Finite Group Databasebeta

← #13173070

Subgroups — #13173070

Order 1728.

886 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 886 conjugacy classes of subgroups — too many to draw. Only its 191 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
1C22×C32×SL(2,𝔽3)C211yesmaximal, characteristic
2C3⋊(C3⋊#859)C212yesmaximal, complemented
3#10593654—312nomaximal
4#10592924—31nomaximal
5C2×C32×SL(2,𝔽3)C412yescomplemented
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7#88864—41nomaximal
8C22×C32×Q8D311yescharacteristic
9C22×C3×SL(2,𝔽3)D3112yes—
10#64135—31no—
11C2×#3748—31no—
12C3⋊#859—324no—
13#63662—32no—
14#63662—32no—
15C23×C33—41no—
16C2×C3⋊(C3⋊Dic3)—42no—
17C32×SL(2,𝔽3)C2×C411yescharacteristic
18#5420—99no—
19#5225—94no—
20C2×C32×Q8—32no—
21C2×C32×Q8—31no—
22C2×C32×Q8Dic312yescomplemented
23C2×C32×Q8C2×D311yescharacteristic
24C22×C32×C4—31no—
25C2×C3×SL(2,𝔽3)Dic3124yescomplemented
26C2×C3×SL(2,𝔽3)C2×D3112yes—
27#3757—61no—
28#3757—1212no—
29C4⋊(C3⋊Dic3)—32no—
30C4×C3⋊Dic3—61no—
31#3748—62no—
32C2×C3⋊(C3⋊C8)—32no—
33C22×C33—41no—
34C22×C33—42no—
35C22×C33—82no—
36C3⋊(C3⋊Dic3)—82no—
37C22×C3×Q8C3⋊D314yes—
38C22×SL(2,𝔽3)C3⋊D319yes—
39#926—94no—
40C2×#208—94no—
41#859—918no—
42#835—98no—
43#835—98no—
44C23×C32—412no—
45C23×C32S411yescharacteristic
46C32×Q8—32no—
47C32×Q8—31no—
48C32×Q8—61no—
49C32×Q8C2×Dic311yescharacteristic
50C2×C32×C4—32no—
51C2×C32×C4—32no—
52C2×C32×C4—31no—
53C2×C32×C4—31no—
54C2×C3⋊Dic3—62no—
55C2×C3⋊Dic3—1224no—
56C3×SL(2,𝔽3)C2×Dic3112yes—
57C3⋊(C3⋊C8)—62no—
58#419—271noSylow
59C2×C33—42no—
60C2×C33—41no—
61C2×C33—82no—
62C2×C3×Q8—34no—
63C2×C3×Q8—38no—
64C2×C3×Q8C3⋊Dic318yescomplemented
65C2×C3×Q8C2×C3⋊D314yes—
66C22×C3×C4—34no—
67C2×SL(2,𝔽3)C3⋊Dic3118yescomplemented
68C2×SL(2,𝔽3)C2×C3⋊D319yes—
69#217—184no—
70#217—369no—
71C4⋊Dic3—98no—
72C4×Dic3—184no—
73#208—188no—
74C2×C3⋊C8—98no—
75C22×C32—22no—
76C22×C32—412no—
77C22×C32—424no—
78C22×C32—824no—
79C22×C32SL(2,ℤ/4)12yes—
80C22×C32C2×S411yescharacteristic
81C32×C4—31no—
82C32×C4—31no—
83C32×C4—32no—
84C3⋊Dic3—122no—
85C3⋊Dic3—2424no—
86C22×Q8C3⋊(C3⋊D3)11yescharacteristic
87C2×M16—271no—
88#117—271no—
89#103—272no—
90#103—272no—
91C33—41noSylow
92C23×C3—49no—
93C23×C3C3⋊S414yes—
94C3×Q8—38no—
95C3×Q8—34no—
96C3×Q8—64no—
97C3×Q8C2×C3⋊Dic314yes—
98C2×C3×C4—34no—
99C2×C3×C4—34no—
100C2×C3×C4—38no—
101C2×C3×C4—38no—
102C2×Dic3—188no—
103C2×Dic3—3618no—
104SL(2,𝔽3)C2×C3⋊Dic319yes—
105C3⋊C8—188no—
106C2×C32—22no—
107C2×C32—424no—
108C2×C32—412no—
109C2×C32—824no—
110C2×C32#98312yes—
111C2×C32C2×SL(2,ℤ/4)11yescharacteristic
112C2×Q8—31no—
113C2×Q8—32no—
114C2×Q8C3⋊(C3⋊Dic3)12yescomplemented
115C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
116C22×C4—31no—
117M16—542no—
118C2×C8—272no—
119C4⋊C4—272no—
120C22⋊C4—541no—
121C42—541no—
122C22×C3—28no—
123C22×C3—49no—
124C22×C3—418no—
125C22×C3—818no—
126C22×C3C3⋊SL(2,ℤ/4)18yes—
127C22×C3C2×C3⋊S414yes—
128C3×C4—34no—
129C3×C4—34no—
130C3×C4—38no—
131Dic3—368no—
132Dic3—7218no—
133C32—412no—
134C32#542011yescharacteristic, complemented
135C23C3⋊(C3⋊S4)11yescharacteristic
136Q8—32no—
137Q8—31no—
138Q8—61no—
139Q8C2×C3⋊(C3⋊Dic3)11yescharacteristic
140C2×C4—31no—
141C2×C4—31no—
142C2×C4—32no—
143C2×C4—32no—
144C2×C4—542no—
145C8—542no—
146C2×C3—28no—
147C2×C3—418no—
148C2×C3—49no—
149C2×C3—818no—
150C2×C3#6426418yes—
151C2×C3C2×C3⋊SL(2,ℤ/4)14yes—
152C22—22no—
153C22C3⋊(C3⋊SL(2,ℤ/4))12yes—
154C22C2×C3⋊(C3⋊S4)11yescharacteristic
155C4—31no—
156C4—31no—
157C4—32no—
158C4—1082no—
159C3—49no—
160C3#1059365414yescomplemented
161C2—22no—
162C2#1172446812yes—
163C2C2×C3⋊(C3⋊SL(2,ℤ/4))11yescharacteristic
subgroupG/H
center, FrattiniC22C2×C3⋊(C3⋊S4)
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC22×C32×Q8D3
socleC22×C32C2×S4
prime radicalC22×C32×SL(2,𝔽3)C2
cosocleC2×C32×SL(2,𝔽3)C22
hypercenterC23C3⋊(C3⋊S4)
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C3⋊(C3⋊#859)C2semidirect
5. C2×C32×SL(2,𝔽3)C4semidirect
22. C2×C32×Q8Dic3semidirect
25. C2×C3×SL(2,𝔽3)Dic3semidirect
64. C2×C3×Q8C3⋊Dic3semidirect
67. C2×SL(2,𝔽3)C3⋊Dic3semidirect
114. C2×Q8C3⋊(C3⋊Dic3)semidirect
134. C32#5420semidirect
160. C3#10593654semidirect

Maximal subgroups

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

subgroupindex
C22×C32×SL(2,𝔽3)2
C3⋊(C3⋊#859)2
#105936543
#105929243
#888644