Finite Group Databasebeta

← #10593665

Subgroups — #10593665

Order 576.

182 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 29 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
1C3×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2#64267C211yesmaximal, characteristic, complemented
3C3⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#5431—33nomaximal
5#5245—31nomaximal
6C3×#41⋊C3C2211yescharacteristic
7C2×C3×SL(2,𝔽3)—21no—
8C3⋊GL(2,𝔽3)—21no—
9#3753—41nomaximal
10C3×2−1+4D311yescharacteristic, complemented
112−1+4⋊C3D313yescomplemented
12#986—33no—
13#949—31no—
14#949—31no—
15#947—31no—
16GU(2,𝔽3)—33no—
17#837—31no—
18#837—31no—
19#836—31no—
20C32×D4—41no—
21C4⋊(C3⋊D3)—41no—
22C3×SL(2,𝔽3)D411yescharacteristic
23C3⋊(C3⋊C8)—41no—
24#454—91noSylow
25C3×#41—31no—
26C3×#41—61no—
27C3×#41C2×D311yescharacteristic
28C2×C3×Q8—21no—
29C2×C3×Q8—31no—
30C2×D12—31no—
31#41⋊C3C2×D313yes—
32C2×SL(2,𝔽3)—23no—
33GL(2,𝔽3)—63no—
34#215—61no—
35#215—61no—
36#213—61no—
37#213—61no—
38#213—123no—
39(C2×D3)⋊C4—62no—
40C4×Dic3—31no—
41#208—31no—
42#208—31no—
43C22×C32—81no—
44C2×C3⋊D3—81no—
45C32×C4—41no—
462−1+4C3⋊D311yescharacteristic, complemented
47AGL(1,ℤ/8)—91no—
48AGL(1,ℤ/8)—91no—
49#124—91no—
50#104—91no—
51#104—91no—
52#101—91no—
53C22×D3—61no—
54C3×Q8—31no—
55C3×Q8—61no—
56C3×Q8#6711yescharacteristic
57C3×D4—31no—
58C3×D4—43no—
59C3×D4—61no—
60C3×D4S411yescharacteristic
61C2×C3×C4—31no—
62C2×C3×C4—32no—
63C2×C3×C4—61no—
64C2×Dic3—61no—
65D12—61no—
66D12—61no—
67D12—123no—
68SL(2,𝔽3)#6713yes—
69C3⋊C8—61no—
70C3⋊C8—61no—
71C3⋊C8—123no—
72C2×C32—41no—
73C2×C32—161no—
74C3⋊D3—161no—
75#41—31no—
76#41—61no—
77#41C2×C3⋊D311yescharacteristic
78C2×Q8—21no—
79C2×Q8—31no—
80C2×D4—91no—
81SD16—181no—
82SD16—181no—
83D8—181no—
84D8—181no—
85M16—91no—
86M16—91no—
87C22⋊C4—182no—
88C42—91no—
89C22×C3—21no—
90C22×C3—31no—
91C22×C3—83no—
92C2×D3—122no—
93C2×D3—121no—
94C2×D3—243no—
95C3×C4—31no—
96C3×C4—43no—
97C3×C4—61no—
98C3×C4C2×S411yescharacteristic
99Dic3—121no—
100C32—41noSylow
101C23—181no—
102Q8—31no—
103Q8—61no—
104Q8#63811yescharacteristic
105D4—31no—
106D4—61no—
107D4—181no—
108D4—181no—
109D4C3⋊S411yescharacteristic
110C2×C4—31no—
111C2×C4—32no—
112C2×C4—61no—
113C2×C4—181no—
114C8—181no—
115C8—181no—
116C2×C3—41no—
117C2×C3—43no—
118C2×C3—61no—
119C2×C3—163no—
120C2×C3GL(2,ℤ/4)11yescharacteristic
121D3—241no—
122D3—483no—
123C22—21no—
124C22—31no—
125C22—362no—
126C22—361no—
127C4—31no—
128C4—61no—
129C4—361no—
130C4C2×C3⋊S411yescharacteristic
131C3—43no—
132C3#543111yescharacteristic, complemented
133C2—41no—
134C2—61no—
135C2—721no—
136C2#6426911yescharacteristic
subgroupG/H
centerC2#64269
commutator, cosocleC3×#41⋊C3C22
FrattiniC4C2×C3⋊S4
FittingC3×2−1+4D3
socleC2×C3GL(2,ℤ/4)
hypercenterD4C3⋊S4
hypocenterC3×SL(2,𝔽3)D4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C3×2−1+4⋊C3C2semidirect
2. #64267C2semidirect
3. C3⋊GU(2,𝔽3)C2semidirect
10. C3×2−1+4D3semidirect
11. 2−1+4⋊C3D3semidirect
46. 2−1+4C3⋊D3semidirect
132. C3#5431semidirect

Maximal subgroups

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

subgroupindex
C3×2−1+4⋊C32
#642672
C3⋊GU(2,𝔽3)2
#54313
#52453
#37534