Finite Group Databasebeta

← C22×C3×C4⋊Dic9 #13153530

Subgroups — C22×C3×C4⋊Dic9 #13153530

Order 1728.

1490 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4×C9C211yesmaximal, characteristic
2C23×C3×Dic9C212yesmaximal
3C2×C3×C4⋊Dic9C2112yesmaximal, direct product factor
4C22×C3×C4⋊Dic3—31nomaximal
5C22×C4⋊Dic9C311yesmaximal, characteristic, direct product factor
6C24×C3×C9C2211yescharacteristic
7C22×C3×C4×C9C418yescomplemented
8C22×C3×C4×C9C2216yes—
9C22×C3×Dic9—28no—
10C22×C3×Dic9C22112yes—
11C3×C4⋊Dic9C22116yesdirect product factor
12C23×C32×C4D311yescharacteristic
13C23×C3×Dic3—32no—
14C2×C3×C4⋊Dic3—312no—
15C23×C4×C9—21no—
16C23×C4×C9C2×C311yescharacteristic
17C23×Dic9C2×C312yes—
18C2×C4⋊Dic9C2×C3112yesdirect product factor
19C23×C3×C9C2×C414yes—
20C23×C3×C9D414yes—
21C23×C3×C9Q814yes—
22C23×C3×C9C2313yes—
23C2×C3×C4×C9C2×C4124yescomplemented
24C2×C3×C4×C9C2314yes—
25C2×C3×Dic9—224no—
26C2×C3×Dic9C2318yes—
27C22×C3×C4⋊C4—91no—
28C22×C4⋊Dic3—31no—
29C24×C32C2×D311yescharacteristic
30C22×C32×C4Dic318yes—
31C22×C32×C4C2×D316yes—
32C22×C3×Dic3—312no—
33C22×C3×Dic3—68no—
34C24×C9—21no—
35C24×C9C22×C311yescharacteristic
36C3×C4⋊Dic3—316no—
37C22×C4×C9—28no—
38C22×C4×C9—26no—
39C22×C4×C9C3×C418yescomplemented
40C22×C4×C9C22×C316yes—
41C22×Dic9—28no—
42C22×Dic9C22×C3112yes—
43C4⋊Dic9C22×C3116yesdirect product factor
44C22×C3×C9C4⋊C4116yescomplemented
45C22×C3×C9C22×C416yes—
46C22×C3×C9C2×D416yes—
47C22×C3×C9C2×Q816yes—
48C22×C3×C9C2411yescharacteristic
49C3×C4×C9C22×C418yescomplemented
50C3×Dic9—28no—
51C23×C3×C4—21no—
52C23×C3×C4—92no—
53C23×C3×C4D911yescharacteristic
54C23×C3×C4C3×D311yescharacteristic
55C23×Dic3—32no—
56C2×C3×C4⋊C4—912no—
57C2×C4⋊Dic3—312no—
58C23×C32Dic614yes—
59C23×C32D1214yes—
60C23×C32C2×Dic314yes—
61C23×C32C22×D313yes—
62C2×C32×C4C2×Dic3124yes—
63C2×C32×C4C22×D314yes—
64C2×C3×Dic3—38no—
65C2×C3×Dic3—624no—
66C23×C9—24no—
67C23×C9—24no—
68C23×C9—24no—
69C23×C9—23no—
70C23×C9C2×C3×C414yes—
71C23×C9C3×D414yes—
72C23×C9C3×Q814yes—
73C23×C9C23×C313yes—
74C2×C4×C9—224no—
75C2×C4×C9—24no—
76C2×C4×C9C2×C3×C4124yescomplemented
77C2×C4×C9C23×C314yes—
78C2×Dic9—224no—
79C2×Dic9C23×C318yes—
80C22×C4⋊C4—91noSylow
81C2×C3×C9C2×C4⋊C4112yescomplemented
82C2×C3×C9C23×C411yescharacteristic
83C2×C3×C9C22×D411yescharacteristic
84C2×C3×C9C22×Q811yescharacteristic
85C24×C3—21no—
86C24×C3C2×D911yescharacteristic
87C24×C3C2×C3×D311yescharacteristic
88C22×C3×C4—28no—
89C22×C3×C4—26no—
90C22×C3×C4—912no—
91C22×C3×C4—188no—
92C22×C3×C4Dic918yescomplemented
93C22×C3×C4C2×D916yes—
94C22×C3×C4C3×Dic318yes—
95C22×C3×C4C2×C3×D316yes—
96C22×Dic3—312no—
97C22×Dic3—68no—
98C3×C4⋊C4—916no—
99C4⋊Dic3—316no—
100C22×C32C4⋊Dic3116yes—
101C22×C32C2×Dic616yes—
102C22×C32C2×D1216yes—
103C22×C32C22×Dic316yes—
104C22×C32C23×D311yescharacteristic
105C32×C4C22×Dic318yes—
106C3×Dic3—68no—
107C22×C9—21no—
108C22×C9—216no—
109C22×C9—26no—
110C22×C9—26no—
111C22×C9—26no—
112C22×C9C3×C4⋊C4116yescomplemented
113C22×C9C22×C3×C416yes—
114C22×C9C2×C3×D416yes—
115C22×C9C2×C3×Q816yes—
