Finite Group Databasebeta

← D3×#63566 #13155358

Subgroups — D3×#63566 #13155358

Order 1728.

850 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×Dic6C211yesmaximal, characteristic
2D3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
3C2×C3×D3×C3⋊C8C211yesmaximal, characteristic
4#11722076C211yesmaximal, characteristic, complemented
5#11722040C211yesmaximal, characteristic, complemented
6#11722038C211yesmaximal, characteristic, complemented
7C3×#63566C211yesmaximal, characteristic, complemented
8D3×#835—31nomaximal
9C2×#63566—31nomaximal
10D3×#816—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×D3×C3⋊Dic3—21no—
15C2×C3×#627—21no—
16C2×C3×D3×Dic3—21no—
17C3×D3×Dic6—22no—
18C3×D3×Dic6C414yescomplemented
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23Dic3⋊(C3⋊Dic3)—21no—
24C3×D3×C3⋊C8—22no—
25C2×C3×D3×Q8—31no—
26C22×C3×Dic6—31no—
27C2×D3×Dic6D311yescharacteristic
28C2×C4⋊(C3⋊Dic3)—31no—
29C22×C3×C3⋊C8—31no—
30C2×C3×D3×C8—31no—
31D3×C4⋊Dic3—31no—
32D3×C4⋊Dic3—31no—
33D3×C4⋊Dic3—61no—
34C2×D3×C3⋊C8D311yescharacteristic
35#63662—31no—
36#63643—31no—
37C3×#835—31no—
38C3×#816—31no—
39#63567—31no—
40#63566—31no—
41#63566—31no—
42#63566—31no—
43#63566—31no—
44#63566—31no—
45#63566—61no—
46#63566—61no—
47#63566D311yescharacteristic, direct product factor
48#63565—31no—
49C22×C32×D3D411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C3×#634—21no—
55C3×#634C2×C412yescomplemented
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C32×C4×D3C2×C412yes—
59C32×C4×D3D412yes—
60C32×Dic6—21no—
61C32×Dic6C2×C412yescomplemented
62D3×C3⋊Dic3—42no—
63C3×#627—24no—
64C3×D3×Dic3—24no—
65C3×C3⋊(C3⋊C8)—21no—
66C32×C3⋊C8—21no—
67C2×#835—91no—
68C2×#816—91no—
69D3×#103—91no—
70C2×C32×Q8—31no—
71C22×C32×C4—31no—
72C22×C3⋊Dic3—61no—
73C2×#634C2×D311yescharacteristic
74C22×C3×Dic3—61no—
75C3×D3×Q8—34no—
76C3×D3×Q8—62no—
77C2×C3×C4×D3—21no—
78C2×C3×C4×D3—61no—
79C2×C3×C4×D3C2×D311yescharacteristic
80C2×C3×C4×D3C2×D311yescharacteristic
81C2×C3×Dic6—21no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—34no—
85C2×C3×Dic6—62no—
86C2×C3×Dic6—61no—
87C2×C3×Dic6—61no—
88C2×C3×Dic6—61no—
89C2×C3×Dic6C2×D311yescharacteristic
90C2×C3×Dic6C2×D311yescharacteristic
91C2×#627—21no—
92C2×D3×Dic3—21no—
93C2×D3×Dic3—62no—
94C2×D3×Dic3—61no—
95C2×D3×Dic3—61no—
96D3×Dic6—22no—
97D3×Dic6Dic314yescomplemented
98C2×C32×C8—31no—
99C4⋊(C3⋊Dic3)—31no—
100C4⋊(C3⋊Dic3)—31no—
101C4⋊(C3⋊Dic3)—31no—
102C4⋊(C3⋊Dic3)—61no—
103C4⋊(C3⋊Dic3)—61no—
104C4⋊(C3⋊Dic3)—61no—
105C4⋊(C3⋊Dic3)—61no—
106C4⋊(C3⋊Dic3)—121no—
107C4⋊(C3⋊Dic3)C2×D311yescharacteristic
108C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
