Finite Group Databasebeta

← C4⋊C4×SL(2,𝔽3) #73176

Subgroups — C4⋊C4×SL(2,𝔽3) #73176

Order 384.

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

normal subgroups only

This group has 193 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
1C2×C4×SL(2,𝔽3)C212yesmaximal
2C2×C4×SL(2,𝔽3)C211yesmaximal, characteristic
3Q8×C4⋊C4C311yesmaximal, characteristic, complemented, Sylow
4C22×SL(2,𝔽3)C2211yescharacteristic
5C2×C3×C4⋊C4—41nomaximal
6C4×SL(2,𝔽3)—22no—
7C4×SL(2,𝔽3)C412yescomplemented
8C2×C4×Q8C2×C312yes—
9C2×C4×Q8C2×C311yescharacteristic
10#389—32no—
11#384—31no—
12C4×C4⋊C4—31no—
13C22×C3×C4—41no—
14C22×C3×C4—42no—
15C2×SL(2,𝔽3)C2×C411yescharacteristic
16C2×SL(2,𝔽3)D411yescharacteristic
17C2×SL(2,𝔽3)Q811yescharacteristic
18C3×C4⋊C4—44no—
19C22×Q8C22×C311yescharacteristic
20C4×Q8—22no—
21C4×Q8—32no—
22C4×Q8—62no—
23C4×Q8C3×C412yescomplemented
24C2×C4⋊C4—31no—
25C2×C4⋊C4—32no—
26C2×C4⋊C4—31no—
27C2×C4⋊C4—32no—
28C2×C4⋊C4A411yescharacteristic
29C2×C42—31no—
30C2×C42—32no—
31#95—32no—
32C23×C3—41no—
33C2×C3×C4—42no—
34C2×C3×C4—42no—
35C2×C3×C4—44no—
36C2×C3×C4—42no—
37C2×C3×C4—82no—
38C2×C3×C4—82no—
39SL(2,𝔽3)C4⋊C411yescharacteristic, direct product factor
40C2×Q8—31no—
41C2×Q8—31no—
42C2×Q8—31no—
43C2×Q8C2×C3×C411yescharacteristic
44C2×Q8C3×D411yescharacteristic
45C2×Q8C3×Q811yescharacteristic
46C22×C4—31no—
47C22×C4—31no—
48C22×C4—32no—
49C22×C4C2×A412yes—
50C22×C4C2×A411yescharacteristic
51C4⋊C4—32no—
52C4⋊C4—32no—
53C4⋊C4—62no—
54C4⋊C4—62no—
55C4⋊C4—62no—
56C4⋊C4—122no—
57C4⋊C4SL(2,𝔽3)14yesdirect product factor
58C42—32no—
59C42—32no—
60C42—62no—
61C42—62no—
62C22×C3—41no—
63C22×C3—41no—
64C22×C3—41no—
65C22×C3—41no—
66C22×C3—41no—
67C22×C3—41no—
68C22×C3—41no—
69C3×C4—44no—
70C3×C4—84no—
71C23C22×A411yescharacteristic
72Q8—31no—
73Q8—31no—
74Q8—31no—
75Q8—32no—
76Q8C3×C4⋊C411yescharacteristic, complemented
77C2×C4—22no—
78C2×C4—22no—
79C2×C4—31no—
80C2×C4—31no—
81C2×C4—31no—
82C2×C4—31no—
83C2×C4—31no—
84C2×C4—31no—
85C2×C4—32no—
86C2×C4—61no—
87C2×C4—61no—
88C2×C4—62no—
89C2×C4—62no—
90C2×C4—62no—
91C2×C4C4×A412yes—
92C2×C4C2×SL(2,𝔽3)14yes—
93C2×C4C2×SL(2,𝔽3)12yes—
94C2×C4#41⋊C312yes—
95C2×C3—41no—
96C2×C3—41no—
97C2×C3—41no—
98C2×C3—41no—
99C2×C3—41no—
100C2×C3—41no—
101C2×C3—41no—
102C22C2×C4×A411yescharacteristic
103C22D4×A411yescharacteristic
104C22C22×SL(2,𝔽3)11yescharacteristic
105C22Q8×A411yescharacteristic
106C22C2×#41⋊C311yescharacteristic
107C22#99411yescharacteristic
108C222−1+4⋊C311yescharacteristic
109C4—24no—
110C4—31no—
111C4—31no—
112C4—31no—
113C4—31no—
114C4—62no—
115C4—122no—
116C4C4×SL(2,𝔽3)14yescomplemented
117C3—41noSylow
118C2A4×C4⋊C411yescharacteristic
119C2C2×C4×SL(2,𝔽3)11yescharacteristic
120C2#520⋊C311yescharacteristic
121C2D4×SL(2,𝔽3)11yescharacteristic
122C2C4⋊(#41⋊C3)11yescharacteristic
123C2#557⋊C311yescharacteristic
124C2Q8×SL(2,𝔽3)11yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×A4
commutatorC2×Q8C2×C3×C4
FittingQ8×C4⋊C4C3
prime radicalC22×SL(2,𝔽3)C22
cosocleC22×Q8C22×C3
hypercenterC2×C4⋊C4A4
hypocenterQ8C3×C4⋊C4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. Q8×C4⋊C4C3semidirect
7. C4×SL(2,𝔽3)C4semidirect
23. C4×Q8C3×C4semidirect
39. SL(2,𝔽3)C4⋊C4direct (N × H)
57. C4⋊C4SL(2,𝔽3)direct (N × H)
76. Q8C3×C4⋊C4semidirect
116. C4C4×SL(2,𝔽3)semidirect

Maximal subgroups

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

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