Finite Group Databasebeta

← C2×C3×#63454 #13152400

Subgroups — C2×C3×#63454 #13152400

Order 1728.

1082 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D9C211yesmaximal, characteristic
2C2×C3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×(C2×D9)⋊C4C212yesmaximal
4C2×C3×C4⋊Dic9C211yesmaximal, characteristic, complemented
5C2×C3×C4×Dic9C212yesmaximal, complemented
6C3×#63454C218yesmaximal, direct product factor
7C2×C3×#892—31nomaximal
8C2×#63454C311yesmaximal, characteristic, direct product factor
9C23×C3×D9C2211yescharacteristic
10C22×C3×C4×C9C2212yes—
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3×Dic9C2212yes—
13C22×C3×Dic9C2211yescharacteristic
14C2×C3×C4×D9C418yescomplemented
15C2×C3×C4×D9C2214yes—
16C3×C9×C4⋊C4C2214yescomplemented
17C3×(C2×D9)⋊C4C2218yes—
18C3×C4⋊Dic9C2214yescomplemented
19C3×C4×Dic9C2218yescomplemented
20C22×C3×C4×D3—31no—
21C2×C32×C4⋊C4D311yescharacteristic
22C2×C3×(C2×D3)⋊C4—32no—
23C2×C3×C4⋊Dic3—31no—
24C2×C3×C4×Dic3—32no—
25C3×#892—38no—
26C22×C4×D9C2×C311yescharacteristic
27C2×C9×C4⋊C4—21no—
28C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
29C2×(C2×D9)⋊C4C2×C312yes—
30C2×C4⋊Dic9C2×C311yescharacteristic, complemented
31C2×C4×Dic9C2×C312yescomplemented
32#63454C2×C318yesdirect product factor
33C23×C3×C9C2311yescharacteristic
34C22×C3×D9—22no—
35C22×C3×D9—22no—
36C22×C3×D9C2×C414yes—
37C22×C3×D9C2312yes—
38C2×C3×C4×C9—24no—
39C2×C3×C4×C9C2×C414yes—
40C2×C3×C4×C9C2314yes—
41C2×C3×C4×C9C2312yes—
42C2×C3×Dic9—24no—
43C2×C3×Dic9C2×C414yes—
44C2×C3×Dic9C2314yes—
45C2×C3×Dic9C2312yes—
46C3×C4×D9C2×C4116yescomplemented
47C2×C3×#117—91no—
48C2×#892—31no—
49C23×C3×D3—31no—
50C22×C32×C4C2×D312yes—
51C22×C32×C4C2×D311yescharacteristic
52C22×C3×Dic3—31no—
53C22×C3×Dic3—32no—
54C2×C3×C4×D3—38no—
55C2×C3×C4×D3—34no—
56C23×D9C22×C311yescharacteristic
57C32×C4⋊C4C2×D314yes—
58C3×(C2×D3)⋊C4—38no—
59C3×C4⋊Dic3—34no—
60C3×C4×Dic3—38no—
61C22×C4×C9—21no—
62C22×C4×C9—22no—
63C22×C4×C9C22×C312yes—
64C22×C4×C9C22×C311yescharacteristic
65C22×Dic9C22×C312yes—
66C22×Dic9C22×C311yescharacteristic
67C2×C4×D9C3×C418yescomplemented
68C2×C4×D9C22×C314yes—
69C9×C4⋊C4—24no—
70C9×C4⋊C4C22×C314yescomplemented
71(C2×D9)⋊C4C22×C318yes—
72C4⋊Dic9C22×C314yescomplemented
73C4×Dic9C22×C318yescomplemented
74C22×C3×C9C22×C412yes—
75C22×C3×C9#4112yes—
76C22×C3×C9#4112yes—
77C22×C3×C9C2411yescharacteristic
78C2×C3×D9—28no—
79C2×C3×D9—22no—
80C2×C3×D9—22no—
81C2×C3×D9C22×C414yes—
82C3×C4×C9—24no—
83C3×C4×C9C22×C414yes—
84C3×Dic9—24no—
85C3×Dic9C22×C414yes—
86C23×C3×C4—91no—
87C22×C4×D3—31no—
88C3×#117—98no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—91no—
91C2×C3×C4⋊C4D911yescharacteristic, complemented
92C2×C3×C4⋊C4C3×D311yescharacteristic
93C2×C3×C22⋊C4—92no—
94C2×C3×C42—92no—
95C2×(C2×D3)⋊C4—32no—
96C2×C4⋊Dic3—31no—
97C2×C4×Dic3—32no—
98#892—38no—
99C23×C32C22×D311yescharacteristic
100C22×C3×D3—34no—
101C22×C3×D3—32no—
102C22×C3×D3—62no—
103C22×C3×D3—62no—
104C2×C32×C4—24no—
105C2×C32×C4C4×D314yes—
106C2×C32×C4C22×D314yes—
107C2×C32×C4C22×D312yes—
108C2×C3×Dic3—34no—
109C2×C3×Dic3—32no—
110C2×C3×Dic3—34no—
111C2×C3×Dic3—64no—
112C3×C4×D3—316no—