109C3×C4⋊Dic3—31no—
110C3×C4⋊Dic3—31no—
111C3×C4⋊Dic3—61no—
112C2×C3×C3⋊C8—21no—
113C2×C3×C3⋊C8—31no—
114C2×C3×C3⋊C8—31no—
115C2×C3×C3⋊C8—62no—
116C2×C3×C3⋊C8—61no—
117C2×C3×C3⋊C8C2×D311yescharacteristic
118C2×C3×C3⋊C8C2×D311yescharacteristic
119C3×D3×C8—62no—
120Dic3⋊Dic3—62no—
121Dic3⋊Dic3—61no—
122Dic3⋊Dic3—61no—
123D3×C3⋊C8—22no—
124C22×C33C2×D411yescharacteristic
125C2×C32×D3C22⋊C412yes—
126C2×C32×D3SD1612yes—
127C2×C32×D3Q1612yes—
128C33×C4C22×C411yescharacteristic
129C33×C4C2×D411yescharacteristic
130C3⋊(C3⋊Dic3)—41no—
131C3×C3⋊Dic3—22no—
132C3×C3⋊Dic3—41no—
133C32×Dic3—22no—
134C32×Dic3C22⋊C412yes—
135C22×C3×Q8—91no—
136C2×D3×Q8—31no—
137C22×Dic6—31no—
138C22×C3×C8—91no—
139C2×C4⋊Dic3—91no—
140C2×C4⋊Dic3—91no—
141C2×C4⋊Dic3—181no—
142C22×C3⋊C8—31no—
143C2×D3×C8—31no—
144D3×C4⋊C4—91no—
145C3×#103—91no—
146#835—31no—
147#835—91no—
148#835—91no—
149#835—91no—
150#835—91no—
151#835—181no—
152#816—31no—
153#816—91no—
154#816—91no—
155#816—91no—
156#816—91no—
157#816—181no—
158#810—91no—
159C23×C32—31no—
160C22×C3×D3—21no—
161C22×C3×D3D1211yescharacteristic
162C22×C3×D3#6711yescharacteristic
163C32×Q8—32no—
164C32×Q8—61no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—31no—
169C2×C32×C4—32no—
170C2×C32×C4—32no—
171C2×C32×C4—41no—
172C2×C32×C4—61no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3—21no—
178C2×C3⋊Dic3—61no—
179C2×C3⋊Dic3—62no—
180C2×C3⋊Dic3—61no—
181C2×C3⋊Dic3—61no—
182C2×C3⋊Dic3—121no—
183C2×C3⋊Dic3—121no—
184C2×C3⋊Dic3—122no—
185C2×C3⋊Dic3—122no—
186#634—21no—
187#634C2×Dic312yescomplemented
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—41no—
192C2×C3×Dic3—64no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—121no—
199C2×C3×Dic3D1211yescharacteristic
200C2×C3×Dic3#6711yescharacteristic
201C3×C4×D3—22no—
202C3×C4×D3—22no—
203C3×C4×D3—64no—
204C3×C4×D3C4×D312yes—
205C3×C4×D3C2×Dic312yes—
206C3×C4×D3#6712yes—
207C3×C4×D3#6712yes—
208C3×Dic6—21no—
209C3×Dic6—21no—
210C3×Dic6—22no—
211C3×Dic6—32no—
212C3×Dic6—32no—
213C3×Dic6—41no—
214C3×Dic6—62no—
215C3×Dic6—64no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218C3×Dic6—64no—
219C3×Dic6—121no—
220C3×Dic6C4×D312yescomplemented
221C3×Dic6C2×Dic312yescomplemented
222#627—24no—
223D3×Dic3—24no—
224D3×Dic3—124no—
225D3×Dic3—122no—
226D3×Dic3—122no—
227C32×C8—61no—
228C3⋊(C3⋊C8)—21no—
229C3×C3⋊C8—21no—
230C3×C3⋊C8—21no—
231C3×C3⋊C8—41no—
232C3×C3⋊C8—61no—
233C3×C3⋊C8—61no—
234C3×C3⋊C8—121no—
235C2×#103—271noSylow
236C2×C33C2×C22⋊C411yescharacteristic
237C2×C33C2×SD1611yescharacteristic
238C2×C33C2×Q1611yescharacteristic
239C32×D3#10314yescomplemented
