Finite Group Databasebeta

← D3×#63491 #13150533

Subgroups — D3×#63491 #13150533

Order 1728.

1104 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D3×C9C211yesmaximal, characteristic, complemented
2C2×D3×#611C211yesmaximal, characteristic
3#11721818C211yesmaximal, characteristic, complemented
4C3×#63491C211yesmaximal, characteristic, complemented
5D3×#3676C211yesmaximal, characteristic, complemented
6#11721259C211yesmaximal, characteristic, complemented
7#11721251C211yesmaximal, characteristic, complemented
8D3×(C2×D9)⋊C4C212yesmaximal, complemented
9#11721032C212yesmaximal, complemented
10D3×Dic9⋊C4C212yesmaximal, complemented
11#11720954C212yesmaximal, complemented
12D3×#929—31nomaximal
13C2×#63491—31nomaximal
14C23×D3×C9C2211yescharacteristic
15C22×D3×D9C2211yescharacteristic
16C22×C3×C4×C9C2211yescharacteristic, complemented
17C2×#6233C2211yescharacteristic
18C2×C3×#611C2211yescharacteristic
19C22×C9×Dic3C2211yescharacteristic, complemented
20C2×C4×D3×C9—24no—
21C2×C4×D3×C9C2212yes—
22C2×C4×D3×C9C2212yescomplemented
23D3×#611—24no—
24C2×#6195C2211yescharacteristic
25C2×#6194C2211yescharacteristic
26C2×D3×Dic9C2211yescharacteristic
27C2×D3×Dic9C2212yes—
28#88551C2211yescharacteristic, complemented
29#88536C2212yescomplemented
30#88534C2212yescomplemented
31C3×#3676C2211yescharacteristic, complemented
32C3×(C2×D9)⋊C4C2212yescomplemented
33C3×Dic9⋊C4C2212yescomplemented
34(C2×D3)⋊Dic9C2212yescomplemented
35#88445C2212yescomplemented
36(C2×D9)⋊Dic3C2212yescomplemented
37#88442C2212yescomplemented
38Dic9⋊Dic3C2212yescomplemented
39C22×C3×C4×D3D311yescharacteristic
40C2×D3×#67—31no—
41#64138—31no—
42C3×#929—31no—
43D3×#217—31no—
44#63966—31no—
45#63961—31no—
46D3×(C2×D3)⋊C4—32no—
47#63905—32no—
48D3×Dic3⋊C4—32no—
49#63850—32no—
50C23×C4×C9—31no—
51C22×#611—31no—
52C2×#3676—31no—
53#63491—31no—
54#63491—31no—
55#63491—31no—
56#63491—32no—
57#63491—32no—
58#63491—61no—
59#63491D311yescharacteristic, direct product factor
60C2×(C2×D9)⋊C4—32no—
61C2×Dic9⋊C4—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9—24no—
65C22×D3×C9D414yes—
66C22×D3×C9C2311yescharacteristic
67C22×D3×C9C2311yescharacteristic
68C22×C3×D9C2311yescharacteristic
69C2×D3×D9—22no—
70C2×D3×D9—24no—
71C2×C3×C4×C9—22no—
72C2×C3×C4×C9C2312yes—
73#6233—22no—
74C2×C3⋊Dic9C2311yescharacteristic
75C2×C3⋊Dic9C2312yes—
76C3×#611—22no—
77C2×C9×Dic3—22no—
78C2×C9×Dic3C2312yes—
79C2×C3×Dic9C2311yescharacteristic
80C2×C3×Dic9C2312yes—
81C4×D3×C9—24no—
82C4×D3×C9—24no—
83#6195—22no—
84#6194—22no—
85D3×Dic9—22no—
86D3×Dic9—24no—
87C2×#929—91no—
88D3×#123—91no—
89C23×C3×D3C2×D311yescharacteristic
90C22×D32—31no—
