Finite Group Databasebeta

← C32×C4⋊#226 #13172944

Subgroups — C32×C4⋊#226 #13172944

Order 1728.

490 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 102 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×#226C212yesmaximal
2C32×C4×SL(2,𝔽3)C211yesmaximal, characteristic
3C3×C4⋊#226C314yesmaximal, direct product factor
4C32×#462—31nomaximal
5C2×C32×SL(2,𝔽3)C2211yescharacteristic
6C32×#226—22no—
7C32×C4⋊Dic3—41nomaximal
8C2×C3×#226C2×C318yes—
9C2×C32×Q16—32no—
10C32×C4⋊Q8—31no—
11C32×C4×Q8D311yescharacteristic
12C3×C4×SL(2,𝔽3)—24no—
13C3×C4×SL(2,𝔽3)C2×C314yes—
14C32×C4⋊C8—31no—
15C32×#103—32no—
16C2×C33×C4—41no—
17C2×C32×Dic3—42no—
18C32×SL(2,𝔽3)D411yescharacteristic
19C4⋊#226C3211yescharacteristic, direct product factor
20C3×#462—34no—
21C2×C32×Q8—32no—
22C2×C32×Q8C2×D311yescharacteristic
23C2×C3×SL(2,𝔽3)—24no—
24C2×C3×SL(2,𝔽3)C22×C314yes—
25C3×#226—28no—
26C32×Q16—62no—
27C32×Q16—62no—
28C2×C32×C8—32no—
29C32×C4⋊C4—31no—
30C32×C4⋊C4—61no—
31C32×C4⋊C4—61no—
32C32×C42—31no—
33C3×C4⋊Dic3—44no—
34C22×C33—41no—
35C33×C4—42no—
36C32×Dic3—82no—
37C2×#226C2×C3212yes—
38C2×C3×Q16—38no—
39C3×C4⋊Q8—34no—
40C3×C4×Q8C3×D314yes—
41C4×SL(2,𝔽3)—24no—
42C4×SL(2,𝔽3)C2×C3211yescharacteristic
43C3×C4⋊C8—34no—
44C3×#103—38no—
45C32×Q8—31no—
46C32×Q8—62no—
47C32×Q8—62no—
48C32×Q8D1211yescharacteristic
49C2×C32×C4—31no—
50C2×C32×C4—31no—
51C2×C32×C4—44no—
52C2×C32×C4—62no—
53C2×C32×C4—84no—
54C2×C32×C4S411yescharacteristic
55C2×C3×Dic3—48no—
56C3×SL(2,𝔽3)—24no—
57C3×SL(2,𝔽3)C3×D414yes—
58C32×C8—62no—
59#462—31noSylow
60C2×C33—41no—
61C2×C33—41no—
62C2×C33—41no—
63C2×C3×Q8—38no—
64C2×C3×Q8C2×C3×D314yes—
65C2×SL(2,𝔽3)—24no—
66C2×SL(2,𝔽3)C22×C3211yescharacteristic
67#226—22no—
68C3×Q16—68no—
69C3×Q16—68no—
70C2×C3×C8—38no—
71C3×C4⋊C4—34no—
72C3×C4⋊C4—64no—
73C3×C4⋊C4—64no—
74C3×C42—34no—
75C4⋊Dic3—41no—
76C22×C32—44no—
77C22×C32—84no—
78C22×C32C2×S411yescharacteristic
79C32×C4—31no—
80C32×C4—31no—
81C32×C4—48no—
82C32×C4—61no—
83C32×C4—88no—
84C32×C4—122no—
85C32×C4#22612yescomplemented
86C3×Dic3—88no—
87C2×Q16—32no—
88C4⋊Q8—31no—
89C4×Q8C32×D311yescharacteristic
90C4⋊C8—31no—
91#103—32no—
92C33—41noSylow
93C3×Q8—34no—
94C3×Q8—68no—
95C3×Q8—68no—
96C3×Q8C3×D1214yes—
97C2×C3×C4—34no—
98C2×C3×C4—34no—
99C2×C3×C4—41no—
100C2×C3×C4—68no—
101C2×C3×C4—84no—
102C2×C3×C4C3×S414yes—
103C2×Dic3—42no—
104SL(2,𝔽3)—24no—
105SL(2,𝔽3)C32×D411yescharacteristic
106C3×C8—68no—
107C2×C32—44no—
108C2×C32—44no—
109C2×C32—44no—
110C2×C32—84no—
111C2×C32—84no—
112C2×C32—84no—
113C2×C32C4⋊S411yescharacteristic
114C2×C32C2×#22611yescharacteristic
115C2×C32#98411yescharacteristic
116C2×Q8—32no—
117C2×Q8C2×C32×D311yescharacteristic
118Q16—62no—
119Q16—62no—
120C2×C8—32no—
121C4⋊C4—31no—
122C4⋊C4—61no—
123C4⋊C4—61no—
124C42—31no—
125C22×C3—41no—
126C22×C3—84no—
127C22×C3C2×C3×S414yes—
128C3×C4—34no—
129C3×C4—34no—
130C3×C4—42no—
131C3×C4—64no—
132C3×C4—88no—
133C3×C4—128no—
134C3×C4C3×#22618yescomplemented
135Dic3—82no—
136C32—44no—
137C32—84no—
138C32C4⋊#22611yescharacteristic, direct product factor
139Q8—31no—
140Q8—62no—
141Q8—62no—
142Q8C32×D1211yescharacteristic
143C2×C4—31no—
144C2×C4—31no—
145C2×C4—62no—
146C2×C4C32×S411yescharacteristic
147C8—62no—
148C2×C3—41no—
149C2×C3—41no—
150C2×C3—41no—
151C2×C3—84no—
152C2×C3—84no—
153C2×C3—84no—
154C2×C3C3×C4⋊S414yes—
155C2×C3C2×C3×#22614yes—
156C2×C3C3×#98414yes—
157C22C2×C32×S411yescharacteristic
158C4—31no—
159C4—31no—
160C4—61no—
161C4—122no—
162C4C32×#22612yescomplemented
163C3—41no—
164C3—84no—
165C3C3×C4⋊#22614yesdirect product factor
166C2C32×C4⋊S411yescharacteristic
167C2C2×C32×#22611yescharacteristic
168C2C32×#98411yescharacteristic
subgroupG/H
center, socleC22×C32C2×S4
commutator, cosocleC2×SL(2,𝔽3)C22×C32
FrattiniC22C2×C32×S4
FittingC32×C4×Q8D3
prime radicalC2×C32×SL(2,𝔽3)C22
hypercenterC2×C32×C4S4
hypocenterSL(2,𝔽3)C32×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C32 ◁ C2×C32×C4

Complements

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

NH ≅ G/Ntype
3. C3×C4⋊#226C3direct (N × H)
19. C4⋊#226C32direct (N × H)
85. C32×C4#226semidirect
134. C3×C4C3×#226semidirect
138. C32C4⋊#226direct (N × H)
162. C4C32×#226semidirect
165. C3C3×C4⋊#226direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#2262
C32×C4×SL(2,𝔽3)2
C3×C4⋊#2263
C32×#4623
C32×C4⋊Dic34