Finite Group Databasebeta

← C22×#543×C83 #35290494

Subgroups — C22×#543×C83 #35290494

Order 21248.

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×#543×C83C2148yesmaximal, direct product factor
2C22×C22⋊Q8×C83C2112yesmaximal, complemented
3C23×C22⋊C4×C83C213yesmaximal
4C26×C83C2211yescharacteristic
5C23×Q8×C83C2214yescomplemented
6C24×C4×C83C2216yes—
7#543×C83C221256yesdirect product factor
8C2×C22⋊Q8×C83C221288yescomplemented
9C22×C4⋊C4×C83C22112yescomplemented
10C22×C22⋊C4×C83C22112yes—
11C22×C22⋊C4×C83C22172yes—
12C25×C83—48no—
13C25×C83Q8116yescomplemented
14C25×C83C2313yes—
15C25×C83C23112yes—
16C22×Q8×C83—416no—
17C22×Q8×C83C23148yescomplemented
18C23×C4×C83—424no—
19C23×C4×C83C23112yes—
20C23×C4×C83C23172yes—
21C22⋊Q8×C83C231768yescomplemented
22C2×C4⋊C4×C83C231144yescomplemented
23C2×C22⋊C4×C83C231144yes—
24C2×C22⋊C4×C83C231192yes—
25C24×C83—224no—
26C24×C83—464no—
27C24×C83—448no—
28C24×C83C2×Q8124yes—
29C24×C83C2×Q8196yescomplemented
30C24×C83C24118yes—
31C24×C83C24116yes—
32C24×C83C2411yescharacteristic
33C2×Q8×C83—496no—
34C2×Q8×C83C24164yescomplemented
35C22×C4×C83—248no—
36C22×C4×C83—4144no—
37C22×C4×C83C24172yes—
38C22×C4×C83C24196yes—
39C4⋊C4×C83C241192yescomplemented
40C22⋊C4×C83C241192yes—
41C23×C83—296no—
42C23×C83—272no—
43C23×C83—4192no—
44C23×C83—432no—
45C23×C83C22×Q8172yes—
46C23×C83C22×Q814yes—
47C23×C83C22×Q8164yescomplemented
48C23×C83H(5,𝔽2)18yes—
49C23×C83C25112yes—
50C23×C83C2513yes—
51Q8×C83—464no—
52C2×C4×C83—2144no—
53C2×C4×C83—496no—
54C2×C4×C83C25148yes—
55C22×C83—224no—
56C22×C83—2144no—
57C22×C83—464no—
58C22×C83#543116yesdirect product factor
59C22×C83C23×Q8124yes—
60C22×C83C23×Q816yes—
61C22×C83C2×H(5,𝔽2)112yes—
62C22×C83C2611yescharacteristic
63C4×C83—248no—
64C22×#543C8311yesmaximal, characteristic, direct product factor, Sylow
65C2×C83—224no—
66C2×C83C2×#543112yesdirect product factor
67C2×C83C24×Q811yescharacteristic
68C2×C83C22×H(5,𝔽2)12yes—
69C2×#543C2×C83148yesdirect product factor
70C22×C22⋊Q8C2×C83112yescomplemented
71C23×C22⋊C4C2×C8313yes—
72C83C22×#54311yescharacteristic, direct product factor, Sylow
73C26C22×C8311yescharacteristic
74C23×Q8C22×C8314yescomplemented
75C24×C4C22×C8316yes—
76#543C22×C831256yesdirect product factor
77C2×C22⋊Q8C22×C831288yescomplemented
78C22×C4⋊C4C22×C83112yescomplemented
79C22×C22⋊C4C22×C83112yes—
80C22×C22⋊C4C22×C83172yes—
81C25—48no—
82C25Q8×C83116yescomplemented
83C25C23×C8313yes—
84C25C23×C83112yes—
85C22×Q8—416no—
86C22×Q8C23×C83148yescomplemented
87C23×C4—424no—
88C23×C4C23×C83112yes—
89C23×C4C23×C83172yes—
90C22⋊Q8C23×C831768yescomplemented
91C2×C4⋊C4C23×C831144yescomplemented
92C2×C22⋊C4C23×C831144yes—
93C2×C22⋊C4C23×C831192yes—
94C24—224no—
95C24—464no—
96C24—448no—
97C24C2×Q8×C83124yes—
98C24C2×Q8×C83196yescomplemented
99C24C24×C83118yes—
100C24C24×C83116yes—
101C24C24×C8311yescharacteristic
102C2×Q8—496no—
103C2×Q8C24×C83164yescomplemented
104C22×C4—248no—
105C22×C4—4144no—
106C22×C4C24×C83172yes—
107C22×C4C24×C83196yes—
108C4⋊C4C24×C831192yescomplemented
109C22⋊C4C24×C831192yes—
110C23—296no—
111C23—272no—
112C23—4192no—
113C23—432no—
114C23C22×Q8×C83172yes—
115C23C22×Q8×C8314yes—
116C23C22×Q8×C83164yescomplemented
117C23H(5,𝔽2)×C8318yes—
118C23C25×C83112yes—
119C23C25×C8313yes—
120Q8—464no—
121C2×C4—2144no—
122C2×C4—496no—
123C2×C4C25×C83148yes—
124C22—224no—
125C22—2144no—
126C22—464no—
127C22#543×C83116yesdirect product factor
128C22C23×Q8×C83124yes—
129C22C23×Q8×C8316yes—
130C22C2×H(5,𝔽2)×C83112yes—
131C22C26×C8311yescharacteristic
132C4—248no—
133C2—224no—
134C2C2×#543×C83112yesdirect product factor
135C2C24×Q8×C8311yescharacteristic
136C2C22×H(5,𝔽2)×C8312yes—
subgroupG/H
center, socleC24×C83C24
commutator, Frattini, cosocleC22C26×C83
prime radicalC26×C83C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#543×C83C2direct (N × H)
2. C22×C22⋊Q8×C83C2semidirect
5. C23×Q8×C83C22semidirect
7. #543×C83C22direct (N × H)
8. C2×C22⋊Q8×C83C22semidirect
9. C22×C4⋊C4×C83C22semidirect
13. C25×C83Q8semidirect
17. C22×Q8×C83C23semidirect
21. C22⋊Q8×C83C23semidirect
22. C2×C4⋊C4×C83C23semidirect
29. C24×C83C2×Q8semidirect
34. C2×Q8×C83C24semidirect
39. C4⋊C4×C83C24semidirect
47. C23×C83C22×Q8semidirect
58. C22×C83#543direct (N × H)
64. C22×#543C83direct (N × H)
66. C2×C83C2×#543direct (N × H)
69. C2×#543C2×C83direct (N × H)
70. C22×C22⋊Q8C2×C83semidirect
72. C83C22×#543direct (N × H)
74. C23×Q8C22×C83semidirect
76. #543C22×C83direct (N × H)
77. C2×C22⋊Q8C22×C83semidirect
78. C22×C4⋊C4C22×C83semidirect
82. C25Q8×C83semidirect
86. C22×Q8C23×C83semidirect
90. C22⋊Q8C23×C83semidirect
91. C2×C4⋊C4C23×C83semidirect
98. C24C2×Q8×C83semidirect
103. C2×Q8C24×C83semidirect
108. C4⋊C4C24×C83semidirect
116. C23C22×Q8×C83semidirect
127. C22#543×C83direct (N × H)
134. C2C2×#543×C83direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#543×C832
C22×C22⋊Q8×C832
C23×C22⋊C4×C832
C22×#54383