91C22×C32×C4C2×D311yescharacteristic
92C2×#638—31no—
93C2×C3×#67—31no—
94C22×C3×Dic3C2×D311yescharacteristic
95C2×C3×C4×D3—24no—
96C2×C3×C4×D3C2×D312yes—
97C2×C3×C4×D3C2×D312yes—
98D3×#67—64no—
99C2×#626—31no—
100C2×#625—31no—
101C2×D3×Dic3—31no—
102C2×D3×Dic3—32no—
103C24×C9—31no—
104C23×D9—31no—
105#3757—31no—
106#3752—32no—
107#3750—32no—
108C3×#217—31no—
109C3×(C2×D3)⋊C4—32no—
110C3×Dic3⋊C4—32no—
111#3724—32no—
112Dic3⋊Dic3—32no—
113#3722—32no—
114(C2×D3)⋊Dic3—32no—
115(C2×D3)⋊Dic3—32no—
116C22×C4×C9—21no—
117C22×C4×C9—31no—
118C22×C4×C9—32no—
119C22×C4×C9—32no—
120C22×C4×C9—64no—
121C22×C4×C9C2×D311yescharacteristic, complemented
122C2×#611—31no—
123C2×#611—31no—
124C2×#611—31no—
125C2×#611—64no—
126C2×#611—61no—
127C2×#611C2×D311yescharacteristic
128C22×Dic9—32no—
129C22×Dic9—31no—
130#3676—31no—
131#3676—32no—
132#3676—61no—
133#3676C2×D311yescharacteristic, complemented
134(C2×D9)⋊C4—32no—
135(C2×D9)⋊C4—32no—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—62no—
138(C2×D9)⋊C4C2×D312yescomplemented
139Dic9⋊C4—32no—
140Dic9⋊C4—32no—
141Dic9⋊C4—32no—
142Dic9⋊C4—62no—
143Dic9⋊C4C2×D312yescomplemented
144C22×C3×C9—22no—
145C22×C3×C9C2×D412yes—
146C22×C3×C9C2411yescharacteristic
147C2×C3⋊D9—21no—
148C2×C3⋊D9—22no—
149C2×D3×C9—22no—
150C2×D3×C9—28no—
151C2×D3×C9C2×D412yes—
152C2×D3×C9C2×D412yes—
153C2×D3×C9#4114yes—
154C2×C3×D9—21no—
155C2×C3×D9—22no—
156D3×D9—44no—
157C3×C4×C9—22no—
158C3⋊Dic9—21no—
159C3⋊Dic9—22no—
160C9×Dic3—22no—
161C3×Dic9—21no—
162C3×Dic9—22no—
163C23×C3×C4—31no—
164C22×#67—91no—
165C2×D3×D4—91no—
166C22×C4×D3D911yescharacteristic, complemented
167C3×#123—91no—
168C2×#217—91no—
169#935—91no—
170#929—31no—
171#929—92no—
172#929—92no—
173#929—91no—
174#929—91no—
175#929—91no—
176#929—91no—
177#929—181no—
178C2×(C2×D3)⋊C4—92no—
179C2×Dic3⋊C4—92no—
180#894—92no—
181D3×C4⋊C4—92no—
182#886—91no—
183#883—92no—
184D3×C22⋊C4—91no—
185D3×C22⋊C4—92no—
186C23×C32C22×D311yescharacteristic
187C22×C3⋊D3—31no—
188C22×C3×D3—24no—
189C22×C3×D3—31no—
190C22×C3×D3#6714yes—
191C22×C3×D3C22×D311yescharacteristic
192C22×C3×D3C22×D311yescharacteristic
193C2×D32—64no—
194C2×D32—62no—
195C2×C32×C4—22no—
196C2×C32×C4C22×D312yes—
197#638—62no—
198C2×C3⋊Dic3—32no—
199C2×C3⋊Dic3—31no—
200C3×#67—62no—
201C2×C3×Dic3—22no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3C22×D312yes—
205C3×C4×D3—24no—
206C3×C4×D3—24no—
207#626—62no—
208#625—62no—
209D3×Dic3—64no—
210D3×Dic3—62no—
211C23×C9—21no—
212C23×C9—31no—
213C23×C9—34no—
214C23×C9—31no—
215C23×C9—64no—
216C23×C9C22×D311yescharacteristic
217C22×D9—31no—
218C22×D9—61no—
219C22×D9—62no—
220C22×D9—64no—
221C22×D9C22×D311yescharacteristic
