Finite Group Databasebeta

← #13137401

Subgroups — #13137401

Order 1728.

471 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#64078⋊C3C212yesmaximal
2C2×#88433C214yesmaximal
3C4×#6180C211yesmaximal, characteristic
4#10590386—32nomaximal
5C2×#634⋊C3C2214yes—
6C42×H(3,𝔽3)C2211yescharacteristic
7C2×#6180C2212yes—
8C4⋊((C3⋊Dic3)⋊C3)—24no—
9#88433—28no—
10#64078D312yes—
11C3×#869—32no—
12C2×#3719—38no—
13C3×C4×C3⋊C8—32no—
14C3×C4×C3⋊C8—61no—
15C2×C4×H(3,𝔽3)D412yes—
16C2×C4×H(3,𝔽3)C2311yescharacteristic
17C2×(C3⋊Dic3)⋊C3—24no—
18#634⋊C3—28no—
19#6180D414yes—
20#5092—92no—
21#4699—91no—
22C2×#634C2×D314yes—
23C2×C3×Dic6—34no—
24C32×C42—21no—
25C32×C42C2×D312yes—
26C4⋊(C3⋊Dic3)—24no—
27C3×C4⋊Dic3—64no—
28C2×C3×C3⋊C8—34no—
29C2×C3×C3⋊C8—62no—
30#3719—616no—
31C22×H(3,𝔽3)C2×D411yescharacteristic
32C4×H(3,𝔽3)Q1614yescomplemented
33C4×H(3,𝔽3)C2×D412yes—
34(C3⋊Dic3)⋊C3—44no—
35C3×C4⋊Q8—92no—
36C2×#216—98no—
37C2×Dic12—94no—
38#869—32no—
39#869—92no—
40C3×C4×C8—91no—
41C4×C3⋊C8—32no—
42C4×C3⋊C8—61no—
43C2×C32×C4—22no—
44C2×C32×C4—21no—
45C2×C32×C4#6714yes—
46C2×C32×C4C22×D312yes—
47C2×C3⋊Dic3—24no—
48#634—28no—
49C2×C3×Dic3—64no—
50C3×Dic6—68no—
51C3×C3⋊C8—38no—
52C3×C3⋊C8—64no—
53#494—271noSylow
54C2×H(3,𝔽3)#12711yescharacteristic
55C2×H(3,𝔽3)C2×Q1612yes—
56C2×C3×Q8—94no—
57C2×Dic6—34no—
58C2×Dic6—94no—
59C2×C3×C8—92no—
60C3×C4⋊C4—184no—
61C3×C42—32no—
62C3×C42—61no—
63C3×C42D3211yescharacteristic
64#216—1816no—
65C4⋊Dic3—64no—
66C4⋊Dic3—184no—
67C2×C3⋊C8—34no—
68C2×C3⋊C8—62no—
69Dic12—188no—
70C22×C32—21no—
71C22×C32C2×#6712yes—
72C32×C4—24no—
73C32×C4—22no—
74C32×C4#21618yescomplemented
75C32×C4D3×D414yes—
76C3⋊Dic3—44no—
77C3×Dic3—124no—
78C2×Q16—274no—
79C4⋊Q8—92no—
80C4×C8—91no—
81H(3,𝔽3)#49411yescharacteristic, complemented, Sylow
82C3×Q8—188no—
83C2×C3×C4—34no—
84C2×C3×C4—32no—
85C2×C3×C4—62no—
86C2×C3×C4—61no—
87C2×C3×C4—184no—
88C2×C3×C4#62512yes—
89C2×C3×C4C2×D3211yescharacteristic
90C2×Dic3—64no—
91C2×Dic3—184no—
92Dic6—68no—
93Dic6—188no—
94C3×C8—94no—
95C3⋊C8—38no—
96C3⋊C8—64no—
97C2×C32—22no—
98C2×C32—21no—
99C2×C32#94012yes—
100C2×C32C2×#21614yes—
101C2×Q8—94no—
102Q16—548no—
103C2×C8—92no—
104C4⋊C4—184no—
105C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
106C22×C3—32no—
107C22×C3—61no—
108C22×C3C2×#62511yescharacteristic
109C3×C4—34no—
110C3×C4—38no—
111C3×C4—62no—
112C3×C4—64no—
113C3×C4—364no—
114C3×C4#371814yes—
115C3×C4#380212yes—
116Dic3—124no—
117Dic3—364no—
118C32—21no—
119C32#509212yescomplemented
120Q8—188no—
121C2×C4—184no—
122C2×C4#619812yes—
123C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
124C8—94no—
125C2×C3—34no—
126C2×C3—32no—
127C2×C3—62no—
128C2×C3—61no—
129C2×C3C2×#371812yes—
130C2×C3#6391411yescharacteristic
131C22C2×#619811yescharacteristic
132C4—364no—
133C4#8843314yescomplemented
134C4#8865512yes—
135C3—32no—
136C3—61no—
137C3#1059038711yescharacteristic
138C2C2×#8843312yes—
139C2#1172114411yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#494
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
32. C4×H(3,𝔽3)Q16semidirect
74. C32×C4#216semidirect
81. H(3,𝔽3)#494semidirect
119. C32#5092semidirect
133. C4#88433semidirect

Maximal subgroups

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

subgroupindex
#64078⋊C32
C2×#884332
C4×#61802
#105903863