Finite Group Databasebeta

← C2×C3×#63460 #13152409

Subgroups — C2×C3×#63460 #13152409

Order 1728.

830 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic
3C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic
4C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic
5C2×C3×C4⋊Dic9C211yesmaximal, characteristic, complemented
6C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
7C2×C3×C4×Dic9C211yesmaximal, characteristic, complemented
8C3×#63460C218yesmaximal, direct product factor
9C2×C3×#898—31nomaximal
10C2×#63460C311yesmaximal, characteristic, direct product factor
11C23×C3×D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3×Dic9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C3×C9×C4⋊C4C2214yescomplemented
19C3×(C2×D9)⋊C4C2214yes—
20C3×(C2×D9)⋊C4C2214yes—
21C3×(C2×D9)⋊C4C2214yes—
22C3×C4⋊Dic9C2214yescomplemented
23C3×Dic9⋊C4C2214yescomplemented
24C3×C4×Dic9C2214yescomplemented
25C2×C32×C4⋊C4D311yescharacteristic
26C2×C3×(C2×D3)⋊C4—31no—
27C2×C3×(C2×D3)⋊C4—31no—
28C2×C3×(C2×D3)⋊C4—31no—
29C2×C3×C4⋊Dic3—31no—
30C2×C3×Dic3⋊C4—31no—
31C2×C3×C4×Dic3—31no—
32C3×#898—38no—
33C2×C9×C4⋊C4—21no—
34C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
35C2×(C2×D9)⋊C4C2×C311yescharacteristic
36C2×(C2×D9)⋊C4C2×C311yescharacteristic
37C2×(C2×D9)⋊C4C2×C311yescharacteristic
38C2×C4⋊Dic9C2×C311yescharacteristic, complemented
39C2×Dic9⋊C4C2×C311yescharacteristic, complemented
40C2×C4×Dic9C2×C311yescharacteristic, complemented
41#63460C2×C318yesdirect product factor
42C23×C3×C9C2311yescharacteristic
43C22×C3×D9—22no—
44C22×C3×D9—22no—
45C22×C3×D9—22no—
46C22×C3×D9C2312yes—
47C2×C3×C4×C9—22no—
48C2×C3×C4×C9—22no—
49C2×C3×C4×C9—22no—
50C2×C3×C4×C9C2312yes—
51C2×C3×C4×C9C2312yes—
52C2×C3×C4×C9C2312yes—
53C2×C3×Dic9—22no—
54C2×C3×Dic9—22no—
55C2×C3×Dic9—22no—
56C2×C3×Dic9C2312yes—
57C2×C3×Dic9C2312yes—
58C2×C3×Dic9C2312yes—
59C2×C3×#126—91no—
60C2×#898—31no—
61C23×C3×D3—31no—
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C2×D311yescharacteristic
65C22×C3×Dic3—31no—
66C22×C3×Dic3—31no—
67C22×C3×Dic3—31no—
68C23×D9C22×C311yescharacteristic
69C32×C4⋊C4C2×D314yes—
70C3×(C2×D3)⋊C4—34no—
71C3×(C2×D3)⋊C4—34no—
72C3×(C2×D3)⋊C4—34no—
73C3×C4⋊Dic3—34no—
74C3×Dic3⋊C4—34no—
75C3×C4×Dic3—34no—
76C22×C4×C9—21no—
77C22×C4×C9—21no—
78C22×C4×C9—21no—
79C22×C4×C9C22×C311yescharacteristic
80C22×C4×C9C22×C311yescharacteristic
81C22×C4×C9C22×C311yescharacteristic
82C22×Dic9C22×C311yescharacteristic
83C22×Dic9C22×C311yescharacteristic
84C22×Dic9C22×C311yescharacteristic
85C9×C4⋊C4—24no—
86C9×C4⋊C4C22×C314yescomplemented
87(C2×D9)⋊C4C22×C314yes—
88(C2×D9)⋊C4C22×C314yes—
89(C2×D9)⋊C4C22×C314yes—
