Finite Group Databasebeta

← C22×A4×C3⋊Dic3 #13181475

Subgroups — C22×A4×C3⋊Dic3 #13181475

Order 1728.

2158 conjugacy classes of subgroups; only the 321 normal subgroups are shown.

normal subgroups only

This group has 2158 conjugacy classes of subgroups — too many to draw. Only its 321 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×A4C211yesmaximal, characteristic
2C2×A4×C3⋊Dic3C216yesmaximal, direct product factor
3C24×C3⋊Dic3C311yesmaximal, characteristic, complemented
4C22×Dic3×A4—34nomaximal
5C22×C32×A4C414yescomplemented
6C22×C32×A4C2213yes—
7C22×C3×C3⋊Dic3—41nomaximal
8A4×C3⋊Dic3C2214yesdirect product factor
9C25×C32C2×C311yescharacteristic
10C23×C3×A4—24no—
11C23×C3×A4D314yes—
12C23×C3⋊Dic3—31no—
13C23×C3⋊Dic3—36no—
14C23×C3⋊Dic3—31no—
15C23×C3⋊Dic3C2×C316yescomplemented
16C2×Dic3×A4—324no—
17C23×C33—41no—
18C2×C32×A4C2×C416yescomplemented
19C2×C32×A4C2311yescharacteristic
20C2×C3×C3⋊Dic3—46no—
21C24×Dic3—34no—
22C22×C4×A4—91no—
23C24×C32—31no—
24C24×C32—33no—
25C24×C32—34no—
26C24×C32C3×C414yescomplemented
27C24×C32C22×C313yes—
28C22×C3×A4—216no—
29C22×C3×A4—212no—
30C22×C3×A4Dic3116yescomplemented
31C22×C3×A4C2×D3112yes—
32C22×C3⋊Dic3—36no—
33C22×C3⋊Dic3—38no—
34C22×C3⋊Dic3—31no—
35C22×C3⋊Dic3—36no—
36C22×C3⋊Dic3—36no—
37C22×C3⋊Dic3—36no—
38C22×C3⋊Dic3—312no—
39C22×C3⋊Dic3A411yescharacteristic, direct product factor
40C22×C3⋊Dic3C22×C314yescomplemented
41C22×C3×Dic3—124no—
42Dic3×A4—316no—
43C22×C33—44no—
44C22×C33—43no—
45C32×A4C22×C411yescharacteristic, complemented
46C3×C3⋊Dic3—44no—
47C25×C3C3×D314yes—
48C23×A4—24no—
49C23×A4C3⋊D311yescharacteristic
50C23×Dic3—324no—
51C23×Dic3—94no—
52C23×Dic3—924no—
53C23×Dic3—94no—
54C2×C4×A4—96no—
55C23×C32—312no—
56C23×C32—36no—
57C23×C32—33no—
58C23×C32—312no—
59C23×C32—32no—
60C23×C32—34no—
61C23×C32—34no—
62C23×C32—33no—
63C23×C32—33no—
64C23×C32—44no—
65C23×C32—84no—
66C23×C32C2×C3×C416yescomplemented
67C23×C32C2×A411yescharacteristic
68C23×C32C23×C311yescharacteristic
69C2×C3×A4—224no—
70C2×C3×A4—24no—
71C2×C3×A4C2×Dic3124yescomplemented
72C2×C3×A4C22×D314yes—
73C2×C3⋊Dic3—312no—
74C2×C3⋊Dic3—36no—
75C2×C3⋊Dic3—312no—
76C2×C3⋊Dic3—34no—
77C2×C3⋊Dic3—34no—
78C2×C3⋊Dic3C2×A416yesdirect product factor
79C2×C3×Dic3—1224no—
80C24×C4—91noSylow
81C2×C33—46no—
82C2×C33—41no—
83C24×C3—316no—
84C24×C3—312no—
85C24×C3—34no—
86C24×C3C3×Dic3116yescomplemented
87C24×C3C2×C3×D3112yes—
88C22×A4—216no—
89C22×A4—212no—
90C22×A4C3⋊Dic314yesdirect product factor
91C22×A4C2×C3⋊D313yes—
92C22×C3×C4—361no—
93C22×Dic3—34no—
94C22×Dic3—316no—
95C22×Dic3—924no—
96C22×Dic3—924no—
97C22×Dic3—932no—
98C22×Dic3—924no—
99C22×Dic3—94no—
100C22×Dic3—948no—
101C22×Dic3—924no—
102C4×A4—94no—
103C22×C32—36no—
104C22×C32—31no—
105C22×C32—38no—
106C22×C32—36no—
107C22×C32—312no—
108C22×C32—36no—
109C22×C32—36no—
110C22×C32—33no—
111C22×C32—31no—
112C22×C32—412no—
113C22×C32—416no—
114C22×C32—812no—
115C22×C32—816no—
116C22×C32C4×A414yescomplemented
117C22×C32C22×C3×C411yescharacteristic, complemented
118C22×C32C22×A413yes—
119C3×A4—24no—
120C3×A4C22×Dic314yescomplemented
121C3⋊Dic3—34no—
122C3⋊Dic3C22×A414yesdirect product factor
123C3×Dic3—1216no—
124C25C3×C3⋊D311yescharacteristic
125C23×C4—96no—
126C23×C4—271no—
127C23×C4—276no—
128C23×C4—271no—
129C33—41noSylow
130C23×C3—312no—
131C23×C3—348no—
132C23×C3—324no—
133C23×C3—38no—
134C23×C3—348no—
135C23×C3—312no—
136C23×C3—312no—
137C23×C3—316no—
138C23×C3—316no—
139C23×C3—41no—
140C23×C3—84no—
141C23×C3C2×C3×Dic3124yescomplemented
142C23×C3D3×A414yes—
