Finite Group Databasebeta

← #13136954

Subgroups — #13136954

Order 1728.

232 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 33 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#11720785C212yesmaximal
3C2×Q8×H(3,𝔽3)C411yescharacteristic
4C2×#6229C411yescharacteristic
5C2×#6188—22no—
6#88540—22no—
7C4×#1084C2211yescharacteristic
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×#333—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—41no—
25#1084SD1612yes—
26#1084Q1612yes—
27C3×C4⋊Q8—91no—
28#944—62no—
29C3×C4⋊C8—92no—
30C32×Q8—44no—
31C2×C32×C4—24no—
32C2×C32×C4—22no—
33C2×C3×Dic3—64no—
34C2×C3×Dic3—64no—
35C3×Dic6—124no—
36#333—91noSylow
37C2×H(3,𝔽3)#10011yescharacteristic
38C2×H(3,𝔽3)#10312yes—
39C2×C3×Q8—62no—
40C2×C3×Q8—91no—
41C2×C3×Q8#15711yescharacteristic
42C2×Dic6—62no—
43C2×C3×C8—182no—
44C3×C4⋊C4—182no—
45C3×C42—91no—
46Dic3⋊C4—68no—
47C4×Dic3—62no—
48C22×C32—22no—
49C32×C4—24no—
50C32×C4—44no—
51C3×Dic3—64no—
52C3×Dic3—64no—
53C3×Dic3—124no—
54C4⋊Q8—91no—
55C4⋊C8—92no—
56H(3,𝔽3)#33311yescharacteristic, complemented, Sylow
57C3×Q8—41no—
58C3×Q8—124no—
59C3×Q8—361no—
60C2×C3×C4—21no—
61C2×C3×C4—64no—
62C2×C3×C4—62no—
63C2×C3×C4—92no—
64C2×C3×C4—181no—
65C2×C3×C4C2×#15711yescharacteristic
66C2×Dic3—64no—
67C2×Dic3—64no—
68Dic6—124no—
69C3×C8—362no—
70C2×C32—24no—
71C2×C32—22no—
72C2×Q8—91no—
73C2×Q8#108811yescharacteristic
74C2×C8—182no—
75C4⋊C4—182no—
76C42—91no—
77C22×C3—62no—
78C22×C3C22⋊#15711yescharacteristic
79C3×C4—21no—
80C3×C4—41no—
81C3×C4—64no—
82C3×C4—92no—
83C3×C4—92no—
84C3×C4—124no—
85C3×C4—361no—
86Dic3—64no—
87Dic3—64no—
88Dic3—124no—
89C32—22no—
90Q8—41no—
91Q8—361no—
92C2×C4—21no—
93C2×C4—92no—
94C2×C4—181no—
95C2×C4C2×#108811yescharacteristic
96C8—362no—
97C2×C3—64no—
98C2×C3—62no—
99C2×C3#6378412yes—
100C2×C3#6378811yescharacteristic
101C22C22⋊#108811yescharacteristic
102C4—21no—
103C4—41no—
104C4—92no—
105C4—92no—
106C4—361no—
107C3—62no—
108C3#1059022011yescharacteristic
109C2#1172079012yes—
110C2#1172079411yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#157
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC2×C3×C4C2×#157
FittingC2×Q8×H(3,𝔽3)C4
prime radicalC22×H(3,𝔽3)C22⋊C4
cosocleC4×#1084C22
hypercenterC2×C3×Q8#157
hypocenterH(3,𝔽3)#333
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
56. H(3,𝔽3)#333semidirect

Maximal subgroups

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

subgroupindex
#117218612
#117207852
C3×#3339