Finite Group Databasebeta

← C23×C9×S4 #13180078

Subgroups — C23×C9×S4 #13180078

Order 1728.

2167 conjugacy classes of subgroups; only the 297 normal subgroups are shown.

normal subgroups only

This group has 2167 conjugacy classes of subgroups — too many to draw. Only its 297 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×C9×A4C211yesmaximal, characteristic, complemented
2C22×C9×S4C2114yesmaximal, direct product factor
3C23×C3×S4C311yesmaximal, characteristic
4C23×D4×C9—31nomaximal
5C23×D3×C9—41nomaximal
6C22×C9×A4C2217yescomplemented
7C2×C9×S4C22128yesdirect product factor
8C23×C3×A4C2×C311yescharacteristic
9C22×C3×S4C2×C3114yes—
10C25×C9—31no—
11C25×C9D311yescharacteristic, complemented
12C23×C22⋊C9—21no—
13C22×D4×C9—314no—
14C22×D4×C9—314no—
15C23×C4×C9—31no—
16C23×C3×C9—41no—
17C22×D3×C9—414no—
18C2×C9×A4C2317yescomplemented
19C9×S4C2318yesdirect product factor
20C23×S4C911yescharacteristic, direct product factor
21C23×C3×D4—31no—
22C23×C3×D3—41no—
23C22×C3×A4C22×C317yes—
24C2×C3×S4C22×C3128yes—
25C24×C9—31no—
26C24×C9—37no—
27C24×C9—314no—
28C24×C9—67no—
29C24×C9—61no—
30C24×C9C2×D317yescomplemented
31C22×C22⋊C9—27no—
32C2×D4×C9—384no—
33C2×D4×C9—328no—
34C22×C4×C9—314no—
35C22×C3×C9—47no—
36C2×D3×C9—428no—
37C9×A4C2411yescharacteristic, complemented
38C25×C3—31no—
39C25×C3C3×D311yescharacteristic
40C23×A4—21no—
41C23×A4C2×C911yescharacteristic, complemented
42C22×S4C2×C9114yesdirect product factor
43C22×C3×D4—314no—
44C22×C3×D4—314no—
45C23×C3×C4—31no—
46C23×C32—41no—
47C22×C3×D3—414no—
48C2×C3×A4C23×C317yes—
49C3×S4C23×C318yes—
50C23×C9—37no—
51C23×C9—37no—
52C23×C9—321no—
53C23×C9—328no—
54C23×C9—67no—
55C23×C9—642no—
56C23×C9—614no—
57C23×C9—81no—
58C23×C9S411yescharacteristic, direct product factor
59C23×C9C22×D317yescomplemented
60C2×C22⋊C9—27no—
61D4×C9—356no—
62D4×C9—38no—
63C2×C4×C9—328no—
64C23×D4—31noSylow
65C2×C3×C9—47no—
66D3×C9—48no—
67C24×C3—31no—
68C24×C3—37no—
69C24×C3—314no—
70C24×C3—67no—
71C24×C3—61no—
72C24×C3C2×C3×D317yes—
73C23×D3—41no—
74C22×A4—27no—
75C22×A4C22×C917yescomplemented
76C2×S4C22×C9128yesdirect product factor
77C2×C3×D4—384no—
78C2×C3×D4—328no—
79C22×C3×C4—314no—
80C22×C32—47no—
81C2×C3×D3—428no—
82C3×A4C24×C311yescharacteristic
83C22×C9—37no—
84C22×C9—37no—
85C22×C9—321no—
86C22×C9—38no—
87C22×C9—67no—
88C22×C9—628no—
89C22×C9—628no—
90C22×C9—87no—
91C22×C9C2×S417yesdirect product factor
92C22×C9C23×D311yescharacteristic, complemented
93C22⋊C9—21no—
94C4×C9—38no—
95C25—31no—
96C25D3×C911yescharacteristic, complemented
97C22×D4—314no—
98C22×D4—314no—
99C23×C4—31no—
100C3×C9—41noSylow
101C23×C3—37no—
102C23×C3—37no—
103C23×C3—321no—
104C23×C3—328no—
105C23×C3—41no—
106C23×C3—67no—
107C23×C3—642no—
108C23×C3—614no—
109C23×C3—81no—
110C23×C3C3×S411yescharacteristic
111C23×C3C22×C3×D317yes—
112C22×D3—414no—
113C2×A4—27no—
114C2×A4C23×C917yescomplemented
115S4C23×C918yesdirect product factor
116C3×D4—356no—
117C3×D4—38no—
118C2×C3×C4—328no—
119C2×C32—47no—
120C3×D3—48no—
121C2×C9—31no—
122C2×C9—37no—
123C2×C9—68no—
124C2×C9—87no—
125C2×C9C22×S417yesdirect product factor
126C24—31no—
127C24—37no—
128C24—314no—
129C24—67no—
130C24—61no—
131C24C2×D3×C917yescomplemented
132C2×D4—384no—
133C2×D4—328no—
134C22×C4—314no—
135C22×C3—37no—
136C22×C3—37no—
137C22×C3—321no—
138C22×C3—38no—
139C22×C3—47no—
140C22×C3—67no—
141C22×C3—628no—
142C22×C3—628no—
143C22×C3—87no—
144C22×C3C2×C3×S417yes—
145C22×C3C23×C3×D311yescharacteristic
146C2×D3—428no—
147A4—21no—
148A4C24×C911yescharacteristic, complemented
149C3×C4—38no—
150C32—41no—
151C9—81no—
152C9C23×S411yescharacteristic, direct product factor
153C23—37no—
154C23—37no—
155C23—321no—
156C23—328no—
157C23—67no—
158C23—642no—
159C23—614no—
160C23C9×S411yescharacteristic, direct product factor
161C23C22×D3×C917yescomplemented
162D4—356no—
163D4—38no—
164C2×C4—328no—
165C2×C3—31no—
166C2×C3—37no—
167C2×C3—47no—
168C2×C3—68no—
169C2×C3—87no—
170C2×C3C22×C3×S417yes—
171D3—48no—
172C22—37no—
173C22—37no—
174C22—321no—
175C22—38no—
176C22—67no—
177C22—628no—
178C22—628no—
179C22C2×C9×S417yesdirect product factor
180C22C23×D3×C911yescharacteristic, complemented
181C4—38no—
182C3—41no—
183C3—81no—
184C3C23×C3×S411yescharacteristic
185C2—31no—
186C2—37no—
187C2—68no—
188C2C22×C9×S417yesdirect product factor
subgroupG/H
center, hypercenterC23×C9S4
commutator, hypocenterA4C24×C9
FrattiniC3C23×C3×S4
FittingC25×C9D3
socleC25×C3C3×D3
prime radicalC23×C3×S4C3
cosocleC3×A4C24×C3
LCSG ▷ A4
UCSC1 ◁ C23×C9