143C23×C3C22×C3×D314yes—
144C2×A4—224no—
145C2×A4—24no—
146C2×A4C2×C3⋊Dic316yesdirect product factor
147C2×A4C22×C3⋊D311yescharacteristic
148C2×C3×C4—366no—
149C2×Dic3—324no—
150C2×Dic3—916no—
151C2×Dic3—948no—
152C2×Dic3—948no—
153C2×Dic3—924no—
154C2×Dic3—916no—
155C2×C32—36no—
156C2×C32—31no—
157C2×C32—31no—
158C2×C32—424no—
159C2×C32—44no—
160C2×C32—824no—
161C2×C32—84no—
162C2×C32C2×C4×A416yescomplemented
163C2×C32C23×A411yescharacteristic
164C24—33no—
165C24—34no—
166C24—31no—
167C24C3×C3⋊Dic314yescomplemented
168C24C2×C3×C3⋊D313yes—
169C22×C4—91no—
170C22×C4—94no—
171C22×C4—276no—
172C22×C4—276no—
173C22×C4—278no—
174C22×C4—276no—
175C22×C4—271no—
176C22×C4—2712no—
177C22×C4—276no—
178C22×C3—324no—
179C22×C3—34no—
180C22×C3—324no—
181C22×C3—332no—
182C22×C3—324no—
183C22×C3—348no—
184C22×C3—324no—
185C22×C3—34no—
186C22×C3—312no—
187C22×C3—43no—
188C22×C3—44no—
189C22×C3—812no—
190C22×C3—816no—
191C22×C3Dic3×A4116yescomplemented
192C22×C3C22×C3×Dic314yescomplemented
193C22×C3C2×D3×A4112yes—
194A4—24no—
195A4C22×C3⋊Dic311yescharacteristic, direct product factor
196C3×C4—364no—
197Dic3—316no—
198Dic3—916no—
199C32—44no—
200C32—84no—
201C32C22×C4×A411yescharacteristic, complemented
202C23—33no—
203C23—312no—
204C23—36no—
205C23—32no—
206C23—312no—
207C23—33no—
208C23—33no—
209C23—34no—
210C23—34no—
211C23C2×C3×C3⋊Dic316yescomplemented
212C23A4×C3⋊D311yescharacteristic
213C23C22×C3×C3⋊D311yescharacteristic
214C2×C4—96no—
215C2×C4—274no—
216C2×C4—2712no—
217C2×C4—2712no—
218C2×C4—276no—
219C2×C4—274no—
220C2×C3—34no—
221C2×C3—324no—
222C2×C3—34no—
223C2×C3—41no—
224C2×C3—46no—
225C2×C3—84no—
226C2×C3—824no—
227C2×C3C2×Dic3×A4124yescomplemented
228C2×C3C22×D3×A414yes—
229C22—36no—
230C22—31no—
231C22—36no—
232C22—38no—
233C22—31no—
234C22—33no—
235C22—36no—
236C22—312no—
237C22—36no—
238C22A4×C3⋊Dic314yesdirect product factor
239C22C22×C3×C3⋊Dic311yescharacteristic, complemented
240C22C2×A4×C3⋊D313yes—
241C4—94no—
242C4—274no—
243C3—41no—
244C3—84no—
245C3C22×Dic3×A414yescomplemented
246C2—31no—
247C2—36no—
248C2—31no—
249C2C2×A4×C3⋊Dic316yesdirect product factor
250C2C22×A4×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC23A4×C3⋊D3
commutator, hypocenterC22×C32C22×C3×C4
FrattiniC2C22×A4×C3⋊D3
Fitting, socleC25×C32C2×C3
prime radicalC23×C32×A4C2
cosocleC23×C32C23×C3
LCSG ▷ C22×C32
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×A4×C3⋊Dic3C2direct (N × H)
3. C24×C3⋊Dic3C3semidirect
5. C22×C32×A4C4semidirect
8. A4×C3⋊Dic3C22direct (N × H)
15. C23×C3⋊Dic3C2×C3semidirect
18. C2×C32×A4C2×C4semidirect
26. C24×C32C3×C4semidirect
30. C22×C3×A4Dic3semidirect
39. C22×C3⋊Dic3A4direct (N × H)
40. C22×C3⋊Dic3C22×C3semidirect
45. C32×A4C22×C4semidirect
66. C23×C32C2×C3×C4semidirect
71. C2×C3×A4C2×Dic3semidirect
78. C2×C3⋊Dic3C2×A4direct (N × H)
86. C24×C3C3×Dic3semidirect
90. C22×A4C3⋊Dic3direct (N × H)
116. C22×C32C4×A4semidirect
117. C22×C32C22×C3×C4semidirect
120. C3×A4C22×Dic3semidirect
122. C3⋊Dic3C22×A4direct (N × H)
141. C23×C3C2×C3×Dic3semidirect
146. C2×A4C2×C3⋊Dic3direct (N × H)
162. C2×C32C2×C4×A4semidirect
167. C24C3×C3⋊Dic3semidirect
191. C22×C3Dic3×A4semidirect
192. C22×C3C22×C3×Dic3semidirect
195. A4C22×C3⋊Dic3direct (N × H)
201. C32C22×C4×A4semidirect
211. C23C2×C3×C3⋊Dic3semidirect
227. C2×C3C2×Dic3×A4semidirect
238. C22A4×C3⋊Dic3direct (N × H)
239. C22C22×C3×C3⋊Dic3semidirect
245. C3C22×Dic3×A4semidirect
249. C2C2×A4×C3⋊Dic3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×A42
C2×A4×C3⋊Dic32
C24×C3⋊Dic33
C22×Dic3×A43
C22×C3×C3⋊Dic34