Finite Group Databasebeta

← C2×#11723949 #13179871

Subgroups — C2×#11723949 #13179871

Order 1728.

4172 conjugacy classes of subgroups; only the 891 normal subgroups are shown.

normal subgroups only

This group has 4172 conjugacy classes of subgroups — too many to draw. Only its 891 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×#89076C211yesmaximal, characteristic, complemented
2C22×C4⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
3C2×C33×M16C211yesmaximal, characteristic, complemented
4#11723949C218yesmaximal, direct product factor
5C2×#88852C212yesmaximal, complemented
6C2×#88851C212yesmaximal, complemented
7C2×#64117—313nomaximal
8C23×C3⋊(C3⋊D3)—21no—
9C22×C33×C4C2211yescharacteristic
10C2×#6312—21no—
11#89076—22no—
12#89076C2214yescomplemented
13C2×C4⋊(C3⋊(C3⋊D3))—22no—
14C2×C4⋊(C3⋊(C3⋊D3))—21no—
15C2×C4⋊(C3⋊(C3⋊D3))C2212yes—
16C2×C4⋊(C3⋊(C3⋊D3))C2214yescomplemented
17C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
18C2×C4×C3⋊(C3⋊D3)—21no—
19C2×#6308C2211yescharacteristic, complemented
20C33×M16C2214yescomplemented
21C2×C33×C8C2212yescomplemented
22#88852C2218yescomplemented
23#88851C2218yescomplemented
24C2×#3828—313no—
25C22×C4⋊(C3⋊D3)—313no—
26C2×C32×M16D3113yescomplemented
27#64117—3104no—
28C2×#3745—326no—
29C2×#3744—326no—
30C23×C33D411yescharacteristic
31C22×C3⋊(C3⋊D3)—22no—
32C22×C3⋊(C3⋊D3)—24no—
33C22×C3⋊(C3⋊D3)—21no—
34C22×C3⋊(C3⋊D3)—44no—
35C2×C33×C4D411yescharacteristic
36C2×C33×C4D412yes—
37C2×C33×C4C2311yescharacteristic
38C2×C33×C4C2312yes—
39#6312—24no—
40C2×C3⋊(C3⋊Dic3)—21no—
41C4⋊(C3⋊(C3⋊D3))—22no—
42C4⋊(C3⋊(C3⋊D3))—24no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))C2314yes—
45C4⋊(C3⋊(C3⋊D3))C2312yescomplemented
46C4×C3⋊(C3⋊D3)—24no—
47#6308—21no—
48#6308C2312yescomplemented
49C33×C8C2314yescomplemented
50C2×#908—913no—
51C23×C3⋊D3—613no—
52C22×C32×C4C2×D3113yes—
53C2×#638—613no—
54#3828—352no—
55#3828—626no—
56C2×C4⋊(C3⋊D3)—326no—
57C2×C4⋊(C3⋊D3)—352no—
58C2×C4⋊(C3⋊D3)—313no—
59C2×C4⋊(C3⋊D3)—613no—
60C2×C4⋊(C3⋊D3)—626no—
61C2×C4×C3⋊D3—613no—
62C2×#634—313no—
63C32×M16C2×D3152yescomplemented
64C2×C32×C8C2×D3126yescomplemented
65#3745—3104no—
66#3744—3104no—
67C22×C33—22no—
68C22×C33C2×D412yes—
69C22×C33C2×D411yescharacteristic
70C2×C3⋊(C3⋊D3)—22no—
71C2×C3⋊(C3⋊D3)—24no—
72C2×C3⋊(C3⋊D3)—48no—
73C2×C3⋊(C3⋊D3)—42no—
74C2×C3⋊(C3⋊D3)—44no—
75C33×C4C2×D411yescharacteristic
76C33×C4C2×D412yes—
77C33×C4C2411yescharacteristic
78C3⋊(C3⋊Dic3)—22no—
79C2×#235—913no—
80C22×D12—913no—
