Finite Group Databasebeta

← C2×3−1+2×C22⋊A4 #72245797

Subgroups — C2×3−1+2×C22⋊A4 #72245797

Order 2592.

1492 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 148 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
13−1+2×C22⋊A4C211yesmaximal, characteristic, direct product factor
2C2×C32×C22⋊A4C311yesmaximal, characteristic
3C25×3−1+2C311yesmaximal, characteristic, complemented
4C2×#88907C312yesmaximal, complemented
5C2×C9⋊(C22⋊A4)C316yesmaximal, complemented
6C2×C9×C22⋊A4C313yesmaximal, complemented
7C2×A4×3−1+2—45nomaximal
8C32×C22⋊A4C2×C311yescharacteristic
9C24×3−1+2—35no—
10C24×3−1+2—35no—
11C24×3−1+2C2×C311yescharacteristic, complemented
12#88907C2×C312yescomplemented
13C9⋊(C22⋊A4)C2×C316yescomplemented
14C9×C22⋊A4C2×C313yescomplemented
15A4×3−1+2—45no—
16C25×C32C3211yescharacteristic
17C2×C3×C22⋊A4—33no—
18C2×C3×C22⋊A4C3212yes—
19C2×C3×C22⋊A4C3211yescharacteristic
20C25×C9C3213yescomplemented
21C2×C22⋊(C22⋊C9)C3216yescomplemented
22C2×C32×A4—45no—
23C23×3−1+2—330no—
24C23×3−1+2—310no—
25C23×3−1+2—35no—
26C23×3−1+2—35no—
27C23×3−1+2A415yescomplemented
28C2×#1094—410no—
29C2×C9⋊A4—430no—
30C2×C9×A4—415no—
31C2×C3×3−1+2—161no—
32C24×C32—35no—
33C24×C32—35no—
34C24×C32C2×C3211yescharacteristic
35C3×C22⋊A4—33no—
36C3×C22⋊A4C2×C3212yes—
37C3×C22⋊A4C2×C3211yescharacteristic
38C24×C9—315no—
39C24×C9—315no—
40C24×C9C2×C3213yescomplemented
41C22⋊(C22⋊C9)C2×C3216yescomplemented
42C32×A4—45no—
43C22×3−1+2—330no—
44C22×3−1+2—310no—
45C22×3−1+2—35no—
46C22×3−1+2—35no—
47C22×3−1+2C2×A415yescomplemented
48#1094—410no—
49C9⋊A4—430no—
50C9×A4—415no—
51C25×C3—31no—
52C25×C3C3311yescharacteristic
53C2×C22⋊A4—32no—
54C2×C22⋊A43−1+213yesdirect product factor
55C3×3−1+2—161noSylow
56C23×C32—330no—
57C23×C32—310no—
58C23×C32—35no—
59C23×C32—35no—
60C23×C32C3×A415yes—
61C2×C3×A4—45no—
62C2×C3×A4—410no—
63C2×C3×A4—1215no—
64C23×C9—390no—
65C23×C9—330no—
66C23×C9—315no—
67C23×C9—315no—
68C23×C9C3×A4115yescomplemented
69C2×C22⋊C9—430no—
70C2×C33—161no—
71C2×3−1+2—35no—
72C2×3−1+2—35no—
73C2×3−1+2—162no—
74C2×3−1+2—166no—
75C2×3−1+2C22⋊A411yescharacteristic, direct product factor
76C2×C3×C9—163no—
77C24×C3—35no—
78C24×C3—35no—
79C24×C3—31no—
80C24×C3—95no—
81C24×C3—95no—
82C24×C3C2×C3311yescharacteristic
83C22⋊A4—32no—
84C22⋊A4C2×3−1+213yesdirect product factor
85C22×C32—35no—
86C22×C32—330no—
87C22×C32—310no—
88C22×C32—35no—
89C22×C32C2×C3×A415yes—
90C3×A4—45no—
91C3×A4—410no—
92C3×A4—1215no—
93C22×C9—315no—
94C22×C9—390no—
95C22×C9—330no—
96C22×C9—315no—
97C22×C9C2×C3×A4115yescomplemented
98C22⋊C9—430no—
99C25C3×3−1+211yescharacteristic, complemented, Sylow
100C33—161no—
1013−1+2—162no—
1023−1+2—166no—
1033−1+2C2×C22⋊A411yescharacteristic, direct product factor
104C3×C9—163no—
105C23×C3—330no—
106C23×C3—310no—
107C23×C3—35no—
108C23×C3—35no—
109C23×C3—35no—
110C23×C3—95no—
111C23×C3—95no—
112C23×C3—910no—
113C23×C3—930no—
114C23×C3C32×A415yes—
115C2×A4—415no—
116C2×A4—1210no—
117C2×C32—35no—
118C2×C32—35no—
119C2×C32—161no—
120C2×C32—162no—
121C2×C32—483no—
122C2×C32C3×C22⋊A411yescharacteristic
123C2×C9—315no—
124C2×C9—315no—
125C2×C9—166no—
126C2×C9C3×C22⋊A413yescomplemented
127C24—35no—
128C24—35no—
129C24C2×C3×3−1+211yescharacteristic, complemented
130C22×C3—35no—
131C22×C3—310no—
132C22×C3—330no—
133C22×C3—35no—
134C22×C3—35no—
135C22×C3—95no—
136C22×C3—95no—
137C22×C3—910no—
138C22×C3—930no—
139C22×C3C2×C32×A415yes—
140A4—415no—
141A4—1210no—
142C32—161no—
143C32—162no—
144C32—483no—
145C32C2×C3×C22⋊A411yescharacteristic
146C9—166no—
147C9C2×C3×C22⋊A413yescomplemented
148C23—330no—
149C23—310no—
150C23—35no—
151C23—35no—
152C23A4×3−1+215yescomplemented
153C2×C3—35no—
154C2×C3—35no—
155C2×C3—31no—
156C2×C3—95no—
157C2×C3—95no—
158C2×C3—163no—
159C2×C3—482no—
160C2×C3C32×C22⋊A411yescharacteristic
161C22—35no—
162C22—310no—
163C22—330no—
164C22—35no—
165C22C2×A4×3−1+215yescomplemented
166C3—31no—
167C3—163no—
168C3—482no—
169C3C2×C32×C22⋊A411yescharacteristic
170C2—35no—
171C2—35no—
172C23−1+2×C22⋊A411yescharacteristic, direct product factor
subgroupG/H
centerC2×C3C32×C22⋊A4
commutator, cosocleC24×C3C2×C33
FrattiniC3C2×C32×C22⋊A4
FittingC25×3−1+2C3
socleC25×C3C33
prime radicalC2×C32×C22⋊A4C3
hypercenterC2×3−1+2C22⋊A4
hypocenterC24C2×C3×3−1+2
LCSG ▷ C24×C3 ▷ C24
UCSC1 ◁ C2×C3 ◁ C2×3−1+2

