Finite Group Databasebeta

← C22×C43×#543 #24982687

Subgroups — C22×C43×#543 #24982687

Order 11008.

9310 conjugacy classes of subgroups; only the 6366 normal subgroups are shown.

normal subgroups only

This group has 9310 conjugacy classes of subgroups — too many to draw. Only its 6366 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C43×#543C2148yesmaximal, direct product factor
2C22×C22⋊Q8×C43C2112yesmaximal, complemented
3C23×C22⋊C4×C43C213yesmaximal
4C26×C43C2211yescharacteristic
5C23×Q8×C43C2214yescomplemented
6C24×C4×C43C2216yes—
7C43×#543C221256yesdirect product factor
8C2×C22⋊Q8×C43C221288yescomplemented
9C22×C4⋊C4×C43C22112yescomplemented
10C22×C22⋊C4×C43C22112yes—
11C22×C22⋊C4×C43C22172yes—
12C25×C43—48no—
13C25×C43Q8116yescomplemented
14C25×C43C2313yes—
15C25×C43C23112yes—
16C22×Q8×C43—416no—
17C22×Q8×C43C23148yescomplemented
18C23×C4×C43—424no—
19C23×C4×C43C23112yes—
20C23×C4×C43C23172yes—
21C22⋊Q8×C43C231768yescomplemented
22C2×C4⋊C4×C43C231144yescomplemented
23C2×C22⋊C4×C43C231144yes—
24C2×C22⋊C4×C43C231192yes—
25C24×C43—224no—
26C24×C43—464no—
27C24×C43—448no—
28C24×C43C2×Q8124yes—
29C24×C43C2×Q8196yescomplemented
30C24×C43C24118yes—
31C24×C43C24116yes—
32C24×C43C2411yescharacteristic
33C2×Q8×C43—496no—
34C2×Q8×C43C24164yescomplemented
35C22×C4×C43—248no—
36C22×C4×C43—4144no—
37C22×C4×C43C24172yes—
38C22×C4×C43C24196yes—
39C4⋊C4×C43C241192yescomplemented
40C22⋊C4×C43C241192yes—
41C23×C43—296no—
42C23×C43—272no—
43C23×C43—4192no—
44C23×C43—432no—
45C23×C43C22×Q8172yes—
46C23×C43C22×Q814yes—
47C23×C43C22×Q8164yescomplemented
48C23×C43H(5,𝔽2)18yes—
49C23×C43C25112yes—
50C23×C43C2513yes—
51Q8×C43—464no—
52C2×C4×C43—2144no—
53C2×C4×C43—496no—
54C2×C4×C43C25148yes—
55C22×#543C4311yesmaximal, characteristic, direct product factor, Sylow
56C22×C43—224no—
57C22×C43—2144no—
58C22×C43—464no—
59C22×C43#543116yesdirect product factor
60C22×C43C23×Q8124yes—
61C22×C43C23×Q816yes—
62C22×C43C2×H(5,𝔽2)112yes—
63C22×C43C2611yescharacteristic
64C4×C43—248no—
65C2×#543C2×C43148yesdirect product factor
66C22×C22⋊Q8C2×C43112yescomplemented
67C23×C22⋊C4C2×C4313yes—
68C2×C43—224no—
69C2×C43C2×#543112yesdirect product factor
70C2×C43C24×Q811yescharacteristic
71C2×C43C22×H(5,𝔽2)12yes—
72C26C22×C4311yescharacteristic
73C23×Q8C22×C4314yescomplemented
74C24×C4C22×C4316yes—
75#543C22×C431256yesdirect product factor
76C2×C22⋊Q8C22×C431288yescomplemented
77C22×C4⋊C4C22×C43112yescomplemented
78C22×C22⋊C4C22×C43112yes—
79C22×C22⋊C4C22×C43172yes—
80C43C22×#54311yescharacteristic, direct product factor, Sylow
81C25—48no—
