Finite Group Databasebeta

← C23×C3×D3×A4 #13181859

Subgroups — C23×C3×D3×A4 #13181859

Order 1728.

5566 conjugacy classes of subgroups; only the 664 normal subgroups are shown.

normal subgroups only

This group has 5566 conjugacy classes of subgroups — too many to draw. Only its 664 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×C3×D3×A4C2114yesmaximal, direct product factor
3C25×C3×D3C311yesmaximal, characteristic, complemented
4C24×C3×A4—31nomaximal
5C23×D3×A4C313yesmaximal, direct product factor
6C23×C32×D3—41nomaximal
7C22×C32×A4C2217yescomplemented
8C2×C3×D3×A4C22128yesdirect product factor
9C25×C32C2×C311yescharacteristic, complemented
10C24×C3×D3—31no—
11C24×C3×D3—31no—
12C24×C3×D3—314no—
13C24×C3×D3C2×C3114yescomplemented
14C23×C3×A4—23no—
15C23×C3×A4—21no—
16C23×C3×A4—314no—
17C23×C3×A4D311yescharacteristic, direct product factor
18C23×C3×A4C2×C313yescomplemented
19C22×D3×A4C2×C3142yesdirect product factor
20C23×C33—41no—
21C22×C32×D3—414no—
22C2×C32×A4C2317yescomplemented
23C3×D3×A4C2318yesdirect product factor
24C26×C3—31no—
25C25×D3C3211yescharacteristic, complemented
26C24×A4—33no—
27C24×C32—31no—
28C24×C32—37no—
29C24×C32—121no—
30C24×C32C22×C317yescomplemented
31C23×C3×D3—314no—
32C23×C3×D3—356no—
33C23×C3×D3—31no—
34C23×C3×D3—314no—
35C23×C3×D3—314no—
36C23×C3×D3—314no—
37C23×C3×D3—384no—
38C23×C3×D3—43no—
39C23×C3×D3A411yescharacteristic, direct product factor
40C23×C3×D3C22×C3128yescomplemented
41C22×C3×A4—27no—
42C22×C3×A4—221no—
43C22×C3×A4—328no—
44C22×C3×A4C2×D317yesdirect product factor
45C22×C3×A4C22×C3121yescomplemented
46C2×D3×A4C22×C3184yesdirect product factor
47C22×C33—47no—
48C2×C32×D3—428no—
49C32×A4C2411yescharacteristic, complemented
50C25×C3—21no—
51C25×C3—314no—
52C25×C3—91no—
53C25×C3—914no—
54C25×C3—91no—
55C25×C3C3×D311yescharacteristic, complemented
56C25×C3C2×C3211yescharacteristic, complemented
57C24×D3—31no—
58C24×D3—31no—
59C24×D3—314no—
60C24×D3C2×C32114yescomplemented
61C23×A4—23no—
62C23×A4—342no—
63C23×A4C3×D313yesdirect product factor
64C23×C32—321no—
65C23×C32—314no—
66C23×C32—37no—
67C23×C32—37no—
68C23×C32—41no—
69C23×C32—43no—
70C23×C32—83no—
71C23×C32—81no—
72C23×C32—1214no—
73C23×C32C2×A411yescharacteristic, complemented
74C23×C32C23×C317yescomplemented
75C22×C3×D3—384no—
76C22×C3×D3—3112no—
77C22×C3×D3—314no—
78C22×C3×D3—384no—
79C22×C3×D3—328no—
80C22×C3×D3—328no—
81C22×C3×D3—356no—
82C22×C3×D3—442no—
83C22×C3×D3C2×A4114yesdirect product factor
84C22×C3×D3C23×C318yescomplemented
85C2×C3×A4—27no—
86C2×C3×A4—221no—
87C2×C3×A4—38no—
88C2×C3×A4C22×D317yesdirect product factor
89C2×C3×A4C23×C3121yescomplemented
90D3×A4C23×C3124yesdirect product factor
91C26—31noSylow
92C2×C33—47no—
93C32×D3—48no—
94C24×C3—27no—
95C24×C3—37no—
96C24×C3—37no—
97C24×C3—31no—
98C24×C3—31no—
99C24×C3—31no—
100C24×C3—328no—
101C24×C3—61no—
102C24×C3—67no—
103C24×C3—914no—
104C24×C3—914no—
105C24×C3—984no—
106C24×C3—914no—
107C24×C3—956no—
108C24×C3—914no—
109C24×C3—91no—
110C24×C3—123no—
111C24×C3C2×C3×D317yescomplemented
112C24×C3C22×C3217yescomplemented
113C23×D3—314no—
114C23×D3—356no—
115C23×D3—31no—
116C23×D3—314no—
117C23×D3—314no—
118C23×D3—314no—
119C23×D3—384no—
