Finite Group Databasebeta

← C22×C32×#41⋊C3 #13181700

Subgroups — C22×C32×#41⋊C3 #13181700

Order 1728.

2008 conjugacy classes of subgroups; only the 610 normal subgroups are shown.

normal subgroups only

This group has 2008 conjugacy classes of subgroups — too many to draw. Only its 610 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
1C2×C32×#41⋊C3C216yesmaximal, direct product factor
2C22×C32×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C22×C32×#41C311yesmaximal, characteristic, complemented
4C22×C3×#41⋊C3C3112yesmaximal, direct product factor
5C22×C33×C4—41nomaximal
6C32×#41⋊C3C2214yesdirect product factor
7C2×C32×SL(2,𝔽3)C2213yescomplemented
8C2×C32×#41—36no—
9C2×C32×#41C2×C316yescomplemented
10C22×C32×Q8C2×C311yescharacteristic
11C22×C32×D4—31no—
12C23×C32×C4—31no—
13C2×C3×#41⋊C3C2×C3172yesdirect product factor
14C22×C3×SL(2,𝔽3)C2×C3112yescomplemented
15C23×C33—41no—
16C2×C33×C4—46no—
17C32×SL(2,𝔽3)C2311yescharacteristic, complemented
18C22×C3×#41C3214yescomplemented
19C22×#41⋊C3C3219yesdirect product factor
20C24×C32—31no—
21C32×#41—38no—
22C32×#41—312no—
23C32×#41C22×C314yescomplemented
24C2×C32×Q8—33no—
25C2×C32×Q8C22×C313yes—
26C2×C32×D4—36no—
27C2×C32×D4—36no—
28C22×C32×C4—31no—
29C22×C32×C4—36no—
30C22×C32×C4—36no—
31C22×C32×C4—412no—
32C22×C32×C4A411yescharacteristic
33C3×#41⋊C3C22×C3148yesdirect product factor
34C2×C3×SL(2,𝔽3)C22×C3136yescomplemented
35C22×C33—44no—
36C22×C33—43no—
37C33×C4—44no—
38C2×C3×#41—324no—
39C2×C3×#41C2×C32124yescomplemented
40C22×C3×Q8C2×C3214yes—
41C22×C3×D4—34no—
42C23×C3×C4—34no—
43C2×#41⋊C3C2×C32154yesdirect product factor
44C22×SL(2,𝔽3)C2×C3219yescomplemented
45C23×C32—36no—
46C23×C32—412no—
47C23×C32—64no—
48C23×C32C2×A411yescharacteristic
49C32×Q8—33no—
50C32×Q8—32no—
51C32×Q8C23×C311yescharacteristic
52C32×D4—34no—
53C32×D4—312no—
54C2×C32×C4—33no—
55C2×C32×C4—33no—
56C2×C32×C4—312no—
57C2×C32×C4—34no—
58C2×C32×C4—472no—
59C2×C32×C4C2×A416yes—
60C3×SL(2,𝔽3)C23×C3112yescomplemented
61C22×#41C3311yescharacteristic, complemented, Sylow
62C2×C33—46no—
63C2×C33—41no—
64C24×C3—34no—
65C3×#41—332no—
66C3×#41—348no—
67C3×#41C22×C32116yescomplemented
68C2×C3×Q8—312no—
69C2×C3×Q8C22×C32112yes—
70C2×C3×D4—324no—
71C2×C3×D4—324no—
72C22×C3×C4—34no—
73C22×C3×C4—324no—
74C22×C3×C4—324no—
75C22×C3×C4—49no—
76C22×C3×C4C3×A414yes—
77#41⋊C3C22×C32136yesdirect product factor
78C2×SL(2,𝔽3)C22×C32127yescomplemented
79C22×C32—34no—
80C22×C32—436no—
81C22×C32—448no—
82C22×C32—612no—
83C22×C32#41⋊C314yesdirect product factor
84C22×C32C22×A413yes—
85C32×C4—33no—
86C32×C4—31no—
87C32×C4—448no—
88C32×C4C22×A414yes—
89C2×#41—36no—
90C2×#41C2×C3316yescomplemented
91C22×Q8C2×C3311yescharacteristic
92C22×D4—31no—
93C23×C4—31no—
94C33—41noSylow
95C23×C3—324no—
96C23×C3—49no—
97C23×C3—616no—
98C23×C3C2×C3×A414yes—
99C3×Q8—312no—
100C3×Q8—38no—
101C3×Q8C23×C3214yes—