90C4⋊Dic9C22×C314yescomplemented
91Dic9⋊C4C22×C314yescomplemented
92C4×Dic9C22×C314yescomplemented
93C22×C3×C9#4112yes—
94C22×C3×C9#4112yes—
95C22×C3×C9#4112yes—
96C22×C3×C9C2411yescharacteristic
97C2×C3×D9—22no—
98C2×C3×D9—22no—
99C2×C3×D9—22no—
100C2×C3×D9—44no—
101C3×C4×C9—22no—
102C3×C4×C9—22no—
103C3×C4×C9—22no—
104C3×Dic9—22no—
105C3×Dic9—22no—
106C3×Dic9—22no—
107C3×#126—98no—
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4—91no—
110C2×C3×C4⋊C4—91no—
111C2×C3×C4⋊C4D911yescharacteristic, complemented
112C2×C3×C4⋊C4C3×D311yescharacteristic
113C2×C3×C22⋊C4—91no—
114C2×C3×C22⋊C4—91no—
115C2×C3×C22⋊C4—91no—
116C2×C3×C42—91no—
117C2×(C2×D3)⋊C4—31no—
118C2×(C2×D3)⋊C4—31no—
119C2×(C2×D3)⋊C4—31no—
120C2×C4⋊Dic3—31no—
121C2×Dic3⋊C4—31no—
122C2×C4×Dic3—31no—
123#898—38no—
124C23×C32C22×D311yescharacteristic
125C22×C3×D3—32no—
126C22×C3×D3—62no—
127C22×C3×D3—62no—
128C22×C3×D3—62no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4C22×D312yes—
133C2×C32×C4C22×D312yes—
134C2×C32×C4C22×D312yes—
135C2×C3×Dic3—32no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—32no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—62no—
141C23×C9—21no—
142C23×C9C23×C311yescharacteristic
143C22×D9—22no—
144C22×D9—22no—
145C22×D9—22no—
146C22×D9C23×C312yes—
147C2×C4×C9—24no—
148C2×C4×C9—22no—
149C2×C4×C9—22no—
150C2×C4×C9—22no—
151C2×C4×C9—22no—
152C2×C4×C9—22no—
153C2×C4×C9—22no—
154C2×C4×C9—42no—
155C2×C4×C9—42no—
156C2×C4×C9C23×C312yes—
157C2×C4×C9C23×C312yes—
158C2×C4×C9C23×C312yes—
159C2×Dic9—22no—
160C2×Dic9—22no—
161C2×Dic9—22no—
162C2×Dic9C23×C312yes—
163C2×Dic9C23×C312yes—
164C2×Dic9C23×C312yes—
165C2×#126—91noSylow
166C2×C3×C9#12614yescomplemented
167C2×C3×C9C2×#4111yescharacteristic
168C2×C3×C9C2×#4111yescharacteristic
169C2×C3×C9C2×#4111yescharacteristic
170C3×D9—42no—
171C24×C3—91no—
172C23×D3—31no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—91no—
177C22×C3×C4—91no—
178C22×C3×C4—91no—
179C22×C3×C4C2×D911yescharacteristic
180C22×C3×C4C2×D911yescharacteristic
181C22×C3×C4C2×D911yescharacteristic
182C22×C3×C4C2×C3×D311yescharacteristic
183C22×C3×C4C2×C3×D311yescharacteristic
184C22×C3×C4C2×C3×D311yescharacteristic
185C22×Dic3—31no—
186C22×Dic3—31no—
187C22×Dic3—31no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—94no—
190C3×C4⋊C4—94no—
191C3×C4⋊C4C2×D914yescomplemented
192C3×C4⋊C4C2×C3×D314yes—
193C3×C22⋊C4—94no—
194C3×C22⋊C4—94no—
195C3×C22⋊C4—94no—
196C3×C42—94no—
197(C2×D3)⋊C4—34no—
198(C2×D3)⋊C4—34no—
199(C2×D3)⋊C4—34no—
200C4⋊Dic3—34no—
201Dic3⋊C4—34no—
202C4×Dic3—34no—
203C22×C32#23512yes—
204C22×C32#23712yes—
205C22×C32#23912yes—