222C2×C4×C9—22no—
223C2×C4×C9—22no—
224C2×C4×C9—32no—
225C2×C4×C9—42no—
226C2×C4×C9—64no—
227C2×C4×C9—64no—
228C2×C4×C9—62no—
229C2×C4×C9C22×D312yes—
230#611—22no—
231#611—62no—
232#611—62no—
233#611—62no—
234#611—122no—
235C2×Dic9—31no—
236C2×Dic9—32no—
237C2×Dic9—61no—
238C2×Dic9—62no—
239C2×Dic9—64no—
240C2×Dic9—62no—
241C2×Dic9C22×D311yescharacteristic
242C2×Dic9C22×D312yes—
243C2×#123—271noSylow
244C2×C3×C9—22no—
245C2×C3×C9C22×D411yescharacteristic
246C2×C3×C9C2×#4112yes—
247C3⋊D9—41no—
248D3×C9—22no—
249D3×C9#12314yescomplemented
250C3×D9—41no—
251C24×C3—31no—
252C23×D3—91no—
253C23×D3—91no—
254C23×D3C2×D911yescharacteristic
255C2×C3×D4—91no—
256C22×C3×C4—21no—
257C22×C3×C4—31no—
258C22×C3×C4—32no—
259C22×C3×C4—32no—
260C22×C3×C4—64no—
261C22×C3×C4C2×D911yescharacteristic, complemented
262C22×C3×C4D3211yescharacteristic
263C2×#67—31no—
264C2×#67—91no—
265C2×#67—91no—
266C2×#67—91no—
267C2×#67—91no—
268C2×#67—91no—
269C2×#67—181no—
270C2×#67—184no—
271C22×Dic3—91no—
272C22×Dic3—92no—
273C22×Dic3C2×D911yescharacteristic, complemented
274D3×D4—184no—
275C2×D12—91no—
276C2×C4×D3—24no—
277C2×C4×D3—92no—
278C2×C4×D3—91no—
279C2×C4×D3C2×D912yes—
280C2×C4×D3C2×D912yescomplemented
281C3×C4⋊C4—92no—
282C3×C22⋊C4—91no—
283C3×C22⋊C4—92no—
284#217—31no—
285#217—92no—
286#217—91no—
287#217—91no—
288#217—92no—
289#217—181no—
290(C2×D3)⋊C4—32no—
291(C2×D3)⋊C4—92no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—182no—
298C4⋊Dic3—92no—
299Dic3⋊C4—32no—
300Dic3⋊C4—92no—
301Dic3⋊C4—92no—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304Dic3⋊C4—92no—
305Dic3⋊C4—182no—
306C22×C32—22no—
307C22×C32C2×#6712yes—
308C22×C32C23×D311yescharacteristic
309C2×C3⋊D3—62no—
310C2×C3⋊D3—61no—
311C2×C3×D3—22no—
312C2×C3×D3—28no—
313C2×C3×D3—62no—
314C2×C3×D3—61no—
315C2×C3×D3#23514yes—
316C2×C3×D3C2×#6712yes—
317C2×C3×D3C2×#6712yes—
318D32—124no—
319C32×C4—22no—
320C3⋊Dic3—62no—
321C3⋊Dic3—61no—
322C3×Dic3—22no—
323C3×Dic3—62no—
324C3×Dic3—61no—
325C22×C9—22no—
326C22×C9—21no—
327C22×C9—22no—
328C22×C9—32no—
329C22×C9—34no—
330C22×C9—32no—
331C22×C9—42no—
332C22×C9—68no—
333C22×C9—62no—
334C22×C9D3×D412yes—
335C22×C9C23×D311yescharacteristic
336C2×D9—21no—
337C2×D9—22no—
338C2×D9—61no—
339C2×D9—62no—
340C2×D9—121no—
341C2×D9—122no—
342C2×D9—124no—
343C4×C9—22no—
344C4×C9—42no—
345C4×C9—62no—
346Dic9—21no—
347Dic9—22no—
348Dic9—61no—
349Dic9—62no—
350Dic9—122no—
351Dic9—121no—
352C22×D4—271no—
353C23×C4—31no—
354#123—91no—
355#123—272no—
356#123—271no—
357#123—272no—
358#123—271no—
359#123—271no—
360C2×C4⋊C4—272no—
361C2×C22⋊C4—272no—
362C2×C22⋊C4—271no—
363C3×C9C2×#12311yescharacteristic, complemented, Sylow
