Finite Group Databasebeta

← C22×C32×C22⋊A4 #13181869

Subgroups — C22×C32×C22⋊A4 #13181869

Order 1728.

6560 conjugacy classes of subgroups; only the 320 normal subgroups are shown.

normal subgroups only

This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 320 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×C22⋊A4C213yesmaximal, direct product factor
2C26×C32C311yesmaximal, characteristic, complemented
3C22×C3×C22⋊A4C3112yesmaximal, direct product factor
4C32×C22⋊A4C2211yescharacteristic, direct product factor
5C22×C32×A4—45nomaximal
6C25×C32—315no—
7C25×C32—35no—
8C25×C32C2×C313yescomplemented
9C2×C3×C22⋊A4C2×C3136yesdirect product factor
10C2×C32×A4—415no—
11C26×C3C3214yescomplemented
12C22×C22⋊A4C3219yesdirect product factor
13C24×C32—390no—
14C24×C32—360no—
15C24×C32—310no—
16C24×C32—310no—
17C24×C32—315no—
18C24×C32—315no—
19C24×C32—315no—
20C24×C32A415yescomplemented
21C24×C32C22×C311yescharacteristic, complemented
22C3×C22⋊A4C22×C3112yesdirect product factor
23C22×C3×A4—460no—
24C22×C33—161no—
25C32×A4—45no—
26C25×C3—360no—
27C25×C3—320no—
28C25×C3C2×C32112yescomplemented
29C2×C22⋊A4C2×C32127yesdirect product factor
30C23×C32—315no—
31C23×C32—390no—
32C23×C32—390no—
33C23×C32—3120no—
34C23×C32—390no—
35C23×C32—330no—
36C23×C32—315no—
37C23×C32—35no—
38C23×C32—35no—
39C23×C32C2×A4115yescomplemented
40C2×C3×A4—4180no—
41C26C3311yescharacteristic, complemented, Sylow
42C2×C33—163no—
43C24×C3—360no—
44C24×C3—3360no—
45C24×C3—3240no—
46C24×C3—340no—
47C24×C3—340no—
48C24×C3—360no—
49C24×C3—360no—
50C24×C3C3×A4120yescomplemented
51C24×C3C22×C3214yescomplemented
52C22⋊A4C22×C3219yesdirect product factor
53C22×A4—445no—
54C22×C32—315no—
55C22×C32—310no—
56C22×C32—360no—
57C22×C32—390no—
58C22×C32—310no—
59C22×C32—315no—
60C22×C32—315no—
61C22×C32—1612no—
62C22×C32C22×A415yescomplemented
63C22×C32C22⋊A411yescharacteristic, direct product factor
64C3×A4—460no—
65C25—315no—
66C25—35no—
67C25C2×C3313yescomplemented
68C33—161noSylow
69C23×C3—360no—
70C23×C3—3360no—
71C23×C3—3360no—
72C23×C3—3480no—
73C23×C3—3120no—
74C23×C3—3360no—
75C23×C3—360no—
76C23×C3—320no—
77C23×C3—320no—
78C23×C3C2×C3×A4160yescomplemented
79C2×A4—4135no—
80C2×C32—315no—
81C2×C32—35no—
82C2×C32—1636no—
83C2×C32C2×C22⋊A413yesdirect product factor
84C24—315no—
85C24—390no—
86C24—360no—
87C24—310no—
88C24—310no—
89C24—315no—
90C24—315no—
91C24C32×A415yescomplemented
92C24C22×C3311yescharacteristic, complemented
93C22×C3—360no—
94C22×C3—340no—
95C22×C3—3360no—
96C22×C3—340no—
97C22×C3—3240no—
98C22×C3—360no—
99C22×C3—360no—
100C22×C3—169no—
101C22×C3C22×C3×A4120yescomplemented
102C22×C3C3×C22⋊A414yesdirect product factor
103A4—445no—
104C32—1612no—
105C32C22×C22⋊A411yescharacteristic, direct product factor
106C23—315no—
107C23—390no—
108C23—390no—
109C23—3120no—
110C23—330no—
111C23—390no—
112C23—35no—
113C23—315no—
114C23—35no—
115C23C2×C32×A4115yescomplemented
116C2×C3—320no—
117C2×C3—360no—
118C2×C3—1627no—
119C2×C3C2×C3×C22⋊A4112yesdirect product factor
120C22—315no—
121C22—310no—
122C22—390no—
123C22—310no—
124C22—360no—
125C22—315no—
126C22—315no—
127C22C22×C32×A415yescomplemented
128C22C32×C22⋊A411yescharacteristic, direct product factor
129C3—169no—
130C3C22×C3×C22⋊A414yesdirect product factor
131C2—35no—
132C2—315no—
133C2C2×C32×C22⋊A413yesdirect product factor
subgroupG/H
center, hypercenterC22×C32C22⋊A4
commutator, cosocle, hypocenterC24C22×C33
FrattiniC1G
Fitting, socleC26×C32C3
LCSG ▷ C24
UCSC1 ◁ C22×C32

Complements

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

NH ≅ G/Ntype
1. C2×C32×C22⋊A4C2direct (N × H)
2. C26×C32C3semidirect
3. C22×C3×C22⋊A4C3direct (N × H)
4. C32×C22⋊A4C22direct (N × H)
8. C25×C32C2×C3semidirect
9. C2×C3×C22⋊A4C2×C3direct (N × H)
11. C26×C3C32semidirect
12. C22×C22⋊A4C32direct (N × H)
20. C24×C32A4semidirect
21. C24×C32C22×C3semidirect
22. C3×C22⋊A4C22×C3direct (N × H)
28. C25×C3C2×C32semidirect
29. C2×C22⋊A4C2×C32direct (N × H)
39. C23×C32C2×A4semidirect
41. C26C33semidirect
50. C24×C3C3×A4semidirect
51. C24×C3C22×C32semidirect
52. C22⋊A4C22×C32direct (N × H)
62. C22×C32C22×A4semidirect
63. C22×C32C22⋊A4direct (N × H)
67. C25C2×C33semidirect
78. C23×C3C2×C3×A4semidirect
83. C2×C32C2×C22⋊A4direct (N × H)
91. C24C32×A4semidirect
92. C24C22×C33semidirect
101. C22×C3C22×C3×A4semidirect
102. C22×C3C3×C22⋊A4direct (N × H)
105. C32C22×C22⋊A4direct (N × H)
115. C23C2×C32×A4semidirect
119. C2×C3C2×C3×C22⋊A4direct (N × H)
127. C22C22×C32×A4semidirect
128. C22C32×C22⋊A4direct (N × H)
130. C3C22×C3×C22⋊A4direct (N × H)
133. C2C2×C32×C22⋊A4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C22⋊A42
C26×C323
C22×C3×C22⋊A43
C22×C32×A44