Finite Group Databasebeta

← #13136931

Subgroups — #13136931

Order 1728.

207 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 27 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
1Q16×#272C211yesmaximal, characteristic
2#11720775C211yesmaximal, characteristic
3#11720770C211yesmaximal, characteristic
4Q8×#272—21no—
5C4⋊#1088—21no—
6Q16×H(3,𝔽3)C411yescharacteristic, complemented
7#88550—21no—
8#88530C411yescharacteristic, complemented
9C8×#272C2211yescharacteristic
10#88410—21no—
11C3×D3×Q16—62no—
12C2×#1088—41no—
13Q8×H(3,𝔽3)—21no—
14C4×#272—41no—
15C4×#272D411yescharacteristic
16#6229—21no—
17#6229—41no—
18C8×H(3,𝔽3)C2×C411yescharacteristic
19#6180D411yescharacteristic
20C3×#358—91nomaximal
21C3×D3×Q8—64no—
22C32×Q16—22no—
23C3×#216—64no—
24C3×Dic12—62no—
25C3×D3×C8—62no—
26C2×#272D811yescharacteristic
27#1088—81no—
28C4×H(3,𝔽3)—41no—
29C4×H(3,𝔽3)C22⋊C411yescharacteristic
30#1084—41no—
31#1084SD1611yescharacteristic
32C2×C3×Q16—91no—
33D3×Q16—62no—
34C2×C3×C16—91no—
35C3×#107—91no—
36C32×Q8—24no—
37C3×C4×D3—62no—
38C3×C4×D3—124no—
39C3×Dic6—64no—
40C3×Dic6—124no—
41C32×C8—22no—
42C3×C3⋊C8—62no—
43#358—91noSylow
44C2×H(3,𝔽3)#10211yescharacteristic
45#272SD3211yescharacteristic
46#272Q3211yescharacteristic
47C2×C3×Q8—181no—
48D3×Q8—64no—
49C3×Q16—62no—
50C3×Q16—91no—
51C3×Q16—181no—
52C3×Q16#15711yescharacteristic
53C2×C3×C8—91no—
54C3×C4⋊C4—181no—
55#216—64no—
56Dic12—62no—
57D3×C8—62no—
58C3×C16—181no—
59C2×C3×D3—62no—
60C32×C4—22no—
61C32×C4—44no—
62C3×Dic3—62no—
63C3×Dic3—124no—
64C2×Q16—91no—
65C2×C16—91no—
66#107—91no—
67H(3,𝔽3)#35811yescharacteristic, complemented, Sylow
68C3×Q8—21no—
69C3×Q8—64no—
70C3×Q8—181no—
71C3×Q8—361no—
72C2×C3×C4—91no—
73C2×C3×C4—361no—
74C2×C3×C4—361no—
75C4×D3—62no—
76C4×D3—124no—
77Dic6—64no—
78Dic6—124no—
79C3×C8—62no—
80C3×C8—91no—
81C3×C8C2×#15711yescharacteristic
82C3⋊C8—62no—
83C2×C32—22no—
84C3×D3—62no—
85C3×D3—62no—
86C2×Q8—181no—
87Q16—91no—
88Q16—181no—
89Q16#108811yescharacteristic, complemented
90C2×C8—91no—
91C4⋊C4—181no—
92C16—181no—
93C22×C3—91no—
94C2×D3—62no—
95C3×C4—41no—
96C3×C4—62no—
97C3×C4—91no—
98C3×C4—124no—
99C3×C4—361no—
100C3×C4—721no—
101C3×C4C22⋊#15711yescharacteristic
102Dic3—62no—
103Dic3—124no—
104C32—22no—
105Q8—21no—
106Q8—181no—
107Q8—361no—
108C2×C4—91no—
109C2×C4—361no—
110C2×C4—361no—
111C8—91no—
112C8C2×#108811yescharacteristic
113C2×C3—62no—
114C2×C3—91no—
115C2×C3—91no—
116C2×C3#6378211yescharacteristic
117D3—62no—
118D3—62no—
119C22—91no—
120C4—41no—
121C4—91no—
122C4—361no—
123C4—721no—
124C4C22⋊#108811yescharacteristic
125C3—62no—
126C3#1059019711yescharacteristic
127C2—91no—
128C2—91no—
129C2#1172078811yescharacteristic
subgroupG/H
center, socleC2×C3#63782
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC3×C8C2×#157
FittingQ16×H(3,𝔽3)C4
prime radicalC2×#272D8
cosocleC8×#272C22
hypercenterC3×Q16#157
hypocenterH(3,𝔽3)#358
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 ◁ C3×Q16

Complements

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

NH ≅ G/Ntype
6. Q16×H(3,𝔽3)C4semidirect
8. #88530C4semidirect
67. H(3,𝔽3)#358semidirect
89. Q16#1088semidirect

Maximal subgroups

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

subgroupindex
Q16×#2722
#117207752
#117207702
C3×#3589