Finite Group Databasebeta

← #519⋊C3 #5439

Subgroups — #519⋊C3 #5439

Order 192.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×SL(2,𝔽3)C211yesmaximal, characteristic
2C4×SL(2,𝔽3)C212yesmaximal, complemented
3#519C311yesmaximal, characteristic, complemented, Sylow
4C2×SL(2,𝔽3)C412yescomplemented
5C2×SL(2,𝔽3)C2211yescharacteristic
6C3×C22⋊C4—41nomaximal
7C22×Q8C2×C311yescharacteristic
8C4×Q8—32no—
9C4×Q8C2×C312yescomplemented
10#117—31no—
11#117—31no—
12C23×C3—41no—
13C2×C3×C4—42no—
14SL(2,𝔽3)C2×C411yescharacteristic
15C2×Q8—31no—
16C2×Q8—32no—
17C2×Q8C3×C412yescomplemented
18C2×Q8C22×C311yescharacteristic
19C22×C4—31no—
20C4⋊C4—32no—
21C4⋊C4—32no—
22C22⋊C4—31no—
23C22⋊C4A411yescharacteristic
24C42—32no—
25C42—32no—
26C22×C3—42no—
27C22×C3—41no—
28C22×C3—82no—
29C3×C4—82no—
30C23C2×A411yescharacteristic
31Q8—32no—
32Q8—31no—
33Q8—32no—
34Q8C2×C3×C411yescharacteristic
35C2×C4—32no—
36C2×C4—32no—
37C2×C4—31no—
38C2×C4—31no—
39C2×C4—32no—
40C2×C4C2×A412yes—
41C2×C3—41no—
42C2×C3—42no—
43C2×C3—82no—
44C22—22no—
45C22C4×A412yes—
46C22C22×A411yescharacteristic
47C4—22no—
48C4—31no—
49C4—32no—
50C4—31no—
51C4—62no—
52C3—41noSylow
53C2—22no—
54C2C2×C4×A411yescharacteristic
55C22−1+4⋊C312yes—
subgroupG/H
center, Frattini, socleC22C22×A4
commutator, hypocenterQ8C2×C3×C4
Fitting#519C3
prime radicalC22×SL(2,𝔽3)C2
cosocleC2×Q8C22×C3
hypercenterC22⋊C4A4
LCSG ▷ Q8
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C4×SL(2,𝔽3)C2semidirect
3. #519C3semidirect
4. C2×SL(2,𝔽3)C4semidirect
9. C4×Q8C2×C3semidirect
17. C2×Q8C3×C4semidirect

Maximal subgroups

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

subgroupindex
C22×SL(2,𝔽3)2
C4×SL(2,𝔽3)2
#5193
C3×C22⋊C44