Finite Group Databasebeta

← #73051

Subgroups — #73051

Order 384.

155 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 155 conjugacy classes of subgroups — too many to draw. Only its 32 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
1A4×C4⋊C4C211yesmaximal, characteristic
2C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic
3C2×A4⋊C8C211yesmaximal, characteristic
4#1853—31nomaximal, Sylow
5C2×C4×A4—21no—
6C2×C4×A4C2211yescharacteristic
7C2×SL(2,ℤ/4)—21no—
8A4⋊C8C412yescomplemented
9#808—41nomaximal
10C22×C4⋊C4D311yescharacteristic
11C2×#106—31no—
12C2×#106—31no—
13C2×C22⋊C8—31no—
14#380—31no—
15#340—31no—
16#340—31no—
17C22×A4D411yescharacteristic
18C4×A4—41no—
19C4×A4C2×C411yescharacteristic
20C4×A4Q811yescharacteristic
21SL(2,ℤ/4)—41no—
22C3×C4⋊C4—41no—
23C4⋊Dic3—41no—
24C2×C3⋊C8—41no—
25C23×C4—21no—
26C23×C4C2×D311yescharacteristic
27C22×C8—31no—
28C22×C8—31no—
29C2×C4⋊C4—31no—
30C2×C4⋊C4—31no—
31C2×C4⋊C4—31no—
32C2×C4⋊C4—31no—
33C2×C4⋊C4—61no—
34C2×C22⋊C4—61no—
35#106—62no—
36#106—61no—
37#106—61no—
38C22⋊C8—32no—
39C22⋊C8—32no—
40#95—61no—
41C2×A4C4⋊C411yescharacteristic
42C2×A4SD1611yescharacteristic
43C2×A4SD1611yescharacteristic
44C2×C3×C4—41no—
45C2×C3×C4—81no—
46C2×Dic3—81no—
47C3⋊C8—42no—
48C24#6711yescharacteristic
49C22×C4—31no—
50C22×C4—31no—
51C22×C4—31no—
52C22×C4—31no—
53C22×C4—41no—
54C22×C4—61no—
55C22×C4—61no—
56C22×C4—61no—
57C22×C4—61no—
58C22×C4—121no—
59C22×C4Dic611yescharacteristic
60C22×C4C4×D311yescharacteristic
61C2×C8—32no—
62C2×C8—32no—
63C2×C8—61no—
64C2×C8—61no—
65C2×C8—61no—
66C2×C8—61no—
67C4⋊C4—32no—
68C4⋊C4—31no—
69C4⋊C4—61no—
70C4⋊C4—61no—
71C4⋊C4—61no—
72C4⋊C4—61no—
73C4⋊C4—61no—
74C4⋊C4S411yescharacteristic
75C22⋊C4—121no—
76C22⋊C4—121no—
77C22×C3—41no—
78A4#10611yescharacteristic, complemented
79C3×C4—41no—
80C3×C4—41no—
81C3×C4—161no—
82Dic3—161no—
83C23—31no—
84C23—31no—
85C23—31no—
86C23—31no—
87C23Dic3⋊C411yescharacteristic
88C23#21411yescharacteristic
89C23#21511yescharacteristic
90C2×C4—21no—
91C2×C4—31no—
92C2×C4—31no—
93C2×C4—31no—
94C2×C4—31no—
95C2×C4—31no—
96C2×C4—31no—
97C2×C4—31no—
98C2×C4—61no—
99C2×C4—61no—
100C2×C4—121no—
101C2×C4—121no—
102C2×C4—121no—
103C2×C4—121no—
104C2×C4—121no—
105C2×C4—121no—
106C2×C4—121no—
107C2×C4—121no—
108C2×C4—121no—
109C2×C4—121no—
110C2×C4C2×S411yescharacteristic
111C8—62no—
112C8—62no—
113C2×C3—41no—
114C2×C3—41no—
115C2×C3—41no—
116C22—31no—
117C22—31no—
118C22—31no—
119C22—31no—
120C22—31no—
121C22—31no—
122C22—31no—
123C22—31no—
124C22—31no—
125C22—61no—
126C22#80811yescharacteristic, complemented
127C22GL(2,ℤ/4)11yescharacteristic
128C4—31no—
129C4—31no—
130C4—41no—
131C4—121no—
132C4—241no—
133C4—241no—
134C4A4⋊Q811yescharacteristic
135C4C4×S411yescharacteristic
136C3—41noSylow
137C2—31no—
138C2—31no—
139C2—31no—
140C2—31no—
141C2SL(2,ℤ/4)⋊C411yescharacteristic
142C2#541411yescharacteristic
143C2#541711yescharacteristic
subgroupG/H
centerC22GL(2,ℤ/4)
commutatorC4×A4C2×C4
FrattiniC2×C4C2×S4
FittingC22×C4⋊C4D3
socleC24#67
prime radicalC22×A4D4
cosocleC2×C4×A4C22
hypercenterC4⋊C4S4
hypocenterA4#106
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
8. A4⋊C8C4semidirect
78. A4#106semidirect
126. C22#808semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊C42
C4⋊SL(2,ℤ/4)2
C2×A4⋊C82
#18533
#8084