Finite Group Databasebeta

← C2×C4×D3×C22⋊C9 #13164604

Subgroups — C2×C4×D3×C22⋊C9 #13164604

Order 1728.

1226 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×C22⋊C9C211yesmaximal, characteristic
2C2×C3×C4×C22⋊C9C211yesmaximal, characteristic, complemented
3C2×Dic3×C22⋊C9C211yesmaximal, characteristic, complemented
4C4×D3×C22⋊C9C214yesmaximal, direct product factor
5C23×C3×C4×D3C311yesmaximal, characteristic
6C22×C4×C22⋊C9—31nomaximal
7C22×C3×C22⋊C9C2211yescharacteristic
8C2×D3×C22⋊C9C414yesdirect product factor
9C2×D3×C22⋊C9C2212yes—
10C2×C4×D3×C9—41nomaximal
11C3×C4×C22⋊C9C2212yescomplemented
12Dic3×C22⋊C9C2212yescomplemented
13C24×C3×D3C2×C311yescharacteristic
14C23×C32×C4C2×C311yescharacteristic
15C23×C3×Dic3C2×C311yescharacteristic
16C22×C3×C4×D3—31no—
17C22×C3×C4×D3—31no—
18C22×C3×C4×D3—31no—
19C22×C3×C4×D3—31no—
20C22×C3×C4×D3—34no—
21C22×C3×C4×D3C2×C314yes—
22C23×C22⋊C9—31no—
23C2×C4×C22⋊C9—21no—
24C2×C4×C22⋊C9—31no—
25C2×C4×C22⋊C9—34no—
26C2×C4×C22⋊C9D311yescharacteristic, direct product factor
27C22×D3×C9—41no—
28C2×C3×C22⋊C9C2×C412yescomplemented
29C2×C3×C22⋊C9C2311yescharacteristic
30D3×C22⋊C9C2×C414yesdirect product factor
31C2×C3×C4×C9—41no—
32C2×C9×Dic3—41no—
33C4×D3×C9—44no—
34C24×C3×C4—31no—
35C23×C4×D3C911yescharacteristic, complemented
36C24×C32C22×C311yescharacteristic
37C23×C3×D3—31no—
38C23×C3×D3—31no—
39C23×C3×D3—32no—
40C23×C3×D3—34no—
41C23×C3×D3C3×C414yes—
42C23×C3×D3C22×C312yes—
43C22×C32×C4—31no—
44C22×C32×C4—31no—
45C22×C32×C4—32no—
46C22×C32×C4C22×C312yes—
47C22×C3×Dic3—31no—
48C22×C3×Dic3—31no—
49C22×C3×Dic3—32no—
50C22×C3×Dic3C22×C312yes—
51C2×C3×C4×D3—32no—
52C2×C3×C4×D3—34no—
53C2×C3×C4×D3—34no—
54C2×C3×C4×D3—34no—
55C2×C3×C4×D3—31no—
56C2×C3×C4×D3—31no—
57C2×C3×C4×D3—31no—
58C2×C3×C4×D3—34no—
59C2×C3×C4×D3—34no—
60C2×C3×C4×D3—34no—
61C2×C3×C4×D3—34no—
62C2×C3×C4×D3—34no—
63C2×C3×C4×D3A411yescharacteristic
64C22×C22⋊C9—21no—
65C22×C22⋊C9—34no—
66C22×C22⋊C9—32no—
67C22×C22⋊C9C2×D311yescharacteristic
68C22×C4×C9—121no—
69C4×C22⋊C9—22no—
70C4×C22⋊C9—32no—
71C4×C22⋊C9C2×D312yesdirect product factor
72C22×C3×C9—41no—
73C2×D3×C9—44no—
74C2×D3×C9—42no—
75C3×C22⋊C9C22×C411yescharacteristic, complemented
76C3×C4×C9—42no—
77C9×Dic3—42no—
78C25×C3—31no—
79C24×D3C2×C911yescharacteristic
80C23×C3×C4—21no—
81C23×C3×C4—31no—
82C23×C3×C4—34no—
83C23×C3×C4—91no—
84C23×C3×C4—91no—
85C23×C3×C4—94no—
86C23×C3×C4—91no—
87C23×C3×C4—91no—
88C23×C3×C4C2×C911yescharacteristic, complemented
89C23×C3×C4C3×D311yescharacteristic
90C23×Dic3C2×C911yescharacteristic, complemented
