Finite Group Databasebeta

← C2×D4×#73026 #72344096

Subgroups — C2×D4×#73026 #72344096

Order 6144.

At least 10657 conjugacy classes of subgroups; only the 659 normal subgroups are shown.

The subgroup lattice has 182 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×B(4,𝔽2)⋊C3C211yesmaximal, characteristic, complemented
2C2×D4×#5935C211yesmaximal, characteristic, complemented
3C2×#72245330C211yesmaximal, characteristic, complemented
4C2×#72245234C212yesmaximal, complemented
5C2×#72245288C211yesmaximal, characteristic, complemented
6C2×C4⋊#5935C2211yescharacteristic, complemented
7C2×#72241382C212yesmaximal, complemented
8C2×#11711630C2212yescomplemented
9C2×#72218504C212yesmaximal, complemented
10C2×#11711631C2212yescomplemented
11C2×#11715701C2212yescomplemented
12C2×#72245160C211yesmaximal, characteristic, complemented
13C2×C4⋊(B(4,𝔽2)⋊C3)C2211yescharacteristic, complemented
14C2×#11711626C2212yescomplemented
15C23×#73026C212yesmaximal, complemented
16C23×B(4,𝔽2)⋊C3C2212yescomplemented
17C2×C4×#73026C211yesmaximal, characteristic, complemented
18C2×#11711579C2211yescharacteristic, complemented
19C2×C4×B(4,𝔽2)⋊C3C2211yescharacteristic, complemented
20C2×#11711591C2211yescharacteristic, complemented
21C2×#11711578C2211yescharacteristic, complemented
22C2×#32898⋊C3C2212yescomplemented
23C2×D4×#5934C211yesmaximal, characteristic, complemented
24C2×D4×#997C2211yescharacteristic, complemented
25C2×C4⋊#5934C2211yescharacteristic, complemented
26C2×#11715716C2212yescomplemented
27D4×#73026C2116yesmaximal, direct product factor
28#72245330C2218yescomplemented
29#72245288C2218yescomplemented
30#72245234C22116yescomplemented
31D4×B(4,𝔽2)⋊C3C2218yescomplemented
32#72245160C2218yescomplemented
33C4×#73026C2218yescomplemented
34C2×C4×#5934C2211yescharacteristic, complemented
35#72241382C22116yescomplemented
36C23×#5934C2212yescomplemented
37C2×D4×B(4,𝔽2)D311yescharacteristic, complemented
38C22×#73026C2211yescharacteristic
39C22×#73026C22116yescomplemented
40C22×#73026C2212yescomplemented
41C22×#73026C2212yescomplemented
42C22×#73026C2212yescomplemented
43D4×#5934C2218yescomplemented
44D4×#5935C2218yescomplemented
45C23×#5935C2212yescomplemented
46C2×C4×#5935C2211yescharacteristic, complemented
47#72218504C22116yescomplemented
48C22×#5934C2311yescharacteristic
49C22×#5934C2318yescomplemented
50C22×#5934C2312yescomplemented
51C22×#5935C2311yescharacteristic
52C22×#5935C2318yescomplemented
53C22×#5935C2312yescomplemented
54C4⋊#5934C2314yescomplemented
55C4⋊#5935C2314yescomplemented
56#11715716C2318yescomplemented
57#11715701C2318yescomplemented
58C4×#5934C2314yescomplemented
59C4×#5935C2314yescomplemented
60C2×#85565C2311yescharacteristic, complemented
61C2×#85564C2311yescharacteristic, complemented
62C2×#73026D418yesdirect product factor
