Finite Group Databasebeta

← #13136926

Subgroups — #13136926

Order 1728.

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

normal subgroups only

This group has 211 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
1#11721794C211yesmaximal, characteristic
2#11720774C211yesmaximal, characteristic, complemented
3#11720771C211yesmaximal, characteristic, complemented
4#88744—21no—
5C4⋊#1088—21no—
6D8×H(3,𝔽3)C411yescharacteristic, complemented
7#88546—21no—
8#88530C411yescharacteristic, complemented
9C8×#272C2211yescharacteristic
10#88410—21no—
11C3×#912—62no—
12C2×#1088—41no—
13D4×H(3,𝔽3)—21no—
14#6235—41no—
15C2×#1084—41no—
16C4×#272D411yescharacteristic
17#6229—21no—
18C8×H(3,𝔽3)C2×C411yescharacteristic
19#6180D411yescharacteristic
20C3×#360—91nomaximal
21C3×#237—64no—
22C32×D8—22no—
23C3×#214—64no—
24C3×Dic12—62no—
25C3×D3×C8—62no—
26C22×H(3,𝔽3)—41no—
27C2×#272D811yescharacteristic
28#1088—81no—
29C4×H(3,𝔽3)C22⋊C411yescharacteristic
30#1084—41no—
31#1084SD1611yescharacteristic
32C3×#135—91no—
33#912—62no—
34C3×M32—91no—
35C3×#106—91no—
36C32×D4—24no—
37C3×#67—124no—
38C2×C3×Dic3—124no—
39C3×C4×D3—62no—
40C3×Dic6—64no—
41C32×C8—22no—
42C3×C3⋊C8—62no—
43#360—91noSylow
44C2×H(3,𝔽3)—81no—
45C2×H(3,𝔽3)#10211yescharacteristic
46#272—21no—
47C3×#41—181no—
48#237—64no—
49C3×Q16—91no—
50C3×SD16—181no—
51C3×D8—62no—
52C3×D8#15711yescharacteristic
53C2×C3×C8—91no—
54C3×C4⋊C4—181no—
55#214—64no—
56Dic12—62no—
57D3×C8—62no—
58C3×C16—181no—
59C22×C32—44no—
60C2×C3×D3—62no—
61C32×C4—22no—
62C3×Dic3—62no—
63C3×Dic3—124no—
64#135—91no—
65M32—91no—
66#106—91no—
67H(3,𝔽3)#36011yescharacteristic, complemented, Sylow
68C3×Q8—181no—
69C3×D4—21no—
70C3×D4—64no—
71C3×D4—361no—
72C2×C3×C4—91no—
73C2×C3×C4—361no—
74C2×C3×C4—361no—
75#67—124no—
76C2×Dic3—124no—
77C4×D3—62no—
78Dic6—64no—
79C3×C8—62no—
80C3×C8—91no—
81C3×C8C2×#15711yescharacteristic
82C3⋊C8—62no—
83C2×C32—22no—
84C2×C32—84no—
85C3×D3—122no—
86#41—181no—
87Q16—91no—
88SD16—181no—
89D8#108811yescharacteristic, complemented
90C2×C8—91no—
91C4⋊C4—181no—
92C16—181no—
93C22×C3—41no—
94C22×C3—91no—
95C22×C3—124no—
96C2×D3—62no—
97C3×C4—62no—
98C3×C4—91no—
99C3×C4—361no—
100C3×C4—721no—
101C3×C4C22⋊#15711yescharacteristic
102Dic3—62no—
103Dic3—124no—
104C32—22no—
105Q8—181no—
106D4—21no—
107D4—361no—
108C2×C4—91no—
109C2×C4—361no—
110C2×C4—361no—
111C8—91no—
112C8C2×#108811yescharacteristic
113C2×C3—62no—
114C2×C3—81no—
115C2×C3—181no—
116C2×C3—244no—
117C2×C3#6378211yescharacteristic
118D3—122no—
119C22—41no—
120C22—91no—
121C4—91no—
122C4—361no—
123C4—721no—
124C4C22⋊#108811yescharacteristic
125C3—62no—
126C3#1059019211yescharacteristic
127C2—81no—
128C2—181no—
129C2#1172078811yescharacteristic
subgroupG/H
center, socleC2×C3#63782
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC3×C8C2×#157
FittingD8×H(3,𝔽3)C4
prime radical#11721794C2
cosocleC8×#272C22
hypercenterC3×D8#157
hypocenterH(3,𝔽3)#360
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. #11720774C2semidirect
3. #11720771C2semidirect
6. D8×H(3,𝔽3)C4semidirect
8. #88530C4semidirect
67. H(3,𝔽3)#360semidirect
89. D8#1088semidirect

Maximal subgroups

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

subgroupindex
#117217942
#117207742
#117207712
C3×#3609