Finite Group Databasebeta

← #13143725

Subgroups — #13143725

Order 1728.

649 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 649 conjugacy classes of subgroups — too many to draw. Only its 151 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
1H(3,𝔽3)×#95C211yesmaximal, characteristic
2#11720574C213yesmaximal
3#11720574C213yesmaximal
4C3×C3⋊#401—34nomaximal
5C22×C4×H(3,𝔽3)C2213yes—
6C22×#1084C2211yescharacteristic
7C22×#1084C2213yes—
8C32×#95D314yes—
9C3×C3⋊#95—312no—
10C3×C3⋊#95—312no—
11C23×H(3,𝔽3)C2311yescharacteristic
12C2×C4×H(3,𝔽3)—23no—
13C2×C4×H(3,𝔽3)—26no—
14C2×#1084—23no—
15C2×#1084—23no—
16C2×#1084—23no—
17C2×#1084—23no—
18C3×#401—91no—
19C3⋊#401—34no—
20C22×C32×C4C2×D3112yes—
21C22×C3×Dic3—34no—
22C22×C3×Dic3—312no—
23C22×H(3,𝔽3)#4113yes—
24C22×H(3,𝔽3)#4113yes—
25C22×H(3,𝔽3)#4111yescharacteristic
26C4×H(3,𝔽3)—43no—
27#1084—41no—
28#1084—43no—
29C3×#95—34no—
30C3×#95—93no—
31C3×#95—93no—
32C3×#95C3⋊D311yescharacteristic
33C3⋊#95—312no—
34C3⋊#95—312no—
35C23×C32C22×D314yes—
36C2×C32×C4—212no—
37C2×C32×C4—224no—
38C2×C3×Dic3—612no—
39C2×C3×Dic3—612no—
40C2×C3×Dic3—612no—
41C2×C3×Dic3—612no—
42#401—91noSylow
43C2×H(3,𝔽3)#12611yescharacteristic
44C2×H(3,𝔽3)#12613yes—
45C2×H(3,𝔽3)#12613yes—
46C22×C3×C4—312no—
47C22×C3×C4—91no—
48C22×C3×C4—93no—
49C22×C3×C4C2×C3⋊D313yes—
50C22×Dic3—34no—
51C22×Dic3—312no—
52C22×C32#235112yes—
53C22×C32#237112yes—
54C22×C32#23914yes—
55C32×C4—412no—
56C3×Dic3—124no—
57C3×Dic3—1212no—
58#95—93no—
59#95—93no—
60#95#27211yescharacteristic
61H(3,𝔽3)#40111yescharacteristic, complemented, Sylow
62C23×C3—34no—
63C23×C3C22×C3⋊D311yescharacteristic
64C2×C3×C4—23no—
65C2×C3×C4—26no—
66C2×C3×C4—612no—
67C2×C3×C4—624no—
68C2×C3×C4—183no—
69C2×C3×C4—183no—
70C2×C3×C4—183no—
71C2×C3×C4—183no—
72C2×Dic3—612no—
73C2×Dic3—612no—
74C2×Dic3—612no—
75C2×Dic3—612no—
76C2×C32#87614yes—
77C2×C32#879112yes—
78C2×C32#898112yes—
79C22×C4—91no—
80C22×C4—93no—
81C22×C4C2×#27213yes—
82C22×C3—312no—
83C22×C3—312no—
84C22×C3—34no—
85C22×C3#382813yes—
86C22×C3#383013yes—
87C22×C3#383211yescharacteristic
88C3×C4—43no—
89C3×C4—1212no—
90C3×C4—361no—
91C3×C4—363no—
92Dic3—1212no—
93Dic3—124no—
94C32C3⋊#40114yescomplemented
95C23C22×#27211yescharacteristic
96C2×C4—23no—
97C2×C4—26no—
98C2×C4—183no—
99C2×C4—183no—
100C2×C4—183no—
101C2×C4—183no—
102C2×C3—312no—
103C2×C3—312no—
104C2×C3—34no—
105C2×C3#6408511yescharacteristic
106C2×C3#6408813yes—
107C2×C3#6410713yes—
108C22#8874013yes—
109C22#8874413yes—
110C22#8874811yescharacteristic
111C4—43no—
112C4—363no—
113C4—361no—
114C3—34no—
115C3#1059236111yescharacteristic
116C2#1172172211yescharacteristic
117C2#1172173513yes—
118C2#1172176613yes—
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, prime radical, cosocleC23×H(3,𝔽3)C23
FittingH(3,𝔽3)×#95C2
hypercenterC3×#95C3⋊D3
hypocenterH(3,𝔽3)#401
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
61. H(3,𝔽3)#401semidirect
94. C32C3⋊#401semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#952
#117205742
C3×C3⋊#4013