Finite Group Databasebeta

← C2×#72240998 #72343774

Subgroups — C2×#72240998 #72343774

Order 6144.

Only the 611 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 205 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#72236903C211yesmaximal, characteristic, complemented
2C2×#72236889C211yesmaximal, characteristic, complemented
3C2×#72236926C211yesmaximal, characteristic, complemented
4C2×#72236891C212yesmaximal, complemented
5C2×#72236892C212yesmaximal, complemented
6C2×#11711589C2212yescomplemented
7C2×#72215741C211yesmaximal, characteristic, complemented
8C2×#72236908C212yesmaximal, complemented
9C2×#11711640C2212yescomplemented
10C2×#11715710C2212yescomplemented
11C2×#11711645C2212yescomplemented
12C22×#11714174C211yesmaximal, characteristic, complemented
13C22×#87522C2211yescharacteristic, complemented
14C2×#72236904C212yesmaximal, complemented
15C2×#11711587C2212yescomplemented
16C2×#11711638C2212yescomplemented
17C2×#72236755C211yesmaximal, characteristic, complemented
18C22×#3027⋊C3C2211yescharacteristic, complemented
19C2×#11711643C2212yescomplemented
20C2×#32894⋊C3C2212yescomplemented
21C2×#32895⋊C3C2212yescomplemented
22C2×#20612⋊C3C2211yescharacteristic, complemented
23C2×#11715708C2212yescomplemented
24C2×#11715667C2211yescharacteristic, complemented
25C2×#72233470C211yesmaximal, characteristic, complemented
26C2×#11715842C2211yescharacteristic, complemented
27C2×#11715840C2211yescharacteristic, complemented
28C2×#11715841C2211yescharacteristic, complemented
29C2×#11715838C2211yescharacteristic, complemented
30C2×#11710937C2211yescharacteristic, complemented
31C2×#6628885D311yescharacteristic, complemented
32C22×#87519C2211yescharacteristic
33#72240998C2116yesmaximal, direct product factor
34#72236926C2218yescomplemented
35#72236908C22116yescomplemented
36#72236904C22116yescomplemented
37#72236903C2218yescomplemented
38#72236892C22116yescomplemented
39#72236891C22116yescomplemented
40#72236889C2218yescomplemented
41C2×#11714174C2212yescomplemented
42C2×#11714174C2212yescomplemented
43C2×#11714174C2218yescomplemented
44#72236755C2218yescomplemented
45#72233470C2218yescomplemented
46C2×#11715839C2211yescharacteristic, complemented
47#72215741C2218yescomplemented
48#11715842C2314yescomplemented
49#11715841C2314yescomplemented
50#11715840C2314yescomplemented
51#11715839C2314yescomplemented
52#11715838C2314yescomplemented
53C2×#87517C2311yescharacteristic, complemented
54C2×#87521C2311yescharacteristic, complemented
55C2×#87519D411yescharacteristic, complemented
56C2×#87519D411yescharacteristic, complemented
57C2×#87519D414yescomplemented
58C2×#87519C2311yescharacteristic
59C2×#87519C2311yescharacteristic
60C2×#87519C2314yes—
61C22×#5932D411yescharacteristic, complemented
62C22×#5934D411yescharacteristic, complemented
63C2×#87522C2312yes—
64C2×#87522C2314yescomplemented
65#11715710C2318yescomplemented
66#11715708C2318yescomplemented
67#11715667C2314yescomplemented
68C2×#85567C2312yescomplemented
69#11714174C2318yescomplemented
70#11714174C2318yescomplemented
71#11711645C2318yescomplemented
72#11711643C2318yescomplemented
73#11711640C2318yescomplemented
74#11711638C2318yescomplemented
75#11711589C2318yescomplemented
76#11711587C2318yescomplemented
77#11710937C2314yescomplemented
78C2×#87548C2311yescharacteristic, complemented
79C2×#87547C2311yescharacteristic, complemented
80C22×#5950C2311yescharacteristic
81C2×#3027⋊C3C2312yes—
82C2×#3027⋊C3C2314yescomplemented
83#32895⋊C3C2318yescomplemented
84#32894⋊C3C2318yescomplemented
85#20612⋊C3C2314yescomplemented
86C2×#1791⋊C3C2312yescomplemented
87C2×#63103C2×D311yescharacteristic, complemented
88C22×#3027C2×D311yescharacteristic, complemented
89#6628885C2×D318yescomplemented
90C2×#32895C2×D312yescomplemented
91C2×#32894C2×D312yescomplemented
