Finite Group Databasebeta

← #13136946

Subgroups — #13136946

Order 1728.

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

normal subgroups only

This group has 248 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#11721846C211yesmaximal, characteristic
2#11720786C212yesmaximal, complemented
3C2×D4×H(3,𝔽3)C411yescharacteristic
4C2×#6229C2211yescharacteristic
5#88631—22no—
6#88552—22no—
7C4×#1084C411yescharacteristic
8C3×#936—62no—
9C23×H(3,𝔽3)—21no—
10D4×H(3,𝔽3)—41no—
11C2×C4×H(3,𝔽3)C2×C411yescharacteristic
12C2×#1084—21no—
13C2×#1084D412yes—
14#6229—41no—
15#6188—42no—
16C3×#355—91nomaximal
17C2×C32×D4—22no—
18C2×C3×Dic6—62no—
19C3×#217—68no—
20C3×C4×Dic3—62no—
21C22×H(3,𝔽3)—42no—
22C22×H(3,𝔽3)—41no—
23C22×H(3,𝔽3)C22⋊C411yescharacteristic
24C4×H(3,𝔽3)—21no—
25#1084—22no—
26#1084—41no—
27C3×#124—91no—
28#936—62no—
29C3×#101—92no—
30C23×C32—24no—
31C32×D4—44no—
32C2×C32×C4—22no—
33C2×C3×Dic3—64no—
34C2×C3×Dic3—64no—
35C3×Dic6—124no—
36#355—91noSylow
37C2×H(3,𝔽3)—21no—
38C2×H(3,𝔽3)—81no—
39C2×H(3,𝔽3)#9911yescharacteristic
40C2×C3×Q8—91no—
41C2×C3×D4—62no—
42C2×C3×D4#15711yescharacteristic
43C2×Dic6—62no—
44C3×M16—182no—
45C3×C22⋊C4—182no—
46C3×C42—91no—
47#217—68no—
48C4×Dic3—62no—
49C22×C32—22no—
50C22×C32—44no—
51C22×C32—48no—
52C32×C4—24no—
53C3×Dic3—124no—
54C3×Dic3—124no—
55#124—91no—
56#101—92no—
57H(3,𝔽3)#35511yescharacteristic, complemented, Sylow
58C23×C3—21no—
59C23×C3—64no—
60C3×Q8—361no—
61C3×D4—41no—
62C3×D4—124no—
63C2×C3×C4—62no—
64C2×C3×C4—92no—
65C2×C3×C4—181no—
66C2×C3×C4C2×#15711yescharacteristic
67C2×Dic3—64no—
68C2×Dic3—64no—
69Dic6—124no—
70C3×C8—362no—
71C2×C32—24no—
72C2×C32—22no—
73C2×C32—84no—
74C2×Q8—91no—
75C2×D4#108811yescharacteristic
76M16—182no—
77C22⋊C4—182no—
78C42—91no—
79C22×C3—42no—
80C22×C3—41no—
81C22×C3—62no—
82C22×C3—124no—
83C22×C3—128no—
84C22×C3C22⋊#15711yescharacteristic
85C3×C4—21no—
86C3×C4—64no—
87C3×C4—182no—
88C3×C4—361no—
89Dic3—124no—
90Dic3—124no—
91C32—22no—
92C23—21no—
93Q8—361no—
94D4—41no—
95C2×C4—92no—
96C2×C4—181no—
97C2×C4C2×#108811yescharacteristic
98C8—362no—
99C2×C3—21no—
100C2×C3—64no—
101C2×C3—62no—
102C2×C3—81no—
103C2×C3—244no—
104C2×C3#6378611yescharacteristic
105C22—42no—
106C22—41no—
107C22C22⋊#108811yescharacteristic
108C4—21no—
109C4—182no—
110C4—361no—
111C3—62no—
112C3#1059021211yescharacteristic
113C2—21no—
114C2—81no—
115C2#1172079211yescharacteristic
subgroupG/H
center, socleC2×C3#63786
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC2×C3×C4C2×#157
Fitting, prime radicalC2×D4×H(3,𝔽3)C4
cosocleC2×#6229C22
hypercenterC2×C3×D4#157
hypocenterH(3,𝔽3)#355
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×D4

Complements

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

NH ≅ G/Ntype
2. #11720786C2semidirect
57. H(3,𝔽3)#355semidirect

Maximal subgroups

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

subgroupindex
#117218462
#117207862
C3×#3559