63C2×#73026C2314yes—
64C2×#73026C2318yescomplemented
65C2×#73026C2318yescomplemented
66C2×#73026C2318yescomplemented
67#11711631C2318yescomplemented
68#11711630C2318yescomplemented
69#11711626C2318yescomplemented
70#11711591C2314yescomplemented
71#11711579C2314yescomplemented
72#11711578C2314yescomplemented
73C23×#997C2312yescomplemented
74D4×#997C2314yescomplemented
75C2×C4×#997C2311yescharacteristic, complemented
76C22×B(4,𝔽2)⋊C3C2311yescharacteristic
77C22×B(4,𝔽2)⋊C3C2318yescomplemented
78C22×B(4,𝔽2)⋊C3C2312yescomplemented
79C4⋊(B(4,𝔽2)⋊C3)C2314yescomplemented
80#32898⋊C3C2318yescomplemented
81C4×B(4,𝔽2)⋊C3C2314yescomplemented
82C2×#1790⋊C3C2311yescharacteristic, complemented
83C2×D4×H(5,𝔽2)C2×D311yescharacteristic, complemented
84C23×B(4,𝔽2)C2×D312yescomplemented
85C2×#32926C2×D311yescharacteristic, complemented
86C2×#32898C2×D312yescomplemented
87D4×B(4,𝔽2)C2×D318yescomplemented
88C2×C4×B(4,𝔽2)C2×D311yescharacteristic, complemented
89C22×#997C2411yescharacteristic
90C22×#997C2414yescomplemented
91C2×#5935C2×D414yescomplemented
92C2×#5935C2412yes—
93C2×#5935C2414yescomplemented
94C2×#5934C2×D414yescomplemented
95C2×#5934C2412yes—
96C2×#5934C2414yescomplemented
97C4×#997C2412yescomplemented
98#85565C2412yescomplemented
99#85564C2412yescomplemented
100C2×B(4,𝔽2)⋊C3C2×D414yescomplemented
101C2×B(4,𝔽2)⋊C3C2412yes—
102C2×B(4,𝔽2)⋊C3C2414yescomplemented
103#73026C2×D4116yesdirect product factor
104#1790⋊C3C2412yescomplemented
105C23×H(5,𝔽2)C22×D312yescomplemented
106C25×D4S411yescharacteristic, complemented
107D4×H(5,𝔽2)C22×D314yescomplemented
108C2×C4×H(5,𝔽2)C22×D311yescharacteristic, complemented
109C22×B(4,𝔽2)C22×D311yescharacteristic
110C22×B(4,𝔽2)C22×D312yescomplemented
111C22×B(4,𝔽2)C22×D318yescomplemented
112#32926C22×D314yescomplemented
113#32898C22×D318yescomplemented
114C4×B(4,𝔽2)C22×D314yescomplemented
115C2×#1790C22×D311yescharacteristic, complemented
116C2×#997C22×D412yescomplemented
117C2×#997C2511yescharacteristic
118#5935C22×D414yescomplemented
119#5934C22×D414yescomplemented
120B(4,𝔽2)⋊C3C22×D414yescomplemented
121C27C2×S412yescomplemented
122C22×H(5,𝔽2)C23×D311yescharacteristic
123C22×H(5,𝔽2)C23×D314yescomplemented
124C24×D4C2×S411yescharacteristic
125C24×D4C2×S412yescomplemented
126C24×D4C2×S411yescharacteristic, complemented
127C24×D4C2×S418yescomplemented
128C25×C4C2×S411yescharacteristic, complemented
129C4×H(5,𝔽2)C23×D312yescomplemented
130C2×B(4,𝔽2)D3×D414yescomplemented
131C2×B(4,𝔽2)C23×D312yes—
132C2×B(4,𝔽2)C23×D314yescomplemented
133#1790C23×D312yescomplemented
134#997C23×D411yescharacteristic, complemented
135C26C22×S412yes—
136C26C22×S412yes—
137C26C22×S411yescharacteristic
138C26C22×S418yescomplemented
139C2×H(5,𝔽2)C2×D3×D412yescomplemented
140C2×H(5,𝔽2)C24×D311yescharacteristic
141C23×D4C22×S414yes—