120C23×D3C3×A411yescharacteristic, direct product factor
121C23×D3C22×C32128yescomplemented
122C22×A4—221no—
123C22×A4—384no—
124C22×A4C2×C3×D3121yesdirect product factor
125C22×C32—37no—
126C22×C32—314no—
127C22×C32—321no—
128C22×C32—37no—
129C22×C32—47no—
130C22×C32—421no—
131C22×C32—821no—
132C22×C32—87no—
133C22×C32—1228no—
134C22×C32C22×A417yescomplemented
135C22×C32C24×C311yescharacteristic, complemented
136C2×C3×D3—356no—
137C2×C3×D3—328no—
138C2×C3×D3—356no—
139C2×C3×D3—38no—
140C2×C3×D3—38no—
141C2×C3×D3—484no—
142C2×C3×D3C22×A4128yesdirect product factor
143C3×A4—21no—
144C3×A4—23no—
145C3×A4C23×D311yescharacteristic, direct product factor
146C3×A4C24×C313yescomplemented
147C25—314no—
148C25—91no—
149C25—914no—
150C25—91no—
151C25C32×D311yescharacteristic, complemented
152C33—41noSylow
153C23×C3—21no—
154C23×C3—27no—
155C23×C3—321no—
156C23×C3—321no—
157C23×C3—314no—
158C23×C3—314no—
159C23×C3—314no—
160C23×C3—37no—
161C23×C3—37no—
162C23×C3—37no—
163C23×C3—37no—
164C23×C3—38no—
165C23×C3—43no—
166C23×C3—67no—
167C23×C3—67no—
168C23×C3—621no—
169C23×C3—614no—
170C23×C3—83no—
171C23×C3—928no—
172C23×C3—984no—
173C23×C3—956no—
174C23×C3—928no—
175C23×C3—9112no—
176C23×C3—984no—
177C23×C3—914no—
178C23×C3—1242no—
179C23×C3D3×A411yescharacteristic, direct product factor
180C23×C3C2×C3×A411yescharacteristic, complemented
181C23×C3C22×C3×D317yescomplemented
182C23×C3C23×C3217yescomplemented
183C22×D3—384no—
184C22×D3—3112no—
185C22×D3—314no—
186C22×D3—384no—
187C22×D3—328no—
188C22×D3—328no—
189C22×D3—356no—
190C22×D3C2×C3×A4114yesdirect product factor
191C22×D3C23×C3218yescomplemented
192C2×A4—221no—
193C2×A4—324no—
194C2×A4C22×C3×D3121yesdirect product factor
195C2×C32—37no—
196C2×C32—31no—
197C2×C32—47no—
198C2×C32—421no—
199C2×C32—821no—
200C2×C32—87no—
201C2×C32—128no—
202C2×C32C23×A417yescomplemented
203C3×D3—38no—
204C3×D3—424no—
205C3×D3C23×A418yesdirect product factor
206C24—37no—
207C24—31no—
208C24—31no—
209C24—328no—
210C24—914no—
211C24—914no—
212C24—984no—
213C24—914no—
214C24—956no—
215C24—914no—
216C24—91no—
217C24C2×C32×D317yescomplemented
218C22×C3—27no—
219C22×C3—21no—
220C22×C3—37no—
221C22×C3—37no—
222C22×C3—314no—
223C22×C3—314no—
224C22×C3—37no—
225C22×C3—321no—
226C22×C3—328no—
227C22×C3—37no—
228C22×C3—321no—
229C22×C3—421no—
230C22×C3—67no—
231C22×C3—621no—
232C22×C3—67no—
233C22×C3—614no—
234C22×C3—821no—
235C22×C3—98no—
236C22×C3—956no—
237C22×C3—98no—
238C22×C3—956no—
239C22×C3—928no—
240C22×C3—1284no—
241C22×C3C2×D3×A417yesdirect product factor
242C22×C3C22×C3×A417yescomplemented
243C22×C3C23×C3×D311yescharacteristic, complemented
244C22×C3C24×C3211yescharacteristic, complemented
245C2×D3—356no—
246C2×D3—328no—
247C2×D3—38no—
248C2×D3—356no—
249C2×D3—38no—
250C2×D3C22×C3×A4128yesdirect product factor
251A4—23no—
252A4C23×C3×D313yesdirect product factor
253C32—41no—
254C32—43no—
255C32—83no—
256C32—81no—
257C32C24×A411yescharacteristic, complemented
258C23—321no—
259C23—314no—
260C23—37no—
261C23—37no—
262C23—314no—
263C23—38no—
264C23—928no—
265C23—984no—
266C23—956no—
267C23—928no—
268C23—9112no—
269C23—984no—
270C23—914no—
271C23C3×D3×A411yescharacteristic, direct product factor