116C22×C9C24×C311yescharacteristic
117C4×C9—28no—
118C4×C9C22×C3×C418yescomplemented
119Dic9—28no—
120C23×C4—92no—
121C23×C4C3×D911yescharacteristic
122C2×C4⋊C4—912no—
123C3×C9C22×C4⋊C411yescharacteristic, complemented, Sylow
124C23×C3—24no—
125C23×C3—24no—
126C23×C3—24no—
127C23×C3—23no—
128C23×C3Dic1814yes—
129C23×C3D3614yes—
130C23×C3C2×Dic914yes—
131C23×C3C22×D913yes—
132C23×C3C3×Dic614yes—
133C23×C3C3×D1214yes—
134C23×C3C2×C3×Dic314yes—
135C23×C3C22×C3×D313yes—
136C2×C3×C4—224no—
137C2×C3×C4—24no—
138C2×C3×C4—98no—
139C2×C3×C4—1824no—
140C2×C3×C4C2×Dic9124yescomplemented
141C2×C3×C4C22×D914yes—
142C2×C3×C4C2×C3×Dic3124yes—
143C2×C3×C4C22×C3×D314yes—
144C2×Dic3—38no—
145C2×Dic3—624no—
146C2×C32C2×C4⋊Dic3112yes—
147C2×C32C22×Dic611yescharacteristic
148C2×C32C22×D1211yescharacteristic
149C2×C32C23×Dic311yescharacteristic
150C2×C9—212no—
151C2×C9—21no—
152C2×C9—21no—
153C2×C9—21no—
154C2×C9C2×C3×C4⋊C4112yescomplemented
155C2×C9C23×C3×C411yescharacteristic
156C2×C9C22×C3×D411yescharacteristic
157C2×C9C22×C3×Q811yescharacteristic
158C24C2×C3×D911yescharacteristic
159C22×C4—912no—
160C22×C4—188no—
161C22×C4C3×Dic918yescomplemented
162C22×C4C2×C3×D916yes—
163C4⋊C4—916no—
164C22×C3—21no—
165C22×C3—26no—
166C22×C3—26no—
167C22×C3—26no—
168C22×C3—216no—
169C22×C3C4⋊Dic9116yesdirect product factor
170C22×C3C2×Dic1816yes—
171C22×C3C2×D3616yes—
172C22×C3C22×Dic916yes—
173C22×C3C3×C4⋊Dic3116yes—
174C22×C3C23×D911yescharacteristic
175C22×C3C2×C3×Dic616yes—
176C22×C3C2×C3×D1216yes—
177C22×C3C22×C3×Dic316yes—
178C22×C3C23×C3×D311yescharacteristic
179C3×C4—28no—
180C3×C4—188no—
181C3×C4C22×Dic918yescomplemented
182C3×C4C22×C3×Dic318yes—
183Dic3—68no—
184C32C22×C4⋊Dic311yescharacteristic
185C9—21no—
186C9C22×C3×C4⋊C411yescharacteristic, complemented
187C23C3×Dic1814yes—
188C23C3×D3614yes—
189C23C2×C3×Dic914yes—
190C23C22×C3×D913yes—
191C2×C4—98no—
192C2×C4—1824no—
193C2×C4C2×C3×Dic9124yescomplemented
194C2×C4C22×C3×D914yes—
195C2×C3—21no—
196C2×C3—21no—
197C2×C3—212no—
198C2×C3—21no—
199C2×C3C2×C4⋊Dic9112yesdirect product factor
200C2×C3C22×Dic1811yescharacteristic
201C2×C3C22×D3611yescharacteristic
202C2×C3C23×Dic911yescharacteristic
203C2×C3C2×C3×C4⋊Dic3112yes—
204C2×C3C22×C3×Dic611yescharacteristic
205C2×C3C22×C3×D1211yescharacteristic
206C2×C3C23×C3×Dic311yescharacteristic
207C22C3×C4⋊Dic9116yesdirect product factor
208C22C2×C3×Dic1816yes—
209C22C2×C3×D3616yes—
210C22C22×C3×Dic916yes—
211C22C23×C3×D911yescharacteristic
212C4—188no—
213C4C22×C3×Dic918yescomplemented
214C3—21no—
215C3C22×C4⋊Dic911yescharacteristic, direct product factor
216C3C22×C3×C4⋊Dic311yescharacteristic
217C2C2×C3×C4⋊Dic9112yesdirect product factor
218C2C22×C3×Dic1811yescharacteristic
219C2C22×C3×D3611yescharacteristic
220C2C23×C3×Dic911yescharacteristic
subgroupG/H
centerC24×C3C2×D9
commutatorC2×C9C23×C3×C4
FrattiniC22×C3C23×C3×D3
FittingC23×C3×C4×C9C2
socle, prime radicalC24×C32C2×D3
cosocleC22×C9C24×C3
hypercenterC23×C3×C4D9
hypocenterC9C22×C3×C4⋊C4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C24×C3 ◁ C23×C3×C4