92C2×#20612C2×D311yescharacteristic, complemented
93#87548C2412yescomplemented
94#87547C2412yescomplemented
95C2×#5950C2×D411yescharacteristic
96C2×#5950C2×D412yes—
97C2×#5950C2411yescharacteristic
98C2×#5950C2412yes—
99C22×#997C2×D411yescharacteristic
100#87522C2414yes—
101#87521C2412yescomplemented
102#87519C2×D412yescomplemented
103#87519C2×D412yescomplemented
104#87519C2×D414yescomplemented
105#87519C2×D414yescomplemented
106#87519C2412yes—
107#87519C2412yes—
108#87517C2412yescomplemented
109C2×#5934C2×D411yescharacteristic, complemented
110C2×#5934C2×D411yescharacteristic, complemented
111C2×#5934C2×D414yescomplemented
112C2×#5932C2×D411yescharacteristic, complemented
113C2×#5932C2×D411yescharacteristic, complemented
114C2×#5932C2×D414yescomplemented
115#85567C2414yescomplemented
116#3027⋊C3C2414yes—
117#1791⋊C3C2414yescomplemented
118#63103C22×D314yescomplemented
119C2×2−1+6C22×D311yescharacteristic, complemented
120C2×H(7,𝔽2)C22×D311yescharacteristic, complemented
121C22×#585C22×D311yescharacteristic
122C23×AGL(1,ℤ/8)S411yescharacteristic, complemented
123C2×#3027C22×D312yes—
124C2×#3027C22×D314yescomplemented
125#32895C22×D318yescomplemented
126#32894C22×D318yescomplemented
127#20612C22×D314yescomplemented
128C2×#1791C22×D312yescomplemented
129#5950C22×D411yescharacteristic
130#5950C22×D412yes—
131#5950C2511yescharacteristic
132C2×#997C22×D411yescharacteristic
133C2×#997C22×D412yes—
134#5934C22×D412yescomplemented
135#5934C22×D412yescomplemented
136#5932C22×D412yescomplemented
137#5932C22×D412yescomplemented
1382−1+6C23×D312yescomplemented
139H(7,𝔽2)C23×D312yescomplemented
140C2×#585D3×D412yes—
141C2×#585D3×D411yescharacteristic
142C2×#585C23×D311yescharacteristic
143C2×#585C23×D312yes—
144C22×H(5,𝔽2)D3×D411yescharacteristic
145C23×#41C2×S411yescharacteristic
146C24×D4C2×S411yescharacteristic, complemented
147C22×AGL(1,ℤ/8)C2×S418yescomplemented
148C23×SD16C2×S412yescomplemented
149C23×D8C2×S412yescomplemented
150C23×M16C2×S411yescharacteristic, complemented
151#3027C23×D314yes—
152#1791C23×D314yescomplemented
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×S414yes—
160C23×Q8C22×S411yescharacteristic
161C23×D4C22×S411yescharacteristic
162C23×D4C22×S412yes—
163C23×D4C22×S414yescomplemented
164C24×C4C22×S411yescharacteristic
165C22×SD16C22×S418yescomplemented
166C22×D8C22×S418yescomplemented
167C22×M16C22×S414yescomplemented
168C23×C8C22×S412yes—
169C25D4×S411yescharacteristic
170H(5,𝔽2)C22×D3×D411yescharacteristic
171C2×#41#1020⋊C211yescharacteristic
172C22×Q8C23×S412yes—
173C22×D4C23×S412yes—
174C22×D4C23×S414yes—
175C23×C4D4×S412yes—
176C23×C4D4×S411yescharacteristic
177C23×C4C23×S411yescharacteristic
178C23×C4C23×S412yes—
179C22×C8C23×S414yes—
180C24C2×D4×S412yes—
181C24C2×D4×S411yescharacteristic
182#41C2×#1020⋊C214yes—
183C2×Q8C2×#1020⋊C211yescharacteristic
184C2×D4C2×#1020⋊C211yescharacteristic
185C22×C4C2×#1020⋊C211yescharacteristic
186C22×C4C2×D4×S411yescharacteristic
187C22×C4C2×D4×S412yes—
188C22×C4C24×S411yescharacteristic
189C23#1171320011yescharacteristic
190C23C22×D4×S411yescharacteristic
191Q8C22×#1020⋊C212yes—
192D4C22×#1020⋊C212yes—
193C2×C4#1171320012yes—
194C2×C4#1171320011yescharacteristic
195C2×C4C22×#1020⋊C212yes—
196C2×C4C22×#1020⋊C211yescharacteristic
197C22C2×#1171320012yes—
198C22C2×#1171320011yescharacteristic
199C4C2×#1171320012yes—
200C4C2×#1171320011yescharacteristic
201C4C23×#1020⋊C211yescharacteristic
202C2#7224099812yesdirect product factor
203C2C22×#1171320011yescharacteristic
subgroupG/H
center, socleC22C2×#11713200
commutator, cosocle#5950C25
FrattiniC22×C4C24×S4
FittingC2×#6628885D3
hypercenterC2×#41#1020⋊C2
hypocenter#997C23×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#72236903C2semidirect
2. C2×#72236889C2semidirect
