Finite Group Databasebeta

← C23×C3×#239 #11919194

Subgroups — C23×C3×#239 #11919194

Order 1152.

13465 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C32×Q8C211yesmaximal, characteristic, complemented
2C22×C3×#239C2156yesmaximal, direct product factor
3C23×C3×D12C213yesmaximal, complemented
4C23×C3×C4×D3C213yesmaximal, complemented
5C23×C3×#41—31nomaximal
6C23×#239C311yesmaximal, characteristic, direct product factor
7C24×C3×D3C2213yes—
8C22×C32×Q8C22128yescomplemented
9C23×C32×C4C2213yes—
10C23×C3×Dic3C2211yescharacteristic
11C2×C3×#239C221448yesdirect product factor
12C22×C3×D12C22184yescomplemented
13C22×C3×C4×D3C22184yescomplemented
14C22×C3×#41—356no—
15C23×C3×Q8—21no—
16C23×C3×Q8D311yescharacteristic, complemented
17C23×C3×Q8C2×C311yescharacteristic, complemented
18C23×C3×D4—33no—
19C24×C3×C4—33no—
20C22×#239C2×C3156yesdirect product factor
21C23×D12C2×C313yescomplemented
22C23×C4×D3C2×C313yescomplemented
23C24×C32C2311yescharacteristic
24C23×C3×D3—224no—
25C23×C3×D3C23142yes—
26C2×C32×Q8C231112yescomplemented
27C22×C32×C4C23142yes—
28C22×C3×Dic3C23114yes—
29C3×#239C231512yesdirect product factor
30C2×C3×D12C231336yescomplemented
31C2×C3×C4×D3C231336yescomplemented
32C23×#41—31noSylow
33C25×C3—33no—
34C24×D3C22×C313yes—
35C2×C3×#41—3448no—
36C22×C3×Q8—228no—
37C22×C3×Q8C2×D3128yescomplemented
38C22×C3×Q8C22×C3128yescomplemented
39C22×C3×D4—384no—
40C23×C3×C4—23no—
41C23×C3×C4—31no—
42C23×C3×C4—384no—
43C23×C3×C4C2×D313yes—
44C23×C3×C4C22×C313yes—
45C23×Dic3C22×C311yescharacteristic
46C2×#239C22×C31448yesdirect product factor
47C22×D12C22×C3184yescomplemented
48C22×C4×D3C22×C3184yescomplemented
49C23×C32#4118yescomplemented
50C23×C32C2417yes—
51C22×C3×D3—2168no—
52C22×C3×D3C24184yes—
53C32×Q8C24164yescomplemented
54C2×C32×C4C24184yes—
55C2×C3×Dic3C24128yes—
56C3×D12C241192yescomplemented
57C3×C4×D3C241192yescomplemented
58C22×#41—356no—
59C23×Q8C3×D311yescharacteristic, complemented
60C23×D4—33no—
61C24×C4—33no—
62C24×C3—21no—
63C24×C3—342no—
64C24×C3—624no—
65C24×C3C22×D311yescharacteristic
66C24×C3C23×C311yescharacteristic
67C23×D3—224no—
68C23×D3C23×C3142yes—
69C3×#41—3512no—
70C2×C3×Q8—2112no—
71C2×C3×Q8C22×D31112yescomplemented
72C2×C3×Q8C23×C31112yescomplemented
73C2×C3×D4—3336no—
74C22×C3×C4—242no—
75C22×C3×C4—314no—
76C22×C3×C4—3336no—
77C22×C3×C4C22×D3142yes—
78C22×C3×C4C23×C3142yes—
79C22×Dic3C23×C3114yes—
80#239C23×C31512yesdirect product factor
81C2×D12C23×C31336yescomplemented
82C2×C4×D3C23×C31336yescomplemented
83C22×C32C2×#41128yescomplemented
84C22×C32C2517yes—
85C2×C3×D3—2168no—
86C2×C3×D3C25124yes—
87C32×C4C25124yes—
88C3×Dic3C2518yes—
89C25—33no—
90C2×#41—3448no—
91C22×Q8C2×C3×D3128yescomplemented
92C22×D4—384no—
93C23×C4—31no—
94C23×C4—384no—
95C23×C4C2×C3×D313yes—
96C23×C3—27no—
97C23×C3—28no—
98C23×C3—384no—
99C23×C3—6168no—
100C23×C3#23918yesdirect product factor
101C23×C3C3×#4118yescomplemented
102C23×C3C23×D317yes—
103C23×C3C24×C317yes—
104C22×D3—2168no—
105C22×D3C24×C3184yes—
106C3×Q8—264no—
107C3×Q8C23×D3164yescomplemented
108C3×Q8C24×C3164yescomplemented
109C3×D4—3192no—
110C2×C3×C4—284no—
111C2×C3×C4—328no—
112C2×C3×C4—3192no—
113C2×C3×C4C23×D3184yes—
114C2×C3×C4C24×C3184yes—
115C2×Dic3C24×C3128yes—
116D12C24×C31192yescomplemented
117C4×D3C24×C31192yescomplemented
118C2×C32C22×#41114yescomplemented