81C2×C3×M16C3⋊D3113yescomplemented
82#908—9104no—
83C2×D24—926no—
84C2×#204—926no—
85C23×C32D12113yes—
86C22×C3⋊D3—652no—
87C22×C3⋊D3—626no—
88C22×C3⋊D3—613no—
89C22×C3⋊D3—1252no—
90C2×C32×C4D12113yes—
91C2×C32×C4D12126yes—
92C2×C32×C4C22×D3113yes—
93C2×C32×C4C22×D3126yes—
94#638—652no—
95C2×C3⋊Dic3—613no—
96C4⋊(C3⋊D3)—352no—
97C4⋊(C3⋊D3)—326no—
98C4⋊(C3⋊D3)—626no—
99C4⋊(C3⋊D3)—652no—
100C4⋊(C3⋊D3)—613no—
101C4×C3⋊D3—652no—
102#634—326no—
103#634—613no—
104C32×C8C22×D3152yescomplemented
105C2×AGL(1,ℤ/8)—271noSylow
106C2×C33—22no—
107C2×C33AGL(1,ℤ/8)12yescomplemented
108C2×C33C22×D411yescharacteristic
109C3⋊(C3⋊D3)—42no—
110C3⋊(C3⋊D3)—44no—
111C23×D3—1813no—
112C22×C3×C4C2×C3⋊D3113yes—
113C2×#67—1813no—
114#235—952no—
115#235—1826no—
116C2×D12—926no—
117C2×D12—913no—
118C2×D12—952no—
119C2×D12—1813no—
120C2×D12—1826no—
121C2×C4×D3—1813no—
122C2×Dic6—913no—
123C3×M16C2×C3⋊D3152yescomplemented
124C2×C3×C8C2×C3⋊D3126yescomplemented
125D24—9104no—
126#204—9104no—
127C22×C32—226no—
128C22×C32C2×D12126yes—
129C22×C32C2×D12113yes—
130C2×C3⋊D3—626no—
131C2×C3⋊D3—652no—
132C2×C3⋊D3—12104no—
133C2×C3⋊D3—1226no—
134C2×C3⋊D3—1252no—
135C32×C4C2×D12113yes—
136C32×C4C2×D12126yes—
137C32×C4C23×D3113yes—
138C3⋊Dic3—626no—
139C2×#41—271no—
140C22×D4—271no—
141AGL(1,ℤ/8)—278no—
142C2×SD16—272no—
143C2×D8—272no—
144C2×M16C3⋊(C3⋊D3)11yescharacteristic, complemented
145C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
146C23×C3C4⋊(C3⋊D3)113yes—
147C22×D3—1813no—
148C22×D3—1852no—
149C22×D3—1826no—
150C22×D3—3652no—
151C2×C3×C4C4⋊(C3⋊D3)113yes—
152C2×C3×C4C4⋊(C3⋊D3)126yes—
153C2×C3×C4C22×C3⋊D3113yes—
154C2×C3×C4C22×C3⋊D3126yes—
155#67—1852no—
156C2×Dic3—1813no—
157D12—952no—
158D12—926no—
159D12—1826no—
160D12—1852no—
161D12—1813no—
162C4×D3—1852no—
163Dic6—926no—
164Dic6—1813no—
165C3×C8C22×C3⋊D3152yescomplemented
166C2×C32—226no—
167C2×C32#908126yescomplemented
168C2×C32C22×D12113yes—
169C3⋊D3—1226no—
170C3⋊D3—1252no—
171C24—541no—
172#41—274no—
173#41—542no—
174C2×Q8—271no—
175C2×D4—271no—
176C2×D4—272no—
177C2×D4—274no—
178C2×D4—541no—
179C2×D4—542no—
180C2×D4—541no—
181C22×C4—541no—
182C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
183SD16—278no—
184D8—278no—
185M16C2×C3⋊(C3⋊D3)14yescomplemented
186C2×C8C2×C3⋊(C3⋊D3)12yescomplemented
187C22×C3—226no—
188C22×C3C2×C4⋊(C3⋊D3)126yes—
189C22×C3C2×C4⋊(C3⋊D3)113yes—
190C2×D3—1826no—
191C2×D3—1852no—
192C2×D3—36104no—
193C2×D3—3652no—
194C2×D3—3626no—
195C3×C4C2×C4⋊(C3⋊D3)113yes—
196C3×C4C2×C4⋊(C3⋊D3)126yes—
