Finite Group Databasebeta

← #13173074

Subgroups — #13173074

Order 1728.

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

normal subgroups only

This group has 842 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
1C2×C32×#41⋊C3C211yesmaximal, characteristic
2C3⋊(C3⋊#859)C212yesmaximal, complemented
3#10593658—312nomaximal
4#10592932—31nomaximal
5C32×#41⋊C3C412yescomplemented
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C4×C3⋊(C3⋊Dic3)—41nomaximal
8C2×C32×#41D311yescharacteristic
9C2×C3×#41⋊C3D3112yes—
10#64135—31no—
11C22×C3⋊(C3⋊C8)—31no—
12C3⋊#859—324no—
13#63662—32no—
14#63659—32no—
15C2×C33×C4—41no—
16C2×C3⋊(C3⋊Dic3)—42no—
17C32×SL(2,𝔽3)C2×C411yescharacteristic
18#5424—99no—
19#5233—94no—
20C32×#41—32no—
21C32×#41Dic312yescomplemented
22C2×C32×Q8C2×D311yescharacteristic
23C2×C32×D4—31no—
24C22×C32×C4—31no—
25C3×#41⋊C3Dic3124yescomplemented
26C2×C3×SL(2,𝔽3)C2×D3112yes—
27#3757—61no—
28C4⋊(C3⋊Dic3)—32no—
29C4×C3⋊Dic3—61no—
30C4×C3⋊Dic3—1212no—
31C2×C3⋊(C3⋊C8)—32no—
32C2×C3⋊(C3⋊C8)—62no—
33C22×C33—41no—
34C33×C4—42no—
35C3⋊(C3⋊Dic3)—44no—
36C2×C3×#41C3⋊D314yes—
37C2×#41⋊C3C3⋊D319yes—
38#926—94no—
39C22×C3⋊C8—94no—
40#859—918no—
41#835—98no—
42#832—98no—
43C23×C32—31no—
44C32×Q8—31no—
45C32×Q8C2×Dic311yescharacteristic
46C32×D4—32no—
47C32×D4—61no—
48C2×C32×C4—31no—
49C2×C32×C4—32no—
50C2×C32×C4—32no—
51C2×C32×C4—412no—
52C2×C32×C4S411yescharacteristic
53C2×C3⋊Dic3—62no—
54C2×C3⋊Dic3—1224no—
55C3×SL(2,𝔽3)C2×Dic3112yes—
56C3⋊(C3⋊C8)—62no—
57#416—271noSylow
58C2×C33—42no—
59C2×C33—41no—
60C3×#41—38no—
61C3×#41C3⋊Dic318yescomplemented
62C2×C3×Q8C2×C3⋊D314yes—
63C2×C3×D4—34no—
64C22×C3×C4—34no—
65#41⋊C3C3⋊Dic3118yescomplemented
66C2×SL(2,𝔽3)C2×C3⋊D319yes—
67#217—184no—
68C4⋊Dic3—98no—
69C4×Dic3—184no—
70C4×Dic3—369no—
71C2×C3⋊C8—98no—
72C2×C3⋊C8—188no—
73C22×C32—32no—
74C22×C32—412no—
75C22×C32—62no—
76C22×C32C2×S411yescharacteristic
77C32×C4—31no—
78C32×C4—31no—
79C32×C4—424no—
80C32×C4SL(2,ℤ/4)12yes—
81C3⋊Dic3—1248no—
82C3⋊Dic3—122no—
83C2×#41C3⋊(C3⋊D3)11yescharacteristic
84C22×C8—271no—
85#117—271no—
86#103—272no—
87#102—272no—
88C33—41noSylow
89C23×C3—34no—
90C3×Q8—34no—
91C3×Q8C2×C3⋊Dic314yes—
92C3×D4—38no—
93C3×D4—64no—
94C2×C3×C4—34no—
95C2×C3×C4—38no—
96C2×C3×C4—38no—
97C2×C3×C4—49no—
98C2×C3×C4C3⋊S414yes—
99C2×Dic3—188no—
100C2×Dic3—3618no—
101SL(2,𝔽3)C2×C3⋊Dic319yes—
102C3⋊C8—188no—
103C2×C32—424no—
104C2×C32—412no—
105C2×C32—62no—
106C2×C32#98512yes—
107C2×C32C2×SL(2,ℤ/4)11yescharacteristic
108#41—32no—
109#41C3⋊(C3⋊Dic3)12yescomplemented
110C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
111C2×D4—31no—
112C22×C4—31no—
113C2×C8—272no—
114C2×C8—542no—
115C4⋊C4—272no—
116C22⋊C4—541no—
117C42—541no—
118C22×C3—38no—
119C22×C3—49no—
120C22×C3—68no—
121C22×C3C2×C3⋊S414yes—
122C3×C4—34no—
123C3×C4—34no—
124C3×C4—418no—
125C3×C4C3⋊SL(2,ℤ/4)18yes—
126Dic3—3636no—
127Dic3—368no—
128C32—412no—
129C32#542411yescharacteristic, complemented
130C23—31no—
131Q8—31no—
132Q8C2×C3⋊(C3⋊Dic3)11yescharacteristic
133D4—32no—
134D4—61no—
135C2×C4—31no—
136C2×C4—32no—
137C2×C4—32no—
138C2×C4—542no—
139C2×C4C3⋊(C3⋊S4)11yescharacteristic
140C8—542no—
141C2×C3—418no—
142C2×C3—49no—
143C2×C3—68no—
144C2×C3#6426618yes—
145C2×C3C2×C3⋊SL(2,ℤ/4)14yes—
146C22—32no—
147C22—62no—
148C22C2×C3⋊(C3⋊S4)11yescharacteristic
149C4—31no—
150C4—31no—
151C4—1082no—
152C4C3⋊(C3⋊SL(2,ℤ/4))12yes—
153C3—49no—
154C3#1059365814yescomplemented
155C2—62no—
156C2#1172447012yes—
157C2C2×C3⋊(C3⋊SL(2,ℤ/4))11yescharacteristic
subgroupG/H
center, hypercenterC2×C4C3⋊(C3⋊S4)
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FrattiniC22C2×C3⋊(C3⋊S4)
FittingC2×C32×#41D3
socleC22×C32C2×S4
prime radicalC2×C32×#41⋊C3C2
cosocleC2×C32×SL(2,𝔽3)C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3⋊(C3⋊#859)C2semidirect
5. C32×#41⋊C3C4semidirect
21. C32×#41Dic3semidirect
25. C3×#41⋊C3Dic3semidirect
61. C3×#41C3⋊Dic3semidirect
65. #41⋊C3C3⋊Dic3semidirect
109. #41C3⋊(C3⋊Dic3)semidirect
129. C32#5424semidirect
154. C3#10593658semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C3⋊(C3⋊#859)2
#105936583
#105929323
C4×C3⋊(C3⋊Dic3)4