Finite Group Databasebeta

← C22×2−1+4⋊C3 #87534

Subgroups — C22×2−1+4⋊C3 #87534

Order 384.

1287 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 1287 conjugacy classes of subgroups — too many to draw. Only its 212 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
1C2×2−1+4⋊C3C2112yesmaximal, direct product factor
2C22×#41⋊C3C211yesmaximal, characteristic, complemented
3C23×SL(2,𝔽3)C212yesmaximal, complemented
4C22×2−1+4C311yesmaximal, characteristic, complemented, Sylow
5C22×C3×D4—41nomaximal
62−1+4⋊C3C22116yesdirect product factor
7C2×#41⋊C3C2216yescomplemented
8C22×SL(2,𝔽3)C2211yescharacteristic
9C22×SL(2,𝔽3)C22112yescomplemented
10C2×2−1+4—312no—
11C2×2−1+4C2×C3112yescomplemented
12C22×#41—31no—
13C22×#41—32no—
14C22×#41C2×C311yescharacteristic, complemented
15C23×Q8—31no—
16C23×Q8C2×C312yescomplemented
17C24×C3—42no—
18C2×C3×D4—412no—
19C22×C3×C4—41no—
20#41⋊C3C2314yescomplemented
21C2×SL(2,𝔽3)C2313yes—
22C2×SL(2,𝔽3)C2318yescomplemented
232−1+4—348no—
242−1+4—332no—
252−1+4C22×C3116yescomplemented
26C2×#41—324no—
27C2×#41—312no—
28C2×#41—312no—
29C2×#41—324no—
30C2×#41—36no—
31C2×#41C22×C316yescomplemented
32C22×Q8—312no—
33C22×Q8—312no—
34C22×Q8—31no—
35C22×Q8—32no—
36C22×Q8—312no—
37C22×Q8C22×C311yescharacteristic
38C22×Q8C22×C3112yescomplemented
39C22×D4—31no—
40C22×D4—32no—
41C22×D4A411yescharacteristic
42C23×C4—32no—
43C23×C4—32no—
44C23×C4—31no—
45C23×C3—412no—
46C23×C3—41no—
47C23×C3—88no—
48C3×D4—416no—
49C2×C3×C4—46no—
50SL(2,𝔽3)C2411yescharacteristic
51C24—31no—
52C24C2×A412yes—
53#41—396no—
54#41—38no—
55#41—348no—
56#41—316no—
57#41—332no—
58#41—312no—
59#41C23×C314yescomplemented
60C2×Q8—316no—
61C2×Q8—312no—
62C2×Q8—348no—
63C2×Q8—316no—
64C2×Q8—312no—
65C2×Q8—33no—
66C2×Q8—312no—
67C2×Q8—324no—
68C2×Q8C23×C313yes—
69C2×Q8C23×C318yescomplemented
70C2×D4—36no—
71C2×D4—36no—
72C2×D4—324no—
73C2×D4C2×A4112yes—
74C22×C4—324no—
75C22×C4—312no—
76C22×C4—312no—
77C22×C4—32no—
78C22×C4—31no—
79C22×C4—36no—
80C22×C4—36no—
81C22×C4C2×A411yescharacteristic
82C22×C3—44no—
83C22×C3—43no—
84C22×C3—48no—
85C22×C3—824no—
86C3×C4—44no—
87C23—28no—
88C23—36no—
89C23—64no—
90C23C22×A4112yes—
91C23C22×A411yescharacteristic
92Q8—33no—
93Q8—38no—
94Q8—324no—
95Q8—32no—
96Q8—316no—
97Q8C24×C311yescharacteristic
98D4—34no—
99D4—312no—
100D4—332no—
101D4C22×A4116yes—
102C2×C4—332no—
103C2×C4—312no—
104C2×C4—33no—
105C2×C4—33no—
106C2×C4—38no—
107C2×C4—324no—
108C2×C4—312no—
109C2×C4—34no—
110C2×C4C22×A416yes—
111C2×C3—46no—
112C2×C3—41no—
113C2×C3—88no—
114C22—224no—
115C22—34no—
116C22—612no—
117C222−1+4⋊C314yesdirect product factor
118C22C23×A418yes—
119C22C23×A413yes—
120C4—38no—
121C4—33no—
122C4—31no—
123C4C23×A414yes—
124C3—41noSylow
125C2—28no—
126C2—64no—
127C2C2×2−1+4⋊C316yesdirect product factor
128C2C24×A411yescharacteristic
subgroupG/H
center, socleC23C22×A4
commutator, cosocle, hypocenterQ8C24×C3
FrattiniC2C24×A4
FittingC22×2−1+4C3
hypercenterC22×D4A4
LCSG ▷ Q8
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×2−1+4⋊C3C2direct (N × H)
2. C22×#41⋊C3C2semidirect
3. C23×SL(2,𝔽3)C2semidirect
4. C22×2−1+4C3semidirect
6. 2−1+4⋊C3C22direct (N × H)
7. C2×#41⋊C3C22semidirect
9. C22×SL(2,𝔽3)C22semidirect
11. C2×2−1+4C2×C3semidirect
14. C22×#41C2×C3semidirect
16. C23×Q8C2×C3semidirect
20. #41⋊C3C23semidirect
22. C2×SL(2,𝔽3)C23semidirect
25. 2−1+4C22×C3semidirect
31. C2×#41C22×C3semidirect
38. C22×Q8C22×C3semidirect
59. #41C23×C3semidirect
69. C2×Q8C23×C3semidirect
117. C222−1+4⋊C3direct (N × H)
127. C2C2×2−1+4⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×2−1+4⋊C32
C22×#41⋊C32
C23×SL(2,𝔽3)2
C22×2−1+43
C22×C3×D44