Finite Group Databasebeta

← #13136906

Subgroups — #13136906

Order 1728.

169 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 169 conjugacy classes of subgroups — too many to draw. Only its 43 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
1C4⋊#6188C212yesmaximal
2C8×#1084C211yesmaximal, characteristic
3C2×#6188—22no—
4C2×C8×H(3,𝔽3)C411yescharacteristic
5C4×#1084C2211yescharacteristic
6C2×#6180C411yescharacteristic
7C3×C8×Dic3—62no—
8C2×C4×H(3,𝔽3)C2×C411yescharacteristic
9C2×#1084D411yescharacteristic
10C2×#1084Q811yescharacteristic
11#6188—42no—
12C8×H(3,𝔽3)C812yescomplemented
13#6180—21no—
14C3×#335—91nomaximal
15C2×C32×C8—22no—
16C3×C4×Dic3—62no—
17C2×C3×C3⋊C8—62no—
18C22×H(3,𝔽3)C4⋊C411yescharacteristic
19C4×H(3,𝔽3)C2×C811yescharacteristic
20C4×H(3,𝔽3)M1611yescharacteristic
21#1084—21no—
22#1084D811yescharacteristic
23#1084Q1611yescharacteristic
24C3×C4⋊C8—92no—
25C3×C4×C8—91no—
26C8×Dic3—62no—
27C2×C32×C4—22no—
28C2×C3×Dic3—62no—
29C2×C3×Dic3—62no—
30C32×C8—24no—
31C3×C3⋊C8—64no—
32#335—91noSylow
33C2×H(3,𝔽3)C4⋊C811yescharacteristic
34C2×H(3,𝔽3)#10711yescharacteristic
35C2×H(3,𝔽3)#10811yescharacteristic
36C2×C3×C8—62no—
37C2×C3×C8—91no—
38C2×C3×C8—182no—
39C2×C3×C8#15711yescharacteristic
40C3×C42—91no—
41C4×Dic3—62no—
42C2×C3⋊C8—62no—
43C22×C32—22no—
44C32×C4—22no—
45C32×C4—22no—
46C3×Dic3—62no—
47C3×Dic3—62no—
48C3×Dic3—64no—
49C4⋊C8—92no—
50C4×C8—91no—
51H(3,𝔽3)#33511yescharacteristic, complemented, Sylow
52C2×C3×C4—62no—
53C2×C3×C4—91no—
54C2×C3×C4—91no—
55C2×C3×C4C2×#15711yescharacteristic
56C2×Dic3—62no—
57C2×Dic3—62no—
58C3×C8—64no—
59C3×C8—181no—
60C3×C8—362no—
61C3×C8#62212yes—
62C3⋊C8—64no—
63C2×C32—22no—
64C2×C32—22no—
65C2×C32—22no—
66C2×C8—91no—
67C2×C8—182no—
68C2×C8#108811yescharacteristic
69C42—91no—
70C22×C3—62no—
71C22×C3C4⋊#15711yescharacteristic
72C3×C4—62no—
73C3×C4—62no—
74C3×C4—91no—
75C3×C4—91no—
76C3×C4—181no—
77C3×C4#378811yescharacteristic
78C3×C4C2×#62211yescharacteristic
79Dic3—62no—
80Dic3—62no—
81Dic3—64no—
82C32—22no—
83C2×C4—91no—
84C2×C4—91no—
85C2×C4C2×#108811yescharacteristic
86C8—181no—
87C8—362no—
88C8#618812yescomplemented
89C2×C3—62no—
90C2×C3—62no—
91C2×C3—62no—
92C2×C3#6376911yescharacteristic
93C2×C3#6377111yescharacteristic
94C2×C3C4⋊#62211yescharacteristic
95C22C4⋊#108811yescharacteristic
96C4—91no—
97C4—91no—
98C4—181no—
99C4#8862711yescharacteristic
100C4C2×#618811yescharacteristic
101C3—62no—
102C3#1059017111yescharacteristic
103C2#1172077511yescharacteristic
104C2#1172077711yescharacteristic
105C2C4⋊#618811yescharacteristic
subgroupG/H
center, socleC22×C3C4⋊#157
commutatorC4×H(3,𝔽3)C2×C8
FrattiniC2×C3×C4C2×#157
FittingC2×C8×H(3,𝔽3)C4
prime radicalC22×H(3,𝔽3)C4⋊C4
cosocleC4×#1084C22
hypercenterC2×C3×C8#157
hypocenterH(3,𝔽3)#335
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
12. C8×H(3,𝔽3)C8semidirect
51. H(3,𝔽3)#335semidirect
88. C8#6188semidirect

Maximal subgroups

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

subgroupindex
C4⋊#61882
C8×#10842
C3×#3359