Finite Group Databasebeta

← C22×C3⋊(C3⋊(#41⋊C3)) #13181462

Subgroups — C22×C3⋊(C3⋊(#41⋊C3)) #13181462

Order 1728.

1488 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 1488 conjugacy classes of subgroups — too many to draw. Only its 189 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⋊(#41⋊C3))C216yesmaximal, direct product factor
3C22×#3832C311yesmaximal, characteristic, complemented
4C22×C3⋊(#41⋊C3)—34nomaximal
5C22×C3×C3⋊Dic3—41nomaximal
6C2×C32×SL(2,𝔽3)C2213yescomplemented
7C3⋊(C3⋊(#41⋊C3))C2214yesdirect product factor
8C22×C32×Q8C2×C311yescharacteristic
9C2×#3832—36no—
10C2×#3832C2×C316yescomplemented
11C22×C4⋊(C3⋊D3)—31no—
12C22×C4×C3⋊D3—31no—
13C22×C3×SL(2,𝔽3)—24no—
14C22×C3×SL(2,𝔽3)D314yescomplemented
15C2×C3⋊(#41⋊C3)—324no—
16C23×C33—41no—
17C2×C3×C3⋊Dic3—46no—
18C32×SL(2,𝔽3)C2311yescharacteristic, complemented
19C22×#239—34no—
20C22×#41⋊C3—91no—
21C23×C3⋊D3—31no—
22C2×C32×Q8—33no—
23C2×C32×Q8C22×C313yes—
24C22×C32×C4—31no—
25C22×C3⋊Dic3A411yescharacteristic
26#3832—38no—
27#3832—312no—
28#3832C22×C314yescomplemented
29C2×C4⋊(C3⋊D3)—36no—
30C2×C4⋊(C3⋊D3)—36no—
31C2×C4×C3⋊D3—36no—
32C2×C4×C3⋊D3—36no—
33C22×C3×Dic3—124no—
34C2×C3×SL(2,𝔽3)—212no—
35C2×C3×SL(2,𝔽3)C2×D3112yescomplemented
36C3⋊(#41⋊C3)—316no—
37C22×C33—44no—
38C22×C33—43no—
39C3×C3⋊Dic3—44no—
40C22×C3×Q8C3×D314yes—
41C2×#239—324no—
42C2×#239—924no—
43C22×D12—94no—
44C22×C4×D3—94no—
45C2×#41⋊C3—96no—
46C22×SL(2,𝔽3)—24no—
47C22×SL(2,𝔽3)C3⋊D311yescharacteristic, complemented
48C23×C32—44no—
49C23×C32—84no—
50C23×C32C2×A411yescharacteristic
51C22×C3⋊D3—36no—
52C22×C3⋊D3—64no—
53C32×Q8—33no—
54C32×Q8—32no—
55C32×Q8C23×C311yescharacteristic
56C2×C32×C4—33no—
57C2×C32×C4—33no—
58C2×C3⋊Dic3C2×A416yes—
59C4⋊(C3⋊D3)—34no—
60C4⋊(C3⋊D3)—312no—
61C4×C3⋊D3—312no—
62C4×C3⋊D3—34no—
63C2×C3×Dic3—1224no—
64C3×SL(2,𝔽3)—24no—
65C3×SL(2,𝔽3)C22×D314yescomplemented
66C22×#41—91noSylow
67C2×C33—46no—
68C2×C33—41no—
69C23×D3—94no—
70C2×C3×Q8—312no—
71C2×C3×Q8C2×C3×D3112yes—
72C22×C3×C4—34no—
73C22×C3×C4—361no—
74C22×Dic3—34no—
75#239—316no—
76#239—932no—
77#239—948no—
78C2×D12—924no—
79C2×D12—924no—
80C2×C4×D3—924no—
81C2×C4×D3—924no—
82#41⋊C3—94no—
83C2×SL(2,𝔽3)—212no—
84C2×SL(2,𝔽3)C2×C3⋊D313yescomplemented
85C22×C32—412no—
86C22×C32—416no—
87C22×C32—812no—
88C22×C32—816no—
89C22×C32#41⋊C314yescomplemented
90C22×C32C22×A413yes—
91C2×C3⋊D3—34no—
92C2×C3⋊D3—612no—
93C32×C4—33no—
94C32×C4—31no—
95C3⋊Dic3C22×A414yes—
96C3×Dic3—1216no—
97C2×#41—96no—
98C2×#41—276no—
99C22×Q8C3×C3⋊D311yescharacteristic
100C22×D4—271no—
101C23×C4—271no—
102C33—41noSylow
103C23×C3—41no—
104C23×C3—84no—
105C23×C3D3×A414yes—
