Finite Group Databasebeta

← #13135805

Subgroups — #13135805

Order 1728.

319 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 75 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#11721779C211yesmaximal, characteristic
2H(3,𝔽3)×#106C211yesmaximal, characteristic, complemented
3#11720545C211yesmaximal, characteristic, complemented
4C3×#4488—34nomaximal
5#88740—21no—
6C2×C8×H(3,𝔽3)C2211yescharacteristic
7C4⋊C4×H(3,𝔽3)—21no—
8#88530C411yescharacteristic, complemented
9#88529C411yescharacteristic, complemented
10#88528—21no—
11#88386—21no—
12C3×#904—34no—
13C32×#106D314yescomplemented
14C3×#812—34no—
15C2×C4×H(3,𝔽3)—41no—
16C2×C4×H(3,𝔽3)D411yescharacteristic
17#6235—41no—
18C4⋊#272—21no—
19C4×#272—41no—
20#6229—21no—
21C8×H(3,𝔽3)C2×C411yescharacteristic
22C8×H(3,𝔽3)D411yescharacteristic
23C3×#360—91no—
24#4488—34no—
25C3×#235—64no—
26C2×C32×C8C2×D314yes—
27C32×C4⋊C4—24no—
28C3×Dic12—34no—
29C3×D24—34no—
30C3×#204—64no—
31C3×C3⋊C16—64no—
32C22×H(3,𝔽3)D811yescharacteristic
33C2×#272—41no—
34C4×H(3,𝔽3)—81no—
35C4×H(3,𝔽3)C22⋊C411yescharacteristic
36C4×H(3,𝔽3)SD1611yescharacteristic
37#1084—41no—
38C3×#135—91no—
39#904—34no—
40C3×M32—91no—
41C3×#106—34no—
42C3×#106C3⋊D311yescharacteristic
43#812—34no—
44C2×C32×C4—44no—
45C2×C32×C4#6714yes—
46C3×#67—124no—
47C3×D12—64no—
48C3×C4×D3—124no—
49C3×Dic6—64no—
50C32×C8C4×D314yes—
51C32×C8D1214yes—
52#360—91noSylow
53C2×H(3,𝔽3)—21no—
54C2×H(3,𝔽3)#10211yescharacteristic
55#272—81no—
56C3×#41—181no—
57#235—64no—
58C3×Q16—91no—
59C3×SD16—181no—
60C3×D8—91no—
61C2×C3×C8—34no—
62C2×C3×C8C2×C3⋊D311yescharacteristic
63C3×C4⋊C4—21no—
64C3×C4⋊C4—64no—
65Dic12—34no—
66D24—34no—
67#204—64no—
68C3×C16—181no—
69C3⋊C16—64no—
70C22×C32#21314yes—
71C2×C3×D3—124no—
72C32×C4—84no—
73C32×C4(C2×D3)⋊C414yes—
74C32×C4#21514yes—
75C3×Dic3—124no—
76#135—91no—
77M32—91no—
78#106#27211yescharacteristic, complemented
79H(3,𝔽3)#36011yescharacteristic, complemented, Sylow
80C3×Q8—181no—
81C3×D4—181no—
82C3×D4—361no—
83C2×C3×C4—34no—
84C2×C3×C4—41no—
85C2×C3×C4—124no—
86C2×C3×C4—361no—
87C2×C3×C4#63811yescharacteristic
88#67—124no—
89D12—64no—
90C4×D3—124no—
91Dic6—64no—
92C3×C8—34no—
93C3×C8—34no—
94C3×C8C4×C3⋊D311yescharacteristic
95C3×C8C4⋊(C3⋊D3)11yescharacteristic
96C2×C32—24no—
97C2×C32#80914yes—
98C3×D3—244no—
99#41—181no—
100Q16—91no—
101SD16—181no—
102D8—91no—
103C2×C8C2×#27211yescharacteristic
104C4⋊C4—21no—
105C16—181no—
106C22×C3—34no—
107C22×C3—361no—
108C22×C3#375311yescharacteristic
109C2×D3—124no—
110C3×C4—34no—
111C3×C4—34no—
112C3×C4—81no—
113C3×C4—244no—
114C3×C4—361no—
115C3×C4#375211yescharacteristic
116C3×C4#375511yescharacteristic
117Dic3—124no—
118C32#448814yescomplemented
119Q8—181no—
120D4—181no—
121D4—361no—
122C2×C4—41no—
123C2×C4—361no—
124C2×C4#623511yescharacteristic
125C8C4×#27211yescharacteristic
126C8C4⋊#27211yescharacteristic
127C2×C3—21no—
128C2×C3—34no—
129C2×C3—64no—
130C2×C3—721no—
131C2×C3#6363611yescharacteristic
132D3—244no—
133C22—361no—
134C22#8854511yescharacteristic
135C4—81no—
136C4—361no—
137C4#8854211yescharacteristic
138C4#8854911yescharacteristic
139C3—34no—
140C3#1058935411yescharacteristic
141C2—21no—
142C2—721no—
143C2#1172053111yescharacteristic
subgroupG/H
center, socleC2×C3#63636
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC2×C3×C8C2×C3⋊D3
FittingH(3,𝔽3)×#106C2
prime radical#11721779C2
cosocleC2×C8×H(3,𝔽3)C22
hypercenterC3×#106C3⋊D3
hypocenterH(3,𝔽3)#360
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C3×#106

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#106C2semidirect
3. #11720545C2semidirect
8. #88530C4semidirect
9. #88529C4semidirect
13. C32×#106D3semidirect
78. #106#272semidirect
79. H(3,𝔽3)#360semidirect
118. C32#4488semidirect

Maximal subgroups

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

subgroupindex
#117217792
H(3,𝔽3)×#1062
#117205452
C3×#44883