Finite Group Databasebeta

← C2×#72240994 #72343687

Subgroups — C2×#72240994 #72343687

Order 6144.

Only the 818 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 219 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#72236877C211yesmaximal, characteristic, complemented
2C2×#72234214C211yesmaximal, characteristic, complemented
3C2×#72236845C211yesmaximal, characteristic, complemented
4C2×#72236871C212yesmaximal, complemented
5C2×#72236882C212yesmaximal, complemented
6C2×#72236888C212yesmaximal, complemented
7C2×#11715688C2212yescomplemented
8C2×#72236876C212yesmaximal, complemented
9C2×#11715735C2212yescomplemented
10C2×#11715713C2212yescomplemented
11C2×#11715736C2212yescomplemented
12C2×#72236781C211yesmaximal, characteristic, complemented
13C2×#11715709C2212yescomplemented
14C2×#11715709C2212yescomplemented
15C2×#11715711C2212yescomplemented
16C2×#11715661C2211yescharacteristic, complemented
17C2×#72215685C211yesmaximal, characteristic, complemented
18C2×#11710873C2211yescharacteristic, complemented
19C2×#11710899C2212yescomplemented
20C22×#11715839C211yesmaximal, characteristic, complemented
21C22×#87521C2211yescharacteristic, complemented
22C2×#72233470C211yesmaximal, characteristic, complemented
23C2×#11715842C2211yescharacteristic, complemented
24C2×#11715840C2211yescharacteristic, complemented
25C2×#11715841C2211yescharacteristic, complemented
26C2×#11715838C2211yescharacteristic, complemented
27C2×#11710937C2211yescharacteristic, complemented
28C22×#87547C2211yescharacteristic, complemented
29C2×#10586999D311yescharacteristic, complemented
30C22×#87519C2211yescharacteristic
31#72240994C2116yesmaximal, direct product factor
32C2×#11715687C2212yescomplemented
33#72236888C22116yescomplemented
34#72236882C22116yescomplemented
35#72236877C2218yescomplemented
36#72236876C22116yescomplemented
37#72236871C22116yescomplemented
38#72236845C2218yescomplemented
39#72236781C2218yescomplemented
40#72234214C2218yescomplemented
41#72233470C2218yescomplemented
42C2×#11710898C2212yescomplemented
43C2×#11715839C2212yescomplemented
44C2×#11715839C2212yescomplemented
45C2×#11715839C2218yescomplemented
46C2×#11715839C2211yescharacteristic, complemented
47#72215685C2218yescomplemented
48#11715842C2314yescomplemented
49#11715841C2314yescomplemented
50#11715840C2314yescomplemented
51#11715839C2318yescomplemented
52#11715839C2318yescomplemented
53#11715839C2314yescomplemented
54#11715838C2314yescomplemented
55C2×#87517C2311yescharacteristic, complemented
56C2×#87521C2312yes—
57C2×#87521C2314yescomplemented
58C2×#87521C2311yescharacteristic, complemented
59C2×#87519D414yes—
60C2×#87519D411yescharacteristic
61C2×#87519D411yescharacteristic
62C2×#87519C2314yes—
63C2×#87519C2311yescharacteristic
64C2×#87519C2311yescharacteristic
65C22×#5932D411yescharacteristic
66C22×#5934D411yescharacteristic
67#11715736C2318yescomplemented
68#11715735C2318yescomplemented
69#11715713C2318yescomplemented
70#11715711C2318yescomplemented
71#11715709C2318yescomplemented
72#11715709C2318yescomplemented
73#11715688C2318yescomplemented
74#11715687C2318yescomplemented
75#11715661C2314yescomplemented
76C2×#85566C2312yescomplemented
77#11710937C2314yescomplemented
78C2×#87548C2311yescharacteristic, complemented
79C2×#87547C2312yes—
80C2×#87547C2314yescomplemented
81C2×#87547C2311yescharacteristic, complemented
82C22×#5950C2311yescharacteristic
83#11710899C2318yescomplemented
84#11710898C2318yescomplemented
85#11710873C2314yescomplemented
86C2×#85666C2312yescomplemented
87C2×#63103C2×D311yescharacteristic, complemented
88C22×H(7,𝔽2)C2×D311yescharacteristic, complemented
89#10586999C2×D318yescomplemented
90C2×#63092C2×D312yescomplemented
91C2×#63091C2×D312yescomplemented
