Finite Group Databasebeta

← #5751⋊C3 #10593693

Subgroups — #5751⋊C3 #10593693

Order 576.

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

normal subgroups only

This group has 153 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
1#1010⋊C3C211yesmaximal, characteristic, complemented
2C3⋊#994C211yesmaximal, characteristic, complemented
3C3×#131⋊C3C211yesmaximal, characteristic, complemented
4#5751C311yesmaximal, characteristic, complemented
5#577⋊C3—31nomaximal
6C3×#41⋊C3C2211yescharacteristic
7D3×SL(2,𝔽3)—21no—
8C3⋊(#41⋊C3)—21no—
9C3×#204—41nomaximal
10#1010C2×C311yescharacteristic
11#1009C2×C311yescharacteristic, complemented
122−1+4⋊C3—31no—
13#994—31no—
14C3×#131C2×C311yescharacteristic, complemented
15#909—31no—
16#908—31no—
17#904—31no—
18C2×#204—31no—
19#131⋊C3—21no—
20#131⋊C3D311yescharacteristic, complemented
21C3×D12—41no—
22C3×Dic6—41no—
23C3×SL(2,𝔽3)D411yescharacteristic
24C32×C8—41no—
25#577—31noSylow
26C3×#41C22×C311yescharacteristic
27#239—21no—
28D3×Q8—21no—
29#237—61no—
30D3×D4—61no—
31#235—31no—
32#235—31no—
33C2×D12—31no—
34C2×Dic6—31no—
35#41⋊C3—21no—
36#41⋊C3—61no—
37#41⋊C3C2×D311yescharacteristic
38C2×SL(2,𝔽3)—61no—
39C3×SD16—121no—
40C3×M16—31no—
41C2×C3×C8—31no—
42Dic12—31no—
43D24—31no—
44#204—32no—
45#204—31no—
46#204A411yescharacteristic
47C2×C3×D3—81no—
48C32×C4—41no—
49C3×Dic3—81no—
502−1+4—31no—
51H(5,𝔽2)—31no—
52#137—91no—
53AGL(1,ℤ/8)—91no—
54#135—91no—
55C2×SD16—91no—
56#131C3×D311yescharacteristic, complemented
57C22×D3—61no—
58C3×Q8—121no—
59C3×Q8C3×D411yescharacteristic
60C3×D4—31no—
61C3×D4—121no—
62C2×C3×C4—31no—
63#67—61no—
64#67—61no—
65C2×Dic3—61no—
66D12—31no—
67D12—61no—
68D12C2×A411yescharacteristic
69C4×D3—61no—
70C4×D3—61no—
71Dic6—31no—
72Dic6—61no—
73Dic6C2×A411yescharacteristic
74SL(2,𝔽3)—21no—
75SL(2,𝔽3)D1211yescharacteristic
76C3×C8—31no—
77C3×C8—41no—
78C3×C8—81no—
79C3×C8C2×A411yescharacteristic
80C2×C32—41no—
81C3×D3—161no—
82#41—61no—
83#41—91no—
84#41—91no—
85#41—181no—
86#41C2×C3×D311yescharacteristic
87C2×Q8—61no—
88C2×Q8—91no—
89C2×D4—91no—
90C2×D4—181no—
91Q16—91no—
92SD16—31no—
93SD16—92no—
94SD16—91no—
95D8—91no—
96M16—31no—
97C2×C8—31no—
98C22×C3—31no—
99C22×C3—241no—
100C2×D3—21no—
101C2×D3—61no—
102C2×D3—241no—
103C3×C4—31no—
104C3×C4—41no—
105C3×C4—81no—
106C3×C4—241no—
107C3×C4C22×A411yescharacteristic
108Dic3—21no—
109Dic3—61no—
110C32—41noSylow
111C23—181no—
112Q8—31no—
113Q8—91no—
114Q8—181no—
115Q8C3×D1211yescharacteristic
116D4—31no—
117D4—31no—
118D4—91no—
119D4—181no—
120D4—181no—
121D4—181no—
122C2×C4—31no—
123C2×C4—181no—
124C2×C4—181no—
125C2×C4—181no—
126C8—31no—
127C8D3×A411yescharacteristic
128C2×C3—41no—
129C2×C3—61no—
130C2×C3—81no—
131C2×C3—481no—
132C2×C3D4×A411yescharacteristic
133D3—41no—
134D3—121no—
135C22—31no—
136C22—61no—
137C22—181no—
138C22—721no—
139C4—31no—
140C4—61no—
141C4—181no—
142C4C2×D3×A411yescharacteristic
143C3—41no—
144C3—81no—
145C3#577⋊C311yescharacteristic, complemented
146C2—61no—
147C2—121no—
148C2—361no—
149C2A4×D1211yescharacteristic
subgroupG/H
centerC2A4×D12
commutator, cosocleC3×#41C22×C3
FrattiniC4C2×D3×A4
FittingC3×#131C2×C3
socleC2×C3D4×A4
hypercenterC8D3×A4
hypocenterC3×Q8C3×D4
LCSG ▷ C3×#41 ▷ C3×Q8
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #1010⋊C3C2semidirect
2. C3⋊#994C2semidirect
3. C3×#131⋊C3C2semidirect
4. #5751C3semidirect
11. #1009C2×C3semidirect
14. C3×#131C2×C3semidirect
20. #131⋊C3D3semidirect
56. #131C3×D3semidirect
145. C3#577⋊C3semidirect

Maximal subgroups

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

subgroupindex
#1010⋊C32
C3⋊#9942
C3×#131⋊C32
#57513
#577⋊C33
C3×#2044