Finite Group Databasebeta

← #72344080

Subgroups — #72344080

Order 3072.

At least 6498 conjugacy classes of subgroups; only the 525 normal subgroups are shown.

The subgroup lattice has 153 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72344078C211yesmaximal, characteristic, complemented
2#72344075C211yesmaximal, characteristic, complemented
3#72344074C214yesmaximal, complemented
4#72344071C214yesmaximal, complemented
5#72344070C212yesmaximal, complemented
6#72344067C214yesmaximal, complemented
7#72236922C212yesmaximal, complemented
8#72236843C214yesmaximal, complemented
9C2×#11714182C214yesmaximal, complemented
10#72217027C214yesmaximal, complemented
11C2×#11715828C211yesmaximal, characteristic, complemented
12#11715841C2214yescomplemented
13#11715839C2214yescomplemented
14#11715838C2212yescomplemented
15C2×#87515C2214yescomplemented
16#11715828C2214yescomplemented
17#11715828C2214yescomplemented
18C2×#87522C2211yescharacteristic, complemented
19C2×#87516C2212yescomplemented
20C2×#87520C2211yescharacteristic, complemented
21C22×#5935C2212yescomplemented
22#11714187C2212yescomplemented
23#11714186C2212yescomplemented
24#11714185C2212yescomplemented
25#11714184C2212yescomplemented
26#11714183C2214yescomplemented
27#11714182C2218yescomplemented
28#11714182C2214yescomplemented
29#11714182C2218yescomplemented
30#11714182C2218yescomplemented
31#11714181C2214yescomplemented
32#11714180C2214yescomplemented
33#11714175C2214yescomplemented
34#11714174C2214yescomplemented
35#11714173C2218yescomplemented
36#11714172C2218yescomplemented
37#11714171C2218yescomplemented
38#11714170C2218yescomplemented
39#11714169C2214yescomplemented
40C2×#73028C2214yescomplemented
41C2×#73032C2214yescomplemented
42#11714162C2214yescomplemented
43C2×#73031C2214yescomplemented
44C2×#73026C2214yescomplemented
45C2×#87547C2211yescharacteristic, complemented
46#60401⋊C3C2214yescomplemented
47#60399⋊C3C2214yescomplemented
48#60398⋊C3C2212yescomplemented
49C2×#3026⋊C3C2214yescomplemented
50#10574873D311yescharacteristic, complemented
51#87547C2314yescomplemented
52C2×#5950C2311yescharacteristic, complemented
53C22×#997C2312yescomplemented
54#87522C2314yescomplemented
55#87521C2312yescomplemented
56#87520C2312yes—
57#87520C2312yescomplemented
58#87519C2314yescomplemented
59#87517C2312yescomplemented
60#87516C2318yescomplemented
61#87515C2318yescomplemented
62#87515C2314yescomplemented
63C2×#5935C2314yes—
64C2×#5935C2311yescharacteristic
65C2×#5935C2314yescomplemented
66C2×#5935C2312yescomplemented
67C2×#5934C2314yescomplemented
68C2×#5933C2311yescharacteristic, complemented
69C2×#5932C2314yescomplemented
70#3028⋊C3C2312yescomplemented
71#3027⋊C3C2312yescomplemented
72#3026⋊C3C2318yescomplemented
73#3026⋊C3C2314yescomplemented
74#3025⋊C3C2314yescomplemented
75C2×#458⋊C3C2314yescomplemented
76C2×B(4,𝔽2)⋊C3C2314yescomplemented
77#73032C2314yescomplemented
78#73032C2318yescomplemented
79#73031C2314yescomplemented
80#73031C2318yescomplemented
81#73030C2314yescomplemented
82#73029C2314yescomplemented
83#73028C2314yescomplemented
84#73028C2318yescomplemented
85#73027C2314yescomplemented
86#73026C2314yescomplemented
87#73026C2318yescomplemented
88#73025C2314yescomplemented
89C2×H(7,𝔽2)C2×D311yescharacteristic, complemented
90#60401C2×D314yescomplemented
91#60399C2×D314yescomplemented
92#60398C2×D312yescomplemented
93C2×#3026C2×D314yescomplemented
94#5950C2412yes—
95C2×#997C2414yes—
96C2×#997C2411yescharacteristic
97#5935C2411yescharacteristic
98#5935C2414yes—
99#5935C2411yescharacteristic
100#5934C2414yes—
101#5933C2412yes—
102#5932C2414yes—
103#458⋊C3C2414yes—
104B(4,𝔽2)⋊C3C2414yes—
105H(7,𝔽2)C22×D314yescomplemented
106C2×#585C22×D311yescharacteristic, complemented
107C22×H(5,𝔽2)S411yescharacteristic, complemented
108C22×H(5,𝔽2)C22×D312yescomplemented
109#3028C22×D312yescomplemented
110#3027C22×D312yescomplemented
111#3026C22×D318yescomplemented
112#3026C22×D314yescomplemented
113#3025C22×D314yescomplemented
114C2×#458C22×D314yescomplemented
115C2×B(4,𝔽2)C22×D314yescomplemented
116#997C2511yescharacteristic
117#585C23×D312yes—
118C2×H(5,𝔽2)C23×D311yescharacteristic
