Finite Group Databasebeta

← C22×C32⋊(C42⋊C3) #13152161

Subgroups — C22×C32⋊(C42⋊C3) #13152161

Order 1728.

633 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 633 conjugacy classes of subgroups — too many to draw. Only its 65 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⋊(C42⋊C3)C213yesmaximal, direct product factor
2C22×C32×C42C311yesmaximal, characteristic, complemented
3C22×C3×C42⋊C3C313yesmaximal, complemented
4C22×C22⋊H(3,𝔽3)—41nomaximal
5C32⋊(C42⋊C3)C2211yescharacteristic, direct product factor
6C23×C32×C4—31no—
7C2×C32×C42—33no—
8C2×C32×C42C2×C313yescomplemented
9C2×C3×C42⋊C3C2×C319yescomplemented
10C2×C22⋊H(3,𝔽3)—43no—
11C22×C3×C42—31no—
12C22×C3×C42C3211yescharacteristic
13C22×C42⋊C3—33no—
14C24×C32A411yescharacteristic
15C22×C3×A4—43no—
16C22×C32×C4—36no—
17C22×C32×C4—32no—
18C22×C32×C4—33no—
19C22×C32×C4—33no—
20C32×C42—32no—
21C32×C42—33no—
22C32×C42C22×C311yescharacteristic, complemented
23C3×C42⋊C3C22×C313yescomplemented
24C22×H(3,𝔽3)—161no—
25C22⋊H(3,𝔽3)—41no—
26C23×C3×C4—33no—
27C23×C3×C4—31no—
28C2×C3×C42—33no—
29C2×C3×C42—39no—
30C2×C3×C42—33no—
31C2×C3×C42C2×C3213yes—
32C2×C42⋊C3—39no—
33C23×C32—33no—
34C23×C32—31no—
35C23×C32C2×A413yes—
36C2×C3×A4—49no—
37C2×C32×C4—36no—
38C2×C32×C4—36no—
39C2×C32×C4—36no—
40C2×C32×C4—36no—
41C2×C32×C4—31no—
42C2×C32×C4—33no—
43C22×C42H(3,𝔽3)11yescharacteristic, complemented, Sylow
44C2×H(3,𝔽3)—163no—
45C24×C3—31no—
46C24×C3C3×A411yescharacteristic
47C22×A4—123no—
48C22×C3×C4—39no—
49C22×C3×C4—39no—
50C22×C3×C4—36no—
51C22×C3×C4—318no—
52C22×C3×C4—36no—
53C22×C3×C4—32no—
54C22×C3×C4—33no—
55C22×C3×C4—33no—
56C3×C42—33no—
57C3×C42—39no—
58C3×C42—36no—
59C3×C42—32no—
60C3×C42—31no—
61C3×C42C22×C3211yescharacteristic
62C42⋊C3—33no—
63C22×C32—33no—
64C22×C32—32no—
65C22×C32—33no—
66C22×C32—33no—
67C22×C32—163no—
68C22×C32C42⋊C311yescharacteristic, complemented
69C22×C32C22×A411yescharacteristic
70C3×A4—43no—
71C32×C4—32no—
72C32×C4—36no—
73C23×C4—31no—
74C2×C42—33no—
75C2×C42C2×H(3,𝔽3)13yescomplemented
76H(3,𝔽3)—161noSylow
77C23×C3—33no—
78C23×C3—39no—
79C23×C3—33no—
80C23×C3—31no—
81C23×C3—33no—
82C23×C3C2×C3×A413yes—
83C2×A4—129no—
84C2×C3×C4—33no—
85C2×C3×C4—39no—
86C2×C3×C4—36no—
87C2×C3×C4—36no—
88C2×C3×C4—318no—
89C2×C3×C4—318no—
90C2×C3×C4—318no—
91C2×C3×C4—318no—
92C2×C3×C4—36no—
93C2×C3×C4—36no—
94C2×C3×C4—31no—
95C2×C3×C4—33no—
96C2×C32—31no—
97C2×C32—33no—
98C2×C32—169no—
99C2×C32C2×C42⋊C313yescomplemented
100C24C22⋊H(3,𝔽3)11yescharacteristic
101C22×C4—36no—
102C22×C4—32no—
103C22×C4—33no—
104C22×C4—33no—
105C42—33no—
106C42—32no—
107C42C22×H(3,𝔽3)11yescharacteristic, complemented
108C22×C3—33no—
109C22×C3—39no—
110C22×C3—39no—
111C22×C3—36no—
112C22×C3—32no—
113C22×C3—39no—
114C22×C3—33no—
115C22×C3—33no—
116C22×C3—31no—
117C22×C3—31no—
118C22×C3—483no—
119C22×C3C3×C42⋊C311yescharacteristic
120C22×C3C22×C3×A411yescharacteristic
121A4—123no—
122C3×C4—36no—
123C3×C4—318no—
124C3×C4—32no—
125C3×C4—36no—
126C32—163no—
127C32C22×C42⋊C311yescharacteristic, complemented
128C23—33no—
129C23—31no—
130C23C2×C22⋊H(3,𝔽3)13yes—
131C2×C4—36no—
132C2×C4—36no—
133C2×C4—36no—
134C2×C4—36no—
135C2×C4—31no—
136C2×C4—33no—
137C2×C3—33no—
138C2×C3—39no—
139C2×C3—31no—
140C2×C3—33no—
141C2×C3—33no—
142C2×C3—489no—
143C2×C3C2×C3×C42⋊C313yes—
144C22—33no—
145C22—32no—
146C22—33no—
147C22—33no—
148C22C32⋊(C42⋊C3)11yescharacteristic, direct product factor
149C22C22×C22⋊H(3,𝔽3)11yescharacteristic
150C4—32no—
151C4—36no—
152C3—31no—
153C3—483no—
154C3C22×C3×C42⋊C311yescharacteristic
155C2—31no—
156C2—33no—
157C2C2×C32⋊(C42⋊C3)13yesdirect product factor
subgroupG/H
centerC22×C3C3×C42⋊C3
commutator, cosocleC3×C42C22×C32
FrattiniC22×C3C22×C3×A4
FittingC22×C32×C42C3
socleC24×C3C3×A4
hypercenterC22×C32C42⋊C3
hypocenterC42C22×H(3,𝔽3)
LCSG ▷ C3×C42 ▷ C42
UCSC1 ◁ C22×C3 ◁ C22×C32

Complements

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

NH ≅ G/Ntype
1. C2×C32⋊(C42⋊C3)C2direct (N × H)
2. C22×C32×C42C3semidirect
3. C22×C3×C42⋊C3C3semidirect
5. C32⋊(C42⋊C3)C22direct (N × H)
8. C2×C32×C42C2×C3semidirect
9. C2×C3×C42⋊C3C2×C3semidirect
22. C32×C42C22×C3semidirect
23. C3×C42⋊C3C22×C3semidirect
43. C22×C42H(3,𝔽3)semidirect
68. C22×C32C42⋊C3semidirect
75. C2×C42C2×H(3,𝔽3)semidirect
99. C2×C32C2×C42⋊C3semidirect
107. C42C22×H(3,𝔽3)semidirect
127. C32C22×C42⋊C3semidirect
148. C22C32⋊(C42⋊C3)direct (N × H)
157. C2C2×C32⋊(C42⋊C3)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32⋊(C42⋊C3)2
C22×C32×C423
C22×C3×C42⋊C33
C22×C22⋊H(3,𝔽3)4