Finite Group Databasebeta

← C23×C9⋊S4 #13180082

Subgroups — C23×C9⋊S4 #13180082

Order 1728.

2639 conjugacy classes of subgroups; only the 259 normal subgroups are shown.

normal subgroups only

This group has 2639 conjugacy classes of subgroups — too many to draw. Only its 259 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⋊S4—31nomaximal
4C23×(C22⋊C9)⋊C2—32nomaximal
5C23×#611—31nomaximal
6C23×C3⋊D9—41nomaximal
7C22×C9×A4C2217yescomplemented
8C2×C9⋊S4C22128yesdirect product factor
9C23×C3×A4D311yescharacteristic
10C22×C3⋊S4—314no—
11C25×C9D311yescharacteristic, complemented
12C24×D9—31no—
13C23×C22⋊C9D312yescomplemented
14C22×(C22⋊C9)⋊C2—328no—
15C22×#611—314no—
16C22×#611—314no—
17C23×Dic9—31no—
18C23×C3×C9—41no—
19C22×C3⋊D9—414no—
20C2×C9×A4C2317yescomplemented
21C9⋊S4C2318yesdirect product factor
22C23×S4—93no—
23C23×#67—91no—
24C23×C3⋊D3—121no—
25C22×C3×A4C2×D317yes—
26C2×C3⋊S4—328no—
27C24×C9—31no—
28C24×C9—37no—
29C24×C9C2×D317yescomplemented
30C23×D9—314no—
31C23×D9—67no—
32C23×D9—61no—
33C23×D9—122no—
34C22×C22⋊C9C2×D3114yescomplemented
35C2×(C22⋊C9)⋊C2—356no—
36C2×#611—384no—
37C2×#611—328no—
38C22×Dic9—314no—
39C22×C3×C9—47no—
40C2×C3⋊D9—428no—
41C9×A4C2411yescharacteristic, complemented
42C25×C3C3⋊D311yescharacteristic
43C24×D3—91no—
44C23×A4D913yescomplemented
45C22×S4—942no—
46C22×#67—914no—
47C22×#67—914no—
48C23×Dic3—91no—
49C23×C32—41no—
50C22×C3⋊D3—1214no—
51C2×C3×A4C22×D317yes—
52C3⋊S4—38no—
53C23×C9—37no—
54C23×C9—37no—
55C23×C9—321no—
56C23×C9—42no—
57C23×C9—67no—
58C23×C9S411yescharacteristic, complemented
59C23×C9C22×D317yescomplemented
60C22×D9—328no—
61C22×D9—642no—
62C22×D9—614no—
63C22×D9—1228no—
64C2×C22⋊C9C22×D3114yescomplemented
65(C22⋊C9)⋊C2—316no—
66#611—356no—
67#611—38no—
68C2×Dic9—328no—
69C23×D4—271noSylow
70C2×C3×C9—47no—
71C3⋊D9—48no—
72C24×C3—31no—
73C24×C3—37no—
74C24×C3C2×C3⋊D317yes—
75C23×D3—914no—
76C23×D3—187no—
77C23×D3—181no—
78C23×D3—363no—
79C22×A4C2×D9121yescomplemented
80C2×S4—984no—
81C2×#67—984no—
82C2×#67—928no—
83C22×Dic3—914no—
84C22×C32—47no—
85C2×C3⋊D3—1228no—
86C3×A4C23×D311yescharacteristic
87C22×C9—37no—
88C22×C9—37no—
89C22×C9—321no—
90C22×C9—414no—
91C22×C9—67no—
92C22×C9C2×S417yescomplemented
93C22×C9C23×D311yescharacteristic, complemented
94C2×D9—38no—
95C2×D9—628no—
96C2×D9—628no—
97C2×D9—1256no—
98C22⋊C9C23×D312yescomplemented
99Dic9—38no—
100C25—271no—
101C25C3⋊D911yescharacteristic, complemented
102C22×D4—2714no—
103C22×D4—2714no—
104C23×C4—271no—
105C3×C9—41noSylow
106C23×C3—37no—
107C23×C3—37no—
108C23×C3—321no—
109C23×C3—43no—
