Finite Group Databasebeta

← C2×#519⋊C3 #85577

Subgroups — C2×#519⋊C3 #85577

Order 384.

403 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C23×SL(2,𝔽3)C211yesmaximal, characteristic
2#519⋊C3C214yesmaximal, direct product factor
3C2×C4×SL(2,𝔽3)C212yesmaximal, complemented
4C2×#519C311yesmaximal, characteristic, complemented, Sylow
5C22×SL(2,𝔽3)C414yescomplemented
6C22×SL(2,𝔽3)C2211yescharacteristic
7C22×SL(2,𝔽3)C2212yes—
8C2×C3×C22⋊C4—41nomaximal
9C4×SL(2,𝔽3)C2214yescomplemented
10C23×Q8C2×C311yescharacteristic
11#519—34no—
12#519C2×C314yescomplemented
13C2×C4×Q8—32no—
14C2×C4×Q8C2×C312yescomplemented
15C2×#117—31no—
16C2×#117—31no—
17C24×C3—41no—
18C22×C3×C4—42no—
19C2×SL(2,𝔽3)C2×C412yes—
20C2×SL(2,𝔽3)C2×C414yescomplemented
21C2×SL(2,𝔽3)C2311yescharacteristic
22C3×C22⋊C4—44no—
23C22×Q8—31no—
24C22×Q8—32no—
25C22×Q8—34no—
26C22×Q8C3×C414yescomplemented
27C22×Q8C22×C311yescharacteristic
28C22×Q8C22×C312yes—
29C23×C4—31no—
30C4×Q8—38no—
31C4×Q8—38no—
32C4×Q8—34no—
33C4×Q8C22×C314yescomplemented
34#117—34no—
35#117—34no—
36#117—34no—
37#117—34no—
38C2×C4⋊C4—32no—
39C2×C4⋊C4—32no—
40C2×C22⋊C4—31no—
41C2×C22⋊C4A411yescharacteristic
42C2×C42—32no—
43C2×C42—32no—
44C23×C3—44no—
45C23×C3—42no—
46C23×C3—41no—
47C23×C3—84no—
48C2×C3×C4—44no—
49C2×C3×C4—84no—
50SL(2,𝔽3)C22×C411yescharacteristic
51C24C2×A411yescharacteristic
52C2×Q8—32no—
53C2×Q8—34no—
54C2×Q8—38no—
55C2×Q8—31no—
56C2×Q8—32no—
57C2×Q8—34no—
58C2×Q8—38no—
59C2×Q8—32no—
60C2×Q8—34no—
61C2×Q8C2×C3×C412yes—
62C2×Q8C2×C3×C414yescomplemented
63C2×Q8C23×C311yescharacteristic
64C22×C4—34no—
65C22×C4—34no—
66C22×C4—32no—
67C22×C4—32no—
68C22×C4—31no—
69C22×C4—31no—
70C22×C4—32no—
71C22×C4C2×A412yes—
72C4⋊C4—34no—
73C4⋊C4—34no—
74C4⋊C4—38no—
75C22⋊C4—34no—
76C22⋊C4C2×A414yes—
77C42—38no—
78C42—34no—
79C42—34no—
80C22×C3—42no—
81C22×C3—44no—
82C22×C3—44no—
83C22×C3—41no—
84C22×C3—84no—
85C22×C3—88no—
86C3×C4—84no—
87C23—24no—
88C23C4×A414yes—
89C23C22×A412yes—
90C23C22×A411yescharacteristic
91Q8—32no—
92Q8—34no—
93Q8—31no—
94Q8—38no—
95Q8—32no—
96Q8—34no—
97Q8C22×C3×C411yescharacteristic
98C2×C4—24no—
99C2×C4—34no—
100C2×C4—38no—
101C2×C4—32no—
102C2×C4—34no—
103C2×C4—32no—
104C2×C4—34no—
105C2×C4—32no—
106C2×C4—31no—
107C2×C4—31no—
108C2×C4—34no—
109C2×C4—64no—
110C2×C4C22×A414yes—
111C2×C3—41no—
112C2×C3—42no—
113C2×C3—44no—
114C2×C3—84no—
115C22—24no—
116C22—28no—
117C22C2×C4×A414yes—
118C22C2×C4×A412yes—
119C222−1+4⋊C314yes—
120C22C23×A411yescharacteristic
121C4—24no—
122C4—31no—
123C4—32no—
124C4—34no—
125C4—31no—
126C4—64no—
127C3—41noSylow
128C2—24no—
129C2#519⋊C314yesdirect product factor
130C2C22×C4×A411yescharacteristic
131C2C2×2−1+4⋊C312yes—
subgroupG/H
center, socleC23C22×A4
commutator, hypocenterQ8C22×C3×C4
FrattiniC22C23×A4
FittingC2×#519C3
prime radicalC23×SL(2,𝔽3)C2
cosocleC2×Q8C23×C3
hypercenterC2×C22⋊C4A4
LCSG ▷ Q8
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #519⋊C3C2direct (N × H)
3. C2×C4×SL(2,𝔽3)C2semidirect
4. C2×#519C3semidirect
5. C22×SL(2,𝔽3)C4semidirect
9. C4×SL(2,𝔽3)C22semidirect
12. #519C2×C3semidirect
14. C2×C4×Q8C2×C3semidirect
20. C2×SL(2,𝔽3)C2×C4semidirect
26. C22×Q8C3×C4semidirect
33. C4×Q8C22×C3semidirect
62. C2×Q8C2×C3×C4semidirect
129. C2#519⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×SL(2,𝔽3)2
#519⋊C32
C2×C4×SL(2,𝔽3)2
C2×#5193
C2×C3×C22⋊C44