206C22×C32C23×D311yescharacteristic
207C2×C3×D3—62no—
208C2×C3×D3—62no—
209C2×C3×D3—62no—
210C2×C3×D3—124no—
211C32×C4—22no—
212C32×C4—22no—
213C32×C4—22no—
214C3×Dic3—62no—
215C3×Dic3—62no—
216C3×Dic3—62no—
217C22×C9—21no—
218C22×C9—22no—
219C22×C9—22no—
220C22×C9—22no—
221C22×C9C3×#4112yes—
222C22×C9C3×#4112yes—
223C22×C9C3×#4112yes—
224C22×C9C24×C311yescharacteristic
225C2×D9—22no—
226C2×D9—22no—
227C2×D9—22no—
228C2×D9—44no—
229C4×C9—24no—
230C4×C9—22no—
231C4×C9—22no—
232C4×C9—22no—
233C4×C9—42no—
234C4×C9—42no—
235Dic9—22no—
236Dic9—22no—
237Dic9—22no—
238#126—98no—
239C2×C4⋊C4—91no—
240C2×C4⋊C4—91no—
241C2×C4⋊C4C3×D911yescharacteristic, complemented
242C2×C22⋊C4—91no—
243C2×C22⋊C4—91no—
244C2×C22⋊C4—91no—
245C2×C42—91no—
246C3×C9C2×#12611yescharacteristic, complemented, Sylow
247C23×C3—21no—
248C23×C3—92no—
249C23×C3—182no—
250C23×C3—182no—
251C23×C3—182no—
252C23×C3C22×D911yescharacteristic
253C23×C3C22×C3×D311yescharacteristic
254C22×D3—32no—
255C22×D3—62no—
256C22×D3—62no—
257C22×D3—62no—
258C2×C3×C4—24no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—92no—
271C2×C3×C4—92no—
272C2×C3×C4—92no—
273C2×C3×C4—182no—
274C2×C3×C4—182no—
275C2×C3×C4—182no—
276C2×C3×C4C22×D912yes—
277C2×C3×C4C22×D912yes—
278C2×C3×C4C22×D912yes—
279C2×C3×C4C22×C3×D312yes—
280C2×C3×C4C22×C3×D312yes—
281C2×C3×C4C22×C3×D312yes—
282C2×Dic3—32no—
283C2×Dic3—32no—
284C2×Dic3—32no—
285C2×Dic3—62no—
286C2×Dic3—62no—
287C2×Dic3—62no—
288C2×C32#89814yes—
289C2×C32C2×#23511yescharacteristic
290C2×C32C2×#23711yescharacteristic
291C2×C32C2×#23911yescharacteristic
292C3×D3—122no—
293C2×C9—24no—
294C2×C9—21no—
295C2×C9—21no—
296C2×C9—21no—
297C2×C9C3×#12614yescomplemented
298C2×C9C2×C3×#4111yescharacteristic
299C2×C9C2×C3×#4111yescharacteristic
300C2×C9C2×C3×#4111yescharacteristic
301D9—42no—
302C24—91no—
303C22×C4—91no—
304C22×C4—91no—
305C22×C4—91no—
306C22×C4C2×C3×D911yescharacteristic
307C22×C4C2×C3×D911yescharacteristic
308C22×C4C2×C3×D911yescharacteristic
309C4⋊C4—94no—
310C4⋊C4—94no—
311C4⋊C4C2×C3×D914yescomplemented
312C22⋊C4—94no—
313C22⋊C4—94no—
314C22⋊C4—94no—
315C42—94no—
316C22×C3—21no—
317C22×C3—22no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3—182no—
321C22×C3—182no—
322C22×C3—182no—
323C22×C3—364no—
324C22×C3#369712yes—
325C22×C3#369912yes—
326C22×C3#370112yes—
327C22×C3C23×D911yescharacteristic
328C22×C3C3×#23512yes—
329C22×C3C3×#23712yes—
330C22×C3C3×#23912yes—
331C22×C3C23×C3×D311yescharacteristic
332C2×D3—62no—
333C2×D3—62no—
334C2×D3—62no—
335C2×D3—124no—
336C3×C4—24no—
337C3×C4—22no—
338C3×C4—22no—
339C3×C4—22no—
340C3×C4—22no—
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—42no—