240C2×C3×Q8—31no—
241C2×C3×Q8—31no—
242C2×C3×Q8—61no—
243C2×C3×Q8—94no—
244C2×C3×Q8—91no—
245C2×C3×Q8—181no—
246C2×C3×Q8—182no—
247C22×C3×C4—31no—
248C22×C3×C4—31no—
249C22×C3×C4—61no—
250C22×C3×C4—181no—
251C22×Dic3—61no—
252C22×Dic3—181no—
253C22×Dic3—181no—
254C22×Dic3—182no—
255D3×Q8—34no—
256D3×Q8—62no—
257C2×C4×D3—61no—
258C2×C4×D3—181no—
259C2×C4×D3D3211yescharacteristic
260C2×Dic6—31no—
261C2×Dic6—31no—
262C2×Dic6—34no—
263C2×Dic6—61no—
264C2×Dic6—62no—
265C2×Dic6—61no—
266C2×Dic6—61no—
267C2×Dic6D3211yescharacteristic
268C2×C3×C8—31no—
269C2×C3×C8—31no—
270C2×C3×C8—61no—
271C2×C3×C8—91no—
272C2×C3×C8—182no—
273C3×C4⋊C4—91no—
274C4⋊Dic3—31no—
275C4⋊Dic3—31no—
276C4⋊Dic3—61no—
277C4⋊Dic3—91no—
278C4⋊Dic3—91no—
279C4⋊Dic3—91no—
280C4⋊Dic3—181no—
281C4⋊Dic3—181no—
282C4⋊Dic3—181no—
283C4⋊Dic3—181no—
284C4⋊Dic3—181no—
285C4⋊Dic3—182no—
286C4⋊Dic3—361no—
287Dic3⋊C4—181no—
288C2×C3⋊C8—31no—
289C2×C3⋊C8—31no—
290C2×C3⋊C8—61no—
291C2×C3⋊C8—62no—
292C2×C3⋊C8D3211yescharacteristic
293D3×C8—62no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—32no—
298C22×C32—32no—
299C22×C32—32no—
300C22×C32—41no—
301C22×C32C2×D1211yescharacteristic
302C22×C32D3×D411yescharacteristic
303C22×C32C2×#6711yescharacteristic
304C2×C3×D3—22no—
305C2×C3×D3—22no—
306C2×C3×D3—22no—
307C2×C3×D3#20412yes—
308C2×C3×D3Dic1212yes—
309C2×C3×D3(C2×D3)⋊C412yes—
310C2×C3×D3#21512yes—
311C2×C3×D3#21612yes—
312C2×C3×D3#21712yes—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—32no—
320C32×C4—41no—
321C32×C4—41no—
322C32×C4—62no—
323C32×C4C2×C4×D311yescharacteristic
324C32×C4C2×C4×D311yescharacteristic
325C32×C4D3×D411yescharacteristic
326C32×C4C22×Dic311yescharacteristic
327C32×C4C2×#6711yescharacteristic
328C32×C4C2×#6711yescharacteristic
329C3⋊Dic3—22no—
330C3⋊Dic3—41no—
331C3⋊Dic3—121no—
332C3⋊Dic3—122no—
333C3⋊Dic3—121no—
334C3⋊Dic3—121no—
335C3⋊Dic3—241no—
336C3⋊Dic3—241no—
337C3⋊Dic3—242no—
338C3×Dic3—22no—
339C3×Dic3—22no—
340C3×Dic3—22no—
341C3×Dic3—42no—
342C3×Dic3—62no—
343C3×Dic3—62no—
344C3×Dic3—122no—
345C3×Dic3—122no—
346C3×Dic3—121no—
347C3×Dic3—121no—
348C3×Dic3(C2×D3)⋊C412yes—
349C3×Dic3#21712yes—
350C22×Q8—91no—
351C22×C8—91no—
352C2×C4⋊C4—271no—
353#103—91no—
354#103—271no—
355#103—271no—
356#103—271no—
357C33C2×#10311yescharacteristic, complemented, Sylow
358C23×C3—31no—
359C23×C3—31no—
360C23×C3—61no—
361C22×D3#62611yescharacteristic
362C3×Q8—32no—
363C3×Q8—32no—
364C3×Q8—62no—
365C3×Q8—61no—
366C3×Q8—61no—
367C3×Q8—92no—
368C3×Q8—121no—
369C3×Q8—184no—
370C3×Q8—181no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—32no—
377C2×C3×C4—32no—
378C2×C3×C4—32no—
379C2×C3×C4—32no—
