Finite Group Databasebeta

← C32×#5930 #13181414

Subgroups — C32×#5930 #13181414

Order 1728.

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

normal subgroups only

This group has 456 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
1C32×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic
2C3×#5930C314yesmaximal, direct product factor
3C32×#451—31nomaximal
4C32×SL(2,ℤ/4)—41nomaximal
5C32×#226—42nomaximal
6C22×C32×Q8D311yescharacteristic
7C3×C22⋊SL(2,𝔽3)—24no—
8C3×C22⋊SL(2,𝔽3)C2×C314yes—
9C2×C32×Q16—32no—
10C32×C22⋊Q8—31no—
11C32×#103—32no—
12C32×C22⋊C8—31no—
13C2×C32×A4—41no—
14C32×SL(2,𝔽3)—42no—
15C32×SL(2,𝔽3)—81no—
16#5930C3211yescharacteristic, direct product factor
17C3×#451—34no—
18C2×C32×Q8—32no—
19C2×C32×Q8—31no—
20C2×C32×Q8—61no—
21C22×C32×C4—31no—
22C3×SL(2,ℤ/4)—44no—
23C3×#226—48no—
24C32×Q16—62no—
25C32×Q16—62no—
26C2×C32×C8—32no—
27C32×C4⋊C4—31no—
28C32×C4⋊C4—61no—
29C32×C22⋊C4—61no—
30C32×A4—41no—
31C32×Dic3—161no—
32C22×C3×Q8C3×D314yes—
33C22⋊SL(2,𝔽3)—24no—
34C22⋊SL(2,𝔽3)C2×C3211yescharacteristic
35C2×C3×Q16—38no—
36C3×C22⋊Q8—34no—
37C3×#103—38no—
38C3×C22⋊C8—34no—
39C23×C32S411yescharacteristic
40C2×C3×A4—44no—
41C2×C3×A4—84no—
42C32×Q8—21no—
43C32×Q8—32no—
44C32×Q8—61no—
45C32×Q8—62no—
46C32×Q8S412yes—
47C2×C32×C4—31no—
48C2×C32×C4—31no—
49C2×C32×C4—62no—
50C2×C32×C4—61no—
51C2×C32×C4—61no—
52C3×SL(2,𝔽3)—48no—
53C3×SL(2,𝔽3)—84no—
54C3×SL(2,𝔽3)—88no—
55C3×SL(2,𝔽3)—88no—
56C32×C8—62no—
57#451—31noSylow
58C2×C33—161no—
59C2×C3×Q8—38no—
60C2×C3×Q8—34no—
61C2×C3×Q8—64no—
62C22×C3×C4—34no—
63SL(2,ℤ/4)—41no—
64#226—42no—
65C3×Q16—68no—
66C3×Q16—68no—
67C2×C3×C8—38no—
68C3×C4⋊C4—34no—
69C3×C4⋊C4—64no—
70C3×C22⋊C4—64no—
71C22×C32—31no—
72C22×C32—31no—
73C22×C32#22611yescharacteristic, complemented
74C3×A4—44no—
75C3×A4—84no—
76C32×C4—32no—
77C32×C4—61no—
78C32×C4—121no—
79C32×C4—121no—
80C3×Dic3—164no—
81C22×Q8C32×D311yescharacteristic
82C2×Q16—32no—
83C22⋊Q8—31no—
84#103—32no—
85C22⋊C8—31no—
86C33—161noSylow
87C23×C3C3×S414yes—
88C2×A4—41no—
89C2×A4—84no—
90C3×Q8—24no—
91C3×Q8—38no—
92C3×Q8—64no—
93C3×Q8—68no—
94C3×Q8C3×S418yes—
95C2×C3×C4—34no—
96C2×C3×C4—34no—
97C2×C3×C4—68no—
98C2×C3×C4—64no—
99C2×C3×C4—64no—
100SL(2,𝔽3)—42no—
101SL(2,𝔽3)—81no—
102SL(2,𝔽3)—88no—
103SL(2,𝔽3)—88no—
104C3×C8—68no—
105C2×C32—31no—
106C2×C32—31no—
107C2×C32—164no—
108C2×C32—324no—
109C2×C32#102011yescharacteristic
110C2×Q8—32no—
111C2×Q8—31no—
112C2×Q8—61no—
113C22×C4—31no—
114Q16—62no—
115Q16—62no—
116C2×C8—32no—
117C4⋊C4—31no—
118C4⋊C4—61no—
119C22⋊C4—61no—
120C22×C3—34no—
121C22×C3—34no—
122C22×C3C3×#22614yescomplemented
123A4—41no—
124A4—84no—
125C3×C4—38no—
126C3×C4—64no—
127C3×C4—124no—
128C3×C4—124no—
129Dic3—161no—
130C32—164no—
131C32—324no—
132C32#593011yescharacteristic, direct product factor
133C23C32×S411yescharacteristic
134Q8—21no—
135Q8—32no—
136Q8—61no—
137Q8—62no—
138Q8C32×S412yes—
139C2×C4—31no—
140C2×C4—31no—
141C2×C4—62no—
142C2×C4—61no—
143C2×C4—61no—
144C8—62no—
145C2×C3—34no—
146C2×C3—34no—
147C2×C3—161no—
148C2×C3—324no—
149C2×C3C3×#102014yes—
150C22—31no—
151C22—31no—
152C22C32×#22611yescharacteristic, complemented
153C4—32no—
154C4—61no—
155C4—121no—
156C4—121no—
157C3—161no—
158C3—324no—
159C3C3×#593014yesdirect product factor
160C2—31no—
161C2—31no—
162C2C32×#102011yescharacteristic
subgroupG/H
center, hypercenterC2×C32#1020
commutator, cosocle, hypocenterC22⋊SL(2,𝔽3)C2×C32
FrattiniC2C32×#1020
FittingC22×C32×Q8D3
socleC23×C32S4
prime radicalC32×C22⋊SL(2,𝔽3)C2
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
2. C3×#5930C3direct (N × H)
16. #5930C32direct (N × H)
73. C22×C32#226semidirect
122. C22×C3C3×#226semidirect
132. C32#5930direct (N × H)
152. C22C32×#226semidirect
159. C3C3×#5930direct (N × H)

Maximal subgroups

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

subgroupindex
C32×C22⋊SL(2,𝔽3)2
C3×#59303
C32×#4513
C32×SL(2,ℤ/4)4
C32×#2264