110C23×C3—67no—
111C23×C3C3⋊S411yescharacteristic
112C23×C3C22×C3⋊D317yes—
113C22×D3—928no—
114C22×D3—1842no—
115C22×D3—1814no—
116C22×D3—3642no—
117C2×A4C22×D9121yescomplemented
118S4—924no—
119#67—956no—
120#67—98no—
121C2×Dic3—928no—
122C2×C32—47no—
123C3⋊D3—128no—
124C2×C9—31no—
125C2×C9—37no—
126C2×C9—414no—
127C2×C9C22×S417yescomplemented
128D9—68no—
129D9—1216no—
130C24—31no—
131C24—37no—
132C24—2714no—
133C24—547no—
134C24—541no—
135C24C2×C3⋊D917yescomplemented
136C2×D4—2784no—
137C2×D4—2728no—
138C22×C4—2714no—
139C22×C3—37no—
140C22×C3—37no—
141C22×C3—321no—
142C22×C3—421no—
143C22×C3—67no—
144C22×C3C2×C3⋊S417yes—
145C22×C3C23×C3⋊D311yescharacteristic
146C2×D3—98no—
147C2×D3—1828no—
148C2×D3—1828no—
149C2×D3—3684no—
150A4C23×D913yescomplemented
151Dic3—98no—
152C32—41no—
153C9—42no—
154C9C23×S411yescharacteristic, complemented
155C23—37no—
156C23—37no—
157C23—321no—
158C23—67no—
159C23—2728no—
160C23—5442no—
161C23—5414no—
162C23C9⋊S411yescharacteristic, direct product factor
163C23C22×C3⋊D917yescomplemented
164D4—2756no—
165D4—278no—
166C2×C4—2728no—
167C2×C3—31no—
168C2×C3—37no—
169C2×C3—421no—
170C2×C3C22×C3⋊S417yes—
171D3—188no—
172D3—3624no—
173C22—37no—
174C22—37no—
175C22—321no—
176C22—67no—
177C22—278no—
178C22—5428no—
179C22—5428no—
180C22C2×C9⋊S417yesdirect product factor
181C22C23×C3⋊D911yescharacteristic, complemented
182C4—278no—
183C3—43no—
184C3C23×C3⋊S411yescharacteristic
185C2—31no—
186C2—37no—
187C2—548no—
188C2C22×C9⋊S417yesdirect product factor
subgroupG/H
center, hypercenterC23C9⋊S4
commutator, cosocle, hypocenterC9×A4C24
FrattiniC3C23×C3⋊S4
FittingC25×C9D3
socleC25×C3C3⋊D3
LCSG ▷ C9×A4
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C9×A4C2semidirect
2. C22×C9⋊S4C2direct (N × H)
7. C22×C9×A4C22semidirect
8. C2×C9⋊S4C22direct (N × H)
11. C25×C9D3semidirect
13. C23×C22⋊C9D3semidirect
20. C2×C9×A4C23semidirect
21. C9⋊S4C23direct (N × H)
29. C24×C9C2×D3semidirect
34. C22×C22⋊C9C2×D3semidirect
41. C9×A4C24semidirect
44. C23×A4D9semidirect
58. C23×C9S4semidirect
59. C23×C9C22×D3semidirect
64. C2×C22⋊C9C22×D3semidirect
79. C22×A4C2×D9semidirect
92. C22×C9C2×S4semidirect
93. C22×C9C23×D3semidirect
98. C22⋊C9C23×D3semidirect
101. C25C3⋊D9semidirect
117. C2×A4C22×D9semidirect
127. C2×C9C22×S4semidirect
135. C24C2×C3⋊D9semidirect
150. A4C23×D9semidirect
154. C9C23×S4semidirect
162. C23C9⋊S4direct (N × H)
163. C23C22×C3⋊D9semidirect
180. C22C2×C9⋊S4direct (N × H)
181. C22C23×C3⋊D9semidirect
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×(C22⋊C9)⋊C23
C23×#6113
C23×C3⋊D94