Finite Group Databasebeta

← C2×S4×3−1+2 #13049270

Subgroups — C2×S4×3−1+2 #13049270

Order 1296.

308 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 63 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×A4×3−1+2C211yesmaximal, characteristic, complemented
2S4×3−1+2C212yesmaximal, direct product factor
3C2×C32×S4C311yesmaximal, characteristic
4C2×C9×S4C313yesmaximal, complemented
5C2×D4×3−1+2—31nomaximal
6C2×D3×3−1+2—41nomaximal
7A4×3−1+2C2211yescharacteristic, complemented
8C2×C32×A4C2×C311yescharacteristic
9C32×S4C2×C312yes—
10C23×3−1+2—31no—
11C23×3−1+2D311yescharacteristic, complemented
12C2×#1094—21no—
13C2×C9⋊A4—23no—
14C2×C9×A4C2×C313yescomplemented
15C9×S4C2×C316yescomplemented
16D4×3−1+2—32no—
17D4×3−1+2—32no—
18C2×C4×3−1+2—31no—
19C2×C3×3−1+2—41no—
20D3×3−1+2—42no—
21C2×C3×S4—31no—
22C2×C3×S4C3211yescharacteristic
23C2×C32×D4—31no—
24C2×D4×C9—33no—
25C2×C32×D3—41no—
26C32×A4C22×C311yescharacteristic
27C22×3−1+2—31no—
28C22×3−1+2—31no—
29C22×3−1+2—32no—
30C22×3−1+2—61no—
31C22×3−1+2—61no—
32C22×3−1+2C2×D311yescharacteristic, complemented
33C2×D3×C9—43no—
34#1094—21no—
35C9⋊A4—23no—
36C9×A4C22×C313yescomplemented
37C4×3−1+2—32no—
38C3×3−1+2—41noSylow
39C23×C32—31no—
40C23×C32C3×D311yescharacteristic
41C2×C3×A4—21no—
42C2×C3×A4—31no—
43C2×C3×A4—61no—
44C2×C3×A4C2×C3211yescharacteristic
45C3×S4—32no—
46C3×S4C2×C3212yes—
47C32×D4—32no—
48C32×D4—32no—
49C2×C32×C4—31no—
50C23×C9—33no—
51C23×C9C3×D313yescomplemented
52C2×C22⋊C9—23no—
53D4×C9—36no—
54D4×C9—36no—
55C2×C4×C9—33no—
56C2×C33—41no—
57C32×D3—42no—
58C2×3−1+2—31no—
59C2×3−1+2—31no—
60C2×3−1+2—62no—
61C2×3−1+2—81no—
62C2×3−1+2—83no—
63C2×3−1+2S411yescharacteristic, direct product factor
64C2×C3×C9—43no—
65D3×C9—46no—
66C2×S43−1+211yescharacteristic, direct product factor
67C2×C3×D4—31no—
68C2×C3×D4—91no—
69C22×C32—31no—
70C22×C32—31no—
71C22×C32—32no—
72C22×C32—61no—
73C22×C32—61no—
74C22×C32C2×C3×D311yescharacteristic
75C2×C3×D3—41no—
76C2×C3×D3—121no—
77C3×A4—21no—
78C3×A4—31no—
79C3×A4—61no—
80C3×A4C22×C3211yescharacteristic
81C32×C4—32no—
82C22×C9—33no—
83C22×C9—33no—
84C22×C9—36no—
85C22×C9—63no—
86C22×C9—63no—
87C22×C9C2×C3×D313yescomplemented
88C22⋊C9—23no—
89C4×C9—36no—
90C33—41no—
913−1+2—81no—
923−1+2—83no—
933−1+2C2×S411yescharacteristic, direct product factor
94C3×C9—43no—
95C23×C3—31no—
96C23×C3—31no—
97C23×C3—91no—
98C23×C3C32×D311yescharacteristic
99C2×A4—21no—
100C2×A4—61no—
101C2×A4C2×3−1+211yescharacteristic, complemented
102S4C2×3−1+212yesdirect product factor
103C3×D4—32no—
