Finite Group Databasebeta

← #73122

Subgroups — #73122

Order 384.

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

normal subgroups only

This group has 129 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
1C2×#131⋊C3C211yesmaximal, characteristic
2#5424C211yesmaximal, characteristic
3C2×GU(2,𝔽3)C211yesmaximal, characteristic
4#1866—31nomaximal, Sylow
5C2×#41⋊C3C2211yescharacteristic
6#131⋊C3—21no—
7GU(2,𝔽3)—21no—
8#859C411yescharacteristic, complemented
9#859C411yescharacteristic, complemented
10Dic3⋊C8—41nomaximal
11C2×#131D311yescharacteristic
12C2×#108—31no—
13#424—31no—
14C2×#104—31no—
15#416—31no—
16#339—31no—
17#336—31no—
18#41⋊C3D411yescharacteristic
19#41⋊C3Q811yescharacteristic
20C2×SL(2,𝔽3)C2×C411yescharacteristic
21C2×C3×C8—41no—
22C4×Dic3—41no—
23C2×C3⋊C8—41no—
24C2×#41C2×D311yescharacteristic
25#131—21no—
26#131—61no—
27C2×M16—31no—
28C2×M16—31no—
29C22×C8—31no—
30C22×C8—31no—
31#117—31no—
32C2×C42—31no—
33#108—62no—
34C4⋊C8—61no—
35C4⋊C8—61no—
36#104—61no—
37#104—61no—
38#103—31no—
39#103—31no—
40#102—31no—
41#102—31no—
42C2×C3×C4—41no—
43C2×Dic3—41no—
44C2×Dic3—41no—
45SL(2,𝔽3)C4⋊C411yescharacteristic
46C3×C8—81no—
47C3⋊C8—81no—
48#41—31no—
49#41—31no—
50#41Dic611yescharacteristic
51#41#6711yescharacteristic
52C2×Q8C4×D311yescharacteristic
53C2×D4—31no—
54C22×C4—31no—
55C22×C4—61no—
56M16—32no—
57M16—61no—
58M16—61no—
59M16—61no—
60C2×C8—31no—
61C2×C8—31no—
62C2×C8—31no—
63C2×C8—61no—
64C2×C8—61no—
65C2×C8—61no—
66C2×C8—61no—
67C2×C8—61no—
68C2×C8S411yescharacteristic
69C4⋊C4—31no—
70C4⋊C4—31no—
71C22⋊C4—61no—
72C42—61no—
73C42—61no—
74C42—61no—
75C22×C3—41no—
76C3×C4—41no—
77C3×C4—41no—
78Dic3—42no—
79Dic3—81no—
80C23—31no—
81Q8—31no—
82Q8Dic3⋊C411yescharacteristic
83D4—31no—
84D4—31no—
85D4—61no—
86C2×C4—31no—
87C2×C4—31no—
88C2×C4—31no—
89C2×C4—31no—
90C2×C4—31no—
91C2×C4—62no—
92C2×C4—61no—
93C2×C4—61no—
94C2×C4—121no—
95C2×C4—121no—
96C2×C4C2×S411yescharacteristic
97C8—21no—
98C8—61no—
99C8—61no—
100C8—61no—
101C8—121no—
102C2×C3—41no—
103C2×C3—41no—
104C2×C3—41no—
105C22—31no—
106C22—31no—
107C22—61no—
108C22—61no—
109C22C4×S411yescharacteristic
110C4—31no—
111C4—31no—
112C4—122no—
113C4—121no—
114C4—121no—
115C4A4⋊Q811yescharacteristic
116C4GL(2,ℤ/4)11yescharacteristic
117C3—41noSylow
118C2—61no—
119C2—61no—
120C2#540411yescharacteristic
121C2#540511yescharacteristic
122C2SL(2,ℤ/4)⋊C411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×S4
commutatorC2×SL(2,𝔽3)C2×C4
FittingC2×#131D3
socleC22C4×S4
prime radical, cosocleC2×#41⋊C3C22
hypercenterC2×C8S4
hypocenterSL(2,𝔽3)C4⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
8. #859C4semidirect
9. #859C4semidirect

Maximal subgroups

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

subgroupindex
C2×#131⋊C32
#54242
C2×GU(2,𝔽3)2
#18663
Dic3⋊C84