Finite Group Databasebeta

← C2×#72246533 #72344163

Subgroups — C2×#72246533 #72344163

Order 6144.

At least 14555 conjugacy classes of subgroups; only the 619 normal subgroups are shown.

The subgroup lattice has 178 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#72245477C211yesmaximal, characteristic, complemented
2C2×#72245709C212yesmaximal, complemented
3C2×#72241395C212yesmaximal, complemented
4C2×#11711634C2212yescomplemented
5C2×#72245509C211yesmaximal, characteristic, complemented
6C2×#11711635C2212yescomplemented
7C2×#72245553C211yesmaximal, characteristic, complemented
8C2×#72241384C211yesmaximal, characteristic, complemented
9C2×#72227799C211yesmaximal, characteristic, complemented
10C2×#20609⋊C3C2211yescharacteristic, complemented
11C2×#72245678C212yesmaximal, complemented
12C2×#32899⋊C3C2212yescomplemented
13C2×#11711594C2212yescomplemented
14C2×#11715717C2212yescomplemented
15C2×#72241382C212yesmaximal, complemented
16C2×#11711630C2212yescomplemented
17C2×#11711631C2212yescomplemented
18C23×#73026C211yesmaximal, characteristic, complemented
19C23×B(4,𝔽2)⋊C3C2211yescharacteristic, complemented
20C22×#11715828C211yesmaximal, characteristic, complemented
21C22×#87522C2211yescharacteristic, complemented
22C22×#87520C2211yescharacteristic, complemented
23C2×#11711591C2212yescomplemented
24C22×#87516C2211yescharacteristic, complemented
25C22×#87516C2211yescharacteristic, complemented
26C2×#32898⋊C3C2212yescomplemented
27C2×#11715658C2211yescharacteristic, complemented
28C22×#87547C2211yescharacteristic, complemented
29C2×#11715716C2212yescomplemented
30#72246533C2116yesmaximal, direct product factor
31#72245709C22116yescomplemented
32#72245678C22116yescomplemented
33#72245553C2218yescomplemented
34#72245509C2218yescomplemented
35#72245477C2218yescomplemented
36#72241395C22116yescomplemented
37#72241384C2218yescomplemented
38#72241382C22116yescomplemented
39C23×#5934C2211yescharacteristic, complemented
40C2×#6628896D311yescharacteristic, complemented
41C22×#73026C2212yescomplemented
42C22×#73026C2212yescomplemented
43C22×#73026C2218yescomplemented
44#72227799C2218yescomplemented
45C23×#5935C2211yescharacteristic
46C23×#5935C2211yescharacteristic, complemented
47C2×#11715828C2218yescomplemented
48C22×#5934C2312yes—
49C22×#5934C2314yescomplemented
50C2×#87522C2314yescomplemented
51C2×#87516C2314yescomplemented
52C2×#87516C2314yescomplemented
53C2×#87520C2314yescomplemented
54C22×#5933C2311yescharacteristic, complemented
55C22×#5935D418yescomplemented
56C22×#5935C2314yes—
57C22×#5935C2311yescharacteristic
58C22×#5935C2311yescharacteristic
59C22×#5935C2314yescomplemented
60C22×#5935C2311yescharacteristic, complemented
61#11715717C2318yescomplemented
62#11715716C2318yescomplemented
63#11715658C2314yescomplemented
64C2×#85564C2312yescomplemented
65C2×#73026C2318yescomplemented
66C2×#73026C2318yescomplemented
67#11711635C2318yescomplemented
68#11711634C2318yescomplemented
69#11711631C2318yescomplemented
70#11711630C2318yescomplemented
71#11711594C2318yescomplemented
72#11711591C2318yescomplemented
73C2×#87547C2314yescomplemented
74C22×#5950C2311yescharacteristic, complemented
75C23×#997C2311yescharacteristic
76C23×#997C2311yescharacteristic, complemented
77C22×B(4,𝔽2)⋊C3C2312yes—
78C22×B(4,𝔽2)⋊C3C2314yescomplemented
79#32899⋊C3C2318yescomplemented
80#32898⋊C3C2318yescomplemented