106C22×D3—924no—
107C22×D3—1816no—
108C3×Q8—312no—
109C3×Q8—38no—
110C3×Q8C22×C3×D314yes—
111C2×C3×C4—312no—
112C2×C3×C4—312no—
113C2×C3×C4—366no—
114C2×Dic3—324no—
115D12—916no—
116D12—948no—
117C4×D3—916no—
118C4×D3—948no—
119SL(2,𝔽3)—24no—
120SL(2,𝔽3)C22×C3⋊D311yescharacteristic, complemented
121C2×C32—424no—
122C2×C32—44no—
123C2×C32—824no—
124C2×C32—84no—
125C2×C32C2×#41⋊C316yescomplemented
126C2×C32C23×A411yescharacteristic
127C3⋊D3—64no—
128C24—271no—
129#41—94no—
130#41—278no—
131#41—2712no—
132C2×Q8—33no—
133C2×Q8C2×C3×C3⋊D313yes—
134C2×D4—276no—
135C2×D4—276no—
136C22×C4—31no—
137C22×C4—91no—
138C22×C4—276no—
139C22×C4—276no—
140C22×C3—43no—
141C22×C3—44no—
142C22×C3—812no—
143C22×C3—816no—
144C22×C3C3⋊(#41⋊C3)116yescomplemented
145C22×C3C2×D3×A4112yes—
146C2×D3—916no—
147C2×D3—1848no—
148C3×C4—34no—
149C3×C4—312no—
150C3×C4—364no—
151Dic3—316no—
152C32—44no—
153C32—84no—
154C32C22×#41⋊C311yescharacteristic, complemented
155C23—276no—
156C23—544no—
157C23A4×C3⋊D311yescharacteristic
158Q8—33no—
159Q8—32no—
160Q8C22×C3×C3⋊D311yescharacteristic
161D4—274no—
162D4—2712no—
163C2×C4—33no—
164C2×C4—33no—
165C2×C4—96no—
166C2×C4—274no—
167C2×C4—2712no—
168C2×C3—46no—
169C2×C3—41no—
170C2×C3—824no—
171C2×C3—84no—
172C2×C3C2×C3⋊(#41⋊C3)124yescomplemented
173C2×C3C22×D3×A414yes—
174D3—1816no—
175C22—274no—
176C22—5412no—
177C22C3⋊(C3⋊(#41⋊C3))14yesdirect product factor
178C22C2×A4×C3⋊D313yes—
179C4—31no—
180C4—33no—
181C4—94no—
182C3—41no—
183C3—84no—
184C3C22×C3⋊(#41⋊C3)14yescomplemented
185C2—544no—
186C2C2×C3⋊(C3⋊(#41⋊C3))16yesdirect product factor
187C2C22×A4×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC23A4×C3⋊D3
commutator, cosocle, hypocenterC32×Q8C23×C3
FrattiniC2C22×A4×C3⋊D3
FittingC22×C32×Q8C2×C3
socleC23×C32C2×A4
LCSG ▷ C32×Q8
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⋊(#41⋊C3))C2direct (N × H)
3. C22×#3832C3semidirect
6. C2×C32×SL(2,𝔽3)C22semidirect
7. C3⋊(C3⋊(#41⋊C3))C22direct (N × H)
10. C2×#3832C2×C3semidirect
14. C22×C3×SL(2,𝔽3)D3semidirect
18. C32×SL(2,𝔽3)C23semidirect
28. #3832C22×C3semidirect
35. C2×C3×SL(2,𝔽3)C2×D3semidirect
47. C22×SL(2,𝔽3)C3⋊D3semidirect
65. C3×SL(2,𝔽3)C22×D3semidirect
84. C2×SL(2,𝔽3)C2×C3⋊D3semidirect
89. C22×C32#41⋊C3semidirect
120. SL(2,𝔽3)C22×C3⋊D3semidirect
125. C2×C32C2×#41⋊C3semidirect
144. C22×C3C3⋊(#41⋊C3)semidirect
154. C32C22×#41⋊C3semidirect
172. C2×C3C2×C3⋊(#41⋊C3)semidirect
177. C22C3⋊(C3⋊(#41⋊C3))direct (N × H)
184. C3C22×C3⋊(#41⋊C3)semidirect
186. C2C2×C3⋊(C3⋊(#41⋊C3))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⋊(#41⋊C3))2
C22×#38323
C22×C3⋊(#41⋊C3)3
C22×C3×C3⋊Dic34