Finite Group Databasebeta

← #13136888

Subgroups — #13136888

Order 1728.

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

normal subgroups only

This group has 258 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#11721742C211yesmaximal, characteristic
2#11720794C211yesmaximal, characteristic
3#11720792C211yesmaximal, characteristic
4C2×#6235C2211yescharacteristic
5C22×#1084C411yescharacteristic, complemented
6C22⋊#1088—21no—
7#88631—21no—
8C22⋊C4×H(3,𝔽3)C411yescharacteristic, complemented
9#88542—21no—
10C4⋊#1084—21no—
11C3×#886—62no—
12C23×H(3,𝔽3)C2×C411yescharacteristic
13C22×#272D411yescharacteristic
14C2×#1088—41no—
15C2×C4×H(3,𝔽3)—21no—
16#6235—41no—
17C2×#1084—21no—
18C2×#1084—41no—
19C2×#1084D411yescharacteristic
20#6188—41no—
21C3×#352—91nomaximal
22C2×C3×#67—62no—
23C22×C3×Dic3—62no—
24C32×C22⋊C4—22no—
25C3×(C2×D3)⋊C4—64no—
26C3×C4⋊Dic3—64no—
27C22×H(3,𝔽3)—21no—
28C22×H(3,𝔽3)—41no—
29C22×H(3,𝔽3)C22⋊C411yescharacteristic
30C2×#272—21no—
31C2×#272—41no—
32#1088—81no—
33C4×H(3,𝔽3)—41no—
34#1084—21no—
35#1084—41no—
36C3×#123—91no—
37#886—62no—
38C3×#100—91no—
39C3×#99—91no—
40C23×C32—22no—
41C22×C3×D3—62no—
42C2×C32×C4—24no—
43C3×#67—124no—
44C2×C3×Dic3—62no—
45C2×C3×Dic3—64no—
46C2×C3×Dic3—124no—
47#352—91noSylow
48C2×H(3,𝔽3)—21no—
49C2×H(3,𝔽3)—41no—
50C2×H(3,𝔽3)#9911yescharacteristic
51#272—41no—
52C2×C3×D4—91no—
53C22×C3×C4—91no—
54C2×#67—62no—
55C22×Dic3—62no—
56C3×M16—181no—
57C3×C4⋊C4—181no—
58C3×C22⋊C4—62no—
59C3×C22⋊C4—181no—
60C3×C22⋊C4—181no—
61C3×C22⋊C4#15711yescharacteristic
62(C2×D3)⋊C4—64no—
63C4⋊Dic3—64no—
64C22×C32—24no—
65C22×C32—22no—
66C22×C32—44no—
67C2×C3×D3—122no—
68C2×C3×D3—124no—
69C32×C4—44no—
70C3×Dic3—122no—
71C3×Dic3—124no—
72#123—91no—
73#100—91no—
74#99—91no—
75H(3,𝔽3)#35211yescharacteristic, complemented, Sylow
76C23×C3—62no—
77C23×C3—91no—
78C23×C3C2×#15711yescharacteristic
79C22×D3—62no—
80C3×D4—361no—
81C2×C3×C4—21no—
82C2×C3×C4—64no—
83C2×C3×C4—91no—
84C2×C3×C4—181no—
85C2×C3×C4—361no—
86C2×C3×C4—361no—
87#67—124no—
88C2×Dic3—62no—
89C2×Dic3—64no—
90C2×Dic3—124no—
91C3×C8—361no—
92C2×C32—24no—
93C2×C32—22no—
94C2×C32—44no—
95C3×D3—242no—
96C2×D4—91no—
97C22×C4—91no—
98M16—181no—
99C4⋊C4—181no—
100C22⋊C4—181no—
101C22⋊C4—181no—
102C22⋊C4#108811yescharacteristic, complemented
103C22×C3—21no—
104C22×C3—41no—
105C22×C3—64no—
106C22×C3—62no—
107C22×C3—124no—
108C22×C3—181no—
109C22×C3—361no—
110C22×C3C22⋊#15711yescharacteristic
111C2×D3—122no—
112C2×D3—124no—
113C3×C4—41no—
114C3×C4—124no—
115C3×C4—181no—
116C3×C4—361no—
117C3×C4—721no—
118Dic3—122no—
119Dic3—124no—
120C32—22no—
121C23—91no—
122C23C2×#108811yescharacteristic
123D4—361no—
124C2×C4—21no—
125C2×C4—91no—
126C2×C4—181no—
127C2×C4—361no—
128C2×C4—361no—
129C8—361no—
130C2×C3—21no—
131C2×C3—41no—
132C2×C3—64no—
133C2×C3—62no—
134C2×C3—124no—
135C2×C3—361no—
136C2×C3#6377411yescharacteristic
137D3—242no—
138C22—21no—
139C22—41no—
140C22—181no—
141C22—361no—
142C22C22⋊#108811yescharacteristic
143C4—41no—
144C4—181no—
145C4—361no—
146C4—721no—
147C3—62no—
148C3#1059015411yescharacteristic
149C2—21no—
150C2—41no—
151C2—361no—
152C2#1172078011yescharacteristic
subgroupG/H
center, socleC2×C3#63774
commutatorC23×H(3,𝔽3)C2×C4
FrattiniC23×C3C2×#157
FittingC22⋊C4×H(3,𝔽3)C4
prime radical, cosocleC2×#6235C22
hypercenterC3×C22⋊C4#157
hypocenterH(3,𝔽3)#352
LCSG ▷ C23×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C23×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
5. C22×#1084C4semidirect
8. C22⋊C4×H(3,𝔽3)C4semidirect
75. H(3,𝔽3)#352semidirect
102. C22⋊C4#1088semidirect

Maximal subgroups

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

subgroupindex
#117217422
#117207942
#117207922
C3×#3529