197C3×C4C23×C3⋊D3113yes—
198Dic3—1826no—
199C32C2×#908113yescomplemented
200C23—544no—
201C23—541no—
202C23—542no—
203C23—1084no—
204C23C4⋊(C3⋊(C3⋊D3))11yescharacteristic
205Q8—272no—
206Q8—541no—
207D4—272no—
208D4—274no—
209D4—544no—
210D4—544no—
211D4—542no—
212D4—541no—
213C2×C4—541no—
214C2×C4—544no—
215C2×C4C4⋊(C3⋊(C3⋊D3))12yes—
216C2×C4C4⋊(C3⋊(C3⋊D3))11yescharacteristic
217C2×C4C22×C3⋊(C3⋊D3)12yes—
218C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
219C8C22×C3⋊(C3⋊D3)14yescomplemented
220C2×C3—226no—
221C2×C3#64117126yescomplemented
222C2×C3C22×C4⋊(C3⋊D3)113yes—
223D3—3652no—
224D3—3626no—
225C22—22no—
226C22—542no—
227C22—544no—
228C22—1088no—
229C22—1084no—
230C22—1082no—
231C22C2×C4⋊(C3⋊(C3⋊D3))12yes—
232C22C2×C4⋊(C3⋊(C3⋊D3))11yescharacteristic
233C4—542no—
234C4C2×C4⋊(C3⋊(C3⋊D3))12yes—
235C4C2×C4⋊(C3⋊(C3⋊D3))11yescharacteristic
236C4C23×C3⋊(C3⋊D3)11yescharacteristic
237C3C2×#64117113yescomplemented
238C2—22no—
239C2—1084no—
240C2—1082no—
241C2#1172394912yesdirect product factor
242C2C22×C4⋊(C3⋊(C3⋊D3))11yescharacteristic
subgroupG/H
centerC22C2×C4⋊(C3⋊(C3⋊D3))
commutator, cosocleC33×C4C24
FrattiniC4C23×C3⋊(C3⋊D3)
FittingC2×C33×M16C2
socleC22×C33C2×D4
hypercenterC2×M16C3⋊(C3⋊D3)
hypocenterC33C2×AGL(1,ℤ/8)
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×#89076C2semidirect
2. C22×C4⋊(C3⋊(C3⋊D3))C2semidirect
3. C2×C33×M16C2semidirect
4. #11723949C2direct (N × H)
5. C2×#88852C2semidirect
6. C2×#88851C2semidirect
12. #89076C22semidirect
16. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
17. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
19. C2×#6308C22semidirect
20. C33×M16C22semidirect
21. C2×C33×C8C22semidirect
22. #88852C22semidirect
23. #88851C22semidirect
26. C2×C32×M16D3semidirect
45. C4⋊(C3⋊(C3⋊D3))C23semidirect
48. #6308C23semidirect
49. C33×C8C23semidirect
63. C32×M16C2×D3semidirect
64. C2×C32×C8C2×D3semidirect
81. C2×C3×M16C3⋊D3semidirect
104. C32×C8C22×D3semidirect
107. C2×C33AGL(1,ℤ/8)semidirect
123. C3×M16C2×C3⋊D3semidirect
124. C2×C3×C8C2×C3⋊D3semidirect
144. C2×M16C3⋊(C3⋊D3)semidirect
145. C33C2×AGL(1,ℤ/8)semidirect
165. C3×C8C22×C3⋊D3semidirect
167. C2×C32#908semidirect
185. M16C2×C3⋊(C3⋊D3)semidirect
186. C2×C8C2×C3⋊(C3⋊D3)semidirect
199. C32C2×#908semidirect
219. C8C22×C3⋊(C3⋊D3)semidirect
221. C2×C3#64117semidirect
237. C3C2×#64117semidirect
241. C2#11723949direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890762
C22×C4⋊(C3⋊(C3⋊D3))2
C2×C33×M162
#117239492
C2×#888522
C2×#888512
C2×#641173