380C2×C3×C4—41no—
381C2×C3×C4—62no—
382C2×C3×C4—62no—
383C2×C3×C4—61no—
384C2×C3×C4—61no—
385C2×C3×C4—61no—
386C2×C3×C4—121no—
387C2×C3×C4—184no—
388C2×C3×C4—181no—
389C2×C3×C4—181no—
390C2×C3×C4C2×D3211yescharacteristic
391C2×C3×C4C2×D3211yescharacteristic
392C2×C3×C4C2×D3211yescharacteristic
393C2×Dic3—21no—
394C2×Dic3—64no—
395C2×Dic3—61no—
396C2×Dic3—61no—
397C2×Dic3—62no—
398C2×Dic3—61no—
399C2×Dic3—61no—
400C2×Dic3—121no—
401C2×Dic3—182no—
402C2×Dic3—181no—
403C2×Dic3—181no—
404C2×Dic3—181no—
405C2×Dic3—361no—
406C2×Dic3—361no—
407C2×Dic3—362no—
408C2×Dic3—362no—
409C2×Dic3—362no—
410C2×Dic3—364no—
411C2×Dic3#62611yescharacteristic
412C4×D3—64no—
413C4×D3—362no—
414C4×D3D3×Dic312yes—
415C4×D3#62512yes—
416Dic6—21no—
417Dic6—32no—
418Dic6—32no—
419Dic6—62no—
420Dic6—64no—
421Dic6—61no—
422Dic6—61no—
423Dic6—64no—
424Dic6—121no—
425Dic6D3×Dic312yescomplemented
426C3×C8—61no—
427C3×C8—61no—
428C3×C8—121no—
429C3×C8—181no—
430C3⋊C8—21no—
431C3⋊C8—61no—
432C3⋊C8—61no—
433C3⋊C8—121no—
434C2×C32—21no—
435C2×C32—21no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—34no—
444C2×C32—41no—
445C2×C32—41no—
446C2×C32—41no—
447C2×C32D3×C22⋊C411yescharacteristic
448C2×C32C2×#20411yescharacteristic
449C2×C32C2×Dic1211yescharacteristic
450C2×C32D3×SD1611yescharacteristic
451C2×C32D3×Q1611yescharacteristic
452C2×C32C2×(C2×D3)⋊C411yescharacteristic
453C2×C32C2×#21511yescharacteristic
454C2×C32C2×#21611yescharacteristic
455C2×C32C2×#21711yescharacteristic
456C3×D3—24no—
457C3×D3#81614yescomplemented
458C3×D3#83514yescomplemented
459C2×Q8—31no—
460C2×Q8—94no—
461C2×Q8—91no—
462C2×Q8—181no—
463C2×Q8—182no—
464C22×C4—31no—
465C22×C4—181no—
466C22×C4—541no—
467C2×C8—31no—
468C2×C8—91no—
469C2×C8—182no—
470C4⋊C4—91no—
471C4⋊C4—271no—
472C4⋊C4—541no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—32no—
477C22×C3—32no—
478C22×C3—32no—
479C22×C3—32no—
480C22×C3—32no—
481C22×C3—32no—
482C22×C3—41no—
483C22×C3—62no—
484C22×C3—62no—
485C22×C3—62no—
486C22×C3D3×D1211yescharacteristic
487C22×C3C2×#62611yescharacteristic
488C22×C3D3×#6711yescharacteristic
489C2×D3#371712yes—
490C2×D3#371912yes—
491C2×D3(C2×D3)⋊Dic312yes—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—32no—
499C3×C4—32no—
500C3×C4—41no—
501C3×C4—41no—
502C3×C4—62no—
503C3×C4—62no—
504C3×C4—62no—
505C3×C4—122no—
506C3×C4—182no—
507C3×C4—361no—
508C3×C4C4×D3211yescharacteristic
509C3×C4C2×D3×Dic311yescharacteristic
510C3×C4C2×D3×Dic311yescharacteristic
511C3×C4C2×#62511yescharacteristic
512C3×C4D3×#6711yescharacteristic
513C3×C4D3×#6711yescharacteristic
514Dic3—22no—
515Dic3—62no—
516Dic3—62no—
517Dic3—122no—
