Finite Group Databasebeta

← C3×#64265 #11724458

Subgroups — C3×#64265 #11724458

Order 864.

233 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 233 conjugacy classes of subgroups — too many to draw. Only its 42 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
1C32×#41⋊C3C211yesmaximal, characteristic
2C3×C3⋊#226C212yesmaximal, complemented
3#64265C311yesmaximal, characteristic, direct product factor
4C3×#984—33nomaximal
5C3×#951—31nomaximal
6C3×#634—41nomaximal
7C32×SL(2,𝔽3)C2211yescharacteristic
8C32×#41D311yescharacteristic
9C2×C3×Dic6—31no—
10C3×#41⋊C3—21no—
11C3×#41⋊C3—23no—
12C3×#41⋊C3D313yes—
13C3×#41⋊C3C2×C311yescharacteristic
14C3⋊#226C2×C312yescomplemented
15C3×#226—36no—
16C3×#216—32no—
17C3×#214—32no—
18C3×#208—31no—
19C33×C4—41no—
20C3×C3⋊Dic3—42no—
21#984—33no—
22C3×#137—91no—
23#951—31no—
24C32×Q8C2×D311yescharacteristic
25C32×D4—31no—
26C2×C32×C4—31no—
27#634—41no—
28C2×C3×Dic3—61no—
29C3×Dic6—32no—
30C3×Dic6—61no—
31C3×Dic6—123no—
32C3×SL(2,𝔽3)—21no—
33C3×SL(2,𝔽3)—23no—
34C3×SL(2,𝔽3)C2×D313yes—
35C3×SL(2,𝔽3)C22×C311yescharacteristic
36C3×C3⋊C8—32no—
37C2×C33—41no—
38C3×#41—21no—
39C3×#41C3×D311yescharacteristic
40C3×#41C3⋊D311yescharacteristic
41C2×C3×Q8—91no—
42C2×Dic6—31no—
43#41⋊C3—23no—
44#41⋊C3C3×D313yes—
45#226—36no—
46C3×Q16—92no—
47C3×SD16—92no—
48C3×M16—91no—
49#216—32no—
50#214—32no—
51#208—31no—
52C22×C32—31no—
53C32×C4—31no—
54C32×C4—41no—
55C32×C4—43no—
56C32×C4—83no—
57C32×C4—81no—
58C32×C4S411yescharacteristic
59C3⋊Dic3—42no—
60C3×Dic3—62no—
61C3×Dic3—126no—
62#137—91noSylow
63C33—41noSylow
64C3×Q8—21no—
65C3×Q8—92no—
66C3×Q8—181no—
67C3×Q8C2×C3×D311yescharacteristic
68C3×Q8C2×C3⋊D311yescharacteristic
69C3×D4—31no—
70C3×D4—31no—
71C3×D4—61no—
72C2×C3×C4—31no—
73C2×C3×C4—31no—
74C2×C3×C4—61no—
75C2×C3×C4—181no—
76C2×Dic3—61no—
77Dic6—32no—
78Dic6—61no—
79Dic6—123no—
80SL(2,𝔽3)—23no—
81SL(2,𝔽3)C2×C3×D313yes—
82C3×C8—92no—
83C3⋊C8—32no—
84C2×C32—41no—
85C2×C32—43no—
86C2×C32—61no—
87C2×C32—83no—
88C2×C32—81no—
89C2×C32C2×S411yescharacteristic
90#41C3×C3⋊D311yescharacteristic
91C2×Q8—91no—
92Q16—92no—
93SD16—92no—
94M16—91no—
95C22×C3—31no—
96C22×C3—31no—
97C22×C3—61no—
98C3×C4—21no—
99C3×C4—31no—
100C3×C4—31no—
101C3×C4—43no—
102C3×C4—61no—
103C3×C4—83no—
104C3×C4—182no—
105C3×C4C3×S411yescharacteristic
106C3×C4C3⋊S411yescharacteristic
107Dic3—62no—
108Dic3—126no—
109C32—41no—
110C32—43no—
111C32—83no—
112C32—81no—
113C32#98411yescharacteristic, complemented
114Q8—92no—
115Q8—181no—
116Q8C2×C3×C3⋊D311yescharacteristic
117D4—31no—
118C2×C4—31no—
119C2×C4—181no—
120C8—92no—
121C2×C3—21no—
122C2×C3—43no—
123C2×C3—61no—
124C2×C3—61no—
125C2×C3—83no—
126C2×C3—121no—
127C2×C3C2×C3×S411yescharacteristic
128C2×C3C2×C3⋊S411yescharacteristic
129C22—31no—
130C4—31no—
131C4—182no—
132C4C3×C3⋊S411yescharacteristic
133C3—21no—
134C3—43no—
135C3—83no—
136C3C3×#98411yescharacteristic, complemented
137C3#6426511yescharacteristic, direct product factor
138C2—61no—
139C2C2×C3×C3⋊S411yescharacteristic
subgroupG/H
centerC2×C3C2×C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C22×C3
FrattiniC2C2×C3×C3⋊S4
FittingC32×#41D3
socleC2×C32C2×S4
prime radicalC32×#41⋊C3C2
hypercenterC3×C4C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×C3⋊#226C2semidirect
3. #64265C3direct (N × H)
14. C3⋊#226C2×C3semidirect
113. C32#984semidirect
136. C3C3×#984semidirect
137. C3#64265direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#41⋊C32
C3×C3⋊#2262
#642653
C3×#9843
C3×#9513
C3×#6344