Finite Group Databasebeta

← #10593659

Subgroups — #10593659

Order 576.

190 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 33 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×C3×#41⋊C3C211yesmaximal, characteristic
2C2×C3⋊#226C211yesmaximal, characteristic, complemented
3C3⋊#859C211yesmaximal, characteristic, complemented
4#5425—33nomaximal
5#5239—31nomaximal
6C3×#41⋊C3—21no—
7C2×C3×SL(2,𝔽3)C2211yescharacteristic
8C3⋊#226—21no—
9#3750—41nomaximal
10C2×C3×#41D311yescharacteristic
11C2×#41⋊C3D313yes—
12C2×#226—33no—
13C2×#216—31no—
14C2×#214—31no—
15#924—31no—
16#859—33no—
17#835—31no—
18#832—31no—
19#830—31no—
20C2×C32×C4—41no—
21C2×C3⋊Dic3—41no—
22C2×C3⋊Dic3—41no—
23C3×SL(2,𝔽3)D411yescharacteristic
24#452—91noSylow
25C3×#41—21no—
26C3×#41—61no—
27C2×C3×Q8C2×D311yescharacteristic
28C2×C3×D4—31no—
29C22×C3×C4—31no—
30C2×Dic6—31no—
31#41⋊C3—23no—
32C2×SL(2,𝔽3)C2×D313yes—
33#226—63no—
34#217—61no—
35#216—61no—
36#216—61no—
37#214—62no—
38C4⋊Dic3—31no—
39Dic3⋊C4—61no—
40Dic3⋊C4—123no—
41C2×C3⋊C8—31no—
42C2×C3⋊C8—31no—
43C22×C32—41no—
44C32×C4—81no—
45C3⋊Dic3—42no—
46C3⋊Dic3—81no—
47C2×#41C3⋊D311yescharacteristic
48C2×Q16—91no—
49C2×SD16—91no—
50C22⋊Q8—91no—
51#103—91no—
52#102—91no—
53C22⋊C8—91no—
54C23×C3—31no—
55C3×Q8—31no—
56C3×Q8#6711yescharacteristic
57C3×D4—32no—
58C3×D4—61no—
59C2×C3×C4—31no—
60C2×C3×C4—43no—
61C2×C3×C4—61no—
62C2×C3×C4—61no—
63C2×C3×C4S411yescharacteristic
64C2×Dic3—61no—
65C2×Dic3—61no—
66C2×Dic3—123no—
67C2×Dic3—123no—
68Dic6—61no—
69Dic6—61no—
70SL(2,𝔽3)#6713yes—
71C3⋊C8—61no—
72C3⋊C8—61no—
73C2×C32—41no—
74C2×C32—41no—
75C2×C32—41no—
76#41—21no—
77#41—61no—
78C2×Q8—91no—
79C2×Q8C2×C3⋊D311yescharacteristic
80C2×D4—31no—
81C22×C4—31no—
82Q16—181no—
83Q16—181no—
84SD16—182no—
85C2×C8—91no—
86C2×C8—91no—
87C4⋊C4—91no—
88C4⋊C4—181no—
89C22⋊C4—181no—
90C22×C3—43no—
91C22×C3—61no—
92C22×C3—61no—
93C22×C3—61no—
94C22×C3C2×S411yescharacteristic
95C3×C4—21no—
96C3×C4—31no—
97C3×C4—31no—
98C3×C4—83no—
99Dic3—126no—
100Dic3—121no—
101Dic3—121no—
102Dic3—243no—
103C32—41noSylow
104C23—31no—
105Q8—31no—
106Q8—181no—
107Q8—181no—
108Q8#63811yescharacteristic
109D4—32no—
110D4—61no—
111C2×C4—31no—
112C2×C4—61no—
113C2×C4—61no—
114C2×C4—181no—
115C2×C4—181no—
116C2×C4C3⋊S411yescharacteristic
117C8—181no—
118C8—181no—
119C2×C3—43no—
120C2×C3—43no—
121C2×C3—43no—
122C2×C3—121no—
123C2×C3#98411yescharacteristic
124C2×C3#98511yescharacteristic
125C2×C3GL(2,ℤ/4)11yescharacteristic
126C22—61no—
127C22—61no—
128C22—61no—
129C22C2×C3⋊S411yescharacteristic
130C4—21no—
131C4—31no—
132C4—31no—
133C4—361no—
134C4—361no—
135C3—43no—
136C3#542511yescharacteristic, complemented
137C2—121no—
138C2#6426511yescharacteristic
139C2#6426611yescharacteristic
140C2#6426911yescharacteristic
subgroupG/H
center, FrattiniC22C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C22
FittingC2×C3×#41D3
socleC22×C3C2×S4
prime radicalC2×C3×#41⋊C3C2
hypercenterC2×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊#226C2semidirect
3. C3⋊#859C2semidirect
136. C3#5425semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#41⋊C32
C2×C3⋊#2262
C3⋊#8592
#54253
#52393
#37504