Finite Group Databasebeta

← (C3⋊Dic3)⋊(C22⋊A4) #13164664

Subgroups — (C3⋊Dic3)⋊(C22⋊A4) #13164664

Order 1728.

1495 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 1495 conjugacy classes of subgroups — too many to draw. Only its 54 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⋊A4)C211yesmaximal, characteristic
2C24×C3⋊Dic3C311yesmaximal, characteristic, complemented
3Dic3×C22⋊A4—31nomaximal
4C32⋊(C22⋊A4)C411yescharacteristic, complemented
5(C3⋊Dic3)⋊A4—45nomaximal
6C25×C32C2×C311yescharacteristic
7C2×C3×C22⋊A4—21no—
8C2×C3×C22⋊A4D311yescharacteristic
9C23×C3⋊Dic3—35no—
10C23×C3⋊Dic3—35no—
11C2×C22⋊H(3,𝔽3)—45no—
12C24×Dic3—31no—
13C24×Dic3—91no—
14C4×C22⋊A4—91no—
15C24×C32—35no—
16C24×C32—35no—
17C24×C32C3×C411yescharacteristic, complemented
18C3×C22⋊A4—21no—
19C3×C22⋊A4Dic311yescharacteristic, complemented
20C22×C3⋊Dic3—330no—
21C22×C3⋊Dic3—310no—
22C22×C3⋊Dic3—35no—
23C22×C3⋊Dic3A415yescomplemented
24Dic3×A4—125no—
25C22⋊H(3,𝔽3)—45no—
26(C3⋊Dic3)⋊C3—161no—
27C25×C3—31no—
28C25×C3C3×D311yescharacteristic
29C2×C22⋊A4—31no—
30C2×C22⋊A4—61no—
31C23×Dic3—95no—
32C23×Dic3—915no—
33C23×Dic3—915no—
34C23×Dic3—95no—
35C23×C32—330no—
36C23×C32—310no—
37C23×C32—35no—
38C23×C32—35no—
39C23×C32C2×A415yes—
40C2×C3×A4—45no—
41C2×C3×A4—85no—
42C2×C3⋊Dic3—330no—
43C2×C3⋊Dic3—35no—
44C2×C3⋊Dic3—35no—
45C24×C4—91noSylow
46C2×H(3,𝔽3)—161no—
47C24×C3—35no—
48C24×C3—315no—
49C24×C3—315no—
50C24×C3—35no—
51C24×C3—31no—
52C24×C3C3×Dic311yescharacteristic
53C22⋊A4—31no—
54C22⋊A4—61no—
55C22×Dic3—35no—
56C22×Dic3—95no—
57C22×Dic3—910no—
58C22×Dic3—930no—
59C22×Dic3—990no—
60C22×Dic3—930no—
61C22×Dic3—915no—
62C22×Dic3—95no—
63C4×A4—365no—
64C22×C32—35no—
65C22×C32—330no—
66C22×C32—310no—
67C22×C32—35no—
68C22×C32C4×A415yescomplemented
69C3×A4—45no—
70C3×A4—85no—
71C3⋊Dic3—35no—
72C3⋊Dic3C22⋊A411yescharacteristic, complemented
73C3×Dic3—481no—
74C25(C3⋊D3)⋊C311yescharacteristic
75C23×C4—275no—
76C23×C4—275no—
77H(3,𝔽3)—161noSylow
78C23×C3—330no—
79C23×C3—310no—
80C23×C3—330no—
81C23×C3—390no—
82C23×C3—315no—
83C23×C3—315no—
84C23×C3—35no—
85C23×C3—35no—
86C23×C3—35no—
87C23×C3D3×A415yes—
88C2×A4—125no—
89C2×A4—245no—
90C2×Dic3—95no—
91C2×Dic3—915no—
92C2×Dic3—990no—
93C2×Dic3—930no—
94C2×Dic3—915no—
95C2×Dic3—95no—
96C2×C32—35no—
97C2×C32—35no—
98C2×C32—161no—
99C2×C32—321no—
100C2×C32C2×C22⋊A411yescharacteristic
101C24—35no—
102C24—35no—
103C24(C3⋊Dic3)⋊C311yescharacteristic, complemented
104C22×C4—95no—
105C22×C4—2710no—
106C22×C4—2730no—
107C22×C4—275no—
108C22×C3—35no—
109C22×C3—315no—
110C22×C3—315no—
111C22×C3—330no—
112C22×C3—390no—
113C22×C3—310no—
114C22×C3—330no—
115C22×C3—35no—
116C22×C3—35no—
117C22×C3Dic3×A415yes—
118A4—125no—
119A4—245no—
120C3×C4—1441no—
121Dic3—31no—
122Dic3—91no—
123Dic3—915no—
124Dic3—95no—
125C32—161no—
126C32—321no—
127C32C4×C22⋊A411yescharacteristic, complemented
128C23—310no—
129C23—330no—
130C23—35no—
131C23—35no—
132C23(C3⋊D3)⋊A415yes—
133C2×C4—275no—
134C2×C4—2730no—
135C2×C4—275no—
136C2×C3—315no—
137C2×C3—315no—
138C2×C3—35no—
139C2×C3—35no—
140C2×C3—31no—
141C2×C3—481no—
142C2×C3—961no—
143C2×C3D3×C22⋊A411yescharacteristic
144C22—35no—
145C22—310no—
146C22—330no—
147C22—35no—
148C22(C3⋊Dic3)⋊A415yescomplemented
149C4—91no—
150C4—275no—
151C3—31no—
152C3—481no—
153C3—961no—
154C3Dic3×C22⋊A411yescharacteristic
155C2—35no—
156C2—35no—
157C2(C3⋊D3)⋊(C22⋊A4)11yescharacteristic
subgroupG/H
center, hypercenterC2(C3⋊D3)⋊(C22⋊A4)
commutator, hypocenterC24×C32C3×C4
FrattiniC2×C3D3×C22⋊A4
Fitting, cosocleC25×C32C2×C3
socleC25×C3C3×D3
prime radicalC2×C32⋊(C22⋊A4)C2
LCSG ▷ C24×C32
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. C24×C3⋊Dic3C3semidirect
4. C32⋊(C22⋊A4)C4semidirect
17. C24×C32C3×C4semidirect
19. C3×C22⋊A4Dic3semidirect
23. C22×C3⋊Dic3A4semidirect
68. C22×C32C4×A4semidirect
72. C3⋊Dic3C22⋊A4semidirect
103. C24(C3⋊Dic3)⋊C3semidirect
127. C32C4×C22⋊A4semidirect
148. C22(C3⋊Dic3)⋊A4semidirect

Maximal subgroups

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

subgroupindex
C2×C32⋊(C22⋊A4)2
C24×C3⋊Dic33
Dic3×C22⋊A43
(C3⋊Dic3)⋊A44