Complements

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

NH ≅ G/Ntype
3. C2×C3×C4⋊Dic9C2direct (N × H)
5. C22×C4⋊Dic9C3direct (N × H)
7. C22×C3×C4×C9C4semidirect
11. C3×C4⋊Dic9C22direct (N × H)
18. C2×C4⋊Dic9C2×C3direct (N × H)
23. C2×C3×C4×C9C2×C4semidirect
39. C22×C4×C9C3×C4semidirect
43. C4⋊Dic9C22×C3direct (N × H)
44. C22×C3×C9C4⋊C4semidirect
49. C3×C4×C9C22×C4semidirect
76. C2×C4×C9C2×C3×C4semidirect
81. C2×C3×C9C2×C4⋊C4semidirect
92. C22×C3×C4Dic9semidirect
112. C22×C9C3×C4⋊C4semidirect
118. C4×C9C22×C3×C4semidirect
123. C3×C9C22×C4⋊C4semidirect
140. C2×C3×C4C2×Dic9semidirect
154. C2×C9C2×C3×C4⋊C4semidirect
161. C22×C4C3×Dic9semidirect
169. C22×C3C4⋊Dic9direct (N × H)
181. C3×C4C22×Dic9semidirect
186. C9C22×C3×C4⋊C4semidirect
193. C2×C4C2×C3×Dic9semidirect
199. C2×C3C2×C4⋊Dic9direct (N × H)
207. C22C3×C4⋊Dic9direct (N × H)
213. C4C22×C3×Dic9semidirect
215. C3C22×C4⋊Dic9direct (N × H)
217. C2C2×C3×C4⋊Dic9direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4×C92
C23×C3×Dic92
C2×C3×C4⋊Dic92
C22×C3×C4⋊Dic33
C22×C4⋊Dic93