91C22×C4×D3—31no—
92C22×C4×D3—31no—
93C22×C4×D3—31no—
94C22×C4×D3—31no—
95C22×C4×D3—34no—
96C22×C4×D3C2×C914yescomplemented
97C23×C32—32no—
98C23×C32—31no—
99C23×C32—31no—
100C23×C32C2×C3×C412yes—
101C23×C32C23×C311yescharacteristic
102C22×C3×D3—34no—
103C22×C3×D3—38no—
104C22×C3×D3—32no—
105C22×C3×D3—31no—
106C22×C3×D3—34no—
107C22×C3×D3—34no—
108C22×C3×D3—32no—
109C22×C3×D3—32no—
110C22×C3×D3—34no—
111C22×C3×D3—38no—
112C22×C3×D3—34no—
113C22×C3×D3—32no—
114C22×C3×D3C2×C3×C414yes—
115C22×C3×D3C2×A411yescharacteristic
116C2×C32×C4—32no—
117C2×C32×C4—31no—
118C2×C32×C4—32no—
119C2×C32×C4—32no—
120C2×C32×C4—32no—
121C2×C32×C4C2×A411yescharacteristic
122C2×C3×Dic3—32no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—32no—
125C2×C3×Dic3—32no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3C2×A411yescharacteristic
128C3×C4×D3—38no—
129C3×C4×D3—34no—
130C3×C4×D3—34no—
131C3×C4×D3—34no—
132C3×C4×D3C2×A414yes—
133C23×C9—121no—
134C2×C22⋊C9—22no—
135C2×C22⋊C9—21no—
136C2×C22⋊C9—34no—
137C2×C22⋊C9C4×D312yesdirect product factor
138C2×C22⋊C9C22×D311yescharacteristic
139C2×C4×C9—41no—
140C2×C4×C9—81no—
141C2×C4×C9—124no—
142C2×C4×C9—121no—
143C24×C4—31noSylow
144C2×C3×C9—42no—
145C2×C3×C9—41no—
146D3×C9—44no—
147C24×C3—21no—
148C24×C3—34no—
149C24×C3—32no—
150C24×C3—91no—
151C24×C3—94no—
152C24×C3—92no—
153C24×C3—91no—
154C24×C3C22×C911yescharacteristic
155C24×C3C2×C3×D311yescharacteristic
156C23×D3—31no—
157C23×D3—31no—
158C23×D3—32no—
159C23×D3—34no—
160C23×D3C4×C914yescomplemented
161C23×D3C22×C912yes—
162C22×C3×C4—22no—
163C22×C3×C4—31no—
164C22×C3×C4—31no—
165C22×C3×C4—31no—
166C22×C3×C4—31no—
167C22×C3×C4—32no—
168C22×C3×C4—31no—
169C22×C3×C4—32no—
170C22×C3×C4—32no—
171C22×C3×C4—61no—
172C22×C3×C4—62no—
173C22×C3×C4—61no—
174C22×C3×C4—94no—
175C22×C3×C4—91no—
176C22×C3×C4—94no—
177C22×C3×C4—91no—
178C22×C3×C4—91no—
179C22×C3×C4—94no—
180C22×C3×C4—94no—
181C22×C3×C4—92no—
182C22×C3×C4—94no—
183C22×C3×C4—94no—
184C22×C3×C4—94no—
185C22×C3×C4—94no—
186C22×C3×C4—91no—
187C22×C3×C4—92no—
188C22×C3×C4—91no—
189C22×C3×C4C22×C912yescomplemented
190C22×C3×C4C2×C3×D312yes—
191C22×Dic3—31no—
192C22×Dic3—31no—
193C22×Dic3—32no—
194C22×Dic3C22×C912yescomplemented
195C2×C4×D3—32no—
196C2×C4×D3—34no—
197C2×C4×D3—34no—
198C2×C4×D3—34no—
199C2×C4×D3—31no—
200C2×C4×D3—31no—
201C2×C4×D3—31no—
202C2×C4×D3—34no—
203C2×C4×D3—34no—
204C2×C4×D3—34no—
205C2×C4×D3—34no—
206C2×C4×D3—34no—
207C2×C4×D3C22⋊C911yescharacteristic, direct product factor
208C22×C32—32no—
