Finite Group Databasebeta

← #13137344

Subgroups — #13137344

Order 1728.

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

normal subgroups only

This group has 387 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
2C4×#6180C211yesmaximal, characteristic
3#11720314C214yesmaximal
4#10590303—32nomaximal
5C2×#634⋊C3—22no—
6C42×H(3,𝔽3)C2211yescharacteristic
7C2×#6180C2212yes—
8C4⋊((C3⋊Dic3)⋊C3)—22no—
9C4⋊((C3⋊Dic3)⋊C3)C2214yes—
10#64078D312yes—
11C3×#869—32no—
12C3×C4×C3⋊C8—32no—
13C3×C4×C3⋊C8—61no—
14#63573—38no—
15C2×C4×H(3,𝔽3)D412yes—
16C2×C4×H(3,𝔽3)C2311yescharacteristic
17C2×(C3⋊Dic3)⋊C3—24no—
18#634⋊C3—44no—
19#6180Q814yescomplemented
20#5080—92no—
21#4680—91no—
22C2×#634—22no—
23C2×C3×Dic6—62no—
24C32×C42—21no—
25C32×C42C2×D312yes—
26C4⋊(C3⋊Dic3)—22no—
27C4⋊(C3⋊Dic3)C2×D314yes—
28C3×C4⋊Dic3—34no—
29C3×C4⋊Dic3—62no—
30C2×C3×C3⋊C8—34no—
31C2×C3×C3⋊C8—62no—
32C22×H(3,𝔽3)C2×D411yescharacteristic
33C4×H(3,𝔽3)SD1614yes—
34C4×H(3,𝔽3)C2×Q812yes—
35(C3⋊Dic3)⋊C3—44no—
36C3×C4⋊Q8—92no—
37#869—32no—
38#869—92no—
39C3×C4×C8—91no—
40#817—94no—
41#808—98no—
42C4×C3⋊C8—32no—
43C4×C3⋊C8—61no—
44C2×C32×C4—22no—
45C2×C32×C4—21no—
46C2×C32×C4#6712yes—
47C2×C32×C4#6712yes—
48C2×C32×C4C22×D312yes—
49C2×C3⋊Dic3—24no—
50#634—44no—
51C2×C3×Dic3—64no—
52C3×Dic6—124no—
53C3×C3⋊C8—38no—
54C3×C3⋊C8—64no—
55#498—271noSylow
56C2×H(3,𝔽3)C4⋊Q811yescharacteristic
57C2×H(3,𝔽3)C2×SD1612yes—
58C2×C3×Q8—182no—
59C2×Dic6—62no—
60C2×Dic6—182no—
61C2×C3×C8—92no—
62C3×C4⋊C4—94no—
63C3×C4⋊C4—182no—
64C3×C42—32no—
65C3×C42—61no—
66C3×C42D3211yescharacteristic
67C4⋊Dic3—34no—
68C4⋊Dic3—62no—
69C4⋊Dic3—94no—
70C4⋊Dic3—182no—
71C2×C3⋊C8—34no—
72C2×C3⋊C8—62no—
73C22×C32—21no—
74C22×C32C2×#6712yes—
75C32×C4—24no—
76C32×C4—22no—
77C32×C4#21414yes—
78C32×C4#21514yes—
79C32×C4D3×Q814yes—
80C3⋊Dic3—44no—
81C3×Dic3—124no—
82C4⋊Q8—92no—
83#106—274no—
84C4×C8—91no—
85H(3,𝔽3)#49811yescharacteristic, complemented, Sylow
86C3×Q8—364no—
87C2×C3×C4—32no—
88C2×C3×C4—32no—
89C2×C3×C4—32no—
90C2×C3×C4—62no—
91C2×C3×C4—61no—
92C2×C3×C4—184no—
93C2×C3×C4#62512yes—
94C2×C3×C4C2×D3211yescharacteristic
95C2×Dic3—64no—
96C2×Dic3—184no—
97Dic6—124no—
98Dic6—364no—
99C3×C8—94no—
100C3⋊C8—38no—
101C3⋊C8—64no—
102C2×C32—22no—
103C2×C32—21no—
104C2×C32C2×#21412yes—
105C2×C32C2×#21512yes—
106C2×C32#94412yes—
107C2×Q8—182no—
108C2×C8—92no—
109C4⋊C4—94no—
110C4⋊C4—182no—
111C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
112C22×C3—32no—
113C22×C3—61no—
114C22×C3C2×#62511yescharacteristic
115C3×C4—34no—
116C3×C4—34no—
117C3×C4—34no—
118C3×C4—62no—
119C3×C4—64no—
120C3×C4—364no—
121C3×C4#371514yes—
122C3×C4#379712yes—
123Dic3—124no—
124Dic3—364no—
125C32—21no—
126C32#508012yescomplemented
127Q8—364no—
128C2×C4—184no—
129C2×C4#619812yes—
130C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
131C8—94no—
132C2×C3—32no—
133C2×C3—32no—
134C2×C3—32no—
135C2×C3—62no—
136C2×C3—61no—
137C2×C3C2×#371512yes—
138C2×C3#6387311yescharacteristic
139C22C2×#619811yescharacteristic
140C4—364no—
141C4#8843114yes—
142C4#8865112yes—
143C3—32no—
144C3—61no—
145C3#1059030911yescharacteristic
146C2C2×#8843112yes—
147C2#1172109911yescharacteristic
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)#498
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
19. #6180Q8semidirect
85. H(3,𝔽3)#498semidirect
126. C32#5080semidirect

Maximal subgroups

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

subgroupindex
#64078⋊C32
C4×#61802
#117203142
#105903033