Finite Group Databasebeta

← #13179873

Subgroups — #13179873

Order 1728.

3668 conjugacy classes of subgroups; only the 835 normal subgroups are shown.

normal subgroups only

This group has 3668 conjugacy classes of subgroups — too many to draw. Only its 835 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×#89076C212yesmaximal, complemented
2C2×C33×M16C211yesmaximal, characteristic, complemented
3#11723950C214yesmaximal, complemented
4#11723949C214yesmaximal, complemented
5#11723945C214yesmaximal, complemented
6#10596219—313nomaximal
7C22×C33×C4C2211yescharacteristic
8C2×#6312—22no—
9#89076—22no—
10#89076—22no—
11#89076C2214yes—
12#89076C2214yescomplemented
13C2×C4⋊(C3⋊(C3⋊D3))C2212yescomplemented
14C2×C4×C3⋊(C3⋊D3)—22no—
15C2×#6308C2212yescomplemented
16C33×M16C2214yescomplemented
17C2×C33×C8C2212yescomplemented
18#88853C2214yescomplemented
19#88852C2214yescomplemented
20#88851C2218yescomplemented
21C2×#3828—326no—
22C2×C32×M16D3113yescomplemented
23#64118—352no—
24#64117—352no—
25#64113—352no—
26C23×C33D411yescharacteristic
27C22×C3⋊(C3⋊D3)—22no—
28C2×C33×C4D412yes—
29C2×C33×C4D411yescharacteristic
30C2×C33×C4C2311yescharacteristic
31C2×C33×C4C2312yes—
32#6312—24no—
33#6312—24no—
34C2×C3⋊(C3⋊Dic3)—22no—
35C4⋊(C3⋊(C3⋊D3))—22no—
36C4⋊(C3⋊(C3⋊D3))C2314yes—
37C4×C3⋊(C3⋊D3)—24no—
38C4×C3⋊(C3⋊D3)—24no—
39#6308—22no—
40#6308C2314yes—
41C33×C8C2314yes—
42#5748—913no—
43C22×C32×C4C2×D3113yes—
44C2×#638—626no—
45#3828—352no—
46#3828—352no—
47#3828—626no—
48#3828—626no—
49C2×C4⋊(C3⋊D3)—326no—
50C2×C4×C3⋊D3—626no—
51C2×#634—326no—
52C32×M16C2×D3152yescomplemented
53C2×C32×C8C2×D3126yescomplemented
54#3746—352no—
55#3745—352no—
56#3744—3104no—
57C22×C33—41no—
58C22×C33C2×D412yes—
59C22×C33C2×D411yescharacteristic
60C2×C3⋊(C3⋊D3)—24no—
61C2×C3⋊(C3⋊D3)—44no—
62C33×C4C2×D412yes—
63C33×C4C2×D411yescharacteristic
64C33×C4C2411yescharacteristic
65C3⋊(C3⋊Dic3)—24no—
66C2×#235—926no—
67C2×C3×M16C3⋊D3113yescomplemented
68#909—952no—
69#908—952no—
70#904—952no—
71C23×C32D12113yes—
72C22×C3⋊D3—626no—
73C2×C32×C4D12126yes—
74C2×C32×C4D12113yes—
75C2×C32×C4C22×D3113yes—
76C2×C32×C4C22×D3126yes—
77#638—652no—
78#638—652no—
79C2×C3⋊Dic3—626no—
80C4⋊(C3⋊D3)—352no—
81C4⋊(C3⋊D3)—626no—
82C4×C3⋊D3—652no—
83C4×C3⋊D3—652no—
84#634—352no—
85#634—626no—
86C32×C8C22×D3152yes—
87#575—271noSylow
88C2×C33—22no—
89C2×C33—21no—
90C2×C33C22×D411yescharacteristic
91C3⋊(C3⋊D3)—44no—
92C22×C3×C4C2×C3⋊D3113yes—
93C2×#67—1826no—
94#235—952no—
95#235—952no—
96#235—1826no—
97#235—1826no—
98C2×D12—926no—
99C2×C4×D3—1826no—
100C2×Dic6—926no—
101C3×M16C2×C3⋊D3152yescomplemented
102C2×C3×C8C2×C3⋊D3126yescomplemented
103Dic12—952no—
104D24—952no—
105#204—9104no—
106C22×C32—413no—
107C22×C32C2×D12126yes—
108C22×C32C2×D12113yes—
109C2×C3⋊D3—652no—
110C2×C3⋊D3—1252no—
111C32×C4C2×D12126yes—
112C32×C4C2×D12113yes—
113C32×C4C23×D3113yes—
114C3⋊Dic3—652no—
115C2×#41—272no—
116#137—274no—
117AGL(1,ℤ/8)—274no—
118#135—274no—
