Finite Group Databasebeta

← C22×Dic3×A4 #10596626

Subgroups — C22×Dic3×A4 #10596626

Order 576.

666 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C23×C3×A4C211yesmaximal, characteristic
2C2×Dic3×A4C216yesmaximal, direct product factor
3C24×Dic3C311yesmaximal, characteristic, complemented
4C22×C4×A4—31nomaximal
5C22×C3×A4C414yescomplemented
6C22×C3×A4C2213yes—
7C22×C3×Dic3—41nomaximal
8Dic3×A4C2214yesdirect product factor
9C25×C3C2×C311yescharacteristic
10C23×A4—21no—
11C23×A4D311yescharacteristic
12C23×Dic3—31no—
13C23×Dic3—31no—
14C23×Dic3—36no—
15C23×Dic3C2×C316yescomplemented
16C2×C4×A4—36no—
17C23×C32—41no—
18C2×C3×A4C2×C416yescomplemented
19C2×C3×A4C2311yescharacteristic
20C2×C3×Dic3—46no—
21C24×C4—31noSylow
22C24×C3—34no—
23C24×C3—33no—
24C24×C3—31no—
25C24×C3C3×C414yescomplemented
26C24×C3C22×C313yes—
27C22×A4—24no—
28C22×A4—23no—
29C22×A4Dic314yesdirect product factor
30C22×A4C2×D313yes—
31C22×C3×C4—121no—
32C22×Dic3—36no—
33C22×Dic3—38no—
34C22×Dic3—31no—
35C22×Dic3—36no—
36C22×Dic3—36no—
37C22×Dic3—36no—
38C22×Dic3—312no—
39C22×Dic3A411yescharacteristic, direct product factor
40C22×Dic3C22×C314yescomplemented
41C4×A4—34no—
42C22×C32—44no—
43C22×C32—43no—
44C3×A4C22×C411yescharacteristic, complemented
45C3×Dic3—44no—
46C25C3×D311yescharacteristic
47C23×C4—36no—
48C23×C4—91no—
49C23×C4—96no—
50C23×C4—91no—
51C23×C3—33no—
52C23×C3—312no—
53C23×C3—36no—
54C23×C3—32no—
55C23×C3—312no—
56C23×C3—34no—
57C23×C3—34no—
58C23×C3—33no—
59C23×C3—33no—
60C23×C3—41no—
61C23×C3—81no—
62C23×C3C2×C3×C416yescomplemented
63C23×C3C2×A411yescharacteristic
64C23×C3C23×C311yescharacteristic
65C2×A4—26no—
66C2×A4—21no—
67C2×A4C2×Dic316yesdirect product factor
68C2×A4C22×D311yescharacteristic
69C2×C3×C4—126no—
70C2×Dic3—312no—
71C2×Dic3—36no—
72C2×Dic3—312no—
73C2×Dic3—34no—
74C2×Dic3—34no—
75C2×Dic3C2×A416yesdirect product factor
76C2×C32—46no—
77C2×C32—41no—
78C24—33no—
79C24—34no—
80C24—31no—
81C24C3×Dic314yescomplemented
82C24C2×C3×D313yes—
83C22×C4—31no—
84C22×C4—34no—
85C22×C4—96no—
86C22×C4—96no—
87C22×C4—98no—
88C22×C4—96no—
89C22×C4—91no—
90C22×C4—912no—
91C22×C4—96no—
92C22×C3—36no—
93C22×C3—31no—
94C22×C3—36no—
95C22×C3—38no—
96C22×C3—312no—
97C22×C3—36no—
98C22×C3—36no—
99C22×C3—33no—
100C22×C3—31no—
101C22×C3—43no—
102C22×C3—44no—
103C22×C3—83no—
104C22×C3—84no—
105C22×C3C4×A414yescomplemented
106C22×C3C22×C3×C411yescharacteristic, complemented
107C22×C3C22×A413yes—
108A4—21no—
109A4C22×Dic311yescharacteristic, direct product factor
110C3×C4—124no—
111Dic3—34no—
112Dic3C22×A414yesdirect product factor
113C32—41noSylow
114C23—33no—
115C23—312no—
116C23—36no—
117C23—32no—
118C23—312no—
119C23—33no—
120C23—33no—
121C23—34no—
122C23—34no—
123C23C2×C3×Dic316yescomplemented
124C23D3×A411yescharacteristic
125C23C22×C3×D311yescharacteristic
126C2×C4—36no—
127C2×C4—94no—
128C2×C4—912no—
129C2×C4—912no—
130C2×C4—96no—
131C2×C4—94no—
132C2×C3—36no—
133C2×C3—31no—
134C2×C3—31no—
135C2×C3—46no—
136C2×C3—41no—
137C2×C3—86no—
138C2×C3—81no—
139C2×C3C2×C4×A416yescomplemented
140C2×C3C23×A411yescharacteristic
141C22—36no—
142C22—31no—
143C22—36no—
144C22—38no—
145C22—36no—
146C22—312no—
147C22—36no—
148C22—31no—
149C22—33no—
150C22Dic3×A414yesdirect product factor
151C22C22×C3×Dic311yescharacteristic, complemented
152C22C2×D3×A413yes—
153C4—34no—
154C4—94no—
155C3—41no—
156C3—81no—
157C3C22×C4×A411yescharacteristic, complemented
158C2—31no—
159C2—36no—
160C2—31no—
161C2C2×Dic3×A416yesdirect product factor
162C2C22×D3×A411yescharacteristic
subgroupG/H
center, hypercenterC23D3×A4
commutator, hypocenterC22×C3C22×C3×C4
FrattiniC2C22×D3×A4
Fitting, socleC25×C3C2×C3
prime radicalC23×C3×A4C2
cosocleC23×C3C23×C3
LCSG ▷ C22×C3
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×A4C2direct (N × H)
3. C24×Dic3C3semidirect
5. C22×C3×A4C4semidirect
8. Dic3×A4C22direct (N × H)
15. C23×Dic3C2×C3semidirect
18. C2×C3×A4C2×C4semidirect
25. C24×C3C3×C4semidirect
29. C22×A4Dic3direct (N × H)
39. C22×Dic3A4direct (N × H)
40. C22×Dic3C22×C3semidirect
44. C3×A4C22×C4semidirect
62. C23×C3C2×C3×C4semidirect
67. C2×A4C2×Dic3direct (N × H)
75. C2×Dic3C2×A4direct (N × H)
81. C24C3×Dic3semidirect
105. C22×C3C4×A4semidirect
106. C22×C3C22×C3×C4semidirect
109. A4C22×Dic3direct (N × H)
112. Dic3C22×A4direct (N × H)
123. C23C2×C3×Dic3semidirect
139. C2×C3C2×C4×A4semidirect
150. C22Dic3×A4direct (N × H)
151. C22C22×C3×Dic3semidirect
157. C3C22×C4×A4semidirect
161. C2C2×Dic3×A4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×A42
C2×Dic3×A42
C24×Dic33
C22×C4×A43
C22×C3×Dic34