81#20609⋊C3C2314yescomplemented
82C2×#1790⋊C3C2312yescomplemented
83C22×H(7,𝔽2)C2×D311yescharacteristic, complemented
84C23×B(4,𝔽2)C2×D311yescharacteristic, complemented
85#6628896C2×D318yescomplemented
86C2×#32899C2×D312yescomplemented
87C2×#32898C2×D312yescomplemented
88C2×#20609C2×D311yescharacteristic, complemented
89C2×#5950C2412yescomplemented
90C22×#997C2×D414yescomplemented
91C22×#997C2411yescharacteristic
92C22×#997C2412yes—
93C22×#997C2412yescomplemented
94C2×#5935C2×D414yescomplemented
95C2×#5935C2×D414yescomplemented
96C2×#5935C2×D4116yescomplemented
97C2×#5935C2412yes—
98C2×#5935C2412yes—
99C2×#5935C2412yescomplemented
100C2×#5934C2414yes—
101C2×#5933C2412yescomplemented
102#85564C2414yescomplemented
103C2×B(4,𝔽2)⋊C3C2414yes—
104#1790⋊C3C2414yescomplemented
105C2×H(7,𝔽2)C22×D314yescomplemented
106C22×#585C22×D311yescharacteristic, complemented
107C23×H(5,𝔽2)C22×D311yescharacteristic
108C23×H(5,𝔽2)C22×D311yescharacteristic, complemented
109C23×#120S411yescharacteristic, complemented
110C22×B(4,𝔽2)C22×D312yes—
111C22×B(4,𝔽2)C22×D314yescomplemented
112#32899C22×D318yescomplemented
113#32898C22×D318yescomplemented
114#20609C22×D314yescomplemented
115C2×#1790C22×D312yescomplemented
116C2×#997C22×D412yes—
117C2×#997C22×D414yescomplemented
118C2×#997C2511yescharacteristic
119#5935C22×D414yescomplemented
120#5935C22×D414yescomplemented
121C27C2×S411yescharacteristic
122C2×#585C23×D312yescomplemented
123C22×H(5,𝔽2)D3×D414yescomplemented
124C22×H(5,𝔽2)C23×D311yescharacteristic
125C22×H(5,𝔽2)C23×D312yes—
126C22×H(5,𝔽2)C23×D312yescomplemented
127C24×D4C2×S411yescharacteristic
128C24×D4C2×S412yescomplemented
129C22×#120C2×S418yescomplemented
130C23×C22⋊C4C2×S411yescharacteristic
131C23×C22⋊C4C2×S412yescomplemented
132C2×B(4,𝔽2)C23×D314yes—
133#1790C23×D314yescomplemented
134#997C23×D411yescharacteristic
135C26C22×S411yescharacteristic
136C26C22×S412yes—
137C26C22×S414yes—
138C26C22×S411yescharacteristic
139C2×H(5,𝔽2)C2×D3×D412yes—
140C2×H(5,𝔽2)C2×D3×D414yescomplemented
141C2×H(5,𝔽2)C24×D311yescharacteristic
142C23×D4C22×S414yes—
143C23×D4C22×S418yescomplemented
144C24×C4C22×S411yescharacteristic
145C24×C4C22×S412yes—
146C22×C22⋊C4C22×S414yes—
147C22×C22⋊C4C22×S418yescomplemented
148C25D4×S414yes—
149C25C23×S412yes—
150C25C23×S414yes—
151C25C23×S411yescharacteristic
152C25C23×S412yes—
153H(5,𝔽2)C22×D3×D411yescharacteristic
154C22×D4#1020⋊C211yescharacteristic
155C23×C4C23×S412yes—
156C23×C4C23×S414yes—
157C24C2×#1020⋊C211yescharacteristic
158C24C2×#1020⋊C211yescharacteristic
159C24C2×D4×S412yes—
160C24C2×D4×S414yes—
161C24C24×S411yescharacteristic
162C2×D4C2×#1020⋊C214yes—
163C22×C4C2×#1020⋊C211yescharacteristic
164C23#1171320014yes—
165C23C22×#1020⋊C212yes—
166C23C22×#1020⋊C212yes—
167C23C22×#1020⋊C211yescharacteristic
168C23C22×D4×S411yescharacteristic
169C2×C4C22×#1020⋊C212yes—
170C22C2×#1171320014yes—
171C22C2×#1171320012yes—
172C22#7221702714yes—
173C22C23×#1020⋊C211yescharacteristic
174C2C2×#7221702712yes—
175C2#7224653314yesdirect product factor
176C2C22×#1171320011yescharacteristic
subgroupG/H
center, socleC23C22×#1020⋊C2
commutator, cosocleC2×#997C25
FrattiniC24C24×S4
FittingC2×#6628896D3
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×#72245477C2semidirect
2. C2×#72245709C2semidirect
3. C2×#72241395C2semidirect
4. C2×#11711634C22semidirect