113C23×C9—21no—
114C23×C9C23×C311yescharacteristic
115C22×D9—22no—
116C22×D9—22no—
117C22×D9C2×C3×C414yes—
118C22×D9C23×C312yes—
119C2×C4×C9—24no—
120C2×C4×C9—24no—
121C2×C4×C9—22no—
122C2×C4×C9—24no—
123C2×C4×C9—44no—
124C2×C4×C9C2×C3×C414yes—
125C2×C4×C9C23×C314yes—
126C2×C4×C9C23×C312yes—
127C2×Dic9—24no—
128C2×Dic9C2×C3×C414yes—
129C2×Dic9C23×C314yes—
130C2×Dic9C23×C312yes—
131C4×D9C2×C3×C4116yescomplemented
132C2×#117—91noSylow
133C2×C3×C9#11714yescomplemented
134C2×C3×C9C23×C411yescharacteristic
135C2×C3×C9C2×#4111yescharacteristic
136C2×C3×C9C2×#4111yescharacteristic
137C3×D9—24no—
138C24×C3—91no—
139C23×D3—31no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—91no—
143C22×C3×C4—98no—
144C22×C3×C4—94no—
145C22×C3×C4—92no—
146C22×C3×C4C2×D912yes—
147C22×C3×C4C2×D911yescharacteristic
148C22×C3×C4C2×C3×D312yes—
149C22×C3×C4C2×C3×D311yescharacteristic
150C22×Dic3—31no—
151C22×Dic3—32no—
152C2×C4×D3—38no—
153C2×C4×D3—34no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—94no—
156C3×C4⋊C4C2×D914yescomplemented
157C3×C4⋊C4C2×C3×D314yes—
158C3×C22⋊C4—98no—
159C3×C42—98no—
160(C2×D3)⋊C4—38no—
161C4⋊Dic3—34no—
162C4×Dic3—38no—
163C22×C32C2×C4×D312yes—
164C22×C32#23712yes—
165C22×C32#23912yes—
166C22×C32C23×D311yescharacteristic
167C2×C3×D3—34no—
168C2×C3×D3—68no—
169C2×C3×D3—62no—
170C2×C3×D3—62no—
171C32×C4—24no—
172C32×C4C2×C4×D314yes—
173C3×Dic3—34no—
174C3×Dic3—64no—
175C22×C9—21no—
176C22×C9—22no—
177C22×C9—22no—
178C22×C9—22no—
179C22×C9C22×C3×C412yes—
180C22×C9C3×#4112yes—
181C22×C9C3×#4112yes—
182C22×C9C24×C311yescharacteristic
183C2×D9—28no—
184C2×D9—22no—
185C2×D9—22no—
186C2×D9C22×C3×C414yes—
187C4×C9—24no—
188C4×C9—24no—
189C4×C9—44no—
190C4×C9C22×C3×C414yes—
191Dic9—24no—
192Dic9C22×C3×C414yes—
193C23×C4—91no—
194#117—98no—
195C2×C4⋊C4—91no—
196C2×C4⋊C4C3×D911yescharacteristic, complemented
197C2×C22⋊C4—92no—
198C2×C42—92no—
199C3×C9C2×#11711yescharacteristic, complemented, Sylow
200C23×C3—21no—
201C23×C3—94no—
202C23×C3—92no—
203C23×C3—182no—
204C23×C3—182no—
205C23×C3C22×D911yescharacteristic
206C23×C3C22×C3×D311yescharacteristic
207C22×D3—32no—
208C22×D3—34no—
209C22×D3—62no—
210C22×D3—62no—
211C2×C3×C4—24no—
212C2×C3×C4—24no—
213C2×C3×C4—24no—
214C2×C3×C4—22no—
215C2×C3×C4—24no—
216C2×C3×C4—44no—
217C2×C3×C4—94no—
218C2×C3×C4—92no—
219C2×C3×C4—916no—
220C2×C3×C4—94no—
221C2×C3×C4—184no—
222C2×C3×C4C4×D914yes—
223C2×C3×C4C22×D914yes—
224C2×C3×C4C22×D912yes—
225C2×C3×C4C3×C4×D314yes—
226C2×C3×C4C22×C3×D314yes—
227C2×C3×C4C22×C3×D312yes—
228C2×Dic3—32no—
229C2×Dic3—34no—
230C2×Dic3—34no—
231C2×Dic3—64no—
232C4×D3—316no—
233C2×C32#89214yes—
234C2×C32C22×C4×D311yescharacteristic
235C2×C32C2×#23711yescharacteristic
236C2×C32C2×#23911yescharacteristic
237C3×D3—64no—
238C2×C9—24no—
239C2×C9—21no—
240C2×C9—21no—
241C2×C9—21no—
242C2×C9C3×#11714yescomplemented
243C2×C9C23×C3×C411yescharacteristic
244C2×C9C2×C3×#4111yescharacteristic
245C2×C9C2×C3×#4111yescharacteristic
246D9—24no—
247C24—91no—
248C22×C4—91no—
249C22×C4—98no—
250C22×C4—94no—
251C22×C4—92no—
252C22×C4C2×C3×D912yes—
253C22×C4C2×C3×D911yescharacteristic
254C4⋊C4—94no—