119C2×C32C2611yescharacteristic
120C3×D3—224no—
121C24—342no—
122C24—624no—
123C24C22×C3×D311yescharacteristic
124#41—3512no—
125C2×Q8C22×C3×D31112yescomplemented
126C2×D4—3336no—
127C22×C4—314no—
128C22×C4—3336no—
129C22×C4C22×C3×D3142yes—
130C22×C3—228no—
131C22×C3—27no—
132C22×C3—324no—
133C22×C3—6168no—
134C22×C3C2×#239128yesdirect product factor
135C22×C3C2×C3×#41128yescomplemented
136C22×C3C24×D317yes—
137C22×C3C25×C317yes—
138C2×D3—2168no—
139C2×D3C25×C3124yes—
140C3×C4—224no—
141C3×C4—38no—
142C3×C4C24×D3124yes—
143C3×C4C25×C3124yes—
144Dic3C25×C318yes—
145C32C23×#4111yescharacteristic, complemented, Sylow
146C23—384no—
147C23—6168no—
148C23C3×#23918yesdirect product factor
149C23C23×C3×D317yes—
150Q8C23×C3×D3164yescomplemented
151D4—3192no—
152C2×C4—328no—
153C2×C4—3192no—
154C2×C4C23×C3×D3184yes—
155C2×C3—214no—
156C2×C3—21no—
157C2×C3—624no—
158C2×C3C22×#239114yesdirect product factor
159C2×C3C22×C3×#41114yescomplemented
160C2×C3C25×D311yescharacteristic
161C2×C3C26×C311yescharacteristic
162D3—224no—
163C22—324no—
164C22—6168no—
165C22C2×C3×#239128yesdirect product factor
166C22C24×C3×D317yes—
167C4—38no—
168C4C24×C3×D3124yes—
169C3—21no—
170C3C23×#23911yescharacteristic, direct product factor
171C3C23×C3×#4111yescharacteristic, complemented
172C2—624no—
173C2C22×C3×#239114yesdirect product factor
174C2C25×C3×D311yescharacteristic
subgroupG/H
centerC24×C3C22×D3
commutator, cosocleC2×C3C26×C3
FrattiniC2C25×C3×D3
FittingC23×C32×Q8C2
socleC24×C32C23
hypercenterC23×C3×Q8D3
hypocenterC3C23×C3×#41
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C24×C3 ◁ C23×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C23×C32×Q8C2semidirect
2. C22×C3×#239C2direct (N × H)
3. C23×C3×D12C2semidirect
4. C23×C3×C4×D3C2semidirect
6. C23×#239C3direct (N × H)
8. C22×C32×Q8C22semidirect
11. C2×C3×#239C22direct (N × H)
12. C22×C3×D12C22semidirect
13. C22×C3×C4×D3C22semidirect
16. C23×C3×Q8D3semidirect
17. C23×C3×Q8C2×C3semidirect
20. C22×#239C2×C3direct (N × H)
21. C23×D12C2×C3semidirect
22. C23×C4×D3C2×C3semidirect
26. C2×C32×Q8C23semidirect
29. C3×#239C23direct (N × H)
30. C2×C3×D12C23semidirect
31. C2×C3×C4×D3C23semidirect
37. C22×C3×Q8C2×D3semidirect
38. C22×C3×Q8C22×C3semidirect
46. C2×#239C22×C3direct (N × H)
47. C22×D12C22×C3semidirect
48. C22×C4×D3C22×C3semidirect
49. C23×C32#41semidirect
53. C32×Q8C24semidirect
56. C3×D12C24semidirect
57. C3×C4×D3C24semidirect
59. C23×Q8C3×D3semidirect
71. C2×C3×Q8C22×D3semidirect
72. C2×C3×Q8C23×C3semidirect
80. #239C23×C3direct (N × H)
81. C2×D12C23×C3semidirect
82. C2×C4×D3C23×C3semidirect
83. C22×C32C2×#41semidirect
91. C22×Q8C2×C3×D3semidirect
100. C23×C3#239direct (N × H)
101. C23×C3C3×#41semidirect
107. C3×Q8C23×D3semidirect
108. C3×Q8C24×C3semidirect
116. D12C24×C3semidirect
117. C4×D3C24×C3semidirect
118. C2×C32C22×#41semidirect
125. C2×Q8C22×C3×D3semidirect
134. C22×C3C2×#239direct (N × H)
135. C22×C3C2×C3×#41semidirect
145. C32C23×#41semidirect
148. C23C3×#239direct (N × H)
150. Q8C23×C3×D3semidirect
158. C2×C3C22×#239direct (N × H)
159. C2×C3C22×C3×#41semidirect
165. C22C2×C3×#239direct (N × H)
170. C3C23×#239direct (N × H)
171. C3C23×C3×#41semidirect
173. C2C22×C3×#239direct (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×C3×#2392
C23×C3×D122
C23×C3×C4×D32
C23×C3×#413
C23×#2393