Finite Group Databasebeta

← C23×C3⋊GL(2,𝔽3) #11923721

Subgroups — C23×C3⋊GL(2,𝔽3) #11923721

Order 1152.

3256 conjugacy classes of subgroups; only the 281 normal subgroups are shown.

normal subgroups only

This group has 3256 conjugacy classes of subgroups — too many to draw. Only its 281 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
1C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C22×C3⋊GL(2,𝔽3)C2114yesmaximal, direct product factor
3C23×GL(2,𝔽3)—33nomaximal
4C23×#215—31nomaximal
5C24×C3⋊D3—41nomaximal
6C22×C3×SL(2,𝔽3)C2217yescomplemented
7C2×C3⋊GL(2,𝔽3)C22128yesdirect product factor
8C23×C3×Q8D311yescharacteristic, complemented
9C23×D12—31no—
10C23×SL(2,𝔽3)D313yescomplemented
11C22×GL(2,𝔽3)—342no—
12C22×#215—314no—
13C22×#215—314no—
14C23×C3⋊C8—31no—
15C24×C32—41no—
16C23×C3⋊D3—414no—
17C23×C3⋊D3—416no—
18C2×C3×SL(2,𝔽3)C2317yescomplemented
19C3⋊GL(2,𝔽3)C2318yesdirect product factor
20C23×SD16—91noSylow
21C24×D3—61no—
22C24×D3—123no—
23C22×C3×Q8—37no—
24C22×C3×Q8C2×D317yescomplemented
25C23×C3×C4—31no—
26C22×D12—314no—
27C22×D12—67no—
28C22×SL(2,𝔽3)C2×D3121yescomplemented
29C2×GL(2,𝔽3)—384no—
30C2×#215—384no—
31C2×#215—328no—
32C22×C3⋊C8—314no—
33C23×C32—48no—
34C23×C32—47no—
35C22×C3⋊D3—428no—
36C22×C3⋊D3—4112no—
37C3×SL(2,𝔽3)C2411yescharacteristic, complemented
38C23×Q8C3⋊D311yescharacteristic, complemented
39C23×D4—91no—
40C22×SD16—914no—
41C22×SD16—914no—
42C23×C8—91no—
43C24×C3—43no—
44C24×C3S411yescharacteristic
45C23×D3—614no—
46C23×D3—1242no—
47C23×D3—1248no—
48C23×D3—128no—
49C2×C3×Q8—321no—
50C2×C3×Q8—67no—
51C2×C3×Q8C22×D317yescomplemented
52C22×C3×C4—37no—
53C22×C3×C4—37no—
54C2×D12—328no—
55C2×D12—642no—
56C2×SL(2,𝔽3)C22×D3121yescomplemented
57GL(2,𝔽3)—324no—
58#215—356no—
59#215—38no—
60C2×C3⋊C8—328no—
61C22×C32—428no—
62C22×C32—47no—
63C2×C3⋊D3—48no—
64C2×C3⋊D3—4112no—
65C25—181no—
66C22×Q8—37no—
67C22×Q8C2×C3⋊D317yescomplemented
68C22×D4—914no—
69C22×D4—187no—
70C23×C4—31no—
71C2×SD16—984no—
72C2×SD16—928no—
73C22×C8—914no—
74C23×C3—421no—
75C23×C3—424no—
76C23×C3GL(2,𝔽3)18yescomplemented
77C23×C3C2×S417yes—
78C22×D3—628no—
79C22×D3—1284no—
80C22×D3—12336no—
81C22×D3—1256no—
82C3×Q8—37no—
83C3×Q8—67no—
84C3×Q8C23×D311yescharacteristic, complemented
85C2×C3×C4—37no—
86C2×C3×C4—321no—
87D12—38no—
88D12—628no—
89SL(2,𝔽3)C23×D313yescomplemented
90C3⋊C8—38no—
91C2×C32—414no—
92C2×C32—41no—
93C3⋊D3—416no—
94C24—1814no—
95C24—368no—
96C24C3⋊S411yescharacteristic
97C2×Q8—321no—
98C2×Q8—67no—
99C2×Q8C22×C3⋊D317yescomplemented
100C2×D4—928no—
101C2×D4—1842no—
102C22×C4—37no—
103C22×C4—37no—
104SD16—956no—
105SD16—98no—
106C2×C8—928no—
107C22×C3—484no—
108C22×C3—421no—
109C22×C3C2×GL(2,𝔽3)128yescomplemented
110C22×C3C22×S417yes—
111C2×D3—68no—
112C2×D3—1224no—
113C2×D3—12336no—
114C2×D3—1256no—
115C3×C4—31no—
116C3×C4—37no—
117C32—41noSylow
118C23—1828no—
119C23—3656no—
120C23C3⋊GL(2,𝔽3)18yesdirect product factor
121C23C2×C3⋊S417yes—
122Q8—37no—
123Q8—67no—
124Q8C23×C3⋊D311yescharacteristic, complemented
125D4—98no—
126D4—1828no—
127C2×C4—37no—
128C2×C4—321no—
129C8—98no—
130C2×C3—43no—
131C2×C3—442no—
132C2×C3C22×GL(2,𝔽3)114yescomplemented
133C2×C3C23×S411yescharacteristic
134D3—1248no—
135D3—128no—
136C22—188no—
137C22—3656no—
138C22C2×C3⋊GL(2,𝔽3)128yesdirect product factor
139C22C22×C3⋊S417yes—
140C4—31no—
141C4—37no—
142C3—43no—
143C3C23×GL(2,𝔽3)11yescharacteristic, complemented
144C2—368no—
145C2C22×C3⋊GL(2,𝔽3)114yesdirect product factor
146C2C23×C3⋊S411yescharacteristic
subgroupG/H
center, hypercenterC24C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C24
FrattiniC2C23×C3⋊S4
FittingC23×C3×Q8D3
socleC24×C3S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×C3×SL(2,𝔽3)C2semidirect
2. C22×C3⋊GL(2,𝔽3)C2direct (N × H)
6. C22×C3×SL(2,𝔽3)C22semidirect
7. C2×C3⋊GL(2,𝔽3)C22direct (N × H)
8. C23×C3×Q8D3semidirect
10. C23×SL(2,𝔽3)D3semidirect
18. C2×C3×SL(2,𝔽3)C23semidirect
19. C3⋊GL(2,𝔽3)C23direct (N × H)
24. C22×C3×Q8C2×D3semidirect
28. C22×SL(2,𝔽3)C2×D3semidirect
37. C3×SL(2,𝔽3)C24semidirect
38. C23×Q8C3⋊D3semidirect
51. C2×C3×Q8C22×D3semidirect
56. C2×SL(2,𝔽3)C22×D3semidirect
67. C22×Q8C2×C3⋊D3semidirect
76. C23×C3GL(2,𝔽3)semidirect
84. C3×Q8C23×D3semidirect
89. SL(2,𝔽3)C23×D3semidirect
99. C2×Q8C22×C3⋊D3semidirect
109. C22×C3C2×GL(2,𝔽3)semidirect
120. C23C3⋊GL(2,𝔽3)direct (N × H)
124. Q8C23×C3⋊D3semidirect
132. C2×C3C22×GL(2,𝔽3)semidirect
138. C22C2×C3⋊GL(2,𝔽3)direct (N × H)
143. C3C23×GL(2,𝔽3)semidirect
145. C2C22×C3⋊GL(2,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×SL(2,𝔽3)2
C22×C3⋊GL(2,𝔽3)2
C23×GL(2,𝔽3)3
C23×#2153
C24×C3⋊D34