Finite Group Databasebeta

← C4×C32⋊(C42⋊C3) #13141219

Subgroups — C4×C32⋊(C42⋊C3) #13141219

Order 1728.

339 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 39 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)C211yesmaximal, characteristic
2C32×C43C311yesmaximal, characteristic, complemented
3C3×C4×C42⋊C3C313yesmaximal, complemented
4C4×C22⋊H(3,𝔽3)—41nomaximal
5C32⋊(C42⋊C3)C411yescharacteristic, direct product factor
6C2×C32×C42—31no—
7C2×C32×C42—31no—
8C2×C32×C42C2×C311yescharacteristic
9C2×C3×C42⋊C3C2×C313yes—
10C2×C22⋊H(3,𝔽3)—41no—
11C3×C43—31no—
12C3×C43C3211yescharacteristic
13C4×C42⋊C3—33no—
14C22×C32×C4—31no—
15C22×C32×C4—31no—
16C22×C32×C4A411yescharacteristic
17C3×C4×A4—43no—
18C32×C42—34no—
19C32×C42—32no—
20C32×C42—32no—
21C32×C42—31no—
22C32×C42C3×C411yescharacteristic, complemented
23C3×C42⋊C3C3×C413yescomplemented
24C22⋊H(3,𝔽3)—41no—
25C4×H(3,𝔽3)—161no—
26C2×C3×C42—31no—
27C2×C3×C42—33no—
28C2×C3×C42—33no—
29C2×C3×C42—31no—
30C2×C3×C42—31no—
31C2×C3×C42C2×C3211yescharacteristic
32C2×C42⋊C3—33no—
33C23×C32C2×A411yescharacteristic
34C2×C3×A4—43no—
35C2×C32×C4—34no—
36C2×C32×C4—32no—
37C2×C32×C4—32no—
38C2×C32×C4—32no—
39C2×C32×C4—31no—
40C2×C32×C4—31no—
41C2×C32×C4—31no—
42C2×C32×C4—31no—
43C43H(3,𝔽3)11yescharacteristic, complemented, Sylow
44C2×H(3,𝔽3)—161no—
45C22×C3×C4—33no—
46C22×C3×C4—33no—
47C22×C3×C4—31no—
48C22×C3×C4—31no—
49C22×C3×C4—31no—
50C22×C3×C4C3×A411yescharacteristic
51C4×A4—123no—
52C3×C42—31no—
53C3×C42—33no—
54C3×C42—34no—
55C3×C42—312no—
56C3×C42—36no—
57C3×C42—36no—
58C3×C42—32no—
59C3×C42—32no—
60C3×C42—31no—
61C3×C42C32×C411yescharacteristic
62C42⋊C3—33no—
63C22×C32—31no—
64C22×C32—31no—
65C22×C32C4×A411yescharacteristic
66C3×A4—43no—
67C32×C4—34no—
68C32×C4—32no—
69C32×C4—32no—
70C32×C4—31no—
71C32×C4—163no—
72C32×C4C42⋊C311yescharacteristic, complemented
73C2×C42—31no—
74C2×C42—31no—
75C2×C42C2×H(3,𝔽3)11yescharacteristic
76H(3,𝔽3)—161noSylow
77C23×C3—31no—
78C23×C3C2×C3×A411yescharacteristic
79C2×A4—123no—
80C2×C3×C4—32no—
81C2×C3×C4—34no—
82C2×C3×C4—36no—
83C2×C3×C4—312no—
84C2×C3×C4—36no—
85C2×C3×C4—36no—
86C2×C3×C4—32no—
87C2×C3×C4—32no—
88C2×C3×C4—33no—
89C2×C3×C4—33no—
90C2×C3×C4—33no—
91C2×C3×C4—33no—
92C2×C3×C4—31no—
93C2×C3×C4—31no—
94C2×C3×C4—31no—
95C2×C3×C4—31no—
96C2×C32—31no—
97C2×C32—31no—
98C2×C32—163no—
99C2×C32C2×C42⋊C311yescharacteristic
100C22×C4—31no—
101C22×C4—31no—
102C22×C4C22⋊H(3,𝔽3)11yescharacteristic
103C42—31no—
104C42—34no—
105C42—32no—
106C42—32no—
107C42C4×H(3,𝔽3)11yescharacteristic, complemented
108C22×C3—31no—
109C22×C3—33no—
110C22×C3—33no—
111C22×C3—31no—
112C22×C3—31no—
113C22×C3C3×C4×A411yescharacteristic
114A4—123no—
115C3×C4—36no—
116C3×C4—36no—
117C3×C4—312no—
118C3×C4—32no—
119C3×C4—32no—
120C3×C4—34no—
121C3×C4—33no—
122C3×C4—31no—
123C3×C4—31no—
124C3×C4—483no—
125C3×C4C3×C42⋊C311yescharacteristic
126C32—163no—
127C32C4×C42⋊C311yescharacteristic, complemented
128C23C2×C22⋊H(3,𝔽3)11yescharacteristic
129C2×C4—32no—
130C2×C4—34no—
131C2×C4—32no—
132C2×C4—32no—
133C2×C4—31no—
134C2×C4—31no—
135C2×C4—31no—
136C2×C4—31no—
137C2×C3—33no—
138C2×C3—33no—
139C2×C3—31no—
140C2×C3—31no—
141C2×C3—31no—
142C2×C3—483no—
143C2×C3C2×C3×C42⋊C311yescharacteristic
144C22—31no—
145C22—31no—
146C22C4×C22⋊H(3,𝔽3)11yescharacteristic
147C4—32no—
148C4—32no—
149C4—34no—
150C4—31no—
151C4C32⋊(C42⋊C3)11yescharacteristic, direct product factor
152C3—31no—
153C3—483no—
154C3C3×C4×C42⋊C311yescharacteristic
155C2—31no—
156C2—31no—
157C2C2×C32⋊(C42⋊C3)11yescharacteristic
subgroupG/H
centerC3×C4C3×C42⋊C3
commutatorC3×C42C32×C4
Frattini, socleC23×C3C2×C3×A4
FittingC32×C43C3
prime radicalC2×C32⋊(C42⋊C3)C2
cosocleC2×C3×C42C2×C32
hypercenterC32×C4C42⋊C3
hypocenterC42C4×H(3,𝔽3)
LCSG ▷ C3×C42 ▷ C42
UCSC1 ◁ C3×C4 ◁ C32×C4

Complements

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

NH ≅ G/Ntype
2. C32×C43C3semidirect
3. C3×C4×C42⋊C3C3semidirect
5. C32⋊(C42⋊C3)C4direct (N × H)
22. C32×C42C3×C4semidirect
23. C3×C42⋊C3C3×C4semidirect
43. C43H(3,𝔽3)semidirect
72. C32×C4C42⋊C3semidirect
107. C42C4×H(3,𝔽3)semidirect
127. C32C4×C42⋊C3semidirect
151. C4C32⋊(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
C32×C433
C3×C4×C42⋊C33
C4×C22⋊H(3,𝔽3)4