209C22×C32—31no—
210C22×C32—32no—
211C22×C32—32no—
212C22×C32—32no—
213C22×C32—31no—
214C22×C32—31no—
215C22×C32C22×C3×C411yescharacteristic
216C22×C32C22×A411yescharacteristic
217C2×C3×D3—34no—
218C2×C3×D3—38no—
219C2×C3×D3—34no—
220C2×C3×D3—32no—
221C2×C3×D3—34no—
222C2×C3×D3—38no—
223C2×C3×D3—34no—
224C2×C3×D3—34no—
225C2×C3×D3C4×A414yes—
226C2×C3×D3C22×A412yes—
227C32×C4—32no—
228C32×C4C22×A412yes—
229C3×Dic3—32no—
230C3×Dic3C22×A412yes—
231C22×C9—41no—
232C22×C9—81no—
233C22×C9—122no—
234C22×C9—124no—
235C22⋊C9—21no—
236C22⋊C9C2×C4×D311yescharacteristic, direct product factor
237C4×C9—42no—
238C4×C9—82no—
239C4×C9—122no—
240C25—31no—
241C23×C4—31no—
242C23×C4—34no—
243C23×C4—91no—
244C23×C4—91no—
245C23×C4—94no—
246C23×C4—91no—
247C23×C4—91no—
248C23×C4D3×C911yescharacteristic, complemented
249C3×C9—41noSylow
250C23×C3—22no—
251C23×C3—21no—
252C23×C3—31no—
253C23×C3—31no—
254C23×C3—32no—
255C23×C3—32no—
256C23×C3—31no—
257C23×C3—31no—
258C23×C3—31no—
259C23×C3—34no—
260C23×C3—61no—
261C23×C3—61no—
262C23×C3—62no—
263C23×C3—92no—
264C23×C3—92no—
265C23×C3—94no—
266C23×C3—94no—
267C23×C3—92no—
268C23×C3—98no—
269C23×C3—94no—
270C23×C3—94no—
271C23×C3—92no—
272C23×C3—98no—
273C23×C3—94no—
274C23×C3—91no—
275C23×C3C2×C4×C912yescomplemented
276C23×C3C23×C911yescharacteristic
277C23×C3C3×C4×D312yes—
278C23×C3C22×C3×D311yescharacteristic
279C22×D3—32no—
280C22×D3—34no—
281C22×D3—38no—
282C22×D3—31no—
283C22×D3—34no—
284C22×D3—34no—
285C22×D3—32no—
286C22×D3—32no—
287C22×D3—32no—
288C22×D3—34no—
289C22×D3—38no—
290C22×D3—34no—
291C22×D3C2×C4×C914yescomplemented
292C22×D3C2×C22⋊C911yescharacteristic
293C2×C3×C4—21no—
294C2×C3×C4—32no—
295C2×C3×C4—32no—
296C2×C3×C4—31no—
297C2×C3×C4—31no—
298C2×C3×C4—31no—
299C2×C3×C4—34no—
300C2×C3×C4—32no—
301C2×C3×C4—32no—
302C2×C3×C4—32no—
303C2×C3×C4—32no—
304C2×C3×C4—32no—
305C2×C3×C4—32no—
306C2×C3×C4—62no—
307C2×C3×C4—62no—
308C2×C3×C4—61no—
309C2×C3×C4—62no—
310C2×C3×C4—62no—
311C2×C3×C4—94no—
312C2×C3×C4—94no—
313C2×C3×C4—92no—
314C2×C3×C4—92no—
315C2×C3×C4—98no—
316C2×C3×C4—92no—
317C2×C3×C4—91no—
318C2×C3×C4—94no—
319C2×C3×C4—92no—
320C2×C3×C4C2×C22⋊C911yescharacteristic, complemented
321C2×C3×C4D3×A411yescharacteristic
322C2×Dic3—32no—
323C2×Dic3—31no—
324C2×Dic3—32no—
325C2×Dic3—32no—
326C2×Dic3—32no—
327C2×Dic3C2×C22⋊C911yescharacteristic, complemented
328C4×D3—38no—
329C4×D3—34no—
330C4×D3—34no—
331C4×D3—34no—
332C4×D3C2×C22⋊C914yesdirect product factor
333C2×C32—32no—
334C2×C32—31no—
335C2×C32—31no—
336C2×C32C2×C4×A412yes—
