Finite Group Databasebeta

← #13136952

Subgroups — #13136952

Order 1728.

226 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 23 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#11721861C211yesmaximal, characteristic
2#11720786C212yesmaximal
3C2×Q8×H(3,𝔽3)C411yescharacteristic
4C2×#6229C2211yescharacteristic
5#88631—22no—
6#88540—22no—
7C4×#1084C411yescharacteristic
8C3×#944—62no—
9Q8×H(3,𝔽3)—41no—
10C2×C4×H(3,𝔽3)—21no—
11C2×C4×H(3,𝔽3)C2×C411yescharacteristic
12C2×#1084—21no—
13C2×#1084D412yes—
14#6229—41no—
15#6188—42no—
16C3×#356—91nomaximal
17C2×C32×Q8—22no—
18C2×C3×Dic6—62no—
19C3×Dic3⋊C4—68no—
20C3×C4×Dic3—62no—
21C22×H(3,𝔽3)C22⋊C411yescharacteristic
22C4×H(3,𝔽3)—21no—
23C4×H(3,𝔽3)—41no—
24#1084—22no—
25#1084—22no—
26C3×C4⋊Q8—91no—
27#944—62no—
28C3×#101—92no—
29C32×Q8—44no—
30C2×C32×C4—24no—
31C2×C32×C4—22no—
32C2×C3×Dic3—64no—
33C2×C3×Dic3—64no—
34C3×Dic6—124no—
35#356—91noSylow
36C2×H(3,𝔽3)—21no—
37C2×H(3,𝔽3)#9911yescharacteristic
38C2×C3×Q8—62no—
39C2×C3×Q8—91no—
40C2×C3×Q8#15711yescharacteristic
41C2×Dic6—62no—
42C3×M16—182no—
43C3×C4⋊C4—182no—
44C3×C42—91no—
45Dic3⋊C4—68no—
46C4×Dic3—62no—
47C22×C32—22no—
48C32×C4—24no—
49C32×C4—44no—
50C3×Dic3—68no—
51C3×Dic3—124no—
52C4⋊Q8—91no—
53#101—92no—
54H(3,𝔽3)#35611yescharacteristic, complemented, Sylow
55C3×Q8—41no—
56C3×Q8—124no—
57C3×Q8—361no—
58C2×C3×C4—21no—
59C2×C3×C4—64no—
60C2×C3×C4—62no—
61C2×C3×C4—92no—
62C2×C3×C4—181no—
63C2×C3×C4C2×#15711yescharacteristic
64C2×Dic3—64no—
65C2×Dic3—64no—
66Dic6—124no—
67C3×C8—362no—
68C2×C32—24no—
69C2×C32—22no—
70C2×Q8—91no—
71C2×Q8#108811yescharacteristic
72M16—182no—
73C4⋊C4—182no—
74C42—91no—
75C22×C3—62no—
76C22×C3C22⋊#15711yescharacteristic
77C3×C4—21no—
78C3×C4—41no—
79C3×C4—64no—
80C3×C4—124no—
81C3×C4—182no—
82C3×C4—182no—
83Dic3—68no—
84Dic3—124no—
85C32—22no—
86Q8—41no—
87Q8—361no—
88C2×C4—21no—
89C2×C4—92no—
90C2×C4—181no—
91C2×C4C2×#108811yescharacteristic
92C8—362no—
93C2×C3—21no—
94C2×C3—64no—
95C2×C3—62no—
96C2×C3#6378611yescharacteristic
97C22C22⋊#108811yescharacteristic
98C4—21no—
99C4—41no—
100C4—182no—
101C4—182no—
102C3—62no—
103C3#1059021811yescharacteristic
104C2—21no—
105C2#1172079211yescharacteristic
subgroupG/H
center, socleC2×C3#63786
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC2×C3×C4C2×#157
FittingC2×Q8×H(3,𝔽3)C4
prime radicalC22×H(3,𝔽3)C22⋊C4
cosocleC2×#6229C22
hypercenterC2×C3×Q8#157
hypocenterH(3,𝔽3)#356
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
54. H(3,𝔽3)#356semidirect

Maximal subgroups

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

subgroupindex
#117218612
#117207862
C3×#3569