255C4⋊C4C2×C3×D914yescomplemented
256C22⋊C4—98no—
257C42—98no—
258C22×C3—21no—
259C22×C3—22no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—94no—
263C22×C3—182no—
264C22×C3—188no—
265C22×C3—182no—
266C22×C3C2×C4×D912yes—
267C22×C3#369912yes—
268C22×C3#370112yes—
269C22×C3C23×D911yescharacteristic
270C22×C3C2×C3×C4×D312yes—
271C22×C3C3×#23712yes—
272C22×C3C3×#23912yes—
273C22×C3C23×C3×D311yescharacteristic
274C2×D3—34no—
275C2×D3—68no—
276C2×D3—62no—
277C2×D3—62no—
278C3×C4—24no—
279C3×C4—24no—
280C3×C4—24no—
281C3×C4—44no—
282C3×C4—94no—
283C3×C4—184no—
284C3×C4C2×C4×D914yes—
285C3×C4C2×C3×C4×D314yes—
286Dic3—34no—
287Dic3—64no—
288C32C2×#89211yescharacteristic
289C9—21no—
290C9C2×C3×#11711yescharacteristic, complemented
291C23—92no—
292C23—94no—
293C23—182no—
294C23—182no—
295C23C22×C3×D911yescharacteristic
296C2×C4—24no—
297C2×C4—92no—
298C2×C4—94no—
299C2×C4—916no—
300C2×C4—94no—
301C2×C4—184no—
302C2×C4C3×C4×D914yes—
303C2×C4C22×C3×D914yes—
304C2×C4C22×C3×D912yes—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—24no—
309C2×C3—184no—
310C2×C3#6345414yesdirect product factor
311C2×C3C22×C4×D911yescharacteristic
312C2×C3C2×#369911yescharacteristic
313C2×C3C2×#370111yescharacteristic
314C2×C3C3×#89214yes—
315C2×C3C22×C3×C4×D311yescharacteristic
316C2×C3C2×C3×#23711yescharacteristic
317C2×C3C2×C3×#23911yescharacteristic
318D3—64no—
319C22—94no—
320C22—182no—
321C22—182no—
322C22—188no—
323C22C2×C3×C4×D912yes—
324C22C3×#369912yes—
325C22C3×#370112yes—
326C22C23×C3×D911yescharacteristic
327C4—24no—
328C4—94no—
329C4—184no—
330C4C2×C3×C4×D914yes—
331C3—21no—
332C3C2×#6345411yescharacteristic, direct product factor
333C3C2×C3×#89211yescharacteristic
334C2—184no—
335C2C3×#6345414yesdirect product factor
336C2C22×C3×C4×D911yescharacteristic
337C2C2×C3×#369911yescharacteristic
338C2C2×C3×#370111yescharacteristic
subgroupG/H
centerC23×C3C22×D9
commutatorC2×C9C23×C3×C4
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socleC23×C32C22×D3
prime radicalC23×C3×D9C22
cosocleC22×C9C24×C3
hypercenterC2×C3×C4⋊C4D9
hypocenterC9C2×C3×#117
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C9×C4⋊C4C2semidirect
4. C2×C3×C4⋊Dic9C2semidirect
5. C2×C3×C4×Dic9C2semidirect
6. C3×#63454C2direct (N × H)
8. C2×#63454C3direct (N × H)
14. C2×C3×C4×D9C4semidirect
16. C3×C9×C4⋊C4C22semidirect
18. C3×C4⋊Dic9C22semidirect
19. C3×C4×Dic9C22semidirect
28. C2×C9×C4⋊C4C2×C3semidirect
30. C2×C4⋊Dic9C2×C3semidirect
31. C2×C4×Dic9C2×C3semidirect
32. #63454C2×C3direct (N × H)
46. C3×C4×D9C2×C4semidirect
67. C2×C4×D9C3×C4semidirect
70. C9×C4⋊C4C22×C3semidirect
72. C4⋊Dic9C22×C3semidirect
73. C4×Dic9C22×C3semidirect
91. C2×C3×C4⋊C4D9semidirect
131. C4×D9C2×C3×C4semidirect
133. C2×C3×C9#117semidirect
156. C3×C4⋊C4C2×D9semidirect
196. C2×C4⋊C4C3×D9semidirect
199. C3×C9C2×#117semidirect
242. C2×C9C3×#117semidirect
255. C4⋊C4C2×C3×D9semidirect
290. C9C2×C3×#117semidirect
310. C2×C3#63454direct (N × H)
332. C3C2×#63454direct (N × H)
335. C2C3×#63454direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D92
C2×C3×C9×C4⋊C42
C2×C3×(C2×D9)⋊C42
C2×C3×C4⋊Dic92
C2×C3×C4×Dic92
C3×#634542
C2×C3×#8923
C2×#634543