364C23×C3—21no—
365C23×C3—31no—
366C23×C3—31no—
367C23×C3—34no—
368C23×C3—64no—
369C23×C3—91no—
370C23×C3C22×D911yescharacteristic
371C23×C3C2×D3211yescharacteristic
372C22×D3—24no—
373C22×D3—31no—
374C22×D3—91no—
375C22×D3—91no—
376C22×D3—181no—
377C22×D3—182no—
378C22×D3—184no—
379C22×D3—184no—
380C22×D3—182no—
381C22×D3#61114yes—
382C22×D3C22×D911yescharacteristic
383C22×D3C22×D911yescharacteristic
384C3×D4—182no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—32no—
389C2×C3×C4—42no—
390C2×C3×C4—64no—
391C2×C3×C4—64no—
392C2×C3×C4—62no—
393C2×C3×C4—91no—
394C2×C3×C4—92no—
395C2×C3×C4C22×D912yes—
396C2×C3×C4C2×D3212yes—
397#67—62no—
398#67—182no—
399#67—182no—
400#67—182no—
401#67—182no—
402#67—182no—
403#67—362no—
404C2×Dic3—22no—
405C2×Dic3—31no—
406C2×Dic3—32no—
407C2×Dic3—91no—
408C2×Dic3—92no—
409C2×Dic3—91no—
410C2×Dic3—92no—
411C2×Dic3—181no—
412C2×Dic3—182no—
413C2×Dic3—184no—
414C2×Dic3—182no—
415C2×Dic3C22×D912yes—
416D12—182no—
417C4×D3—24no—
418C4×D3—24no—
419C4×D3—184no—
420C4×D3—182no—
421C2×C32—22no—
422C2×C32C2×#23512yes—
423C2×C32C22×#6711yescharacteristic
424C3⋊D3—121no—
425C3×D3—22no—
426C3×D3—121no—
427C3×D3#92914yes—
428C2×C9—22no—
429C2×C9—22no—
430C2×C9—21no—
431C2×C9—34no—
432C2×C9—42no—
433C2×C9—62no—
434C2×C9C2×D3×D411yescharacteristic
435C2×C9D3×#4112yes—
436D9—41no—
437D9—121no—
438D9—241no—
439C24—31no—
440C24—271no—
441C2×D4—91no—
442C2×D4—271no—
443C2×D4—271no—
444C2×D4—271no—
445C2×D4—544no—
446C22×C4—31no—
447C22×C4—32no—
448C22×C4—32no—
449C22×C4—64no—
450C22×C4—272no—
451C22×C4—271no—
452C22×C4D3×D911yescharacteristic, complemented
453C4⋊C4—92no—
454C4⋊C4—272no—
455C4⋊C4—272no—
456C4⋊C4—272no—
457C22⋊C4—91no—
458C22⋊C4—92no—
459C22⋊C4—272no—
460C22⋊C4—272no—
461C22⋊C4—272no—
462C22⋊C4—271no—
463C22⋊C4—272no—
464C22×C3—22no—
465C22×C3—22no—
466C22×C3—21no—
467C22×C3—22no—
468C22×C3—34no—
469C22×C3—32no—
470C22×C3—32no—
471C22×C3—42no—
472C22×C3—68no—
473C22×C3—62no—
474C22×C3—182no—
475C22×C3—181no—
476C22×C3C2×#61112yes—
477C22×C3C23×D911yescharacteristic
478C22×C3D3×#6712yes—
479C22×C3C22×D3211yescharacteristic
480C2×D3—22no—
481C2×D3—28no—
482C2×D3—62no—
483C2×D3—61no—
484C2×D3—182no—
485C2×D3—181no—
486C2×D3—182no—
487C2×D3—181no—
488C2×D3—361no—
489C2×D3—362no—
490C2×D3—364no—
491C2×D3—364no—
492C2×D3#369714yes—
493C2×D3C2×#61112yes—
494C2×D3C2×#61112yes—
495C3×C4—22no—
496C3×C4—22no—
497C3×C4—42no—
498C3×C4—62no—
499C3×C4—182no—
500C3×C4—181no—
501Dic3—22no—
502Dic3—62no—
503Dic3—61no—
504Dic3—182no—
505Dic3—181no—
506Dic3—182no—
507Dic3—181no—
508Dic3—361no—
509Dic3—362no—
510C32C2×#92911yescharacteristic
511C9—21no—
