Finite Group Databasebeta

← C22×C42⋊C27 #13136171

Subgroups — C22×C42⋊C27 #13136171

Order 1728.

294 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 50 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×C42⋊C27C213yesmaximal, direct product factor
2C22×C42×C9C311yesmaximal, characteristic
3C22×C22⋊C27—41nomaximal
4C42⋊C27C2211yescharacteristic, direct product factor
5C23×C4×C9—31no—
6C2×C42×C9—33no—
7C2×C42×C9C2×C313yes—
8C2×C22⋊C27—43no—
9C22×C3×C42C911yescharacteristic
10C24×C9A411yescharacteristic
11C22×C4×C9—36no—
12C22×C4×C9—32no—
13C22×C4×C9—33no—
14C22×C4×C9—33no—
15C42×C9—32no—
16C42×C9—33no—
17C42×C9C22×C311yescharacteristic
18C22×C27—161no—
19C22⋊C27—41no—
20C23×C3×C4—31no—
21C2×C3×C42—33no—
22C2×C3×C42C2×C913yes—
23C23×C9—33no—
24C23×C9—31no—
25C23×C9C2×A413yes—
26C2×C4×C9—36no—
27C2×C4×C9—36no—
28C2×C4×C9—36no—
29C2×C4×C9—36no—
30C2×C4×C9—31no—
31C2×C4×C9—33no—
32C22×C42C2711yescharacteristic, complemented, Sylow
33C2×C27—163no—
34C24×C3C22⋊C911yescharacteristic
35C22×C3×C4—36no—
36C22×C3×C4—32no—
37C22×C3×C4—33no—
38C22×C3×C4—33no—
39C3×C42—33no—
40C3×C42—32no—
41C3×C42C22×C911yescharacteristic
42C22×C9—33no—
43C22×C9—32no—
44C22×C9—33no—
45C22×C9—33no—
46C22×C9C42⋊C311yescharacteristic
47C22×C9C22×A411yescharacteristic
48C4×C9—32no—
49C4×C9—36no—
50C23×C4—31no—
51C2×C42—33no—
52C2×C42C2×C2713yescomplemented
53C27—161noSylow
54C23×C3—33no—
55C23×C3—31no—
56C23×C3C2×C22⋊C913yes—
57C2×C3×C4—36no—
58C2×C3×C4—36no—
59C2×C3×C4—36no—
60C2×C3×C4—36no—
61C2×C3×C4—31no—
62C2×C3×C4—33no—
63C2×C9—31no—
64C2×C9—33no—
65C2×C9C2×C42⋊C313yes—
66C24C22⋊C2711yescharacteristic
67C22×C4—36no—
68C22×C4—32no—
69C22×C4—33no—
70C22×C4—33no—
71C42—33no—
72C42—32no—
73C42C22×C2711yescharacteristic, complemented
74C22×C3—33no—
75C22×C3—32no—
76C22×C3—33no—
77C22×C3—33no—
78C22×C3C42⋊C911yescharacteristic
79C22×C3C22×C22⋊C911yescharacteristic
80C3×C4—32no—
81C3×C4—36no—
82C9C22×C42⋊C311yescharacteristic
83C23—33no—
84C23—31no—
85C23C2×C22⋊C2713yes—
86C2×C4—36no—
87C2×C4—36no—
88C2×C4—36no—
89C2×C4—36no—
90C2×C4—31no—
91C2×C4—33no—
92C2×C3—31no—
93C2×C3—33no—
94C2×C3C2×C42⋊C913yes—
95C22—33no—
96C22—32no—
97C22—33no—
98C22—33no—
99C22C42⋊C2711yescharacteristic, direct product factor
100C22C22×C22⋊C2711yescharacteristic
101C4—32no—
102C4—36no—
103C3C22×C42⋊C911yescharacteristic
104C2—31no—
105C2—33no—
106C2C2×C42⋊C2713yesdirect product factor
subgroupG/H
center, hypercenterC22×C9C42⋊C3
commutator, hypocenterC42C22×C27
FrattiniC22×C9C22×A4
FittingC22×C42×C9C3
socle, prime radicalC24×C3C22⋊C9
cosocleC42×C9C22×C3
LCSG ▷ C42
UCSC1 ◁ C22×C9

Complements

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

NH ≅ G/Ntype
1. C2×C42⋊C27C2direct (N × H)
4. C42⋊C27C22direct (N × H)
32. C22×C42C27semidirect
52. C2×C42C2×C27semidirect
73. C42C22×C27semidirect
99. C22C42⋊C27direct (N × H)
106. C2C2×C42⋊C27direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C42⋊C272
C22×C42×C93
C22×C22⋊C274