5. C2×#72245509C2semidirect
6. C2×#11711635C22semidirect
7. C2×#72245553C2semidirect
8. C2×#72241384C2semidirect
9. C2×#72227799C2semidirect
10. C2×#20609⋊C3C22semidirect
11. C2×#72245678C2semidirect
12. C2×#32899⋊C3C22semidirect
13. C2×#11711594C22semidirect
14. C2×#11715717C22semidirect
15. C2×#72241382C2semidirect
16. C2×#11711630C22semidirect
17. C2×#11711631C22semidirect
18. C23×#73026C2semidirect
19. C23×B(4,𝔽2)⋊C3C22semidirect
20. C22×#11715828C2semidirect
21. C22×#87522C22semidirect
22. C22×#87520C22semidirect
23. C2×#11711591C22semidirect
24. C22×#87516C22semidirect
25. C22×#87516C22semidirect
26. C2×#32898⋊C3C22semidirect
27. C2×#11715658C22semidirect
28. C22×#87547C22semidirect
29. C2×#11715716C22semidirect
30. #72246533C2direct (N × H)
31. #72245709C22semidirect
32. #72245678C22semidirect
33. #72245553C22semidirect
34. #72245509C22semidirect
35. #72245477C22semidirect
36. #72241395C22semidirect
37. #72241384C22semidirect
38. #72241382C22semidirect
39. C23×#5934C22semidirect
40. C2×#6628896D3semidirect
41. C22×#73026C22semidirect
42. C22×#73026C22semidirect
43. C22×#73026C22semidirect
44. #72227799C22semidirect
46. C23×#5935C22semidirect
47. C2×#11715828C22semidirect
49. C22×#5934C23semidirect
50. C2×#87522C23semidirect
51. C2×#87516C23semidirect
52. C2×#87516C23semidirect
53. C2×#87520C23semidirect
54. C22×#5933C23semidirect
55. C22×#5935D4semidirect
59. C22×#5935C23semidirect
60. C22×#5935C23semidirect
61. #11715717C23semidirect
62. #11715716C23semidirect
63. #11715658C23semidirect
64. C2×#85564C23semidirect
65. C2×#73026C23semidirect
66. C2×#73026C23semidirect
67. #11711635C23semidirect
68. #11711634C23semidirect
69. #11711631C23semidirect
70. #11711630C23semidirect
71. #11711594C23semidirect
72. #11711591C23semidirect
73. C2×#87547C23semidirect
74. C22×#5950C23semidirect
76. C23×#997C23semidirect
78. C22×B(4,𝔽2)⋊C3C23semidirect
79. #32899⋊C3C23semidirect
80. #32898⋊C3C23semidirect
81. #20609⋊C3C23semidirect
82. C2×#1790⋊C3C23semidirect
83. C22×H(7,𝔽2)C2×D3semidirect
84. C23×B(4,𝔽2)C2×D3semidirect
85. #6628896C2×D3semidirect
86. C2×#32899C2×D3semidirect
87. C2×#32898C2×D3semidirect
88. C2×#20609C2×D3semidirect
89. C2×#5950C24semidirect
90. C22×#997C2×D4semidirect
93. C22×#997C24semidirect
94. C2×#5935C2×D4semidirect
95. C2×#5935C2×D4semidirect
96. C2×#5935C2×D4semidirect
99. C2×#5935C24semidirect
101. C2×#5933C24semidirect
102. #85564C24semidirect
104. #1790⋊C3C24semidirect
105. C2×H(7,𝔽2)C22×D3semidirect
106. C22×#585C22×D3semidirect
108. C23×H(5,𝔽2)C22×D3semidirect
109. C23×#120S4semidirect
111. C22×B(4,𝔽2)C22×D3semidirect
112. #32899C22×D3semidirect
113. #32898C22×D3semidirect
114. #20609C22×D3semidirect
115. C2×#1790C22×D3semidirect
117. C2×#997C22×D4semidirect
119. #5935C22×D4semidirect
120. #5935C22×D4semidirect
122. C2×#585C23×D3semidirect
123. C22×H(5,𝔽2)D3×D4semidirect
126. C22×H(5,𝔽2)C23×D3semidirect
128. C24×D4C2×S4semidirect
129. C22×#120C2×S4semidirect
131. C23×C22⋊C4C2×S4semidirect
133. #1790C23×D3semidirect
140. C2×H(5,𝔽2)C2×D3×D4semidirect
143. C23×D4C22×S4semidirect
147. C22×C22⋊C4C22×S4semidirect
175. C2#72246533direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#722454772
C2×#722457092
C2×#722413952
C2×#722455092
C2×#722455532
C2×#722413842
C2×#722277992
C2×#722456782
C2×#722413822
C23×#730262
C22×#117158282
C2×#722341463
C2×S4×#1204
#722465332