Finite Group Databasebeta

← #13136737

Subgroups — #13136737

Order 1728.

189 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 45 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×#88631C212yesmaximal
2C22×#6188C211yesmaximal, characteristic
3C22×#1084C2211yescharacteristic
4#88631—22no—
5#88631C414yes—
6C2×#6188—22no—
7C2×#6188—22no—
8C2×#6188C412yes—
9C23×H(3,𝔽3)Q811yescharacteristic
10C2×#1084C2×C411yescharacteristic
11C2×#1084C2×C412yes—
12C2×#1084D411yescharacteristic
13C2×#1084D412yes—
14#6188—22no—
15#6188—24no—
16C3×#341—91nomaximal
17C22×C3×Dic3—121no—
18C22×H(3,𝔽3)—22no—
19C22×H(3,𝔽3)C4⋊C412yes—
20C22×H(3,𝔽3)C4⋊C411yescharacteristic
21#1084C4211yescharacteristic
22#1084C22⋊C411yescharacteristic
23#1084C22⋊C412yes—
24C2×C3×M16—92no—
25C22×C3×C8—91no—
26C23×C32—41no—
27C2×C3×Dic3—122no—
28C2×C3×Dic3—121no—
29C2×C3×Dic3—122no—
30C2×C3×Dic3—121no—
31#341—91noSylow
32C2×H(3,𝔽3)—22no—
33C2×H(3,𝔽3)#9511yescharacteristic
34C2×H(3,𝔽3)#10812yes—
35C22×C3×C4—91no—
36C22×Dic3—121no—
37C3×M16—94no—
38C3×M16—182no—
39C2×C3×C8—92no—
40C2×C3×C8—182no—
41C2×C3×C8—182no—
42C22×C32—44no—
43C22×C32—42no—
44C22×C32—41no—
45C3×Dic3—121no—
46C3×Dic3—121no—
47C3×Dic3—122no—
48C2×M16—92no—
49C22×C8—91no—
50H(3,𝔽3)#34111yescharacteristic, complemented, Sylow
51C23×C3—121no—
52C23×C3PSU(3,𝔽2)11yescharacteristic
53C2×C3×C4—92no—
54C2×C3×C4—92no—
55C2×C3×C4—91no—
56C2×C3×C4—91no—
57C2×Dic3—121no—
58C2×Dic3—121no—
59C2×Dic3—122no—
60C2×Dic3—122no—
61C3×C8—182no—
62C3×C8—184no—
63C2×C32—44no—
64C2×C32—41no—
65C2×C32—42no—
66C22×C4—91no—
67M16—94no—
68M16—182no—
69C2×C8—92no—
70C2×C8—182no—
71C2×C8—182no—
72C22×C3—22no—
73C22×C3—124no—
74C22×C3—122no—
75C22×C3—121no—
76C22×C3#377712yes—
77C22×C3#377711yescharacteristic
78C3×C4—91no—
79C3×C4—92no—
80C3×C4—91no—
81Dic3—121no—
82Dic3—122no—
83Dic3—121no—
84C32—41no—
85C23SU(3,𝔽2)11yescharacteristic
86C2×C4—91no—
87C2×C4—91no—
88C2×C4—92no—
89C2×C4—92no—
90C8—182no—
91C8—184no—
92C2×C3—22no—
93C2×C3—124no—
94C2×C3—121no—
95C2×C3—122no—
96C2×C3#6374711yescharacteristic
97C2×C3#6374812yes—
98C22—22no—
99C22#8859212yes—
100C22#8859211yescharacteristic
101C4—91no—
102C4—92no—
103C4—91no—
104C3—121no—
105C3#1059009711yescharacteristic
106C2—22no—
107C2#1172071111yescharacteristic
108C2#1172071212yes—
subgroupG/H
center, socleC22×C3#3777
commutator#1084C42
Frattini, hypercenterC23×C3PSU(3,𝔽2)
Fitting, prime radicalC23×H(3,𝔽3)Q8
cosocleC22×#1084C22
hypocenterH(3,𝔽3)#341
LCSG ▷ #1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
50. H(3,𝔽3)#341semidirect

Maximal subgroups

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

subgroupindex
C2×#886312
C22×#61882
C3×#3419