142C23×D4C22×S418yescomplemented
143C23×D4C22×S414yescomplemented
144C24×C4C22×S411yescharacteristic
145C24×C4C22×S411yescharacteristic
146C24×C4C22×S414yescomplemented
147B(4,𝔽2)C2×D3×D414yescomplemented
148C25D4×S414yescomplemented
149C25C23×S414yes—
150C25C23×S414yes—
151C25C23×S411yescharacteristic
152C25C23×S412yes—
153H(5,𝔽2)C22×D3×D411yescharacteristic, complemented
154C22×D4#1020⋊C211yescharacteristic
155C23×C4C23×S412yes—
156C23×C4C23×S412yes—
157C24C2×#1020⋊C212yes—
158C24C2×D4×S412yes—
159C24C2×D4×S414yescomplemented
160C24C24×S411yescharacteristic
161C2×D4#7302618yesdirect product factor
162C2×D4C2×#1020⋊C214yes—
163C22×C4C2×#1020⋊C211yescharacteristic
164C23C2×#7302618yescomplemented
165C23C22×#1020⋊C214yes—
166C23C22×#1020⋊C211yescharacteristic
167C23C22×D4×S411yescharacteristic
168D4C2×#73026116yesdirect product factor
169C2×C4C2×#7302614yescomplemented
170C2×C4C22×#1020⋊C212yes—
171C22#7221702712yes—
172C22D4×#1020⋊C212yes—
173C22C23×#1020⋊C211yescharacteristic
174C22C22×#7302612yes—
175C22C22×#7302618yescomplemented
176C4C22×#7302614yescomplemented
177C2C2×#7221702711yescharacteristic
178C2C2×D4×#1020⋊C211yescharacteristic
179C2D4×#7302614yesdirect product factor
180C2C23×#7302611yescharacteristic
subgroupG/H
center, socleC23C22×#1020⋊C2
commutator, cosocleC2×#997C25
FrattiniC24C24×S4
FittingC2×D4×B(4,𝔽2)D3
hypercenterC22×D4#1020⋊C2
hypocenter#997C23×D4
LCSG ▷ C2×#997 ▷ #997
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×B(4,𝔽2)⋊C3C2semidirect
2. C2×D4×#5935C2semidirect
3. C2×#72245330C2semidirect
4. C2×#72245234C2semidirect
5. C2×#72245288C2semidirect
6. C2×C4⋊#5935C22semidirect
7. C2×#72241382C2semidirect
8. C2×#11711630C22semidirect
9. C2×#72218504C2semidirect
10. C2×#11711631C22semidirect
11. C2×#11715701C22semidirect
12. C2×#72245160C2semidirect
13. C2×C4⋊(B(4,𝔽2)⋊C3)C22semidirect
14. C2×#11711626C22semidirect
15. C23×#73026C2semidirect
16. C23×B(4,𝔽2)⋊C3C22semidirect
17. C2×C4×#73026C2semidirect
18. C2×#11711579C22semidirect
19. C2×C4×B(4,𝔽2)⋊C3C22semidirect
20. C2×#11711591C22semidirect
21. C2×#11711578C22semidirect
22. C2×#32898⋊C3C22semidirect
23. C2×D4×#5934C2semidirect
24. C2×D4×#997C22semidirect
25. C2×C4⋊#5934C22semidirect
26. C2×#11715716C22semidirect
27. D4×#73026C2direct (N × H)
28. #72245330C22semidirect
29. #72245288C22semidirect
30. #72245234C22semidirect
31. D4×B(4,𝔽2)⋊C3C22semidirect
32. #72245160C22semidirect
33. C4×#73026C22semidirect
34. C2×C4×#5934C22semidirect
35. #72241382C22semidirect
36. C23×#5934C22semidirect
37. C2×D4×B(4,𝔽2)D3semidirect
39. C22×#73026C22semidirect
40. C22×#73026C22semidirect
41. C22×#73026C22semidirect
42. C22×#73026C22semidirect
43. D4×#5934C22semidirect
44. D4×#5935C22semidirect
45. C23×#5935C22semidirect
46. C2×C4×#5935C22semidirect
47. #72218504C22semidirect
49. C22×#5934C23semidirect