337C2×C32C23×A411yescharacteristic
338C3×D3—34no—
339C3×D3C2×C4×A414yes—
340C2×C9—42no—
341C2×C9—41no—
342C2×C9—82no—
343C2×C9—81no—
344C2×C9—124no—
345C24—34no—
346C24—32no—
347C24—91no—
348C24—92no—
349C24—91no—
350C24—94no—
351C24C2×D3×C911yescharacteristic
352C22×C4—31no—
353C22×C4—31no—
354C22×C4—31no—
355C22×C4—32no—
356C22×C4—32no—
357C22×C4—94no—
358C22×C4—91no—
359C22×C4—94no—
360C22×C4—91no—
361C22×C4—91no—
362C22×C4—94no—
363C22×C4—94no—
364C22×C4—92no—
365C22×C4—94no—
366C22×C4—94no—
367C22×C4—94no—
368C22×C4—94no—
369C22×C4—91no—
370C22×C4—92no—
371C22×C4—91no—
372C22×C4C2×D3×C912yescomplemented
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—32no—
376C22×C3—31no—
377C22×C3—32no—
378C22×C3—31no—
379C22×C3—32no—
380C22×C3—32no—
381C22×C3—32no—
382C22×C3—32no—
383C22×C3—34no—
384C22×C3—31no—
385C22×C3—31no—
386C22×C3—32no—
387C22×C3—31no—
388C22×C3—31no—
389C22×C3—32no—
390C22×C3—32no—
391C22×C3—61no—
392C22×C3—61no—
393C22×C3—62no—
394C22×C3—62no—
395C22×C3—62no—
396C22×C3—61no—
397C22×C3—62no—
398C22×C3—94no—
399C22×C3—94no—
400C22×C3—98no—
401C22×C3—94no—
402C22×C3—94no—
403C22×C3—98no—
404C22×C3—94no—
405C22×C3—92no—
406C22×C3C22×C4×C911yescharacteristic, complemented
407C22×C3C22×C22⋊C911yescharacteristic
408C22×C3C2×C3×C4×D311yescharacteristic
409C22×C3C2×D3×A411yescharacteristic
410C2×D3—38no—
411C2×D3—34no—
412C2×D3—32no—
413C2×D3—34no—
414C2×D3—34no—
415C2×D3—34no—
416C2×D3—38no—
417C2×D3—34no—
418C2×D3C4×C22⋊C914yesdirect product factor
419C2×D3C22×C22⋊C912yes—
420C3×C4—22no—
421C3×C4—32no—
422C3×C4—32no—
423C3×C4—32no—
424C3×C4—62no—
425C3×C4—92no—
426C3×C4C22×C22⋊C912yescomplemented
427C3×C4C2×D3×A412yes—
428Dic3—32no—
429Dic3C22×C22⋊C912yescomplemented
430C32C22×C4×A411yescharacteristic
431C9—41no—
432C9—81no—
433C23—31no—
434C23—32no—
435C23—31no—
436C23—31no—
437C23—34no—
438C23—92no—
439C23—92no—
440C23—94no—
441C23—94no—
442C23—92no—
443C23—98no—
444C23—94no—
445C23—94no—
446C23—92no—
447C23—98no—
448C23—94no—
449C23—91no—
450C23C4×D3×C912yescomplemented
451C23C22×D3×C911yescharacteristic
452C2×C4—32no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—32no—
456C2×C4—32no—
457C2×C4—34no—
458C2×C4—32no—
459C2×C4—94no—
460C2×C4—92no—
461C2×C4—94no—
462C2×C4—92no—
463C2×C4—98no—
464C2×C4—92no—
465C2×C4—91no—
466C2×C4—94no—
467C2×C4—92no—
468C2×C4D3×C22⋊C911yescharacteristic, direct product factor
469C2×C3—21no—
470C2×C3—22no—
471C2×C3—31no—
472C2×C3—32no—
473C2×C3—31no—
474C2×C3—34no—
475C2×C3—31no—
476C2×C3—32no—
477C2×C3—31no—