Complements

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

NH ≅ G/Ntype
1. C23×C9×A4C2semidirect
2. C22×C9×S4C2direct (N × H)
6. C22×C9×A4C22semidirect
7. C2×C9×S4C22direct (N × H)
11. C25×C9D3semidirect
18. C2×C9×A4C23semidirect
19. C9×S4C23direct (N × H)
20. C23×S4C9direct (N × H)
30. C24×C9C2×D3semidirect
37. C9×A4C24semidirect
41. C23×A4C2×C9semidirect
42. C22×S4C2×C9direct (N × H)
58. C23×C9S4direct (N × H)
59. C23×C9C22×D3semidirect
75. C22×A4C22×C9semidirect
76. C2×S4C22×C9direct (N × H)
91. C22×C9C2×S4direct (N × H)
92. C22×C9C23×D3semidirect
96. C25D3×C9semidirect
114. C2×A4C23×C9semidirect
115. S4C23×C9direct (N × H)
125. C2×C9C22×S4direct (N × H)
131. C24C2×D3×C9semidirect
148. A4C24×C9semidirect
152. C9C23×S4direct (N × H)
160. C23C9×S4direct (N × H)
161. C23C22×D3×C9semidirect
179. C22C2×C9×S4direct (N × H)
180. C22C23×D3×C9semidirect
188. C2C22×C9×S4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C9×A42
C22×C9×S42
C23×C3×S43
C23×D4×C93
C23×D3×C94