344C3×C4—42no—
345C3×C4—182no—
346C3×C4—182no—
347C3×C4—182no—
348Dic3—62no—
349Dic3—62no—
350Dic3—62no—
351C32C2×#89811yescharacteristic
352C9—21no—
353C9C2×C3×#12611yescharacteristic, complemented
354C23—92no—
355C23—182no—
356C23—182no—
357C23—182no—
358C23C22×C3×D911yescharacteristic
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—92no—
363C2×C4—92no—
364C2×C4—92no—
365C2×C4—182no—
366C2×C4—182no—
367C2×C4—182no—
368C2×C4C22×C3×D912yes—
369C2×C4C22×C3×D912yes—
370C2×C4C22×C3×D912yes—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—24no—
375C2×C3—362no—
376C2×C3#6346014yesdirect product factor
377C2×C3C2×#369711yescharacteristic
378C2×C3C2×#369911yescharacteristic
379C2×C3C2×#370111yescharacteristic
380C2×C3C3×#89814yes—
381C2×C3C2×C3×#23511yescharacteristic
382C2×C3C2×C3×#23711yescharacteristic
383C2×C3C2×C3×#23911yescharacteristic
384D3—122no—
385C22—182no—
386C22—182no—
387C22—182no—
388C22—364no—
389C22C3×#369712yes—
390C22C3×#369912yes—
391C22C3×#370112yes—
392C22C23×C3×D911yescharacteristic
393C4—22no—
394C4—22no—
395C4—22no—
396C4—182no—
397C4—182no—
398C4—182no—
399C3—21no—
400C3C2×#6346011yescharacteristic, direct product factor
401C3C2×C3×#89811yescharacteristic
402C2—362no—
403C2C3×#6346014yesdirect product factor
404C2C2×C3×#369711yescharacteristic
405C2C2×C3×#369911yescharacteristic
406C2C2×C3×#370111yescharacteristic
subgroupG/H
centerC23×C3C22×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socleC23×C32C22×D3
prime radicalC23×C3×D9C22
hypercenterC2×C3×C4⋊C4D9
hypocenterC9C2×C3×#126
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×C4⋊C4C2semidirect
5. C2×C3×C4⋊Dic9C2semidirect
6. C2×C3×Dic9⋊C4C2semidirect
7. C2×C3×C4×Dic9C2semidirect
8. C3×#63460C2direct (N × H)
10. C2×#63460C3direct (N × H)
18. C3×C9×C4⋊C4C22semidirect
22. C3×C4⋊Dic9C22semidirect
23. C3×Dic9⋊C4C22semidirect
24. C3×C4×Dic9C22semidirect
34. C2×C9×C4⋊C4C2×C3semidirect
38. C2×C4⋊Dic9C2×C3semidirect
39. C2×Dic9⋊C4C2×C3semidirect
40. C2×C4×Dic9C2×C3semidirect
41. #63460C2×C3direct (N × H)
86. C9×C4⋊C4C22×C3semidirect
90. C4⋊Dic9C22×C3semidirect
91. Dic9⋊C4C22×C3semidirect
92. C4×Dic9C22×C3semidirect
111. C2×C3×C4⋊C4D9semidirect
166. C2×C3×C9#126semidirect
191. C3×C4⋊C4C2×D9semidirect
241. C2×C4⋊C4C3×D9semidirect
246. C3×C9C2×#126semidirect
297. C2×C9C3×#126semidirect
311. C4⋊C4C2×C3×D9semidirect
353. C9C2×C3×#126semidirect
376. C2×C3#63460direct (N × H)
400. C3C2×#63460direct (N × H)
403. C2C3×#63460direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C9×C4⋊C42
C2×C3×(C2×D9)⋊C42
C2×C3×C4⋊Dic92
C2×C3×Dic9⋊C42
C2×C3×C4×Dic92
C3×#634602
C2×C3×#8983
C2×#634603