Finite Group Databasebeta

← C23×A4×C3⋊D3 #13181860

Subgroups — C23×A4×C3⋊D3 #13181860

Order 1728.

7246 conjugacy classes of subgroups; only the 441 normal subgroups are shown.

normal subgroups only

This group has 7246 conjugacy classes of subgroups — too many to draw. Only its 441 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, complemented
2C22×A4×C3⋊D3C2114yesmaximal, direct product factor
3C25×C3⋊D3C311yesmaximal, characteristic, complemented
4C23×D3×A4—34nomaximal
5C23×C3×C3⋊D3—41nomaximal
6C22×C32×A4C2217yescomplemented
7C2×A4×C3⋊D3C22128yesdirect product factor
8C25×C32C2×C311yescharacteristic, complemented
9C24×C3⋊D3—31no—
10C24×C3⋊D3—31no—
11C24×C3⋊D3—314no—
12C24×C3⋊D3C2×C3114yescomplemented
13C23×C3×A4—24no—
14C23×C3×A4D314yescomplemented
15C22×D3×A4—356no—
16C23×C33—41no—
17C22×C3×C3⋊D3—414no—
18C2×C32×A4C2317yescomplemented
19A4×C3⋊D3C2318yesdirect product factor
20C25×D3—34no—
21C24×A4—91no—
22C24×C32—31no—
23C24×C32—37no—
24C24×C32C22×C317yescomplemented
25C23×C3⋊D3—314no—
26C23×C3⋊D3—356no—
27C23×C3⋊D3—31no—
28C23×C3⋊D3—314no—
29C23×C3⋊D3—314no—
30C23×C3⋊D3—314no—
31C23×C3⋊D3—384no—
32C23×C3⋊D3A411yescharacteristic, direct product factor
33C23×C3⋊D3C22×C3128yescomplemented
34C23×C3×D3—124no—
35C22×C3×A4—228no—
36C22×C3×A4C2×D3128yescomplemented
37C2×D3×A4—3112no—
38C22×C33—47no—
39C2×C3×C3⋊D3—428no—
40C32×A4C2411yescharacteristic, complemented
41C25×C3C3×D314yescomplemented
42C24×D3—356no—
43C24×D3—94no—
44C24×D3—956no—
45C24×D3—94no—
46C23×A4—24no—
47C23×A4—914no—
48C23×A4C3⋊D311yescharacteristic, direct product factor
49C23×C32—321no—
50C23×C32—314no—
51C23×C32—37no—
52C23×C32—37no—
53C23×C32—44no—
54C23×C32—84no—
55C23×C32C2×A411yescharacteristic, complemented
56C23×C32C23×C317yescomplemented
57C22×C3⋊D3—384no—
58C22×C3⋊D3—3112no—
59C22×C3⋊D3—314no—
60C22×C3⋊D3—384no—
61C22×C3⋊D3—328no—
62C22×C3⋊D3—328no—
63C22×C3⋊D3—356no—
64C22×C3⋊D3C2×A4114yesdirect product factor
65C22×C3⋊D3C23×C318yescomplemented
66C22×C3×D3—1256no—
67C2×C3×A4—228no—
68C2×C3×A4C22×D3128yescomplemented
69D3×A4—332no—
70C26—91noSylow
71C2×C33—47no—
72C3×C3⋊D3—48no—
73C24×C3—328no—
74C24×C3—34no—
75C24×C3—361no—
76C24×C3C2×C3×D3128yescomplemented
77C23×D3—34no—
78C23×D3—3112no—
79C23×D3—956no—
80C23×D3—956no—
81C23×D3—9224no—
82C23×D3—956no—
83C23×D3—94no—
84C23×D3—9336no—
85C23×D3—956no—
86C22×A4—228no—
87C22×A4—928no—
88C22×A4C2×C3⋊D317yesdirect product factor
89C22×C32—37no—
90C22×C32—314no—
91C22×C32—321no—
92C22×C32—37no—
93C22×C32—428no—
94C22×C32—828no—
95C22×C32C22×A417yescomplemented
96C22×C32C24×C311yescharacteristic, complemented
97C2×C3⋊D3—356no—
98C2×C3⋊D3—328no—
99C2×C3⋊D3—356no—
100C2×C3⋊D3—38no—
101C2×C3⋊D3—38no—
102C2×C3⋊D3C22×A4128yesdirect product factor
103C2×C3×D3—12112no—
104C3×A4—24no—
105C3×A4C23×D314yescomplemented
106C25—914no—
107C25—271no—
108C25—2714no—
109C25—271no—
110C25C3×C3⋊D311yescharacteristic, complemented
111C33—41noSylow
112C23×C3—384no—
113C23×C3—356no—
114C23×C3—328no—
115C23×C3—328no—
116C23×C3—41no—
117C23×C3—84no—
118C23×C3—3614no—
119C23×C3D3×A414yescomplemented
120C23×C3C22×C3×D3128yescomplemented
121C22×D3—356no—
122C22×D3—332no—
123C22×D3—9112no—
124C22×D3—9336no—
125C22×D3—9448no—
126C22×D3—9336no—
127C22×D3—956no—
128C22×D3—9224no—
129C22×D3—9112no—
130C2×A4—228no—
131C2×A4—98no—