518Dic3—122no—
519Dic3—121no—
520Dic3—121no—
521Dic3—362no—
522Dic3—361no—
523Dic3—361no—
524Dic3—361no—
525Dic3—721no—
526Dic3—721no—
527Dic3—722no—
528Dic3(C2×D3)⋊Dic312yes—
529C32—21no—
530C32—21no—
531C32—21no—
532C32—41no—
533C32D3×#10311yescharacteristic, complemented
534C32C2×#81611yescharacteristic, complemented
535C32C2×#83511yescharacteristic, complemented
536C23—31no—
537Q8—32no—
538Q8—61no—
539Q8—92no—
540Q8—184no—
541Q8—181no—
542C2×C4—31no—
543C2×C4—32no—
544C2×C4—32no—
545C2×C4—61no—
546C2×C4—184no—
547C2×C4—181no—
548C2×C4—181no—
549C2×C4—541no—
550C2×C4—1082no—
551C2×C4D3311yescharacteristic
552C8—61no—
553C8—181no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—34no—
564C2×C3—34no—
565C2×C3—41no—
566C2×C3—41no—
567C2×C3—41no—
568C2×C3—64no—
569C2×C3D3×#20411yescharacteristic
570C2×C3D3×Dic1211yescharacteristic
571C2×C3C2×#371711yescharacteristic
572C2×C3C2×#371911yescharacteristic
573C2×C3D3×(C2×D3)⋊C411yescharacteristic
574C2×C3D3×#21511yescharacteristic
575C2×C3D3×#21611yescharacteristic
576C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
577C2×C3D3×#21711yescharacteristic
578D3#6356614yesdirect product factor
579C22—32no—
580C22—32no—
581C22—32no—
582C22D3×#62611yescharacteristic
583C4—32no—
584C4—62no—
585C4—182no—
586C4—361no—
587C4—1081no—
588C4D32×Dic311yescharacteristic
589C4D3×#62511yescharacteristic
590C3—21no—
591C3—21no—
592C3—21no—
593C3—41no—
594C3D3×#81611yescharacteristic, complemented
595C3C2×#6356611yescharacteristic, complemented
596C3D3×#83511yescharacteristic, complemented
597C2—34no—
598C2D3×#371711yescharacteristic
599C2D3×#371911yescharacteristic
600C2D3×(C2×D3)⋊Dic311yescharacteristic
subgroupG/H
centerC22D3×#626
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypocenterC33C2×#103
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722076C2semidirect
5. #11722040C2semidirect
6. #11722038C2semidirect
7. C3×#63566C2semidirect
18. C3×D3×Dic6C4semidirect
47. #63566D3direct (N × H)
55. C3×#634C2×C4semidirect
61. C32×Dic6C2×C4semidirect
97. D3×Dic6Dic3semidirect
187. #634C2×Dic3semidirect
220. C3×Dic6C4×D3semidirect
221. C3×Dic6C2×Dic3semidirect
239. C32×D3#103semidirect
357. C33C2×#103semidirect
425. Dic6D3×Dic3semidirect
457. C3×D3#816semidirect
458. C3×D3#835semidirect
533. C32D3×#103semidirect
534. C32C2×#816semidirect
535. C32C2×#835semidirect
578. D3#63566direct (N × H)
594. C3D3×#816semidirect
595. C3C2×#63566semidirect
596. C3D3×#835semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×Dic62
D3×C4⋊(C3⋊Dic3)2
C2×C3×D3×C3⋊C82
#117220762
#117220402
#117220382
C3×#635662
D3×#8353
C2×#635663
D3×#8163