Finite Group Databasebeta

← #13136913

Subgroups — #13136913

Order 1728.

223 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 31 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
1C2×#88527C211yesmaximal, characteristic
2#11720779C212yesmaximal, complemented
3C2×C4×#272C411yescharacteristic
4C2×C8×H(3,𝔽3)C411yescharacteristic
5C2×#6180C2211yescharacteristic
6#88527—22no—
7#88410—22no—
8C2×C3×ΓL(1,𝔽25)—62no—
9C22×#272C811yescharacteristic
10C2×C4×H(3,𝔽3)C2×C411yescharacteristic
11C2×#1084C811yescharacteristic
12C4×#272—21no—
13C4×#272—21no—
14C8×H(3,𝔽3)—21no—
15#6180D412yes—
16C3×#349—91nomaximal
17C2×C3×C4×D3—62no—
18C2×C32×C8—22no—
19C2×C3×C3⋊C8—62no—
20C3×ΓL(1,𝔽25)—68no—
21C22×H(3,𝔽3)C2×C811yescharacteristic
22C2×#272—21no—
23C2×#272—41no—
24C4×H(3,𝔽3)C22⋊C411yescharacteristic
25C4×H(3,𝔽3)M1611yescharacteristic
26#1084—21no—
27C2×C3×M16—91no—
28C2×ΓL(1,𝔽25)—62no—
29C3×M32—92no—
30C22×C3×D3—62no—
31C2×C32×C4—22no—
32C2×C3×Dic3—62no—
33C3×C4×D3—64no—
34C3×C4×D3—64no—
35C32×C8—24no—
36C3×C3⋊C8—64no—
37#349—91noSylow
38C2×H(3,𝔽3)—21no—
39C2×H(3,𝔽3)C22⋊C811yescharacteristic
40#272—41no—
41C22×C3×C4—91no—
42C2×C4×D3—62no—
43C3×M16—182no—
44C2×C3×C8—62no—
45C2×C3×C8—91no—
46C2×C3×C8#15711yescharacteristic
47C2×C3⋊C8—62no—
48ΓL(1,𝔽25)—68no—
49C3×C16—182no—
50C22×C32—22no—
51C2×C3×D3—64no—
52C2×C3×D3—124no—
53C32×C4—22no—
54C32×C4—22no—
55C3×Dic3—64no—
56C2×M16—91no—
57M32—92no—
58H(3,𝔽3)#34911yescharacteristic, complemented, Sylow
59C23×C3—91no—
60C22×D3—62no—
61C2×C3×C4—62no—
62C2×C3×C4—91no—
63C2×C3×C4—181no—
64C2×C3×C4—181no—
65C2×C3×C4C2×#15711yescharacteristic
66C2×Dic3—62no—
67C4×D3—64no—
68C4×D3—64no—
69C3×C8—21no—
70C3×C8—64no—
71C3×C8—92no—
72C3⋊C8—64no—
73C2×C32—24no—
74C2×C32—22no—
75C3×D3—124no—
76C22×C4—91no—
77M16—182no—
78C2×C8—91no—
79C2×C8#108811yescharacteristic
80C16—182no—
81C22×C3—62no—
82C22×C3—181no—
83C22×C3—361no—
84C22×C3#378711yescharacteristic
85C2×D3—64no—
86C2×D3—124no—
87C3×C4—62no—
88C3×C4—62no—
89C3×C4—181no—
90C3×C4#378811yescharacteristic
91C3×C4C22⋊#15711yescharacteristic
92Dic3—64no—
93C32—22no—
94C23—91no—
95C2×C4—91no—
96C2×C4—181no—
97C2×C4—181no—
98C2×C4C2×#108811yescharacteristic
99C8—21no—
100C8—92no—
101C2×C3—21no—
102C2×C3—64no—
103C2×C3—62no—
104C2×C3—361no—
105C2×C3#6377811yescharacteristic
106D3—124no—
107C22—181no—
108C22—361no—
109C22#8862611yescharacteristic
110C4—181no—
111C4#8862711yescharacteristic
112C4C22⋊#108811yescharacteristic
113C3—62no—
114C3#1059018011yescharacteristic
115C2—21no—
116C2—361no—
117C2#1172078411yescharacteristic
subgroupG/H
centerC3×C4C22⋊#157
commutatorC22×H(3,𝔽3)C2×C8
FrattiniC2×C3×C4C2×#157
FittingC2×C8×H(3,𝔽3)C4
socleC2×C3#63778
prime radicalC22×#272C8
cosocleC2×#6180C22
hypercenterC2×C3×C8#157
hypocenterH(3,𝔽3)#349
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
2. #11720779C2semidirect
58. H(3,𝔽3)#349semidirect

Maximal subgroups

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

subgroupindex
C2×#885272
#117207792
C3×#3499