Finite Group Databasebeta

← #10596623

Subgroups — #10596623

Order 576.

401 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 65 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×#994C211yesmaximal, characteristic, complemented
2#1010⋊C3C213yesmaximal, complemented
3D3×#41⋊C3C213yesmaximal, complemented
4#5966C311yesmaximal, characteristic, complemented
5#5945—31nomaximal
6C3×#239—41nomaximal
7C3×#41⋊C3C2213yescomplemented
8D3×SL(2,𝔽3)C2213yescomplemented
9C3⋊(#41⋊C3)C2211yescharacteristic, complemented
10C3×H(5,𝔽2)C2×C311yescharacteristic, complemented
11#1010—31no—
12#1010C2×C313yescomplemented
13D3×#41—31no—
14D3×#41C2×C313yescomplemented
15#1004—33no—
16C2×#237—33no—
172−1+4⋊C3—33no—
18#994—21no—
19#994D311yescharacteristic, complemented
20C2×#41⋊C3—33no—
21C32×Q8—41no—
22C3×D12—43no—
23C3×C4×D3—43no—
24C3×SL(2,𝔽3)C2311yescharacteristic
25#585—31noSylow
26C3×#41—31no—
27C3×#41—121no—
28C3×#41C22×C313yes—
29C2×C3×D4—33no—
30C2×#67—36no—
31C22×Dic3—32no—
32#239A411yescharacteristic
33#239C22×C311yescharacteristic
34D3×Q8—31no—
35D3×Q8C22×C313yes—
36#237—36no—
37#237—36no—
38#237—33no—
39#237—33no—
40D3×D4—33no—
41D3×D4—33no—
42#235—33no—
43#235—33no—
44C2×C4×D3—33no—
45C2×Dic6—33no—
46#41⋊C3—23no—
47#41⋊C3—31no—
48#41⋊C3C2×D313yescomplemented
49C2×SL(2,𝔽3)—33no—
50C2×C3×D3—43no—
51C32×C4—43no—
52C3×Dic3—41no—
532−1+4—33no—
542−1+4—91no—
55H(5,𝔽2)—93no—
56H(5,𝔽2)C3×D311yescharacteristic, complemented
57C2×#41—33no—
58C2×#41—93no—
59C2×#41—91no—
60C23×C3—32no—
61C22×D3—33no—
62C3×Q8—41no—
63C3×Q8—81no—
64C3×Q8C2×A411yescharacteristic
65C3×Q8C23×C311yescharacteristic
66C3×D4—33no—
67C3×D4—33no—
68C3×D4—123no—
69C2×C3×C4—33no—
70C2×C3×C4—123no—
71#67—36no—
72#67—36no—
73C2×Dic3—36no—
74C2×Dic3—33no—
75D12—31no—
76D12C2×A413yes—
77C4×D3—33no—
78C4×D3—33no—
79C4×D3—31no—
80C4×D3C2×A413yes—
81Dic6—33no—
82Dic6—33no—
83SL(2,𝔽3)—21no—
84SL(2,𝔽3)C22×D311yescharacteristic
85C2×C32—41no—
86C3×D3—83no—
87#41—31no—
88#41—31no—
89#41—31no—
90#41—93no—
91#41—93no—
92#41—93no—
93#41—96no—
94#41—96no—
95#41—93no—
96#41C2×C3×D313yes—
97C2×Q8—33no—
98C2×Q8—93no—
99C2×Q8—91no—
100C2×D4—33no—
101C2×D4—93no—
102C2×D4—93no—
103C2×D4—96no—
104C22×C4—93no—
105C22×C4—92no—
106C22×C3—33no—
107C22×C3—123no—
108C22×C3—122no—
109C2×D3—31no—
110C2×D3—123no—
111C2×D3C22×A413yes—
112C3×C4—31no—
113C3×C4—43no—
114C3×C4—83no—
115C3×C4—121no—
116C3×C4C22×A413yes—
117Dic3—33no—
118Dic3C22×A411yescharacteristic
119C32—41noSylow
120C23—32no—
121C23—93no—
122Q8—93no—
123Q8—93no—
124Q8D3×A411yescharacteristic
125Q8C22×C3×D311yescharacteristic
126D4—33no—
127D4—33no—
128D4—33no—
129D4—91no—
130D4—96no—
131D4—96no—
132C2×C4—33no—
133C2×C4—33no—
134C2×C4—91no—
135C2×C4—93no—
136C2×C4—93no—
137C2×C4—96no—
138C2×C4—93no—
139C2×C3—41no—
140C2×C3—63no—
141C2×C3—81no—
142C2×C3—243no—
143C2×C3C23×A411yescharacteristic
144D3—23no—
145D3—61no—
146C22—33no—
147C22—33no—
148C22—91no—
149C22—122no—
150C22—363no—
151C4—31no—
152C4—31no—
153C4—93no—
154C4C2×D3×A413yes—
155C3—41no—
156C3—81no—
157C3#594511yescharacteristic, complemented
158C2—63no—
159C2—63no—
160C2—181no—
161C2C22×D3×A411yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×A4
commutator, cosocle, hypocenterC3×Q8C23×C3
FittingC3×H(5,𝔽2)C2×C3
socleC2×C3C23×A4
hypercenterQ8D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C3×#994C2semidirect
2. #1010⋊C3C2semidirect
3. D3×#41⋊C3C2semidirect
4. #5966C3semidirect
7. C3×#41⋊C3C22semidirect
8. D3×SL(2,𝔽3)C22semidirect
9. C3⋊(#41⋊C3)C22semidirect
10. C3×H(5,𝔽2)C2×C3semidirect
12. #1010C2×C3semidirect
14. D3×#41C2×C3semidirect
19. #994D3semidirect
48. #41⋊C3C2×D3semidirect
56. H(5,𝔽2)C3×D3semidirect
157. C3#5945semidirect

Maximal subgroups

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

subgroupindex
C3×#9942
#1010⋊C32
D3×#41⋊C32
#59663
#59453
C3×#2394