no—
85C3×C4—41no—
86C3×C4—81no—
87Dic3—82no—
88C23—21no—
89C23—21no—
90C23—41no—
91C23—41no—
92C23#21511yescharacteristic
93C23#21611yescharacteristic
94C23#21711yescharacteristic
95Q8—42no—
96Q8—42no—
97C2×C4—22no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—122no—
108C2×C4—241no—
109C2×C4—241no—
110C8—241no—
111C2×C3—21no—
112C2×C3—21no—
113C2×C3—41no—
114C2×C3—41no—
115C2×C3#32811yescharacteristic
116C2×C3AΣL(1,𝔽16)11yescharacteristic
117C2×C3#35411yescharacteristic
118C22—21no—
119C22—21no—
120C22—21no—
121C22—21no—
122C22—41no—
123C22—41no—
124C22—41no—
125C22—41no—
126C22—81no—
127C22#83411yescharacteristic
128C22#83511yescharacteristic
129C22#83711yescharacteristic
130C4—42no—
131C4—41no—
132C4—41no—
133C4—81no—
134C4—242no—
135C3#135111yescharacteristic, complemented, Sylow
136C2—21no—
137C2—21no—
138C2—41no—
139C2—41no—
140C2#453611yescharacteristic
141C2#453811yescharacteristic
142C2#454111yescharacteristic
subgroupG/H
centerC22#834
commutatorC22×C3×C4C2×C4
FrattiniC2×C22⋊C4C2×D3
FittingC3×#393C2
socleC22×C3#99
prime radicalC24×C3D4
cosocleC2×C3×C22⋊C4C22
hypercenter#393D3
hypocenterC3#1351
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #393

Complements

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

NH ≅ G/Ntype
5. C22×C3×Q8C4semidirect
10. C3×#95C4semidirect
28. C22×Q8Dic3semidirect
34. #95Dic3semidirect
135. C3#1351semidirect

Maximal subgroups

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

subgroupindex
C3×#3932
#45262
#45252
#13513