512C9D3×#12311yescharacteristic, complemented
513C23—31no—
514C23—31no—
515C23—34no—
516C23—64no—
517C23—91no—
518C23—271no—
519C23—542no—
520C23—544no—
521C23C2×D3×D911yescharacteristic
522D4—182no—
523D4—542no—
524D4—542no—
525D4—542no—
526C2×C4—22no—
527C2×C4—32no—
528C2×C4—64no—
529C2×C4—64no—
530C2×C4—62no—
531C2×C4—92no—
532C2×C4—91no—
533C2×C4—272no—
534C2×C4—271no—
535C2×C4—542no—
536C2×C4—544no—
537C2×C4C2×D3×D912yes—
538C2×C3—22no—
539C2×C3—22no—
540C2×C3—21no—
541C2×C3—22no—
542C2×C3—34no—
543C2×C3—42no—
544C2×C3—62no—
545C2×C3—361no—
546C2×C3C2×#369712yes—
547C2×C3C22×#61111yescharacteristic
548C2×C3D3×#23512yes—
549C2×C3C2×D3×#6711yescharacteristic
550D3—22no—
551D3—121no—
552D3—361no—
553D3—361no—
554D3—721no—
555D3#6349114yesdirect product factor
556C22—22no—
557C22—32no—
558C22—34no—
559C22—32no—
560C22—68no—
561C22—62no—
562C22—181no—
563C22—182no—
564C22—541no—
565C22—542no—
566C22—1084no—
567C22D3×#61112yes—
568C22C22×D3×D911yescharacteristic
569C4—22no—
570C4—62no—
571C4—181no—
572C4—182no—
573C4—541no—
574C4—542no—
575C3—21no—
576C3C2×#6349111yescharacteristic, complemented
577C3D3×#92911yescharacteristic
578C2—22no—
579C2—34no—
580C2—62no—
581C2—361no—
582C2—1081no—
583C2D3×#369712yes—
584C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC2×D3×#611C2
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×#123
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×D3×C9C2semidirect
3. #11721818C2semidirect
4. C3×#63491C2semidirect
5. D3×#3676C2semidirect
6. #11721259C2semidirect
7. #11721251C2semidirect
8. D3×(C2×D9)⋊C4C2semidirect
9. #11721032C2semidirect
10. D3×Dic9⋊C4C2semidirect
11. #11720954C2semidirect
16. C22×C3×C4×C9C22semidirect
19. C22×C9×Dic3C22semidirect
22. C2×C4×D3×C9C22semidirect
28. #88551C22semidirect
29. #88536C22semidirect
30. #88534C22semidirect
31. C3×#3676C22semidirect
32. C3×(C2×D9)⋊C4C22semidirect
33. C3×Dic9⋊C4C22semidirect
34. (C2×D3)⋊Dic9C22semidirect
35. #88445C22semidirect
36. (C2×D9)⋊Dic3C22semidirect
37. #88442C22semidirect
38. Dic9⋊Dic3C22semidirect
59. #63491D3direct (N × H)
121. C22×C4×C9C2×D3semidirect
133. #3676C2×D3semidirect
138. (C2×D9)⋊C4C2×D3semidirect
143. Dic9⋊C4C2×D3semidirect
166. C22×C4×D3D9semidirect
249. D3×C9#123semidirect
261. C22×C3×C4C2×D9semidirect
273. C22×Dic3C2×D9semidirect
280. C2×C4×D3C2×D9semidirect
363. C3×C9C2×#123semidirect
452. C22×C4D3×D9semidirect
512. C9D3×#123semidirect
555. D3#63491direct (N × H)
576. C3C2×#63491semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D3×C92
C2×D3×#6112
#117218182
C3×#634912
D3×#36762
#117212592
#117212512
D3×(C2×D9)⋊C42
#117210322
D3×Dic9⋊C42
#117209542
D3×#9293
C2×#634913