3. C2×#72236926C2semidirect
4. C2×#72236891C2semidirect
5. C2×#72236892C2semidirect
6. C2×#11711589C22semidirect
7. C2×#72215741C2semidirect
8. C2×#72236908C2semidirect
9. C2×#11711640C22semidirect
10. C2×#11715710C22semidirect
11. C2×#11711645C22semidirect
12. C22×#11714174C2semidirect
13. C22×#87522C22semidirect
14. C2×#72236904C2semidirect
15. C2×#11711587C22semidirect
16. C2×#11711638C22semidirect
17. C2×#72236755C2semidirect
18. C22×#3027⋊C3C22semidirect
19. C2×#11711643C22semidirect
20. C2×#32894⋊C3C22semidirect
21. C2×#32895⋊C3C22semidirect
22. C2×#20612⋊C3C22semidirect
23. C2×#11715708C22semidirect
24. C2×#11715667C22semidirect
25. C2×#72233470C2semidirect
26. C2×#11715842C22semidirect
27. C2×#11715840C22semidirect
28. C2×#11715841C22semidirect
29. C2×#11715838C22semidirect
30. C2×#11710937C22semidirect
31. C2×#6628885D3semidirect
33. #72240998C2direct (N × H)
34. #72236926C22semidirect
35. #72236908C22semidirect
36. #72236904C22semidirect
37. #72236903C22semidirect
38. #72236892C22semidirect
39. #72236891C22semidirect
40. #72236889C22semidirect
41. C2×#11714174C22semidirect
42. C2×#11714174C22semidirect
43. C2×#11714174C22semidirect
44. #72236755C22semidirect
45. #72233470C22semidirect
46. C2×#11715839C22semidirect
47. #72215741C22semidirect
48. #11715842C23semidirect
49. #11715841C23semidirect
50. #11715840C23semidirect
51. #11715839C23semidirect
52. #11715838C23semidirect
53. C2×#87517C23semidirect
54. C2×#87521C23semidirect
55. C2×#87519D4semidirect
56. C2×#87519D4semidirect
57. C2×#87519D4semidirect
61. C22×#5932D4semidirect
62. C22×#5934D4semidirect
64. C2×#87522C23semidirect
65. #11715710C23semidirect
66. #11715708C23semidirect
67. #11715667C23semidirect
68. C2×#85567C23semidirect
69. #11714174C23semidirect
70. #11714174C23semidirect
71. #11711645C23semidirect
72. #11711643C23semidirect
73. #11711640C23semidirect
74. #11711638C23semidirect
75. #11711589C23semidirect
76. #11711587C23semidirect
77. #11710937C23semidirect
78. C2×#87548C23semidirect
79. C2×#87547C23semidirect
82. C2×#3027⋊C3C23semidirect
83. #32895⋊C3C23semidirect
84. #32894⋊C3C23semidirect
85. #20612⋊C3C23semidirect
86. C2×#1791⋊C3C23semidirect
87. C2×#63103C2×D3semidirect
88. C22×#3027C2×D3semidirect
89. #6628885C2×D3semidirect
90. C2×#32895C2×D3semidirect
91. C2×#32894C2×D3semidirect
92. C2×#20612C2×D3semidirect
93. #87548C24semidirect
94. #87547C24semidirect
101. #87521C24semidirect
102. #87519C2×D4semidirect
103. #87519C2×D4semidirect
104. #87519C2×D4semidirect
105. #87519C2×D4semidirect
108. #87517C24semidirect
109. C2×#5934C2×D4semidirect
110. C2×#5934C2×D4semidirect
111. C2×#5934C2×D4semidirect
112. C2×#5932C2×D4semidirect
113. C2×#5932C2×D4semidirect
114. C2×#5932C2×D4semidirect
115. #85567C24semidirect
117. #1791⋊C3C24semidirect
118. #63103C22×D3semidirect
119. C2×2−1+6C22×D3semidirect
120. C2×H(7,𝔽2)C22×D3semidirect
122. C23×AGL(1,ℤ/8)S4semidirect
124. C2×#3027C22×D3semidirect
125. #32895C22×D3semidirect
126. #32894C22×D3semidirect
127. #20612C22×D3semidirect
128. C2×#1791C22×D3semidirect
134. #5934C22×D4semidirect
135. #5934C22×D4semidirect
136. #5932C22×D4semidirect
137. #5932C22×D4semidirect
138. 2−1+6C23×D3semidirect
139. H(7,𝔽2)C23×D3semidirect
146. C24×D4C2×S4semidirect
147. C22×AGL(1,ℤ/8)C2×S4semidirect
148. C23×SD16C2×S4semidirect
149. C23×D8C2×S4semidirect
150. C23×M16C2×S4semidirect
152. #1791C23×D3semidirect
163. C23×D4C22×S4semidirect
165. C22×SD16C22×S4semidirect
166. C22×D8C22×S4semidirect
167. C22×M16C22×S4semidirect
202. C2#72240998direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#722369032
C2×#722368892
C2×#722369262
C2×#722368912
C2×#722368922
C2×#722157412
C2×#722369082
C22×#117141742
C2×#722369042
C2×#722367552
C2×#722375293
C2×#722334702
#722409982
C2×S4×AGL(1,ℤ/8)4