119C2×H(5,𝔽2)C23×D314yes—
120C22×#41C2×S412yescomplemented
121C22×#41C2×S414yescomplemented
122C23×D4C2×S411yescharacteristic
123C23×D4C2×S414yescomplemented
124C23×D4C2×S414yescomplemented
125#458C23×D314yes—
126B(4,𝔽2)C23×D314yes—
127C25C22×S412yes—
128C25C22×S414yes—
129H(5,𝔽2)C24×D311yescharacteristic
130C22×Q8C22×S412yes—
131C22×D4C22×S414yes—
132C22×D4C22×S414yes—
133C22×D4C22×S418yes—
134C22×D4C22×S412yes—
135C23×C4C22×S411yescharacteristic
136C23×C4C22×S414yes—
137C23×C4C22×S414yes—
138C24C23×S414yes—
139C24C23×S411yescharacteristic
140C24C23×S414yes—
141C2×D4#1020⋊C211yescharacteristic
142C22×C4C23×S412yes—
143C22×C4C23×S414yes—
144C23C2×#1020⋊C212yes—
145C23C24×S411yescharacteristic
146D4C2×#1020⋊C214yes—
147C2×C4C2×#1020⋊C211yescharacteristic
148C22C22×#1020⋊C211yescharacteristic
149C22C22×#1020⋊C214yes—
150C4C22×#1020⋊C212yes—
151C2C23×#1020⋊C211yescharacteristic
subgroupG/H
center, socleC2C23×#1020⋊C2
commutator, cosocle, hypocenter#997C25
FrattiniC23C24×S4
Fitting#10574873D3
hypercenterC2×D4#1020⋊C2
LCSG ▷ #997
UCSC1 ◁ C2 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #72344078C2semidirect
2. #72344075C2semidirect
3. #72344074C2semidirect
4. #72344071C2semidirect
5. #72344070C2semidirect
6. #72344067C2semidirect
7. #72236922C2semidirect
8. #72236843C2semidirect
9. C2×#11714182C2semidirect
10. #72217027C2semidirect
11. C2×#11715828C2semidirect
12. #11715841C22semidirect
13. #11715839C22semidirect
14. #11715838C22semidirect
15. C2×#87515C22semidirect
16. #11715828C22semidirect
17. #11715828C22semidirect
18. C2×#87522C22semidirect
19. C2×#87516C22semidirect
20. C2×#87520C22semidirect
21. C22×#5935C22semidirect
22. #11714187C22semidirect
23. #11714186C22semidirect
24. #11714185C22semidirect
25. #11714184C22semidirect
26. #11714183C22semidirect
27. #11714182C22semidirect
28. #11714182C22semidirect
29. #11714182C22semidirect
30. #11714182C22semidirect
31. #11714181C22semidirect
32. #11714180C22semidirect
33. #11714175C22semidirect
34. #11714174C22semidirect
35. #11714173C22semidirect
36. #11714172C22semidirect
37. #11714171C22semidirect
38. #11714170C22semidirect
39. #11714169C22semidirect
40. C2×#73028C22semidirect
41. C2×#73032C22semidirect
42. #11714162C22semidirect
43. C2×#73031C22semidirect
44. C2×#73026C22semidirect
45. C2×#87547C22semidirect
46. #60401⋊C3C22semidirect
47. #60399⋊C3C22semidirect
48. #60398⋊C3C22semidirect
49. C2×#3026⋊C3C22semidirect
50. #10574873D3semidirect
51. #87547C23semidirect
52. C2×#5950C23semidirect
53. C22×#997C23semidirect
54. #87522C23semidirect
55. #87521C23semidirect
57. #87520C23semidirect
58. #87519C23semidirect
59. #87517C23semidirect
60. #87516C23semidirect
61. #87515C23semidirect
62. #87515C23semidirect
65. C2×#5935C23semidirect
66. C2×#5935C23semidirect
67. C2×#5934C23semidirect
68. C2×#5933C23semidirect
69. C2×#5932C23semidirect
70. #3028⋊C3C23semidirect
71. #3027⋊C3C23semidirect
72. #3026⋊C3C23semidirect
73. #3026⋊C3C23semidirect
74. #3025⋊C3C23semidirect
75. C2×#458⋊C3C23semidirect
76. C2×B(4,𝔽2)⋊C3C23semidirect
77. #73032C23semidirect
78. #73032C23semidirect
79. #73031C23semidirect
80. #73031C23semidirect
81. #73030C23semidirect
82. #73029C23semidirect
83. #73028C23semidirect
84. #73028C23semidirect
85. #73027C23semidirect
86. #73026C23semidirect
87. #73026C23semidirect
88. #73025C23semidirect
89. C2×H(7,𝔽2)C2×D3semidirect
90. #60401C2×D3semidirect
91. #60399C2×D3semidirect
92. #60398C2×D3semidirect
93. C2×#3026C2×D3semidirect
105. H(7,𝔽2)C22×D3semidirect
106. C2×#585C22×D3semidirect
107. C22×H(5,𝔽2)S4semidirect
108. C22×H(5,𝔽2)C22×D3semidirect
109. #3028C22×D3semidirect
110. #3027C22×D3semidirect
111. #3026C22×D3semidirect
112. #3026C22×D3semidirect
113. #3025C22×D3semidirect
114. C2×#458C22×D3semidirect
115. C2×B(4,𝔽2)C22×D3semidirect
120. C22×#41C2×S4semidirect
121. C22×#41C2×S4semidirect
123. C23×D4C2×S4semidirect
124. C23×D4C2×S4semidirect

Maximal subgroups

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

subgroupindex
#723440782
#723440752
#723440742
#723440712
#723440702
#723440672
#722496543
#722369222
#722368432
C2×#117141822
#722170272
C2×#117158282
S4×H(5,𝔽2)4