Complements

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

NH ≅ G/Ntype
1. 3−1+2×C22⋊A4C2direct (N × H)
3. C25×3−1+2C3semidirect
4. C2×#88907C3semidirect
5. C2×C9⋊(C22⋊A4)C3semidirect
6. C2×C9×C22⋊A4C3semidirect
11. C24×3−1+2C2×C3semidirect
12. #88907C2×C3semidirect
13. C9⋊(C22⋊A4)C2×C3semidirect
14. C9×C22⋊A4C2×C3semidirect
20. C25×C9C32semidirect
21. C2×C22⋊(C22⋊C9)C32semidirect
27. C23×3−1+2A4semidirect
40. C24×C9C2×C32semidirect
41. C22⋊(C22⋊C9)C2×C32semidirect
47. C22×3−1+2C2×A4semidirect
54. C2×C22⋊A43−1+2direct (N × H)
68. C23×C9C3×A4semidirect
75. C2×3−1+2C22⋊A4direct (N × H)
84. C22⋊A4C2×3−1+2direct (N × H)
97. C22×C9C2×C3×A4semidirect
99. C25C3×3−1+2semidirect
103. 3−1+2C2×C22⋊A4direct (N × H)
126. C2×C9C3×C22⋊A4semidirect
129. C24C2×C3×3−1+2semidirect
147. C9C2×C3×C22⋊A4semidirect
152. C23A4×3−1+2semidirect
165. C22C2×A4×3−1+2semidirect
172. C23−1+2×C22⋊A4direct (N × H)

Maximal subgroups

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

subgroupindex
3−1+2×C22⋊A42
C2×C32×C22⋊A43
C25×3−1+23
C2×#889073
C2×C9⋊(C22⋊A4)3
C2×C9×C22⋊A43
C2×A4×3−1+24