Finite Group Databasebeta

← C22×C3⋊(C3⋊GL(2,𝔽3)) #13181400

Subgroups — C22×C3⋊(C3⋊GL(2,𝔽3)) #13181400

Order 1728.

2606 conjugacy classes of subgroups; only the 247 normal subgroups are shown.

normal subgroups only

This group has 2606 conjugacy classes of subgroups — too many to draw. Only its 247 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
1C22×C32×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3⋊(C3⋊GL(2,𝔽3))C216yesmaximal, direct product factor
3C22×C3⋊GL(2,𝔽3)—312nomaximal
4C22×#3755—31nomaximal
5C23×C3⋊(C3⋊D3)—41nomaximal
6C2×C32×SL(2,𝔽3)C2213yescomplemented
7C3⋊(C3⋊GL(2,𝔽3))C2214yesdirect product factor
8C22×C32×Q8D311yescharacteristic, complemented
9C22×C4⋊(C3⋊D3)—31no—
10C22×C3×SL(2,𝔽3)D3112yescomplemented
11C2×C3⋊GL(2,𝔽3)—372no—
12C2×#3755—36no—
13C2×#3755—36no—
14C22×C3⋊(C3⋊C8)—31no—
15C23×C33—41no—
16C22×C3⋊(C3⋊D3)—46no—
17C22×C3⋊(C3⋊D3)—48no—
18C32×SL(2,𝔽3)C2311yescharacteristic, complemented
19C22×GL(2,𝔽3)—99no—
20C22×#215—94no—
21C23×C3⋊D3—61no—
22C23×C3⋊D3—1212no—
23C2×C32×Q8—33no—
24C2×C32×Q8C2×D313yescomplemented
25C22×C32×C4—31no—
26C2×C4⋊(C3⋊D3)—36no—
27C2×C4⋊(C3⋊D3)—63no—
28C2×C3×SL(2,𝔽3)C2×D3136yescomplemented
29C3⋊GL(2,𝔽3)—348no—
30#3755—312no—
31#3755—34no—
32C2×C3⋊(C3⋊C8)—36no—
33C22×C33—44no—
34C22×C33—43no—
35C2×C3⋊(C3⋊D3)—44no—
36C2×C3⋊(C3⋊D3)—424no—
37C22×C3×Q8C3⋊D314yescomplemented
38C22×D12—94no—
39C22×SL(2,𝔽3)C3⋊D319yescomplemented
40C2×GL(2,𝔽3)—954no—
41C2×#215—924no—
42C2×#215—924no—
43C22×C3⋊C8—94no—
44C23×C32—412no—
45C23×C32S411yescharacteristic
46C22×C3⋊D3—66no—
47C22×C3⋊D3—1272no—
48C22×C3⋊D3—1296no—
49C22×C3⋊D3—124no—
50C32×Q8—33no—
51C32×Q8—61no—
52C32×Q8C22×D311yescharacteristic, complemented
53C2×C32×C4—33no—
54C2×C32×C4—33no—
55C4⋊(C3⋊D3)—34no—
56C4⋊(C3⋊D3)—66no—
57C3×SL(2,𝔽3)C22×D3112yescomplemented
58C3⋊(C3⋊C8)—34no—
59C22×SD16—271noSylow
60C2×C33—46no—
61C2×C33—41no—
62C3⋊(C3⋊D3)—48no—
63C23×D3—184no—
64C23×D3—369no—
65C2×C3×Q8—312no—
66C2×C3×Q8C2×C3⋊D3112yescomplemented
67C22×C3×C4—34no—
68C2×D12—924no—
69C2×D12—1812no—
70C2×SL(2,𝔽3)C2×C3⋊D3127yescomplemented
71GL(2,𝔽3)—936no—
72#215—948no—
73#215—916no—
74C2×C3⋊C8—924no—
75C22×C32—436no—
76C22×C32—448no—
77C22×C32GL(2,𝔽3)14yescomplemented
78C22×C32C2×S413yes—
79C2×C3⋊D3—64no—
80C2×C3⋊D3—1248no—
81C2×C3⋊D3—12288no—
82C2×C3⋊D3—1212no—
83C32×C4—31no—
84C32×C4—33no—
85C22×Q8C3⋊(C3⋊D3)11yescharacteristic, complemented
86C22×D4—271no—
87C2×SD16—276no—
88C2×SD16—276no—
89C22×C8—271no—
90C33—41noSylow
91C23×C3—49no—
92C23×C3C3⋊S414yes—
93C22×D3—1824no—
94C22×D3—3654no—
95C22×D3—3672no—
96C22×D3—3616no—
97C3×Q8—312no—
98C3×Q8—64no—