92C2×#63076C2×D311yescharacteristic, complemented
93#87548C2412yescomplemented
94#87547C2414yes—
95#87547C2412yescomplemented
96C2×#5950C2×D411yescharacteristic
97C2×#5950C2×D412yes—
98C2×#5950C2411yescharacteristic
99C2×#5950C2412yes—
100C22×#997C2×D411yescharacteristic
101#87521C2414yes—
102#87521C2412yescomplemented
103#87519C2×D414yes—
104#87519C2×D414yes—
105#87519C2×D412yes—
106#87519C2×D412yes—
107#87519C2412yes—
108#87519C2412yes—
109#87517C2412yescomplemented
110C2×#5934C2×D414yes—
111C2×#5934C2×D411yescharacteristic
112C2×#5934C2×D411yescharacteristic
113C2×#5932C2×D414yes—
114C2×#5932C2×D411yescharacteristic
115C2×#5932C2×D411yescharacteristic
116#85666C2414yescomplemented
117#85566C2414yescomplemented
118#63103C22×D314yescomplemented
119C2×2−1+6C22×D311yescharacteristic, complemented
120C2×H(7,𝔽2)C22×D312yes—
121C2×H(7,𝔽2)C22×D314yescomplemented
122C2×H(7,𝔽2)C22×D311yescharacteristic, complemented
123C22×#585C22×D311yescharacteristic
124#63092C22×D318yescomplemented
125#63091C22×D318yescomplemented
126C23×AGL(1,ℤ/8)S413yescomplemented
127#63076C22×D314yescomplemented
128C2×#3573C22×D312yescomplemented
129#5950C22×D411yescharacteristic
130#5950C22×D412yes—
131#5950C2511yescharacteristic
132C2×#997C22×D411yescharacteristic
133C2×#997C22×D412yes—
134#5934C22×D412yes—
135#5934C22×D412yes—
136#5932C22×D412yes—
137#5932C22×D412yes—
1382−1+6C23×D312yescomplemented
139H(7,𝔽2)C23×D314yes—
140H(7,𝔽2)C23×D312yescomplemented
141C2×#585D3×D412yes—
142C2×#585D3×D411yescharacteristic
143C2×#585C23×D311yescharacteristic
144C2×#585C23×D312yes—
145C22×H(5,𝔽2)D3×D411yescharacteristic
146C23×#41C2×S413yescomplemented
147C24×D4C2×S413yescomplemented
148C22×AGL(1,ℤ/8)C2×S4124yescomplemented
149C23×SD16C2×S416yescomplemented
150C23×D8C2×S416yescomplemented
151#3573C23×D314yescomplemented
152C23×M16C2×S413yescomplemented
153#997C23×D411yescharacteristic
154#585C2×D3×D411yescharacteristic
155#585C2×D3×D412yes—
156#585C24×D311yescharacteristic
157C2×H(5,𝔽2)C2×D3×D412yes—
158C2×H(5,𝔽2)C2×D3×D411yescharacteristic
159C22×#41C22×S4112yescomplemented
160C23×Q8C22×S413yescomplemented
161C23×D4C22×S416yes—
162C23×D4C22×S4112yescomplemented
163C23×D4C22×S413yescomplemented
164C24×C4C22×S413yes—
165C2×AGL(1,ℤ/8)#102011yescharacteristic
166C22×SD16C22×S4124yescomplemented
167C22×D8C22×S4124yescomplemented
168C22×M16C22×S4112yescomplemented
169C23×C8C22×S416yescomplemented
170C25D4×S413yes—
171H(5,𝔽2)C22×D3×D411yescharacteristic
172C2×#41C2×#102011yescharacteristic
173C22×Q8C23×S416yescomplemented
174C22×D4C23×S4112yes—
175C22×D4C23×S416yescomplemented
176C22×D4C2×#102011yescharacteristic
177C23×C4D4×S416yes—
178C23×C4D4×S413yes—
179C23×C4C23×S413yes—
180C23×C4C23×S416yes—
181AGL(1,ℤ/8)C2×#102018yes—
182C2×SD16C2×#102012yes—
183C2×D8C2×#102012yes—
184C2×M16C2×#102011yescharacteristic
185C22×C8C23×S4112yescomplemented
186C24C2×D4×S416yes—
187C24C2×D4×S413yes—
188#41C22×#102014yes—
189C2×Q8C22×#102011yescharacteristic
190C2×D4C22×#102012yes—
191C2×D4C22×#102011yescharacteristic
192C2×D4C22×#102014yes—
193C22×C4C2×D4×S413yes—
194C22×C4C2×D4×S416yes—
195C22×C4C24×S413yes—
196C22×C4C22×#102011yescharacteristic
197SD16C22×#102018yes—
198D8C22×#102018yes—
199M16C22×#102014yes—
200C2×C8C22×#102012yes—
201C23C22×D4×S413yes—
202C23D4×#102011yescharacteristic
203Q8C23×#102012yes—
204D4C23×#102012yes—
205D4C23×#102014yes—
206C2×C4D4×#102012yes—
207C2×C4D4×#102011yescharacteristic
208C2×C4C23×#102012yes—
209C2×C4C23×#102011yescharacteristic
210C8C23×#102014yes—
211C22C2×D4×#102012yes—
212C22C2×D4×#102011yescharacteristic
213C4C24×#102011yescharacteristic
214C4C2×D4×#102012yes—
215C4C2×D4×#102011yescharacteristic
216C2#7224099412yesdirect product factor
217C2C22×D4×#102011yescharacteristic
