Finite Group Databasebeta

← #13152209

Subgroups — #13152209

Order 1728.

356 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 53 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#11721347C213yesmaximal, complemented
2C32×H(3,𝔽4)C311yesmaximal, characteristic
3#10589552C313yesmaximal, complemented
4#88907C2211yescharacteristic, complemented
5C22×#1094—41nomaximal
6C32×#124—33no—
7C32×#120—31no—
8C32×#120C2×C313yes—
9#120⋊C9C2×C319yescomplemented
10C2×#1094—43no—
11C3×H(3,𝔽4)—31no—
12C3×H(3,𝔽4)C3211yescharacteristic
13C24×C32A411yescharacteristic
14C24×C32C22×C311yescharacteristic
15C2×C32×Q8—31no—
16C2×C32×D4—33no—
17C22⋊(C22⋊C9)C22×C313yescomplemented
18C22×C22⋊C9—43no—
19C32×C22⋊C4—33no—
20C32×C22⋊C4—33no—
21C32×C42—31no—
22C22×3−1+2—161no—
23#1094—41no—
24#1094—161no—
25C3×#124—39no—
26C3×#124—33no—
27C3×#120—33no—
28C3×#120—31no—
29C3×#120—33no—
30C3×#120C2×C3213yes—
31C23×C32—31no—
32C23×C32—121no—
33C23×C32—121no—
34C23×C32C2×A413yes—
35C32×Q8—121no—
36C32×D4—123no—
37C2×C32×C4—33no—
38C2×C22⋊C9—49no—
39H(3,𝔽4)3−1+211yescharacteristic, complemented, Sylow
40C2×3−1+2—163no—
41C24×C3—31no—
42C24×C3—31no—
43C24×C3C3×A411yescharacteristic
44C24×C3C22×C3211yescharacteristic
45C2×C3×Q8—33no—
46C2×C3×Q8—31no—
47C2×C3×D4—39no—
48C2×C3×D4—33no—
49C3×C22⋊C4—39no—
50C3×C22⋊C4—39no—
51C3×C22⋊C4—33no—
52C3×C22⋊C4—33no—
53C3×C42—33no—
54C3×C42—31no—
55C22×C32—41no—
56C22×C32—41no—
57C22×C32—63no—
58C22×C32—63no—
59C22×C32—121no—
60C22×C32—121no—
61C22×C32C22×A411yescharacteristic
62C32×C4—63no—
63C22×C9—163no—
64C22⋊C9—43no—
65C22⋊C9—163no—
66#124—33no—
67#120—31no—
68#120C2×3−1+213yescomplemented
693−1+2—161noSylow
70C23×C3—33no—
71C23×C3—31no—
72C23×C3—33no—
73C23×C3—121no—
74C23×C3—121no—
75C23×C3—123no—
76C23×C3—123no—
77C23×C3C2×C3×A413yes—
78C3×Q8—123no—
79C3×Q8—121no—
80C3×D4—123no—
81C3×D4—129no—
82C2×C3×C4—39no—
83C2×C3×C4—33no—
84C2×C32—31no—
85C2×C32—43no—
86C2×C32—121no—
87C2×C9—169no—
88C24#109411yescharacteristic, complemented
89C24C22×3−1+211yescharacteristic, complemented
90C2×Q8—31no—
91C2×D4—33no—
92C22⋊C4—33no—
93C22⋊C4—33no—
94C42—31no—
95C22×C3—31no—
96C22×C3—41no—
97C22×C3—41no—
98C22×C3—63no—
99C22×C3—69no—
100C22×C3—69no—
101C22×C3—63no—
102C22×C3—121no—
103C22×C3—121no—
104C22×C3—121no—
105C22×C3—121no—
106C22×C3—123no—
107C22×C3—123no—
108C22×C3C22×C3×A411yescharacteristic
109C3×C4—69no—
110C3×C4—63no—
111C32#545011yescharacteristic
112C9—163no—
113C23—31no—
114C23—121no—
115C23—121no—
116C23C2×#109413yes—
117Q8—121no—
118D4—123no—
119C2×C4—33no—
120C2×C3—33no—
121C2×C3—31no—
122C2×C3—43no—
123C2×C3—123no—
124C2×C3—123no—
125C2×C3—121no—
126C22—41no—
127C22—41no—
128C22—63no—
129C22—63no—
130C22—121no—
131C22—121no—
132C22C22×#109411yescharacteristic
133C4—63no—
134C3—31no—
135C3C3×#545011yescharacteristic
136C2—31no—
137C2—43no—
138C2—121no—
subgroupG/H
centerC3C3×#5450
commutator, cosocleC24×C3C22×C32
Frattini, socleC22×C3C22×C3×A4
Fitting, prime radicalC32×H(3,𝔽4)C3
hypercenterC32#5450
hypocenterC24C22×3−1+2
LCSG ▷ C24×C3 ▷ C24
UCSC1 ◁ C3 ◁ C32

Complements

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

NH ≅ G/Ntype
1. #11721347C2semidirect
3. #10589552C3semidirect
4. #88907C22semidirect
9. #120⋊C9C2×C3semidirect
17. C22⋊(C22⋊C9)C22×C3semidirect
39. H(3,𝔽4)3−1+2semidirect
68. #120C2×3−1+2semidirect
88. C24#1094semidirect
89. C24C22×3−1+2semidirect

Maximal subgroups

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

subgroupindex
#117213472
C32×H(3,𝔽4)3
#105895523
C22×#10944