Finite Group Databasebeta

← C3×Dic3×#214 #13158444

Subgroups — C3×Dic3×#214 #13158444

Order 1728.

754 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×D4×Dic3C211yesmaximal, characteristic
2C2×C32×#214C211yesmaximal, characteristic
3C3×Dic3×Dic6C211yesmaximal, characteristic, complemented
4C3×#63659C211yesmaximal, characteristic
5C3×#63573C211yesmaximal, characteristic, complemented
6C3×#63565C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8C3×Dic3×SD16—31nomaximal
9C3×C4×#214—31nomaximal
10Dic3×#214C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C22×C32×Dic3—21no—
13C2×C32×Dic6C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×#217—21no—
17C32×#214C414yescomplemented
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C3×Dic32—21no—
23C2×C32×SD16—31no—
24C32×C4×D4—31no—
25C3×Q8×Dic3—31no—
26C3×D4×Dic3—21no—
27C3×D4×Dic3D311yescharacteristic
28C3×D4×Dic3C2×C311yescharacteristic
29C2×C3×#214—21no—
30C2×C3×#214D311yescharacteristic
31C2×C3×#214C2×C311yescharacteristic
32C3×C4×Dic6—31no—
33Dic3×Dic6C2×C311yescharacteristic, complemented
34#63659C2×C311yescharacteristic
35C3×#835—31no—
36C3×#832—31no—
37C3×#832—31no—
38C3×#832—61no—
39C3×#817—31no—
40C3×C8×Dic3—31no—
41C3×#810—31no—
42C3×#808—31no—
43C3×C4×C3⋊C8—31no—
44#63573C2×C311yescharacteristic, complemented
45#63565C2×C311yescharacteristic, complemented
46Dic3×C3⋊C8C2×C311yescharacteristic, complemented
47C23×C33—21no—
48C33×D4—21no—
49C33×D4C2×C412yes—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—42no—
55C2×C32×Dic3D411yescharacteristic
56C2×C32×Dic3D411yescharacteristic
57C32×Dic6—21no—
58C32×Dic6C2×C412yescomplemented
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8C2×C412yescomplemented
61C3×C4×SD16—91no—
62Dic3×SD16—31no—
63C4×#214—31no—
64C2×C32×Q8—31no—
65C2×C32×D4—21no—
66C2×C32×D4—21no—
67C2×C32×D4—21no—
68C2×C32×D4—41no—
69C2×C32×D4C2×D311yescharacteristic
70C2×C32×D4C2×D311yescharacteristic
71C2×C32×D4C22×C311yescharacteristic
72C22×C32×C4—61no—
73C22×C3×Dic3—22no—
74C22×C3×Dic3—21no—
75C22×C3×Dic3—21no—
76C2×C3×Dic6—21no—
77C2×C3×Dic6C2×D311yescharacteristic
78C2×C3×Dic6C22×C311yescharacteristic
79C32×SD16—34no—
80C2×C32×C8—31no—
81C32×C4⋊C4—31no—
82C32×C22⋊C4—61no—
83C32×C42—31no—
84C4⋊(C3⋊Dic3)C22×C311yescharacteristic
85C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
86C3×#217—22no—
87C3×#217—21no—
88C3×#217—21no—
89C3×#214—24no—
90C3×#214Dic314yesdirect product factor
91C3×#214C3×C414yescomplemented
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—61no—
96C3×C4⋊Dic3—61no—
97C3×C4⋊Dic3C2×D311yescharacteristic
98C3×C4⋊Dic3C22×C311yescharacteristic
99C3×Dic3⋊C4—61no—
100C3×C4×Dic3—21no—
101C3×C4×Dic3—61no—
102C3×C4×Dic3—61no—
103C3×C4×Dic3C2×D311yescharacteristic
104C3×C4×Dic3C22×C311yescharacteristic
105C2×C3×C3⋊C8—21no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—61no—
109C2×C3×C3⋊C8C2×D311yescharacteristic
110C2×C3×C3⋊C8C22×C311yescharacteristic
111Dic3⋊Dic3—21no—
112Dic32—21no—
113C22×C33—22no—
114C22×C33—41no—
115C22×C33—41no—
116C22×C33C2×D411yescharacteristic
117C33×C4C22×C411yescharacteristic
118C33×C4#4111yescharacteristic