subgroupG/H
center, socleC22C2×D4×#1020
commutator, cosocle#5950C25
FrattiniC4C24×#1020
FittingC2×#10586999D3
hypercenterC2×AGL(1,ℤ/8)#1020
hypocenter#997C23×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#72236877C2semidirect
2. C2×#72234214C2semidirect
3. C2×#72236845C2semidirect
4. C2×#72236871C2semidirect
5. C2×#72236882C2semidirect
6. C2×#72236888C2semidirect
7. C2×#11715688C22semidirect
8. C2×#72236876C2semidirect
9. C2×#11715735C22semidirect
10. C2×#11715713C22semidirect
11. C2×#11715736C22semidirect
12. C2×#72236781C2semidirect
13. C2×#11715709C22semidirect
14. C2×#11715709C22semidirect
15. C2×#11715711C22semidirect
16. C2×#11715661C22semidirect
17. C2×#72215685C2semidirect
18. C2×#11710873C22semidirect
19. C2×#11710899C22semidirect
20. C22×#11715839C2semidirect
21. C22×#87521C22semidirect
22. C2×#72233470C2semidirect
23. C2×#11715842C22semidirect
24. C2×#11715840C22semidirect
25. C2×#11715841C22semidirect
26. C2×#11715838C22semidirect
27. C2×#11710937C22semidirect
28. C22×#87547C22semidirect
29. C2×#10586999D3semidirect
31. #72240994C2direct (N × H)
32. C2×#11715687C22semidirect
33. #72236888C22semidirect
34. #72236882C22semidirect
35. #72236877C22semidirect
36. #72236876C22semidirect
37. #72236871C22semidirect
38. #72236845C22semidirect
39. #72236781C22semidirect
40. #72234214C22semidirect
41. #72233470C22semidirect
42. C2×#11710898C22semidirect
43. C2×#11715839C22semidirect
44. C2×#11715839C22semidirect
45. C2×#11715839C22semidirect
46. C2×#11715839C22semidirect
47. #72215685C22semidirect
48. #11715842C23semidirect
49. #11715841C23semidirect
50. #11715840C23semidirect
51. #11715839C23semidirect
52. #11715839C23semidirect
53. #11715839C23semidirect
54. #11715838C23semidirect
55. C2×#87517C23semidirect
57. C2×#87521C23semidirect
58. C2×#87521C23semidirect
67. #11715736C23semidirect
68. #11715735C23semidirect
69. #11715713C23semidirect
70. #11715711C23semidirect
71. #11715709C23semidirect
72. #11715709C23semidirect
73. #11715688C23semidirect
74. #11715687C23semidirect
75. #11715661C23semidirect
76. C2×#85566C23semidirect
77. #11710937C23semidirect
78. C2×#87548C23semidirect
80. C2×#87547C23semidirect
81. C2×#87547C23semidirect
83. #11710899C23semidirect
84. #11710898C23semidirect
85. #11710873C23semidirect
86. C2×#85666C23semidirect
87. C2×#63103C2×D3semidirect
88. C22×H(7,𝔽2)C2×D3semidirect
89. #10586999C2×D3semidirect
90. C2×#63092C2×D3semidirect
91. C2×#63091C2×D3semidirect
92. C2×#63076C2×D3semidirect
93. #87548C24semidirect
95. #87547C24semidirect
102. #87521C24semidirect
109. #87517C24semidirect
116. #85666C24semidirect
117. #85566C24semidirect
118. #63103C22×D3semidirect
119. C2×2−1+6C22×D3semidirect
121. C2×H(7,𝔽2)C22×D3semidirect
122. C2×H(7,𝔽2)C22×D3semidirect
124. #63092C22×D3semidirect
125. #63091C22×D3semidirect
126. C23×AGL(1,ℤ/8)S4semidirect
127. #63076C22×D3semidirect
128. C2×#3573C22×D3semidirect
138. 2−1+6C23×D3semidirect
140. H(7,𝔽2)C23×D3semidirect
146. C23×#41C2×S4semidirect
147. C24×D4C2×S4semidirect
148. C22×AGL(1,ℤ/8)C2×S4semidirect
149. C23×SD16C2×S4semidirect
150. C23×D8C2×S4semidirect
151. #3573C23×D3semidirect
152. C23×M16C2×S4semidirect
159. C22×#41C22×S4semidirect
160. C23×Q8C22×S4semidirect
162. C23×D4C22×S4semidirect
163. C23×D4C22×S4semidirect
166. C22×SD16C22×S4semidirect
167. C22×D8C22×S4semidirect
168. C22×M16C22×S4semidirect
169. C23×C8C22×S4semidirect
173. C22×Q8C23×S4semidirect
175. C22×D4C23×S4semidirect
185. C22×C8C23×S4semidirect
216. C2#72240994direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#722368772
C2×#722342142
C2×#722368452
C2×#722368712
C2×#722368822
C2×#722368882
C2×#722368762
C2×#722367812
C2×#722156852
C22×#117158392
C2×#722334702
C2×#722368733
#722409942
C2×S4×AGL(1,ℤ/8)4