478C2×C3—61no—
479C2×C3—61no—
480C2×C3—62no—
481C2×C3—94no—
482C2×C3C2×C4×C22⋊C912yescomplemented
483C2×C3C23×C22⋊C911yescharacteristic
484C2×C3C4×D3×A412yes—
485C2×C3C22×D3×A411yescharacteristic
486D3—34no—
487D3C2×C4×C22⋊C914yesdirect product factor
488C22—32no—
489C22—31no—
490C22—32no—
491C22—31no—
492C22—31no—
493C22—32no—
494C22—32no—
495C22—32no—
496C22—34no—
497C22—94no—
498C22—94no—
499C22—98no—
500C22—94no—
501C22—94no—
502C22—98no—
503C22—92no—
504C22—94no—
505C22C2×C4×D3×C911yescharacteristic, complemented
506C22C2×D3×C22⋊C911yescharacteristic
507C4—32no—
508C4—32no—
509C4—92no—
510C4C2×D3×C22⋊C912yesdirect product factor
511C3—21no—
512C3C22×C4×C22⋊C911yescharacteristic, complemented
513C3C2×C4×D3×A411yescharacteristic
514C2—31no—
515C2—32no—
516C2—31no—
517C2—34no—
518C2—94no—
519C2C4×D3×C22⋊C912yesdirect product factor
520C2C22×D3×C22⋊C911yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4D3×A4
commutator, hypocenterC22×C3C22×C4×C9
FrattiniC2×C3C22×D3×A4
FittingC23×C32×C4C2×C3
socleC24×C32C22×C3
prime radicalC24×C3×D3C2×C3
cosocleC23×C32C23×C3
LCSG ▷ C22×C3
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×C22⋊C9C2semidirect
3. C2×Dic3×C22⋊C9C2semidirect
4. C4×D3×C22⋊C9C2direct (N × H)
8. C2×D3×C22⋊C9C4direct (N × H)
11. C3×C4×C22⋊C9C22semidirect
12. Dic3×C22⋊C9C22semidirect
26. C2×C4×C22⋊C9D3direct (N × H)
28. C2×C3×C22⋊C9C2×C4semidirect
30. D3×C22⋊C9C2×C4direct (N × H)
35. C23×C4×D3C9semidirect
71. C4×C22⋊C9C2×D3direct (N × H)
75. C3×C22⋊C9C22×C4semidirect
88. C23×C3×C4C2×C9semidirect
90. C23×Dic3C2×C9semidirect
96. C22×C4×D3C2×C9semidirect
137. C2×C22⋊C9C4×D3direct (N × H)
160. C23×D3C4×C9semidirect
189. C22×C3×C4C22×C9semidirect
194. C22×Dic3C22×C9semidirect
207. C2×C4×D3C22⋊C9direct (N × H)
236. C22⋊C9C2×C4×D3direct (N × H)
248. C23×C4D3×C9semidirect
275. C23×C3C2×C4×C9semidirect
291. C22×D3C2×C4×C9semidirect
320. C2×C3×C4C2×C22⋊C9semidirect
327. C2×Dic3C2×C22⋊C9semidirect
332. C4×D3C2×C22⋊C9direct (N × H)
372. C22×C4C2×D3×C9semidirect
406. C22×C3C22×C4×C9semidirect
418. C2×D3C4×C22⋊C9direct (N × H)
426. C3×C4C22×C22⋊C9semidirect
429. Dic3C22×C22⋊C9semidirect
450. C23C4×D3×C9semidirect
468. C2×C4D3×C22⋊C9direct (N × H)
482. C2×C3C2×C4×C22⋊C9semidirect
487. D3C2×C4×C22⋊C9direct (N × H)
505. C22C2×C4×D3×C9semidirect
510. C4C2×D3×C22⋊C9direct (N × H)
512. C3C22×C4×C22⋊C9semidirect
519. C2C4×D3×C22⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×C22⋊C92
C2×C3×C4×C22⋊C92
C2×Dic3×C22⋊C92
C4×D3×C22⋊C92
C23×C3×C4×D33
C22×C4×C22⋊C93
C2×C4×D3×C94