Finite Group Databasebeta

← C23×#3832 #11919193

Subgroups — C23×#3832 #11919193

Order 1152.

19560 conjugacy classes of subgroups; only the 5001 normal subgroups are shown.

normal subgroups only

This group has 19560 conjugacy classes of subgroups — too many to draw. Only its 5001 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
1C23×C32×Q8C211yesmaximal, characteristic, complemented
2C22×#3832C2156yesmaximal, direct product factor
3C23×C4⋊(C3⋊D3)C213yesmaximal, complemented
4C23×C4×C3⋊D3C213yesmaximal, complemented
5C23×#239—34nomaximal
6C24×C3⋊D3C2213yes—
7C22×C32×Q8C22128yescomplemented
8C23×C32×C4C2213yes—
9C23×C3⋊Dic3C2211yescharacteristic
10C2×#3832C221448yesdirect product factor
11C22×C4⋊(C3⋊D3)C22184yescomplemented
12C22×C4×C3⋊D3C22184yescomplemented
13C23×C3×Q8D314yescomplemented
14C22×#239—3224no—
15C23×D12—312no—
16C23×C4×D3—312no—
17C24×C32C2311yescharacteristic
18C23×C3⋊D3—224no—
19C23×C3⋊D3C23142yes—
20C2×C32×Q8C231112yescomplemented
21C22×C32×C4C23142yes—
22C22×C3⋊Dic3C23114yes—
23#3832C231512yesdirect product factor
24C2×C4⋊(C3⋊D3)C231336yescomplemented
25C2×C4×C3⋊D3C231336yescomplemented
26C23×#41—91noSylow
27C24×D3—312no—
28C22×C3×Q8C2×D31112yescomplemented
29C23×C3×C4C2×D3112yes—
30C23×Dic3—34no—
31C2×#239—31792no—
32C22×D12—3336no—
33C22×C4×D3—3336no—
34C23×C32#4118yescomplemented
35C23×C32C2417yes—
36C22×C3⋊D3—2168no—
37C22×C3⋊D3C24184yes—
38C32×Q8C24164yescomplemented
39C2×C32×C4C24184yes—
40C2×C3⋊Dic3C24128yes—
41C4⋊(C3⋊D3)C241192yescomplemented
42C4×C3⋊D3C241192yescomplemented
43C22×#41—956no—
44C23×Q8C3⋊D311yescharacteristic, complemented
45C23×D4—93no—
46C24×C4—93no—
47C24×C3C22×D314yes—
48C23×D3—3168no—
49C23×D3—696no—
50C2×C3×Q8C22×D31448yescomplemented
51C22×C3×C4C22×D31168yes—
52C22×Dic3—356no—
53#239—32048no—
54C2×D12—31344no—
55C2×C4×D3—31344no—
56C22×C32C2×#41128yescomplemented
57C22×C32C2517yes—
58C2×C3⋊D3—2168no—
59C2×C3⋊D3C25124yes—
60C32×C4C25124yes—
61C3⋊Dic3C2518yes—
62C25—93no—
63C2×#41—9448no—
64C22×Q8C2×C3⋊D3128yescomplemented
65C22×D4—984no—
66C23×C4—91no—
67C23×C4—984no—
68C23×C4C2×C3⋊D313yes—
69C23×C3#239132yescomplemented
70C23×C3C23×D3128yes—
71C22×D3—3336no—
72C22×D3—6672no—
73C3×Q8C23×D31256yescomplemented
74C2×C3×C4C23×D31336yes—
75C2×Dic3—3112no—
76D12—3768no—
77C4×D3—3768no—
78C2×C32C22×#41114yescomplemented
79C2×C32C2611yescharacteristic
80C3⋊D3—224no—
81C24—942no—
82C24—1824no—
83C24C22×C3⋊D311yescharacteristic
84#41—9512no—
85C2×Q8C22×C3⋊D31112yescomplemented
86C2×D4—9336no—
87C22×C4—914no—
88C22×C4—9336no—
89C22×C4C22×C3⋊D3142yes—
90C22×C3C2×#2391112yescomplemented
91C22×C3C24×D3128yes—
92C2×D3—396no—
93C2×D3—6672no—
94C3×C4C24×D3196yes—
95Dic3—332no—
96C32C23×#4111yescharacteristic, complemented, Sylow
97C23—984no—
98C23—18168no—
99C23#383218yesdirect product factor
100C23C23×C3⋊D317yes—
101Q8C23×C3⋊D3164yescomplemented
102D4—9192no—
103C2×C4—928no—
104C2×C4—9192no—
105C2×C4C23×C3⋊D3184yes—
106C2×C3C22×#239156yescomplemented
107C2×C3C25×D314yes—
108D3—696no—
109C22—924no—
110C22—18168no—
111C22C2×#3832128yesdirect product factor
112C22C24×C3⋊D317yes—
113C4—98no—
114C4C24×C3⋊D3124yes—
115C3C23×#23914yescomplemented
116C2—1824no—
117C2C22×#3832114yesdirect product factor
118C2C25×C3⋊D311yescharacteristic
subgroupG/H
centerC24C22×C3⋊D3
commutator, cosocleC2×C32C26
FrattiniC2C25×C3⋊D3
FittingC23×C32×Q8C2
socleC24×C32C23
hypercenterC23×Q8C3⋊D3
hypocenterC32C23×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C24 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C23×C32×Q8C2semidirect
2. C22×#3832C2direct (N × H)
3. C23×C4⋊(C3⋊D3)C2semidirect
4. C23×C4×C3⋊D3C2semidirect
7. C22×C32×Q8C22semidirect
10. C2×#3832C22direct (N × H)
11. C22×C4⋊(C3⋊D3)C22semidirect
12. C22×C4×C3⋊D3C22semidirect
13. C23×C3×Q8D3semidirect
20. C2×C32×Q8C23semidirect
23. #3832C23direct (N × H)
24. C2×C4⋊(C3⋊D3)C23semidirect
25. C2×C4×C3⋊D3C23semidirect
28. C22×C3×Q8C2×D3semidirect
34. C23×C32#41semidirect
38. C32×Q8C24semidirect
41. C4⋊(C3⋊D3)C24semidirect
42. C4×C3⋊D3C24semidirect
44. C23×Q8C3⋊D3semidirect
50. C2×C3×Q8C22×D3semidirect
56. C22×C32C2×#41semidirect
64. C22×Q8C2×C3⋊D3semidirect
69. C23×C3#239semidirect
73. C3×Q8C23×D3semidirect
78. C2×C32C22×#41semidirect
85. C2×Q8C22×C3⋊D3semidirect
90. C22×C3C2×#239semidirect
96. C32C23×#41semidirect
99. C23#3832direct (N × H)
101. Q8C23×C3⋊D3semidirect
106. C2×C3C22×#239semidirect
111. C22C2×#3832direct (N × H)
115. C3C23×#239semidirect
117. C2C22×#3832direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×Q82
C22×#38322
C23×C4⋊(C3⋊D3)2
C23×C4×C3⋊D32
C23×#2393