132C2×A4C22×C3⋊D317yesdirect product factor
133C2×C32—37no—
134C2×C32—31no—
135C2×C32—428no—
136C2×C32—828no—
137C2×C32C23×A417yescomplemented
138C3⋊D3—38no—
139C3⋊D3C23×A418yesdirect product factor
140C3×D3—1232no—
141C24—37no—
142C24—31no—
143C24—91no—
144C24—928no—
145C24—2714no—
146C24—2714no—
147C24—2784no—
148C24—2714no—
149C24—2756no—
150C24—2714no—
151C24—271no—
152C24C2×C3×C3⋊D317yescomplemented
153C22×C3—328no—
154C22×C3—356no—
155C22×C3—328no—
156C22×C3—384no—
157C22×C3—47no—
158C22×C3—828no—
159C22×C3—3628no—
160C22×C3C2×D3×A4128yescomplemented
161C22×C3C23×C3×D314yescomplemented
162C2×D3—3112no—
163C2×D3—932no—
164C2×D3—9224no—
165C2×D3—9224no—
166C2×D3—9112no—
167C2×D3—932no—
168A4—24no—
169A4C23×C3⋊D311yescharacteristic, direct product factor
170C32—44no—
171C32—84no—
172C32C24×A411yescharacteristic, complemented
173C23—321no—
174C23—314no—
175C23—37no—
176C23—37no—
177C23—914no—
178C23—98no—
179C23—2728no—
180C23—2784no—
181C23—2756no—
182C23—2728no—
183C23—27112no—
184C23—2784no—
185C23—2714no—
186C23A4×C3⋊D311yescharacteristic, direct product factor
187C23C22×C3×C3⋊D317yescomplemented
188C2×C3—34no—
189C2×C3—328no—
190C2×C3—47no—
191C2×C3—828no—
192C2×C3—368no—
193C2×C3C22×D3×A4128yescomplemented
194D3—332no—
195D3—932no—
196C22—37no—
197C22—314no—
198C22—37no—
199C22—321no—
200C22—928no—
201C22—278no—
202C22—2756no—
203C22—278no—
204C22—2756no—
205C22—2728no—
206C22C2×A4×C3⋊D317yesdirect product factor
207C22C23×C3×C3⋊D311yescharacteristic, complemented
208C3—41no—
209C3—84no—
210C3C23×D3×A414yescomplemented
211C2—31no—
212C2—37no—
213C2—98no—
214C2—278no—
215C2C22×A4×C3⋊D317yesdirect product factor
subgroupG/H
center, hypercenterC23A4×C3⋊D3
commutator, cosocle, hypocenterC22×C32C24×C3
FrattiniC1G
Fitting, socleC25×C32C2×C3
LCSG ▷ C22×C32
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C32×A4C2semidirect
2. C22×A4×C3⋊D3C2direct (N × H)
3. C25×C3⋊D3C3semidirect
6. C22×C32×A4C22semidirect
7. C2×A4×C3⋊D3C22direct (N × H)
8. C25×C32C2×C3semidirect
12. C24×C3⋊D3C2×C3semidirect
14. C23×C3×A4D3semidirect
18. C2×C32×A4C23semidirect
19. A4×C3⋊D3C23direct (N × H)
24. C24×C32C22×C3semidirect
32. C23×C3⋊D3A4direct (N × H)
33. C23×C3⋊D3C22×C3semidirect
36. C22×C3×A4C2×D3semidirect
40. C32×A4C24semidirect
41. C25×C3C3×D3semidirect
48. C23×A4C3⋊D3direct (N × H)
55. C23×C32C2×A4semidirect
56. C23×C32C23×C3semidirect
64. C22×C3⋊D3C2×A4direct (N × H)
65. C22×C3⋊D3C23×C3semidirect
68. C2×C3×A4C22×D3semidirect
76. C24×C3C2×C3×D3semidirect
88. C22×A4C2×C3⋊D3direct (N × H)
95. C22×C32C22×A4semidirect
96. C22×C32C24×C3semidirect
102. C2×C3⋊D3C22×A4direct (N × H)
105. C3×A4C23×D3semidirect
110. C25C3×C3⋊D3semidirect
119. C23×C3D3×A4semidirect
120. C23×C3C22×C3×D3semidirect
132. C2×A4C22×C3⋊D3direct (N × H)
137. C2×C32C23×A4semidirect
139. C3⋊D3C23×A4direct (N × H)
152. C24C2×C3×C3⋊D3semidirect
160. C22×C3C2×D3×A4semidirect
161. C22×C3C23×C3×D3semidirect
169. A4C23×C3⋊D3direct (N × H)
172. C32C24×A4semidirect
186. C23A4×C3⋊D3direct (N × H)
187. C23C22×C3×C3⋊D3semidirect
193. C2×C3C22×D3×A4semidirect
206. C22C2×A4×C3⋊D3direct (N × H)
207. C22C23×C3×C3⋊D3semidirect
210. C3C23×D3×A4semidirect
215. C2C22×A4×C3⋊D3direct (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
C22×A4×C3⋊D32
C25×C3⋊D33
C23×D3×A43
C23×C3×C3⋊D34