102C3×D4—316no—
103C3×D4—348no—
104C2×C3×C4—312no—
105C2×C3×C4—312no—
106C2×C3×C4—348no—
107C2×C3×C4—316no—
108C2×C3×C4—454no—
109C2×C3×C4C2×C3×A4124yes—
110SL(2,𝔽3)C23×C3219yescomplemented
111C2×C32—472no—
112C2×C32—412no—
113C2×C32—64no—
114C2×C32C2×#41⋊C316yesdirect product factor
115C2×C32C23×A411yescharacteristic
116C24—31no—
117#41—38no—
118#41—312no—
119#41C22×C3314yescomplemented
120C2×Q8—33no—
121C2×Q8C22×C3313yes—
122C2×D4—36no—
123C2×D4—36no—
124C22×C4—31no—
125C22×C4—36no—
126C22×C4—36no—
127C22×C4C32×A411yescharacteristic
128C22×C3—316no—
129C22×C3—427no—
130C22×C3—436no—
131C22×C3—648no—
132C22×C3C3×#41⋊C3116yesdirect product factor
133C22×C3C22×C3×A4112yes—
134C3×C4—34no—
135C3×C4—312no—
136C3×C4—436no—
137C3×C4C22×C3×A4116yes—
138C32—412no—
139C32C22×#41⋊C311yescharacteristic, direct product factor
140C23—36no—
141C23—64no—
142C23C2×C32×A411yescharacteristic
143Q8—33no—
144Q8—32no—
145Q8C23×C3311yescharacteristic
146D4—34no—
147D4—312no—
148C2×C4—33no—
149C2×C4—33no—
150C2×C4—34no—
151C2×C4—312no—
152C2×C4C2×C32×A416yes—
153C2×C3—454no—
154C2×C3—49no—
155C2×C3—616no—
156C2×C3C2×C3×#41⋊C3124yesdirect product factor
157C2×C3C23×C3×A414yes—
158C22—34no—
159C22—612no—
160C22C32×#41⋊C314yesdirect product factor
161C22C22×C32×A413yes—
162C4—31no—
163C4—33no—
164C4C22×C32×A414yes—
165C3—49no—
166C3C22×C3×#41⋊C314yesdirect product factor
167C2—64no—
168C2C2×C32×#41⋊C316yesdirect product factor
169C2C23×C32×A411yescharacteristic
subgroupG/H
center, hypercenterC22×C32×C4A4
commutator, cosocle, hypocenterQ8C23×C33
FrattiniC2C23×C32×A4
FittingC22×C32×#41C3
socleC23×C32C2×A4
LCSG ▷ Q8
UCSC1 ◁ C22×C32×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2direct (N × H)
2. C22×C32×SL(2,𝔽3)C2semidirect
3. C22×C32×#41C3semidirect
4. C22×C3×#41⋊C3C3direct (N × H)
6. C32×#41⋊C3C22direct (N × H)
7. C2×C32×SL(2,𝔽3)C22semidirect
9. C2×C32×#41C2×C3semidirect
13. C2×C3×#41⋊C3C2×C3direct (N × H)
14. C22×C3×SL(2,𝔽3)C2×C3semidirect
17. C32×SL(2,𝔽3)C23semidirect
18. C22×C3×#41C32semidirect
19. C22×#41⋊C3C32direct (N × H)
23. C32×#41C22×C3semidirect
33. C3×#41⋊C3C22×C3direct (N × H)
34. C2×C3×SL(2,𝔽3)C22×C3semidirect
39. C2×C3×#41C2×C32semidirect
43. C2×#41⋊C3C2×C32direct (N × H)
44. C22×SL(2,𝔽3)C2×C32semidirect
60. C3×SL(2,𝔽3)C23×C3semidirect
61. C22×#41C33semidirect
67. C3×#41C22×C32semidirect
77. #41⋊C3C22×C32direct (N × H)
78. C2×SL(2,𝔽3)C22×C32semidirect
83. C22×C32#41⋊C3direct (N × H)
90. C2×#41C2×C33semidirect
110. SL(2,𝔽3)C23×C32semidirect
114. C2×C32C2×#41⋊C3direct (N × H)
119. #41C22×C33semidirect
132. C22×C3C3×#41⋊C3direct (N × H)
139. C32C22×#41⋊C3direct (N × H)
156. C2×C3C2×C3×#41⋊C3direct (N × H)
160. C22C32×#41⋊C3direct (N × H)
166. C3C22×C3×#41⋊C3direct (N × H)
168. C2C2×C32×#41⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C22×C32×SL(2,𝔽3)2
C22×C32×#413
C22×C3×#41⋊C33
C22×C33×C44