Finite Group Databasebeta

← C22×C32×C42⋊C3 #13178810

Subgroups — C22×C32×C42⋊C3 #13178810

Order 1728.

888 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 200 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×C32×C42⋊C3C213yesmaximal, direct product factor
2C22×C32×C42C311yesmaximal, characteristic, complemented
3C22×C3×C42⋊C3C3112yesmaximal, direct product factor
4C22×C32×A4—41nomaximal
5C32×C42⋊C3C2211yescharacteristic, direct product factor
6C23×C32×C4—31no—
7C2×C32×C42—33no—
8C2×C32×C42C2×C313yescomplemented
9C2×C3×C42⋊C3C2×C3136yesdirect product factor
10C2×C32×A4—43no—
11C22×C3×C42C3214yescomplemented
12C22×C42⋊C3C3219yesdirect product factor
13C24×C32A411yescharacteristic
14C22×C3×A4—412no—
15C22×C32×C4—36no—
16C22×C32×C4—32no—
17C22×C32×C4—33no—
18C22×C32×C4—33no—
19C32×C42—32no—
20C32×C42—33no—
21C32×C42C22×C311yescharacteristic, complemented
22C3×C42⋊C3C22×C3112yesdirect product factor
23C22×C33—161no—
24C32×A4—41no—
25C23×C3×C4—34no—
26C2×C3×C42—312no—
27C2×C3×C42C2×C32112yescomplemented
28C2×C42⋊C3C2×C32127yesdirect product factor
29C23×C32—33no—
30C23×C32—31no—
31C23×C32C2×A413yes—
32C2×C3×A4—436no—
33C2×C32×C4—36no—
34C2×C32×C4—36no—
35C2×C32×C4—36no—
36C2×C32×C4—36no—
37C2×C32×C4—31no—
38C2×C32×C4—33no—
39C22×C42C3311yescharacteristic, complemented, Sylow
40C2×C33—163no—
41C24×C3C3×A414yes—
42C22×A4—49no—
43C22×C3×C4—324no—
44C22×C3×C4—38no—
45C22×C3×C4—312no—
46C22×C3×C4—312no—
47C3×C42—312no—
48C3×C42—38no—
49C3×C42C22×C3214yescomplemented
50C42⋊C3C22×C3219yesdirect product factor
51C22×C32—33no—
52C22×C32—32no—
53C22×C32—33no—
54C22×C32—33no—
55C22×C32—1612no—
56C22×C32C42⋊C311yescharacteristic, direct product factor
57C22×C32C22×A411yescharacteristic
58C3×A4—412no—
59C32×C4—32no—
60C32×C4—36no—
61C23×C4—31no—
62C2×C42—33no—
63C2×C42C2×C3313yescomplemented
64C33—161noSylow
65C23×C3—312no—
66C23×C3—34no—
67C23×C3C2×C3×A4112yes—
68C2×A4—427no—
69C2×C3×C4—324no—
70C2×C3×C4—324no—
71C2×C3×C4—324no—
72C2×C3×C4—324no—
73C2×C3×C4—34no—
74C2×C3×C4—312no—
75C2×C32—31no—
76C2×C32—33no—
77C2×C32—1636no—
78C2×C32C2×C42⋊C313yesdirect product factor
79C24C32×A411yescharacteristic
80C22×C4—36no—
81C22×C4—32no—
82C22×C4—33no—
83C22×C4—33no—
84C42—33no—
85C42—32no—
86C42C22×C3311yescharacteristic, complemented
87C22×C3—312no—
88C22×C3—38no—
89C22×C3—312no—
90C22×C3—312no—
91C22×C3—169no—
92C22×C3C3×C42⋊C314yesdirect product factor
93C22×C3C22×C3×A414yes—
94A4—49no—
95C3×C4—38no—
96C3×C4—324no—
97C32—1612no—
98C32C22×C42⋊C311yescharacteristic, direct product factor
99C23—33no—
100C23—31no—
101C23C2×C32×A413yes—
102C2×C4—36no—
103C2×C4—36no—
104C2×C4—36no—
105C2×C4—36no—
106C2×C4—31no—
107C2×C4—33no—
108C2×C3—34no—
109C2×C3—312no—
110C2×C3—1627no—
111C2×C3C2×C3×C42⋊C3112yesdirect product factor
112C22—33no—
113C22—32no—
114C22—33no—
115C22—33no—
116C22C32×C42⋊C311yescharacteristic, direct product factor
117C22C22×C32×A411yescharacteristic
118C4—32no—
119C4—36no—
120C3—169no—
121C3C22×C3×C42⋊C314yesdirect product factor
122C2—31no—
123C2—33no—
124C2C2×C32×C42⋊C313yesdirect product factor
subgroupG/H
center, hypercenterC22×C32C42⋊C3
commutator, cosocle, hypocenterC42C22×C33
FrattiniC22C22×C32×A4
FittingC22×C32×C42C3
socleC24×C32A4
LCSG ▷ C42
UCSC1 ◁ C22×C32

Complements

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

NH ≅ G/Ntype
1. C2×C32×C42⋊C3C2direct (N × H)
2. C22×C32×C42C3semidirect
3. C22×C3×C42⋊C3C3direct (N × H)
5. C32×C42⋊C3C22direct (N × H)
8. C2×C32×C42C2×C3semidirect
9. C2×C3×C42⋊C3C2×C3direct (N × H)
11. C22×C3×C42C32semidirect
12. C22×C42⋊C3C32direct (N × H)
21. C32×C42C22×C3semidirect
22. C3×C42⋊C3C22×C3direct (N × H)
27. C2×C3×C42C2×C32semidirect
28. C2×C42⋊C3C2×C32direct (N × H)
39. C22×C42C33semidirect
49. C3×C42C22×C32semidirect
50. C42⋊C3C22×C32direct (N × H)
56. C22×C32C42⋊C3direct (N × H)
63. C2×C42C2×C33semidirect
78. C2×C32C2×C42⋊C3direct (N × H)
86. C42C22×C33semidirect
92. C22×C3C3×C42⋊C3direct (N × H)
98. C32C22×C42⋊C3direct (N × H)
111. C2×C3C2×C3×C42⋊C3direct (N × H)
116. C22C32×C42⋊C3direct (N × H)
121. C3C22×C3×C42⋊C3direct (N × H)
124. C2C2×C32×C42⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C42⋊C32
C22×C32×C423
C22×C3×C42⋊C33
C22×C32×A44