Finite Group Databasebeta

← #13135785

Subgroups — #13135785

Order 1728.

330 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 61 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#11721861C211yesmaximal, characteristic
2H(3,𝔽3)×#101C211yesmaximal, characteristic
3#11720584C211yesmaximal, characteristic
4C3×#4478—34nomaximal
5C2×Q8×H(3,𝔽3)C2211yescharacteristic
6C2×#6229C411yescharacteristic
7M16×H(3,𝔽3)—21no—
8#88540—21no—
9#88540—21no—
10C4×#1084C411yescharacteristic
11#88538—21no—
12C3×#944—34no—
13C32×#101D314yes—
14C3×#836—34no—
15Q8×H(3,𝔽3)—41no—
16C2×C4×H(3,𝔽3)C2×C411yescharacteristic
17C2×C4×H(3,𝔽3)D411yescharacteristic
18C2×C4×H(3,𝔽3)D411yescharacteristic
19C2×#1084—21no—
20C2×#1084—21no—
21#6229—41no—
22C8×H(3,𝔽3)—41no—
23#6180—41no—
24C3×#356—91no—
25#4478—34no—
26C2×C32×Q8C2×D314yes—
27C2×C3×Dic6—34no—
28C32×M16—24no—
29C3×Dic3⋊C4—64no—
30C3×Dic3⋊C4—64no—
31C3×C4×Dic3—34no—
32C3×#208—64no—
33C22×H(3,𝔽3)C22⋊C411yescharacteristic
34C4×H(3,𝔽3)—21no—
35C4×H(3,𝔽3)—21no—
36C4×H(3,𝔽3)—21no—
37#1084—22no—
38#1084—41no—
39C3×C4⋊Q8—91no—
40#944—34no—
41C3×#101—34no—
42C3×#101—91no—
43C3×#101C3⋊D311yescharacteristic
44#836—34no—
45C32×Q8—44no—
46C2×C32×C4C4×D314yes—
47C2×C32×C4D1214yes—
48C2×C32×C4#6714yes—
49C2×C3×Dic3—64no—
50C2×C3×Dic3—64no—
51C3×Dic6—124no—
52C32×C8—44no—
53C3×C3⋊C8—124no—
54#356—91noSylow
55C2×H(3,𝔽3)—21no—
56C2×H(3,𝔽3)#9911yescharacteristic
57C2×C3×Q8—34no—
58C2×C3×Q8—91no—
59C2×C3×Q8C2×C3⋊D311yescharacteristic
60C2×Dic6—34no—
61C3×M16—21no—
62C3×M16—64no—
63C3×M16—181no—
64C3×C4⋊C4—181no—
65C3×C4⋊C4—181no—
66C3×C42—91no—
67Dic3⋊C4—64no—
68Dic3⋊C4—64no—
69C4×Dic3—34no—
70#208—64no—
71C22×C32(C2×D3)⋊C414yes—
72C32×C4—24no—
73C32×C4—24no—
74C32×C4—24no—
75C3×Dic3—68no—
76C3×Dic3—124no—
77C4⋊Q8—91no—
78#101—91no—
79#101#27211yescharacteristic
80H(3,𝔽3)#35611yescharacteristic, complemented, Sylow
81C3×Q8—41no—
82C3×Q8—124no—
83C3×Q8—361no—
84C2×C3×C4—34no—
85C2×C3×C4—34no—
86C2×C3×C4—34no—
87C2×C3×C4—181no—
88C2×C3×C4—181no—
89C2×C3×C4C4×C3⋊D311yescharacteristic
90C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
91C2×C3×C4#63811yescharacteristic
92C2×Dic3—64no—
93C2×Dic3—64no—
94Dic6—124no—
95C3×C8—41no—
96C3×C8—124no—
97C3×C8—361no—
98C3⋊C8—124no—
99C2×C32—24no—
100C2×C32#80614yes—
101C2×Q8—91no—
102C2×Q8C2×#27211yescharacteristic
103M16—21no—
104M16—181no—
105C4⋊C4—181no—
106C4⋊C4—181no—
107C42—91no—
108C22×C3—34no—
109C22×C3#375211yescharacteristic
110C3×C4—21no—
111C3×C4—21no—
112C3×C4—21no—
113C3×C4—64no—
114C3×C4—64no—
115C3×C4—64no—
116C3×C4—182no—
117C3×C4—361no—
118Dic3—68no—
119Dic3—124no—
120C32#447814yescomplemented
121Q8—41no—
122Q8—361no—
123C2×C4—181no—
124C2×C4—181no—
125C2×C4C4×#27211yescharacteristic
126C2×C4C4⋊#27211yescharacteristic
127C2×C4#623511yescharacteristic
128C8—41no—
129C8—361no—
130C2×C3—21no—
131C2×C3—34no—
132C2×C3—64no—
133C2×C3#6363311yescharacteristic
134C22#8854211yescharacteristic
135C4—21no—
136C4—21no—
137C4—21no—
138C4—182no—
139C4—361no—
140C3—34no—
141C3#1058934411yescharacteristic
142C2—21no—
143C2#1172052411yescharacteristic
subgroupG/H
center, socleC2×C3#63633
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC2×C3×Q8C2×C3⋊D3
FittingH(3,𝔽3)×#101C2
prime radicalC22×H(3,𝔽3)C22⋊C4
cosocleC2×Q8×H(3,𝔽3)C22
hypercenterC3×#101C3⋊D3
hypocenterH(3,𝔽3)#356
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ C3×#101

Complements

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

NH ≅ G/Ntype
80. H(3,𝔽3)#356semidirect
120. C32#4478semidirect

Maximal subgroups

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

subgroupindex
#117218612
H(3,𝔽3)×#1012
#117205842
C3×#44783