104C3×D4—32no—
105C3×D4—92no—
106C3×D4—92no—
107C2×C3×C4—31no—
108C2×C3×C4—91no—
109C2×C32—31no—
110C2×C32—31no—
111C2×C32—41no—
112C2×C32—62no—
113C2×C32—81no—
114C2×C32—121no—
115C2×C32—241no—
116C2×C32C3×S411yescharacteristic
117C3×D3—42no—
118C3×D3—122no—
119C2×C9—33no—
120C2×C9—33no—
121C2×C9—66no—
122C2×C9—83no—
123C2×C9C3×S413yescomplemented
124C2×D4—31noSylow
125C22×C3—31no—
126C22×C3—31no—
127C22×C3—32no—
128C22×C3—31no—
129C22×C3—61no—
130C22×C3—61no—
131C22×C3—91no—
132C22×C3—91no—
133C22×C3—92no—
134C22×C3—181no—
135C22×C3—181no—
136C22×C3C2×C32×D311yescharacteristic
137C2×D3—41no—
138A4—21no—
139A4—61no—
140A4C22×3−1+211yescharacteristic, complemented
141C3×C4—32no—
142C3×C4—92no—
143C32—41no—
144C32—81no—
145C32—121no—
146C32—241no—
147C32C2×C3×S411yescharacteristic
148C9—83no—
149C9C2×C3×S413yescomplemented
150C23—31no—
151C23D3×3−1+211yescharacteristic, complemented
152D4—32no—
153D4—32no—
154C2×C4—31no—
155C2×C3—31no—
156C2×C3—31no—
157C2×C3—31no—
158C2×C3—41no—
159C2×C3—62no—
160C2×C3—81no—
161C2×C3—91no—
162C2×C3—91no—
163C2×C3—182no—
164C2×C3—241no—
165C2×C3C32×S411yescharacteristic
166D3—42no—
167C22—31no—
168C22—31no—
169C22—32no—
170C22—61no—
171C22—61no—
172C22C2×D3×3−1+211yescharacteristic, complemented
173C4—32no—
174C3—31no—
175C3—41no—
176C3—81no—
177C3—241no—
178C3C2×C32×S411yescharacteristic
179C2—31no—
180C2—31no—
181C2—62no—
182C2S4×3−1+211yescharacteristic, direct product factor
subgroupG/H
centerC2×C3C32×S4
commutator, cosocleC3×A4C22×C32
FrattiniC3C2×C32×S4
FittingC23×3−1+2D3
socleC23×C3C32×D3
prime radicalC2×C32×S4C3
hypercenterC2×3−1+2S4
hypocenterA4C22×3−1+2
LCSG ▷ C3×A4 ▷ A4
UCSC1 ◁ C2×C3 ◁ C2×3−1+2

Complements

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

NH ≅ G/Ntype
1. C2×A4×3−1+2C2semidirect
2. S4×3−1+2C2direct (N × H)
4. C2×C9×S4C3semidirect
7. A4×3−1+2C22semidirect
11. C23×3−1+2D3semidirect
14. C2×C9×A4C2×C3semidirect
15. C9×S4C2×C3semidirect
32. C22×3−1+2C2×D3semidirect
36. C9×A4C22×C3semidirect
51. C23×C9C3×D3semidirect
63. C2×3−1+2S4direct (N × H)
66. C2×S43−1+2direct (N × H)
87. C22×C9C2×C3×D3semidirect
93. 3−1+2C2×S4direct (N × H)
101. C2×A4C2×3−1+2semidirect
102. S4C2×3−1+2direct (N × H)
123. C2×C9C3×S4semidirect
140. A4C22×3−1+2semidirect
149. C9C2×C3×S4semidirect
151. C23D3×3−1+2semidirect
172. C22C2×D3×3−1+2semidirect
182. C2S4×3−1+2direct (N × H)

Maximal subgroups

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

subgroupindex
C2×A4×3−1+22
S4×3−1+22
C2×C32×S43
C2×C9×S43
C2×D4×3−1+23
C2×D3×3−1+24