119C2×M16C3⋊(C3⋊D3)11yescharacteristic, complemented
120C33#57511yescharacteristic, complemented, Sylow
121C23×C3C4⋊(C3⋊D3)113yes—
122C22×D3—1826no—
123C2×C3×C4C4⋊(C3⋊D3)126yes—
124C2×C3×C4C4⋊(C3⋊D3)113yes—
125C2×C3×C4C22×C3⋊D3113yes—
126C2×C3×C4C22×C3⋊D3126yes—
127#67—1852no—
128#67—1852no—
129C2×Dic3—1826no—
130D12—952no—
131D12—1826no—
132C4×D3—1852no—
133C4×D3—1852no—
134Dic6—952no—
135Dic6—1826no—
136C3×C8C22×C3⋊D3152yes—
137C2×C32—226no—
138C2×C32—213no—
139C2×C32C22×D12113yes—
140C3⋊D3—1252no—
141#41—274no—
142#41—274no—
143#41—542no—
144#41—542no—
145C2×Q8—272no—
146C2×D4—272no—
147C2×D4—542no—
148C22×C4—542no—
149C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
150Q16—274no—
151SD16—278no—
152D8—274no—
153M16C2×C3⋊(C3⋊D3)14yescomplemented
154C2×C8C2×C3⋊(C3⋊D3)12yescomplemented
155C22×C3—413no—
156C22×C3C2×C4⋊(C3⋊D3)126yes—
157C22×C3C2×C4⋊(C3⋊D3)113yes—
158C2×D3—1852no—
159C2×D3—3652no—
160C3×C4C2×C4⋊(C3⋊D3)126yes—
161C3×C4C2×C4⋊(C3⋊D3)113yes—
162C3×C4C23×C3⋊D3113yes—
163Dic3—1852no—
164C32#5748113yescomplemented
165C23—542no—
166C23C4⋊(C3⋊(C3⋊D3))11yescharacteristic
167Q8—274no—
168Q8—542no—
169D4—274no—
170D4—544no—
171D4—544no—
172D4—542no—
173C2×C4—542no—
174C2×C4—544no—
175C2×C4—544no—
176C2×C4C4⋊(C3⋊(C3⋊D3))11yescharacteristic
177C2×C4C4⋊(C3⋊(C3⋊D3))12yes—
178C2×C4C22×C3⋊(C3⋊D3)12yes—
179C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
180C8C22×C3⋊(C3⋊D3)14yes—
181C2×C3—226no—
182C2×C3—213no—
183C2×C3C22×C4⋊(C3⋊D3)113yes—
184D3—3652no—
185C22—41no—
186C22—544no—
187C22—1084no—
188C22C2×C4⋊(C3⋊(C3⋊D3))12yes—
189C22C2×C4⋊(C3⋊(C3⋊D3))11yescharacteristic
190C4—544no—
191C4C2×C4⋊(C3⋊(C3⋊D3))12yes—
192C4C2×C4⋊(C3⋊(C3⋊D3))11yescharacteristic
193C4C23×C3⋊(C3⋊D3)11yescharacteristic
194C3#10596219113yescomplemented
195C2—22no—
196C2—21no—
197C2—1084no—
198C2C22×C4⋊(C3⋊(C3⋊D3))11yescharacteristic
subgroupG/H
centerC4C2×C4⋊(C3⋊(C3⋊D3))
commutator, cosocleC33×C4C24
FrattiniC4C23×C3⋊(C3⋊D3)
FittingC2×C33×M16C2
socleC2×C33C22×D4
hypercenterC2×M16C3⋊(C3⋊D3)
hypocenterC33#575
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×#89076C2semidirect
2. C2×C33×M16C2semidirect
3. #11723950C2semidirect
4. #11723949C2semidirect
5. #11723945C2semidirect
12. #89076C22semidirect
13. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
15. C2×#6308C22semidirect
16. C33×M16C22semidirect
17. C2×C33×C8C22semidirect
18. #88853C22semidirect
19. #88852C22semidirect
20. #88851C22semidirect
22. C2×C32×M16D3semidirect
52. C32×M16C2×D3semidirect
53. C2×C32×C8C2×D3semidirect
67. C2×C3×M16C3⋊D3semidirect
101. C3×M16C2×C3⋊D3semidirect
102. C2×C3×C8C2×C3⋊D3semidirect
119. C2×M16C3⋊(C3⋊D3)semidirect
120. C33#575semidirect
153. M16C2×C3⋊(C3⋊D3)semidirect
154. C2×C8C2×C3⋊(C3⋊D3)semidirect
164. C32#5748semidirect
194. C3#10596219semidirect

Maximal subgroups

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

subgroupindex
C2×#890762
C2×C33×M162
#117239502
#117239492
#117239452
#105962193