50. C22×#5934C23semidirect
52. C22×#5935C23semidirect
53. C22×#5935C23semidirect
54. C4⋊#5934C23semidirect
55. C4⋊#5935C23semidirect
56. #11715716C23semidirect
57. #11715701C23semidirect
58. C4×#5934C23semidirect
59. C4×#5935C23semidirect
60. C2×#85565C23semidirect
61. C2×#85564C23semidirect
62. C2×#73026D4direct (N × H)
64. C2×#73026C23semidirect
65. C2×#73026C23semidirect
66. C2×#73026C23semidirect
67. #11711631C23semidirect
68. #11711630C23semidirect
69. #11711626C23semidirect
70. #11711591C23semidirect
71. #11711579C23semidirect
72. #11711578C23semidirect
73. C23×#997C23semidirect
74. D4×#997C23semidirect
75. C2×C4×#997C23semidirect
77. C22×B(4,𝔽2)⋊C3C23semidirect
78. C22×B(4,𝔽2)⋊C3C23semidirect
79. C4⋊(B(4,𝔽2)⋊C3)C23semidirect
80. #32898⋊C3C23semidirect
81. C4×B(4,𝔽2)⋊C3C23semidirect
82. C2×#1790⋊C3C23semidirect
83. C2×D4×H(5,𝔽2)C2×D3semidirect
84. C23×B(4,𝔽2)C2×D3semidirect
85. C2×#32926C2×D3semidirect
86. C2×#32898C2×D3semidirect
87. D4×B(4,𝔽2)C2×D3semidirect
88. C2×C4×B(4,𝔽2)C2×D3semidirect
90. C22×#997C24semidirect
91. C2×#5935C2×D4semidirect
93. C2×#5935C24semidirect
94. C2×#5934C2×D4semidirect
96. C2×#5934C24semidirect
97. C4×#997C24semidirect
98. #85565C24semidirect
99. #85564C24semidirect
100. C2×B(4,𝔽2)⋊C3C2×D4semidirect
102. C2×B(4,𝔽2)⋊C3C24semidirect
103. #73026C2×D4direct (N × H)
104. #1790⋊C3C24semidirect
105. C23×H(5,𝔽2)C22×D3semidirect
106. C25×D4S4semidirect
107. D4×H(5,𝔽2)C22×D3semidirect
108. C2×C4×H(5,𝔽2)C22×D3semidirect
110. C22×B(4,𝔽2)C22×D3semidirect
111. C22×B(4,𝔽2)C22×D3semidirect
112. #32926C22×D3semidirect
113. #32898C22×D3semidirect
114. C4×B(4,𝔽2)C22×D3semidirect
115. C2×#1790C22×D3semidirect
116. C2×#997C22×D4semidirect
118. #5935C22×D4semidirect
119. #5934C22×D4semidirect
120. B(4,𝔽2)⋊C3C22×D4semidirect
121. C27C2×S4semidirect
123. C22×H(5,𝔽2)C23×D3semidirect
125. C24×D4C2×S4semidirect
126. C24×D4C2×S4semidirect
127. C24×D4C2×S4semidirect
128. C25×C4C2×S4semidirect
129. C4×H(5,𝔽2)C23×D3semidirect
130. C2×B(4,𝔽2)D3×D4semidirect
132. C2×B(4,𝔽2)C23×D3semidirect
133. #1790C23×D3semidirect
134. #997C23×D4semidirect
138. C26C22×S4semidirect
139. C2×H(5,𝔽2)C2×D3×D4semidirect
142. C23×D4C22×S4semidirect
143. C23×D4C22×S4semidirect
146. C24×C4C22×S4semidirect
147. B(4,𝔽2)C2×D3×D4semidirect
148. C25D4×S4semidirect
153. H(5,𝔽2)C22×D3×D4semidirect
159. C24C2×D4×S4semidirect
161. C2×D4#73026direct (N × H)
164. C23C2×#73026semidirect
168. D4C2×#73026direct (N × H)
169. C2×C4C2×#73026semidirect
175. C22C22×#73026semidirect
176. C4C22×#73026semidirect
179. C2D4×#73026direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D4×B(4,𝔽2)⋊C32
C2×D4×#59352
C2×#722453302
C2×#722452342
C2×#722452882
C2×#722413822
C2×#722185042
C2×#722451602
C23×#730262
C2×C4×#730262
C2×D4×#21963
C2×D4×#59342
C23×D4×S44
D4×#730262