272C23C22×C32×D317yescomplemented
273C2×C3—27no—
274C2×C3—31no—
275C2×C3—37no—
276C2×C3—38no—
277C2×C3—31no—
278C2×C3—37no—
279C2×C3—421no—
280C2×C3—61no—
281C2×C3—67no—
282C2×C3—821no—
283C2×C3—98no—
284C2×C3—1224no—
285C2×C3C22×D3×A417yesdirect product factor
286C2×C3C23×C3×A417yescomplemented
287D3—38no—
288D3C23×C3×A418yesdirect product factor
289C22—37no—
290C22—314no—
291C22—37no—
292C22—321no—
293C22—328no—
294C22—98no—
295C22—956no—
296C22—98no—
297C22—956no—
298C22—928no—
299C22C2×C3×D3×A417yesdirect product factor
300C22C23×C32×D311yescharacteristic, complemented
301C3—21no—
302C3—43no—
303C3—83no—
304C3C23×D3×A411yescharacteristic, direct product factor
305C3C24×C3×A411yescharacteristic, complemented
306C2—31no—
307C2—37no—
308C2—38no—
309C2—98no—
310C2C22×C3×D3×A417yesdirect product factor
subgroupG/H
center, hypercenterC23×C3D3×A4
commutator, cosocle, hypocenterC22×C3C24×C32
FrattiniC1G
Fitting, socleC25×C32C2×C3
LCSG ▷ C22×C3
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C23×C32×A4C2semidirect
2. C22×C3×D3×A4C2direct (N × H)
3. C25×C3×D3C3semidirect
5. C23×D3×A4C3direct (N × H)
7. C22×C32×A4C22semidirect
8. C2×C3×D3×A4C22direct (N × H)
9. C25×C32C2×C3semidirect
13. C24×C3×D3C2×C3semidirect
17. C23×C3×A4D3direct (N × H)
18. C23×C3×A4C2×C3semidirect
19. C22×D3×A4C2×C3direct (N × H)
22. C2×C32×A4C23semidirect
23. C3×D3×A4C23direct (N × H)
25. C25×D3C32semidirect
30. C24×C32C22×C3semidirect
39. C23×C3×D3A4direct (N × H)
40. C23×C3×D3C22×C3semidirect
44. C22×C3×A4C2×D3direct (N × H)
45. C22×C3×A4C22×C3semidirect
46. C2×D3×A4C22×C3direct (N × H)
49. C32×A4C24semidirect
55. C25×C3C3×D3semidirect
56. C25×C3C2×C32semidirect
60. C24×D3C2×C32semidirect
63. C23×A4C3×D3direct (N × H)
73. C23×C32C2×A4semidirect
74. C23×C32C23×C3semidirect
83. C22×C3×D3C2×A4direct (N × H)
84. C22×C3×D3C23×C3semidirect
88. C2×C3×A4C22×D3direct (N × H)
89. C2×C3×A4C23×C3semidirect
90. D3×A4C23×C3direct (N × H)
111. C24×C3C2×C3×D3semidirect
112. C24×C3C22×C32semidirect
120. C23×D3C3×A4direct (N × H)
121. C23×D3C22×C32semidirect
124. C22×A4C2×C3×D3direct (N × H)
134. C22×C32C22×A4semidirect
135. C22×C32C24×C3semidirect
142. C2×C3×D3C22×A4direct (N × H)
145. C3×A4C23×D3direct (N × H)
146. C3×A4C24×C3semidirect
151. C25C32×D3semidirect
179. C23×C3D3×A4direct (N × H)
180. C23×C3C2×C3×A4semidirect
181. C23×C3C22×C3×D3semidirect
182. C23×C3C23×C32semidirect
190. C22×D3C2×C3×A4direct (N × H)
191. C22×D3C23×C32semidirect
194. C2×A4C22×C3×D3direct (N × H)
202. C2×C32C23×A4semidirect
205. C3×D3C23×A4direct (N × H)
217. C24C2×C32×D3semidirect
241. C22×C3C2×D3×A4direct (N × H)
242. C22×C3C22×C3×A4semidirect
243. C22×C3C23×C3×D3semidirect
244. C22×C3C24×C32semidirect
250. C2×D3C22×C3×A4direct (N × H)
252. A4C23×C3×D3direct (N × H)
257. C32C24×A4semidirect
271. C23C3×D3×A4direct (N × H)
272. C23C22×C32×D3semidirect
285. C2×C3C22×D3×A4direct (N × H)
286. C2×C3C23×C3×A4semidirect
288. D3C23×C3×A4direct (N × H)
299. C22C2×C3×D3×A4direct (N × H)
300. C22C23×C32×D3semidirect
304. C3C23×D3×A4direct (N × H)
305. C3C24×C3×A4semidirect
310. C2C22×C3×D3×A4direct (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×C3×D3×A42
C25×C3×D33
C24×C3×A43
C23×D3×A43
C23×C32×D34