Finite Group Databasebeta

← #13136929

Subgroups — #13136929

Order 1728.

245 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 25 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
1D8×#272C211yesmaximal, characteristic
2#11720776C211yesmaximal, characteristic, complemented
3#11720771C211yesmaximal, characteristic, complemented
4D4×#272—21no—
5#88627—21no—
6D8×H(3,𝔽3)C411yescharacteristic
7#88545—21no—
8#88529C411yescharacteristic
9C8×#272C2211yescharacteristic
10#88410—21no—
11C3×D3×D8—62no—
12C22×#272—41no—
13D4×H(3,𝔽3)—21no—
14#6235—41no—
15C4⋊#272—21no—
16C4×#272D411yescharacteristic
17#6188—41no—
18C8×H(3,𝔽3)C2×C411yescharacteristic
19#6180D411yescharacteristic
20C3×#361—91nomaximal
21C3×D3×D4—64no—
22C32×D8—22no—
23C3×#213—64no—
24C3×D24—62no—
25C3×D3×C8—62no—
26C22×H(3,𝔽3)—41no—
27C2×#272—41no—
28C2×#272—82no—
29C2×#272SD1611yescharacteristic
30C4×H(3,𝔽3)C22⋊C411yescharacteristic
31#1084D811yescharacteristic
32C2×C3×D8—91no—
33D3×D8—62no—
34C3×M32—91no—
35C3×#108—91no—
36C22×C3×D3—124no—
37C32×D4—24no—
38C3×#67—124no—
39C3×D12—64no—
40C3×C4×D3—62no—
41C32×C8—22no—
42C3×C3⋊C8—62no—
43#361—91noSylow
44C2×H(3,𝔽3)—81no—
45C2×H(3,𝔽3)#10211yescharacteristic
46#272—21no—
47#272—81no—
48C2×C3×D4—181no—
49D3×D4—64no—
50C3×D8—62no—
51C3×D8—91no—
52C3×D8—181no—
53C3×D8#15711yescharacteristic
54C3×M16—181no—
55C2×C3×C8—91no—
56#213—64no—
57D24—62no—
58D3×C8—62no—
59C3×C16—181no—
60C22×C32—44no—
61C2×C3×D3—62no—
62C2×C3×D3—124no—
63C2×C3×D3—248no—
64C32×C4—22no—
65C3×Dic3—62no—
66C2×D8—91no—
67M32—91no—
68#108—91no—
69H(3,𝔽3)#36111yescharacteristic, complemented, Sylow
70C23×C3—361no—
71C22×D3—124no—
72C3×D4—21no—
73C3×D4—64no—
74C3×D4—181no—
75C3×D4—361no—
76C2×C3×C4—91no—
77#67—124no—
78D12—64no—
79C4×D3—62no—
80C3×C8—62no—
81C3×C8—91no—
82C3×C8—361no—
83C3×C8C2×#15711yescharacteristic
84C3⋊C8—62no—
85C2×C32—22no—
86C2×C32—84no—
87C3×D3—64no—
88C3×D3—244no—
89C2×D4—181no—
90D8—91no—
91D8—181no—
92D8#108811yescharacteristic
93M16—181no—
94C2×C8—91no—
95C16—181no—
96C22×C3—41no—
97C22×C3—91no—
98C22×C3—124no—
99C22×C3—361no—
100C22×C3—722no—
101C2×D3—62no—
102C2×D3—124no—
103C2×D3—248no—
104C3×C4—62no—
105C3×C4—91no—
106C3×C4C22⋊#15711yescharacteristic
107Dic3—62no—
108C32—22no—
109C23—361no—
110D4—21no—
111D4—181no—
112D4—361no—
113C2×C4—91no—
114C8—91no—
115C8—361no—
116C8C2×#108811yescharacteristic
117C2×C3—62no—
118C2×C3—81no—
119C2×C3—181no—
120C2×C3—244no—
121C2×C3—721no—
122C2×C3#6378211yescharacteristic
123D3—64no—
124D3—244no—
125C22—41no—
126C22—91no—
127C22—361no—
128C22—722no—
129C4—91no—
130C4C22⋊#108811yescharacteristic
131C3—62no—
132C3#1059019511yescharacteristic
133C2—81no—
134C2—181no—
135C2—721no—
136C2#1172078811yescharacteristic
subgroupG/H
center, socleC2×C3#63782
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC3×C8C2×#157
FittingD8×H(3,𝔽3)C4
prime radicalD8×#272C2
cosocleC8×#272C22
hypercenterC3×D8#157
hypocenterH(3,𝔽3)#361
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 ◁ C3×D8

Complements

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

NH ≅ G/Ntype
2. #11720776C2semidirect
3. #11720771C2semidirect
69. H(3,𝔽3)#361semidirect

Maximal subgroups

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

subgroupindex
D8×#2722
#117207762
#117207712
C3×#3619