Finite Group Databasebeta

← #64265⋊C3 #11722357

Subgroups — #64265⋊C3 #11722357

Order 864.

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

normal subgroups only

This group has 148 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
1#41⋊H(3,𝔽3)C211yesmaximal, characteristic
2(C3⋊#226)⋊C3C212yesmaximal, complemented
3#64265C311yesmaximal, characteristic, complemented
4C3×#951—31nomaximal
5#634⋊C3—41nomaximal
6Q8⋊H(3,𝔽3)C2211yescharacteristic
7C32×#41D311yescharacteristic
8C2×C3×Dic6—31no—
9C3×#41⋊C3—21no—
10C3×#41⋊C3C2×C311yescharacteristic
11C3⋊#226C2×C312yescomplemented
12C3×#216—32no—
13C3×#214—32no—
14C3×#208—31no—
15C4×H(3,𝔽3)—41no—
16(C3⋊Dic3)⋊C3—42no—
17#984—91no—
18C3×#137—91no—
19#951—31no—
20C32×Q8C2×D311yescharacteristic
21C32×D4—31no—
22C2×C32×C4—31no—
23#634—41no—
24C2×C3×Dic3—61no—
25C3×Dic6—32no—
26C3×Dic6—61no—
27C3×SL(2,𝔽3)—21no—
28C3×SL(2,𝔽3)C22×C311yescharacteristic
29C3×C3⋊C8—32no—
30C2×H(3,𝔽3)—41no—
31C3×#41—31no—
32C3×#41C3×D311yescharacteristic
33C2×C3×Q8—91no—
34C2×Dic6—31no—
35#41⋊C3—31no—
36#41⋊C3—61no—
37#226—92no—
38C3×Q16—92no—
39C3×SD16—92no—
40C3×M16—91no—
41#216—32no—
42#214—32no—
43#208—31no—
44C22×C32—31no—
45C32×C4—31no—
46C32×C4—41no—
47C32×C4—81no—
48C32×C4S411yescharacteristic
49C3⋊Dic3—42no—
50C3×Dic3—62no—
51#137—91noSylow
52H(3,𝔽3)—41noSylow
53C3×Q8—31no—
54C3×Q8—92no—
55C3×Q8—181no—
56C3×Q8C2×C3×D311yescharacteristic
57C3×D4—31no—
58C3×D4—31no—
59C3×D4—61no—
60C2×C3×C4—31no—
61C2×C3×C4—31no—
62C2×C3×C4—61no—
63C2×C3×C4—181no—
64C2×Dic3—61no—
65Dic6—32no—
66Dic6—61no—
67Dic6—361no—
68SL(2,𝔽3)—31no—
69SL(2,𝔽3)—61no—
70C3×C8—92no—
71C3⋊C8—32no—
72C2×C32—41no—
73C2×C32—61no—
74C2×C32—81no—
75C2×C32C2×S411yescharacteristic
76#41(C3⋊D3)⋊C311yescharacteristic
77C2×Q8—91no—
78Q16—92no—
79SD16—92no—
80M16—91no—
81C22×C3—31no—
82C22×C3—31no—
83C22×C3—61no—
84C3×C4—31no—
85C3×C4—31no—
86C3×C4—31no—
87C3×C4—61no—
88C3×C4—121no—
89C3×C4—182no—
90C3×C4—241no—
91C3×C4C3×S411yescharacteristic
92Dic3—62no—
93Dic3—362no—
94C32—41no—
95C32—81no—
96C32#98411yescharacteristic, complemented
97Q8—92no—
98Q8—181no—
99Q8C2×(C3⋊D3)⋊C311yescharacteristic
100D4—31no—
101C2×C4—31no—
102C2×C4—181no—
103C8—92no—
104C2×C3—31no—
105C2×C3—61no—
106C2×C3—61no—
107C2×C3—121no—
108C2×C3—121no—
109C2×C3—241no—
110C2×C3C2×C3×S411yescharacteristic
111C22—31no—
112C4—31no—
113C4—182no—
114C4(C3⋊S4)⋊C311yescharacteristic
115C3—31no—
116C3—121no—
117C3—241no—
118C3C3×#98411yescharacteristic
119C2—61no—
120C2C2×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
centerC2C2×(C3⋊S4)⋊C3
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C22×C3
Frattini, socleC2×C3C2×C3×S4
FittingC32×#41D3
prime radical#41⋊H(3,𝔽3)C2
hypercenterC4(C3⋊S4)⋊C3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. (C3⋊#226)⋊C3C2semidirect
3. #64265C3semidirect
11. C3⋊#226C2×C3semidirect
96. C32#984semidirect

Maximal subgroups

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

subgroupindex
#41⋊H(3,𝔽3)2
(C3⋊#226)⋊C32
#642653
C3×#9513
#634⋊C34