Finite Group Databasebeta

← C22⋊C4×SL(2,𝔽3) #73174

Subgroups — C22⋊C4×SL(2,𝔽3) #73174

Order 384.

275 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 60 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
2C2×C4×SL(2,𝔽3)C212yesmaximal, complemented
3Q8×C22⋊C4C311yesmaximal, characteristic, complemented, Sylow
4C22×SL(2,𝔽3)—22no—
5C22×SL(2,𝔽3)C412yescomplemented
6C22×SL(2,𝔽3)C2211yescharacteristic
7C2×C3×C22⋊C4—41nomaximal
8C4×SL(2,𝔽3)—22no—
9C23×Q8C2×C311yescharacteristic
10C2×C4×Q8C2×C312yescomplemented
11#391—32no—
12#380—31no—
13C4×C22⋊C4—31no—
14C24×C3—41no—
15C22×C3×C4—42no—
16C2×SL(2,𝔽3)—22no—
17C2×SL(2,𝔽3)C2×C411yescharacteristic
18C2×SL(2,𝔽3)D412yes—
19C3×C22⋊C4—44no—
20C22×Q8—22no—
21C22×Q8—32no—
22C22×Q8—31no—
23C22×Q8—62no—
24C22×Q8C3×C412yescomplemented
25C22×Q8C22×C311yescharacteristic
26C23×C4—31no—
27C4×Q8—22no—
28C4×Q8—62no—
29C2×C4⋊C4—32no—
30C2×C22⋊C4—31no—
31C2×C22⋊C4A411yescharacteristic
32C2×C42—32no—
33#95—32no—
34#95—32no—
35C23×C3—42no—
36C23×C3—42no—
37C23×C3—42no—
38C23×C3—41no—
39C23×C3—82no—
40C23×C3—82no—
41C2×C3×C4—44no—
42C2×C3×C4—82no—
43C2×C3×C4—82no—
44SL(2,𝔽3)C22⋊C411yescharacteristic, direct product factor
45C24C2×A411yescharacteristic
46C2×Q8—22no—
47C2×Q8—32no—
48C2×Q8—32no—
49C2×Q8—31no—
50C2×Q8—32no—
51C2×Q8—62no—
52C2×Q8—62no—
53C2×Q8—64no—
54C2×Q8—62no—
55C2×Q8—64no—
56C2×Q8C2×C3×C411yescharacteristic
57C2×Q8C3×D412yes—
58C22×C4—31no—
59C22×C4—32no—
60C22×C4—32no—
61C22×C4—31no—
62C22×C4—32no—
63C22×C4—62no—
64C22×C4—62no—
65C22×C4C2×A412yes—
66C4⋊C4—62no—
67C4⋊C4—62no—
68C22⋊C4—62no—
69C22⋊C4SL(2,𝔽3)14yesdirect product factor
70C42—62no—
71C42—62no—
72C22×C3—44no—
73C22×C3—41no—
74C22×C3—42no—
75C22×C3—41no—
76C22×C3—41no—
77C22×C3—42no—
78C22×C3—82no—
79C22×C3—84no—
80C22×C3—84no—
81C22×C3—82no—
82C3×C4—84no—
83C23—22no—
84C23—22no—
85C23C4×A412yes—
86C23C2×SL(2,𝔽3)12yes—
87C23#41⋊C312yes—
88C23C22×A411yescharacteristic
89Q8—31no—
90Q8—32no—
91Q8—32no—
92Q8—62no—
93Q8—62no—
94Q8—64no—
95Q8C3×C22⋊C411yescharacteristic, complemented
96C2×C4—22no—
97C2×C4—22no—
98C2×C4—31no—
99C2×C4—31no—
100C2×C4—32no—
101C2×C4—32no—
102C2×C4—32no—
103C2×C4—62no—
104C2×C4—64no—
105C2×C4—62no—
106C2×C4—62no—
107C2×C4—62no—
108C2×C4—62no—
109C2×C4—62no—
110C2×C4C2×SL(2,𝔽3)14yescomplemented
111C2×C3—41no—
112C2×C3—41no—
113C2×C3—41no—
114C2×C3—42no—
115C2×C3—42no—
116C2×C3—84no—
117C22—22no—
118C22—22no—
119C22—24no—
120C22—24no—
121C22C4×SL(2,𝔽3)14yescomplemented
122C22C2×C4×A411yescharacteristic
123C22D4×A412yes—
124C22C22×SL(2,𝔽3)11yescharacteristic
125C22C2×#41⋊C311yescharacteristic
126C222−1+4⋊C312yes—
127C4—24no—
128C4—32no—
129C4—31no—
130C4—31no—
131C4—62no—
132C4—122no—
133C3—41noSylow
134C2—24no—
135C2A4×C22⋊C411yescharacteristic
136C2C2×C4×SL(2,𝔽3)11yescharacteristic
137C2#519⋊C311yescharacteristic
138C2#548⋊C312yes—
139C2D4×SL(2,𝔽3)12yes—
subgroupG/H
center, Frattini, socleC23C22×A4
commutatorC2×Q8C2×C3×C4
FittingQ8×C22⋊C4C3
prime radicalC23×SL(2,𝔽3)C2
cosocleC22×Q8C22×C3
hypercenterC2×C22⋊C4A4
hypocenterQ8C3×C22⋊C4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×SL(2,𝔽3)C2semidirect
3. Q8×C22⋊C4C3semidirect
5. C22×SL(2,𝔽3)C4semidirect
10. C2×C4×Q8C2×C3semidirect
24. C22×Q8C3×C4semidirect
44. SL(2,𝔽3)C22⋊C4direct (N × H)
69. C22⋊C4SL(2,𝔽3)direct (N × H)
95. Q8C3×C22⋊C4semidirect
110. C2×C4C2×SL(2,𝔽3)semidirect
121. C22C4×SL(2,𝔽3)semidirect

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
C2×C4×SL(2,𝔽3)2
Q8×C22⋊C43
C2×C3×C22⋊C44