Finite Group Databasebeta

← #13047026

Subgroups — #13047026

Order 1296.

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

normal subgroups only

This group has 236 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
1C2×#66368C211yesmaximal, characteristic
2C3×#3757—31nomaximal
3(C3⋊SL(2,ℤ/4))⋊C3—31nomaximal
4#1094⋊C4—32nomaximal
5#66368C411yescharacteristic, complemented
6#66330—41nomaximal
7C23×C33D311yescharacteristic
8C2×C3×C3⋊Dic3—31no—
9C2×C3×C3⋊Dic3—31no—
10C2×C22⋊H(3,𝔽3)D311yescharacteristic
11C2×#1094D312yes—
12C2×AΣL(1,𝔽27)—41no—
13C3⋊SL(2,ℤ/4)—31no—
14#3757—31no—
15C3×#217—93no—
16C3×#217—91no—
17(C22⋊C9)⋊C4—32no—
18C22×C33—31no—
19C22×C33—31no—
20C22×C33Dic311yescharacteristic, complemented
21C3×C3⋊Dic3—61no—
22C3×C3⋊Dic3—61no—
23C22⋊H(3,𝔽3)Dic311yescharacteristic, complemented
24#1094Dic312yescomplemented
253−1+2⋊C4—122no—
26(C3⋊Dic3)⋊C3—121no—
27AΣL(1,𝔽27)—41noSylow
28C23×C32—31no—
29C23×C32—33no—
30C23×C32C3⋊D311yescharacteristic
31C2×C3×A4—31no—
32C2×C3⋊Dic3—31no—
33C2×C3⋊Dic3—31no—
34C2×C3×Dic3—93no—
35C2×C3×Dic3—93no—
36C2×C3×Dic3—91no—
37C2×C3×Dic3—91no—
38C2×C22⋊C9—32no—
39C2×C33—31no—
40C2×C33—31no—
41C2×C33S411yescharacteristic
42C2×3−1+2—42no—
43C2×H(3,𝔽3)—41no—
44SL(2,ℤ/4)—271no—
45C3×C22⋊C4—271no—
46#217—93no—
47#217—91no—
48C22×C32—31no—
49C22×C32—31no—
50C22×C32—31no—
51C22×C32—33no—
52C22×C32—31no—
53C22×C32—33no—
54C22×C32—31no—
55C22×C32—33no—
56C22×C32—61no—
57C22×C32—63no—
58C22×C32—61no—
59C22×C32—63no—
60C22×C32C3⋊Dic311yescharacteristic
61C3×A4—31no—
62C3⋊Dic3—61no—
63C3⋊Dic3—61no—
64C3⋊Dic3—121no—
65C3×Dic3—183no—
66C3×Dic3—183no—
67C3×Dic3—181no—
68C3×Dic3—181no—
69C22⋊C9—32no—
70Dic9—122no—
71C33SL(2,ℤ/4)11yescharacteristic, complemented
723−1+2—42no—
73H(3,𝔽3)—41no—
74C23×C3—31no—
75C23×C3—33no—
76C23×C3#27211yescharacteristic
77C2×A4—91no—
78C2×C3×C4—271no—
79C2×C3×C4—271no—
80C2×Dic3—93no—
81C2×Dic3—93no—
82C2×Dic3—91no—
83C2×Dic3—91no—
84C2×C32—31no—
85C2×C32—31no—
86C2×C32—31no—
87C2×C32—33no—
88C2×C32—33no—
89C2×C32—33no—
90C2×C32—31no—
91C2×C32—31no—
92C2×C32—63no—
93C2×C32—63no—
94C2×C32—61no—
95C2×C32—61no—
96C2×C32—121no—
97C2×C32C3⋊S411yescharacteristic
98C2×C9—122no—
99C22⋊C4—271noSylow
100C22×C3—31no—
101C22×C3—33no—
102C22×C3—31no—
103C22×C3—31no—
104C22×C3—31no—
105C22×C3—33no—
106C22×C3—33no—
107C22×C3—31no—
108C22×C3—63no—
109C22×C3—61no—
110C22×C3—63no—
111C22×C3—61no—
112C22×C3#108411yescharacteristic
113A4—91no—
114C3×C4—541no—
115C3×C4—541no—
116Dic3—183no—
117Dic3—181no—
118Dic3—183no—
119Dic3—181no—
120Dic3—1081no—
121C32—33no—
122C32—31no—
123C32—121no—
124C32C3⋊SL(2,ℤ/4)11yescharacteristic
125C9—122no—
126C23#418311yescharacteristic
127C2×C4—271no—
128C2×C4—271no—
129C2×C3—33no—
130C2×C3—31no—
131C2×C3—31no—
132C2×C3—33no—
133C2×C3—31no—
134C2×C3—31no—
135C2×C3—33no—
136C2×C3—31no—
137C2×C3—63no—
138C2×C3—61no—
139C2×C3—63no—
140C2×C3—61no—
141C2×C3—361no—
142C2×C3#625811yescharacteristic
143C22—31no—
144C22—31no—
145C22#6633011yescharacteristic, complemented
146C4—541no—
147C4—541no—
148C3—33no—
149C3—31no—
150C3—361no—
151C3#8861311yescharacteristic
152C2—31no—
153C2—31no—
154C2#1061976211yescharacteristic
subgroupG/H
center, hypercenterC2#10619762
commutator, hypocenter#66368C4
FrattiniC2×C32C3⋊S4
FittingC23×C33D3
socleC23×C3#272
prime radical, cosocleC2×#66368C2
LCSG ▷ #66368
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
5. #66368C4semidirect
20. C22×C33Dic3semidirect
23. C22⋊H(3,𝔽3)Dic3semidirect
24. #1094Dic3semidirect
71. C33SL(2,ℤ/4)semidirect
145. C22#66330semidirect

Maximal subgroups

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

subgroupindex
C2×#663682
C3×#37573
(C3⋊SL(2,ℤ/4))⋊C33
#1094⋊C43
#663304