82C25Q8×C43116yescomplemented
83C25C23×C4313yes—
84C25C23×C43112yes—
85C22×Q8—416no—
86C22×Q8C23×C43148yescomplemented
87C23×C4—424no—
88C23×C4C23×C43112yes—
89C23×C4C23×C43172yes—
90C22⋊Q8C23×C431768yescomplemented
91C2×C4⋊C4C23×C431144yescomplemented
92C2×C22⋊C4C23×C431144yes—
93C2×C22⋊C4C23×C431192yes—
94C24—224no—
95C24—448no—
96C24—464no—
97C24C2×Q8×C43124yes—
98C24C2×Q8×C43196yescomplemented
99C24C24×C43118yes—
100C24C24×C43116yes—
101C24C24×C4311yescharacteristic
102C2×Q8—496no—
103C2×Q8C24×C43164yescomplemented
104C22×C4—248no—
105C22×C4—4144no—
106C22×C4C24×C43172yes—
107C22×C4C24×C43196yes—
108C4⋊C4C24×C431192yescomplemented
109C22⋊C4C24×C431192yes—
110C23—296no—
111C23—272no—
112C23—4192no—
113C23—432no—
114C23C22×Q8×C43172yes—
115C23C22×Q8×C4314yes—
116C23C22×Q8×C43164yescomplemented
117C23H(5,𝔽2)×C4318yes—
118C23C25×C43112yes—
119C23C25×C4313yes—
120Q8—464no—
121C2×C4—2144no—
122C2×C4—496no—
123C2×C4C25×C43148yes—
124C22—2144no—
125C22—224no—
126C22—464no—
127C22C43×#543116yesdirect product factor
128C22C23×Q8×C43124yes—
129C22C23×Q8×C4316yes—
130C22C2×H(5,𝔽2)×C43112yes—
131C22C26×C4311yescharacteristic
132C4—248no—
133C2—224no—
134C2C2×C43×#543112yesdirect product factor
135C2C24×Q8×C4311yescharacteristic
136C2C22×H(5,𝔽2)×C4312yes—
subgroupG/H
center, socleC24×C43C24
commutator, Frattini, cosocleC22C26×C43
prime radicalC26×C43C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C43 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C43×#543C2direct (N × H)
2. C22×C22⋊Q8×C43C2semidirect
5. C23×Q8×C43C22semidirect
7. C43×#543C22direct (N × H)
8. C2×C22⋊Q8×C43C22semidirect
9. C22×C4⋊C4×C43C22semidirect
13. C25×C43Q8semidirect
17. C22×Q8×C43C23semidirect
21. C22⋊Q8×C43C23semidirect
22. C2×C4⋊C4×C43C23semidirect
29. C24×C43C2×Q8semidirect
34. C2×Q8×C43C24semidirect
39. C4⋊C4×C43C24semidirect
47. C23×C43C22×Q8semidirect
55. C22×#543C43direct (N × H)
59. C22×C43#543direct (N × H)
65. C2×#543C2×C43direct (N × H)
66. C22×C22⋊Q8C2×C43semidirect
69. C2×C43C2×#543direct (N × H)
73. C23×Q8C22×C43semidirect
75. #543C22×C43direct (N × H)
76. C2×C22⋊Q8C22×C43semidirect
77. C22×C4⋊C4C22×C43semidirect
80. C43C22×#543direct (N × H)
82. C25Q8×C43semidirect
86. C22×Q8C23×C43semidirect
90. C22⋊Q8C23×C43semidirect
91. C2×C4⋊C4C23×C43semidirect
98. C24C2×Q8×C43semidirect
103. C2×Q8C24×C43semidirect
108. C4⋊C4C24×C43semidirect
116. C23C22×Q8×C43semidirect
127. C22C43×#543direct (N × H)
134. C2C2×C43×#543direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C43×#5432
C22×C22⋊Q8×C432
C23×C22⋊C4×C432
C22×#54343