99C3×Q8C22×C3⋊D314yescomplemented
100C2×C3×C4—312no—
101C2×C3×C4—312no—
102D12—916no—
103D12—1824no—
104SL(2,𝔽3)C22×C3⋊D319yescomplemented
105C3⋊C8—916no—
106C2×C32—472no—
107C2×C32—412no—
108C2×C32C2×GL(2,𝔽3)16yescomplemented
109C2×C32C22×S411yescharacteristic
110C3⋊D3—1296no—
111C3⋊D3—124no—
112C24—541no—
113C2×Q8—33no—
114C2×Q8C2×C3⋊(C3⋊D3)13yescomplemented
115C2×D4—276no—
116C2×D4—543no—
117C22×C4—31no—
118SD16—2712no—
119SD16—274no—
120C2×C8—276no—
121C22×C3—427no—
122C22×C3—436no—
123C22×C3C3⋊GL(2,𝔽3)116yescomplemented
124C22×C3C2×C3⋊S4112yes—
125C2×D3—1816no—
126C2×D3—3636no—
127C2×D3—36216no—
128C2×D3—3648no—
129C3×C4—34no—
130C3×C4—312no—
131C32—412no—
132C32C22×GL(2,𝔽3)11yescharacteristic, complemented
133C23—546no—
134C23—1084no—
135C23C3⋊(C3⋊S4)11yescharacteristic
136Q8—33no—
137Q8—61no—
138Q8C22×C3⋊(C3⋊D3)11yescharacteristic, complemented
139D4—274no—
140D4—546no—
141C2×C4—33no—
142C2×C4—33no—
143C8—274no—
144C2×C3—454no—
145C2×C3—49no—
146C2×C3C2×C3⋊GL(2,𝔽3)124yescomplemented
147C2×C3C22×C3⋊S414yes—
148D3—3672no—
149D3—3616no—
150C22—544no—
151C22—10812no—
152C22C3⋊(C3⋊GL(2,𝔽3))14yesdirect product factor
153C22C2×C3⋊(C3⋊S4)13yes—
154C4—31no—
155C4—33no—
156C3—49no—
157C3C22×C3⋊GL(2,𝔽3)14yescomplemented
158C2—1084no—
159C2C2×C3⋊(C3⋊GL(2,𝔽3))16yesdirect product factor
160C2C22×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
center, hypercenterC23C3⋊(C3⋊S4)
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C23
FrattiniC2C22×C3⋊(C3⋊S4)
FittingC22×C32×Q8D3
socleC23×C32S4
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C32×SL(2,𝔽3)C2semidirect
2. C2×C3⋊(C3⋊GL(2,𝔽3))C2direct (N × H)
6. C2×C32×SL(2,𝔽3)C22semidirect
7. C3⋊(C3⋊GL(2,𝔽3))C22direct (N × H)
8. C22×C32×Q8D3semidirect
10. C22×C3×SL(2,𝔽3)D3semidirect
18. C32×SL(2,𝔽3)C23semidirect
24. C2×C32×Q8C2×D3semidirect
28. C2×C3×SL(2,𝔽3)C2×D3semidirect
37. C22×C3×Q8C3⋊D3semidirect
39. C22×SL(2,𝔽3)C3⋊D3semidirect
52. C32×Q8C22×D3semidirect
57. C3×SL(2,𝔽3)C22×D3semidirect
66. C2×C3×Q8C2×C3⋊D3semidirect
70. C2×SL(2,𝔽3)C2×C3⋊D3semidirect
77. C22×C32GL(2,𝔽3)semidirect
85. C22×Q8C3⋊(C3⋊D3)semidirect
99. C3×Q8C22×C3⋊D3semidirect
104. SL(2,𝔽3)C22×C3⋊D3semidirect
108. C2×C32C2×GL(2,𝔽3)semidirect
114. C2×Q8C2×C3⋊(C3⋊D3)semidirect
123. C22×C3C3⋊GL(2,𝔽3)semidirect
132. C32C22×GL(2,𝔽3)semidirect
138. Q8C22×C3⋊(C3⋊D3)semidirect
146. C2×C3C2×C3⋊GL(2,𝔽3)semidirect
152. C22C3⋊(C3⋊GL(2,𝔽3))direct (N × H)
157. C3C22×C3⋊GL(2,𝔽3)semidirect
159. C2C2×C3⋊(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
C22×C32×SL(2,𝔽3)2
C2×C3⋊(C3⋊GL(2,𝔽3))2
C22×C3⋊GL(2,𝔽3)3
C22×#37553
C23×C3⋊(C3⋊D3)4