Finite Group Databasebeta

← #13136905

Subgroups — #13136905

Order 1728.

169 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 169 conjugacy classes of subgroups — too many to draw. Only its 39 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)—21no—
13#6180C812yescomplemented
14C3×#334—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#1084SD1612yes—
23C3×C4⋊C8—92no—
24C3×C4×C8—91no—
25C8×Dic3—62no—
26C2×C32×C4—22no—
27C2×C3×Dic3—62no—
28C2×C3×Dic3—62no—
29C32×C8—24no—
30C3×C3⋊C8—64no—
31#334—91noSylow
32C2×H(3,𝔽3)C4⋊C811yescharacteristic
33C2×H(3,𝔽3)#10611yescharacteristic
34C2×H(3,𝔽3)#10811yescharacteristic
35C2×C3×C8—62no—
36C2×C3×C8—91no—
37C2×C3×C8—182no—
38C2×C3×C8#15711yescharacteristic
39C3×C42—91no—
40C4×Dic3—62no—
41C2×C3⋊C8—62no—
42C22×C32—22no—
43C32×C4—22no—
44C32×C4—22no—
45C3×Dic3—64no—
46C3×Dic3—64no—
47C4⋊C8—92no—
48C4×C8—91no—
49H(3,𝔽3)#33411yescharacteristic, complemented, Sylow
50C2×C3×C4—62no—
51C2×C3×C4—91no—
52C2×C3×C4—91no—
53C2×C3×C4C2×#15711yescharacteristic
54C2×Dic3—62no—
55C2×Dic3—62no—
56C3×C8—21no—
57C3×C8—64no—
58C3×C8—92no—
59C3×C8—362no—
60C3⋊C8—64no—
61C2×C32—22no—
62C2×C32—22no—
63C2×C32—22no—
64C2×C8—91no—
65C2×C8—182no—
66C2×C8#108811yescharacteristic
67C42—91no—
68C22×C3—62no—
69C22×C3C4⋊#15711yescharacteristic
70C3×C4—62no—
71C3×C4—62no—
72C3×C4—92no—
73C3×C4—181no—
74C3×C4#378711yescharacteristic
75C3×C4#379211yescharacteristic
76Dic3—64no—
77Dic3—64no—
78C32—22no—
79C2×C4—91no—
80C2×C4—91no—
81C2×C4C2×#108811yescharacteristic
82C8—21no—
83C8—92no—
84C8—362no—
85C2×C3—62no—
86C2×C3—62no—
87C2×C3—62no—
88C2×C3#6376811yescharacteristic
89C2×C3#6377111yescharacteristic
90C2×C3#6377911yescharacteristic
91C22C4⋊#108811yescharacteristic
92C4—92no—
93C4—181no—
94C4#8862611yescharacteristic
95C4#8863111yescharacteristic
96C3—62no—
97C3#1059017211yescharacteristic
98C2#1172077411yescharacteristic
99C2#1172077711yescharacteristic
100C2#1172078511yescharacteristic
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)#334
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
13. #6180C8semidirect
49. H(3,𝔽3)#334semidirect

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×#3349