Finite Group Databasebeta

← #13144845

Subgroups — #13144845

Order 1728.

1001 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1001 conjugacy classes of subgroups — too many to draw. Only its 199 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
1C22×Q8×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#88540C213yesmaximal
3#11720574C213yesmaximal
4C3×#5230—34nomaximal
5C2×Q8×H(3,𝔽3)—26no—
6C22×C4×H(3,𝔽3)C2213yes—
7C22×#1084C2211yescharacteristic
8C22×#1084C2213yes—
9#88540—26no—
10C22×C32×Q8D314yes—
11C2×C3×Dic3⋊C4—312no—
12C3×C3⋊#95—312no—
13C23×H(3,𝔽3)C2311yescharacteristic
14Q8×H(3,𝔽3)—28no—
15C2×C4×H(3,𝔽3)—26no—
16C2×C4×H(3,𝔽3)D416yes—
17C2×#1084—23no—
18C2×#1084—23no—
19C2×#1084—23no—
20C2×#1084Q816yescomplemented
21C3×#395—91no—
22#5230—34no—
23C2×C32×Q8—224no—
24C22×C32×C4C2×D3112yes—
25C22×C3×Dic3—312no—
26C22×C3×Dic3—34no—
27C3×Dic3⋊C4—624no—
28C22×H(3,𝔽3)C2×D413yes—
29C22×H(3,𝔽3)C2×Q813yes—
30C22×H(3,𝔽3)#4111yescharacteristic
31C4×H(3,𝔽3)—26no—
32#1084—26no—
33#1084—41no—
34C22×C3×Q8—34no—
35C22×C3×Q8C3⋊D311yescharacteristic
36C2×C3×C4⋊C4—93no—
37C2×Dic3⋊C4—312no—
38C3×#95—93no—
39C3⋊#95—312no—
40C23×C32C22×D314yes—
41C32×Q8—232no—
42C2×C32×C4—224no—
43C2×C32×C4#67124yes—
44C2×C3×Dic3—324no—
45C2×C3×Dic3—612no—
46C2×C3×Dic3—612no—
47C2×C3×Dic3—612no—
48#395—91noSylow
49C2×H(3,𝔽3)#12011yescharacteristic
50C2×H(3,𝔽3)C22⋊Q813yes—
51C2×H(3,𝔽3)C4⋊Q813yes—
52C2×C3×Q8—26no—
53C2×C3×Q8—624no—
54C22×C3×C4—312no—
55C22×C3×C4—93no—
56C22×C3×C4—91no—
57C22×C3×C4C2×C3⋊D313yes—
58C22×Dic3—312no—
59C22×Dic3—34no—
60C3×C4⋊C4—186no—
61Dic3⋊C4—624no—
62C22×C32D3×Q8112yes—
63C22×C32#23914yes—
64C22×C32C2×#67112yes—
65C32×C4—224no—
66C3×Dic3—624no—
67C3×Dic3—124no—
68C22×Q8#27211yescharacteristic
69C2×C4⋊C4—93no—
70#95—93no—
71H(3,𝔽3)#39511yescharacteristic, complemented, Sylow
72C23×C3—34no—
73C23×C3C22×C3⋊D311yescharacteristic
74C3×Q8—28no—
75C3×Q8—632no—
76C2×C3×C4—26no—
77C2×C3×C4—324no—
78C2×C3×C4—624no—
79C2×C3×C4—96no—
80C2×C3×C4—183no—
81C2×C3×C4—183no—
82C2×C3×C4—183no—
83C2×C3×C4#63816yes—
84C2×Dic3—324no—
85C2×Dic3—612no—
86C2×Dic3—612no—
87C2×Dic3—612no—
88C2×C32#944112yes—
89C2×C32#946112yes—
90C2×C32#95314yes—
91C2×Q8—26no—
92C22×C4—93no—
93C22×C4—91no—
94C22×C4C2×#27213yes—
95C4⋊C4—186no—
96C22×C3—312no—
97C22×C3—34no—
98C22×C3—312no—
99C22×C3Q8×C3⋊D313yes—
100C22×C3#383211yescharacteristic
101C22×C3C2×#63813yes—
102C3×C4—26no—
103C3×C4—624no—
104C3×C4—186no—
105C3×C4—361no—
106Dic3—624no—
107Dic3—124no—
108C32#523014yescomplemented
109C23C22×#27211yescharacteristic
110Q8—28no—
111C2×C4—26no—
112C2×C4—96no—
113C2×C4—183no—
114C2×C4—183no—
115C2×C4—183no—
116C2×C4#623516yes—
117C2×C3—312no—
118C2×C3—312no—
119C2×C3—34no—
120C2×C3#6415313yes—
121C2×C3#6415513yes—
122C2×C3#6416211yescharacteristic
123C22Q8×#27213yes—
124C22#8874811yescharacteristic
125C22C2×#623513yes—
126C4—26no—
127C4—186no—
128C4—361no—
129C3—34no—
130C3#1059292911yescharacteristic
131C2#1172186113yes—
132C2#1172186313yes—
133C2#1172187611yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, prime radical, cosocleC23×H(3,𝔽3)C23
FittingC22×Q8×H(3,𝔽3)C2
hypercenterC22×C3×Q8C3⋊D3
hypocenterH(3,𝔽3)#395
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C22×C3×Q8

Complements

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

NH ≅ G/Ntype
20. C2×#1084Q8semidirect
71. H(3,𝔽3)#395semidirect
108. C32#5230semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×H(3,𝔽3)2
C2×#885402
#117205742
C3×#52303