Finite Group Databasebeta

← #13136920

Subgroups — #13136920

Order 1728.

265 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 41 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#11722445C212yesmaximal
2C2×#88527C211yesmaximal, characteristic
3C2×C4×#272C2211yescharacteristic
4#88631—22no—
5#88627—22no—
6#88626C414yes—
7C2×C8×H(3,𝔽3)C411yescharacteristic
8C2×#6180C411yescharacteristic
9#88527—22no—
10C2×C3×ΓL(1,𝔽25)—62no—
11C22×#272Q811yescharacteristic
12C2×C4×H(3,𝔽3)C2×C411yescharacteristic
13C2×#1084D411yescharacteristic
14C4×#272C2×C412yes—
15C4×#272D412yes—
16#6188—24no—
17C8×H(3,𝔽3)—21no—
18#6180—21no—
19C3×#338—91nomaximal
20C2×C3×C4×D3—62no—
21C2×C32×C8—22no—
22C2×C3×C3⋊C8—62no—
23C3×ΓL(1,𝔽25)—68no—
24C22×H(3,𝔽3)C4⋊C411yescharacteristic
25C2×#272—41no—
26C2×#272C4⋊C412yes—
27C4×H(3,𝔽3)C4211yescharacteristic
28C4×H(3,𝔽3)C22⋊C411yescharacteristic
29#1084C22⋊C412yes—
30C2×C3×M16—92no—
31C2×C3×M16—91no—
32C2×ΓL(1,𝔽25)—62no—
33C22×C3×D3—62no—
34C2×C32×C4—22no—
35C2×C3×Dic3—62no—
36C3×C4×D3—64no—
37C3×C4×D3—64no—
38C32×C8—24no—
39C3×C3⋊C8—64no—
40#338—91noSylow
41C2×H(3,𝔽3)—21no—
42C2×H(3,𝔽3)#9511yescharacteristic
43#272—22no—
44C22×C3×C4—91no—
45C2×C4×D3—62no—
46C3×M16—182no—
47C3×M16—182no—
48C3×M16—182no—
49C2×C3×C8—62no—
50C2×C3×C8—94no—
51C2×C3×C8—91no—
52C2×C3×C8#15711yescharacteristic
53C2×C3⋊C8—62no—
54ΓL(1,𝔽25)—68no—
55C22×C32—22no—
56C2×C3×D3—64no—
57C2×C3×D3—124no—
58C32×C4—22no—
59C32×C4—22no—
60C3×Dic3—64no—
61C2×M16—92no—
62C2×M16—91no—
63H(3,𝔽3)#33811yescharacteristic, complemented, Sylow
64C23×C3—91no—
65C22×D3—62no—
66C2×C3×C4—62no—
67C2×C3×C4—91no—
68C2×C3×C4—92no—
69C2×C3×C4—92no—
70C2×C3×C4C2×#15711yescharacteristic
71C2×Dic3—62no—
72C4×D3—64no—
73C4×D3—64no—
74C3×C8—21no—
75C3×C8—64no—
76C3×C8—184no—
77C3×C8—181no—
78C3⋊C8—64no—
79C2×C32—24no—
80C2×C32—22no—
81C3×D3—124no—
82C22×C4—91no—
83M16—182no—
84M16—182no—
85M16—182no—
86C2×C8—94no—
87C2×C8—91no—
88C2×C8#108811yescharacteristic
89C22×C3—62no—
90C22×C3—92no—
91C22×C3—361no—
92C22×C3C4⋊#15711yescharacteristic
93C2×D3—64no—
94C2×D3—124no—
95C3×C4—62no—
96C3×C4—62no—
97C3×C4—92no—
98C3×C4C4×#15711yescharacteristic
99C3×C4C22⋊#15711yescharacteristic
100Dic3—64no—
101C32—22no—
102C23—91no—
103C2×C4—91no—
104C2×C4—92no—
105C2×C4—92no—
106C2×C4C2×#108811yescharacteristic
107C8—21no—
108C8—184no—
109C8—181no—
110C2×C3—21no—
111C2×C3—64no—
112C2×C3—62no—
113C2×C3—182no—
114C2×C3#6378111yescharacteristic
115D3—124no—
116C22—92no—
117C22—361no—
118C22C4⋊#108811yescharacteristic
119C4—92no—
120C4C4×#108811yescharacteristic
121C4C22⋊#108811yescharacteristic
122C3—62no—
123C3#1059018611yescharacteristic
124C2—21no—
125C2—182no—
126C2#1172078711yescharacteristic
subgroupG/H
centerC3×C4C4×#157
commutatorC4×H(3,𝔽3)C42
FrattiniC2×C3×C4C2×#157
FittingC2×C8×H(3,𝔽3)C4
socleC2×C3#63781
prime radicalC22×#272Q8
cosocleC2×C4×#272C22
hypercenterC2×C3×C8#157
hypocenterH(3,𝔽3)#338
LCSG ▷ C4×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
63. H(3,𝔽3)#338semidirect

Maximal subgroups

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

subgroupindex
#117224452
C2×#885272
C3×#3389