Finite Group Databasebeta

← #73114

Subgroups — #73114

Order 384.

126 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 26 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×M16C211yesmaximal, characteristic
2#5409C211yesmaximal, characteristic
3C2×A4⋊C8C211yesmaximal, characteristic
4#1855—31nomaximal, Sylow
5C2×C4×A4C2211yescharacteristic
6C8×A4—21no—
7A4⋊C8—21no—
8A4⋊C8C412yes—
9#822—41nomaximal
10C22×M16D311yescharacteristic
11C2×#108—31no—
12C2×#108—31no—
13#407—31no—
14C2×C22⋊C8—31no—
15#341—31no—
16#341—31no—
17C22×A4Q811yescharacteristic
18C4×A4C2×C411yescharacteristic
19C4×A4D411yescharacteristic
20C3×M16—41no—
21#208—41no—
22C2×C3⋊C8—41no—
23C23×C4C2×D311yescharacteristic
24C2×M16—31no—
25C2×M16—31no—
26C2×M16—31no—
27C2×M16—31no—
28C2×M16—61no—
29C22×C8—21no—
30C22×C8—31no—
31C22×C8—31no—
32#108—62no—
33#108—61no—
34#108—61no—
35C22⋊C8—32no—
36C22⋊C8—32no—
37C22⋊C8—61no—
38C22⋊C8—61no—
39C2×A4—21no—
40C2×A4C4⋊C411yescharacteristic
41C2×C3×C4—41no—
42C3×C8—81no—
43C3⋊C8—42no—
44C3⋊C8—81no—
45C24Dic611yescharacteristic
46C22×C4—31no—
47C22×C4—31no—
48C22×C4—31no—
49C22×C4—31no—
50C22×C4C4×D311yescharacteristic
51C22×C4#6711yescharacteristic
52M16—32no—
53M16—31no—
54M16—61no—
55M16—61no—
56M16—61no—
57M16—61no—
58M16—61no—
59M16S411yescharacteristic
60C2×C8—32no—
61C2×C8—32no—
62C2×C8—61no—
63C2×C8—61no—
64C2×C8—61no—
65C2×C8—61no—
66C2×C8—61no—
67C2×C8—61no—
68C2×C8—61no—
69C2×C8—61no—
70C22×C3—41no—
71A4#10811yescharacteristic, complemented
72C3×C4—41no—
73C3×C4—41no—
74C23—21no—
75C23—31no—
76C23—31no—
77C23—61no—
78C23Dic3⋊C411yescharacteristic
79C2×C4—31no—
80C2×C4—31no—
81C2×C4—31no—
82C2×C4—31no—
83C2×C4—31no—
84C2×C4—31no—
85C2×C4—31no—
86C2×C4—61no—
87C2×C4C2×S411yescharacteristic
88C8—21no—
89C8—62no—
90C8—62no—
91C8—61no—
92C8—121no—
93C8—121no—
94C2×C3—41no—
95C2×C3—81no—
96C22—31no—
97C22—31no—
98C22—31no—
99C22—61no—
100C22—61no—
101C22—61no—
102C22—61no—
103C22#82211yescharacteristic, complemented
104C22A4⋊Q811yescharacteristic
105C4—31no—
106C4—31no—
107C4C4×S411yescharacteristic
108C4GL(2,ℤ/4)11yescharacteristic
109C3—41noSylow
110C2—21no—
111C2—31no—
112C2—31no—
113C2—61no—
114C2SL(2,ℤ/4)⋊C411yescharacteristic
subgroupG/H
centerC4GL(2,ℤ/4)
commutatorC4×A4C2×C4
FrattiniC2×C4C2×S4
FittingC22×M16D3
socleC23Dic3⋊C4
prime radicalC22×A4Q8
cosocleC2×C4×A4C22
hypercenterM16S4
hypocenterA4#108
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
71. A4#108semidirect
103. C22#822semidirect

Maximal subgroups

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

subgroupindex
A4×M162
#54092
C2×A4⋊C82
#18553
#8224