119C3×C3⋊Dic3—41no—
120C32×Dic3—22no—
121C32×Dic3—21no—
122C32×Dic3—41no—
123C32×Dic3SD1612yescomplemented
124C2×C3×SD16—31no—
125C2×C3×SD16—31no—
126C2×C3×SD16—61no—
127C3×C4×Q8—91no—
128C3×C4×D4—31no—
129C3×C4×D4—31no—
130C3×C4×D4—61no—
131Q8×Dic3—31no—
132D4×Dic3C3×D311yescharacteristic
133C2×#214C3×D311yescharacteristic
134C4×Dic6—31no—
135C3×#106—91no—
136C3×#103—91no—
137C3×#102—91no—
138C3×C4×C8—91no—
139#835—31no—
140#832—31no—
141#832—31no—
142#832—61no—
143#817—31no—
144C8×Dic3—31no—
145#810—31no—
146#808—31no—
147C4×C3⋊C8—31no—
148C23×C32—21no—
149C23×C32—22no—
150C23×C32—21no—
151C23×C32—21no—
152C23×C32—41no—
153C23×C32—41no—
154C23×C32—42no—
155C32×Q8—32no—
156C32×Q8—61no—
157C32×D4—21no—
158C32×D4—22no—
159C32×D4—21no—
160C32×D4—21no—
161C32×D4—22no—
162C32×D4—22no—
163C32×D4—22no—
164C32×D4—42no—
165C32×D4—41no—
166C32×D4—41no—
167C32×D4—42no—
168C32×D4C4×D312yes—
169C32×D4C2×Dic312yes—
170C32×D4C2×C3×C412yes—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—31no—
175C2×C32×C4—31no—
176C2×C32×C4—41no—
177C2×C32×C4—61no—
178C2×C32×C4—61no—
179C2×C32×C4—122no—
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C23×C311yescharacteristic
183C2×C3⋊Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—41no—
192C2×C3×Dic3—44no—
193C2×C3×Dic3—42no—
194C2×C3×Dic3—42no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—121no—
198C2×C3×Dic3#6711yescharacteristic
199C2×C3×Dic3#6711yescharacteristic
200C2×C3×Dic3C3×D411yescharacteristic
201C2×C3×Dic3C3×D411yescharacteristic
202C3×Dic6—21no—
203C3×Dic6—21no—
204C3×Dic6—22no—
205C3×Dic6—41no—
206C3×Dic6C2×Dic312yescomplemented
207C3×Dic6C2×C3×C412yescomplemented
208C32×C8—32no—
209C3⋊(C3⋊C8)—21no—
210C3×C3⋊C8—22no—
211C3×C3⋊C8—61no—
212C3×C3⋊C8—61no—
213C3×C3⋊C8—121no—
214C3×C3⋊C8C2×Dic312yescomplemented
215C3×C3⋊C8C2×C3×C412yescomplemented
216C4×SD16—91noSylow
217C2×C33—42no—
218C2×C33C4×D411yescharacteristic
219C2×C33C2×SD1611yescharacteristic
220C2×C33#13511yescharacteristic
221C2×C3×Q8—31no—
222C2×C3×Q8—31no—
223C2×C3×Q8—61no—
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—21no—
227C2×C3×D4—41no—
228C2×C3×D4D3211yescharacteristic
229C2×C3×D4C2×C3×D311yescharacteristic
230C2×C3×D4C2×C3×D311yescharacteristic
231C22×C3×C4—61no—
232C22×C3×C4—61no—
233C22×C3×C4—62no—
234C22×Dic3—21no—
235C2×Dic6C2×C3×D311yescharacteristic
236C3×SD16—34no—
237C3×SD16—34no—
238C3×SD16—64no—
239C2×C3×C8—31no—
240C2×C3×C8—31no—
241C2×C3×C8—61no—
242C2×C3×C8—91no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—61no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—181no—
248C3×C22⋊C4—61no—
249C3×C22⋊C4—61no—
250C3×C22⋊C4—62no—
251C3×C42—31no—
252C3×C42—31no—
253C3×C42—61no—
254C3×C42—181no—
255#217—21no—
256#214C3×Dic314yesdirect product factor
257C4⋊Dic3—31no—
258C4⋊Dic3—31no—
259C4⋊Dic3—61no—
260C4⋊Dic3—61no—
261C4⋊Dic3C2×C3×D311yescharacteristic
262Dic3⋊C4—61no—
263C4×Dic3—61no—
264C4×Dic3—61no—
265C4×Dic3C2×C3×D311yescharacteristic
266C2×C3⋊C8—31no—
267C2×C3⋊C8—31no—
268C2×C3⋊C8—61no—
269C2×C3⋊C8C2×C3×D311yescharacteristic
270C22×C32—22no—
271C22×C32—24no—
272C22×C32—22no—
273C22×C32—22no—
274C22×C32—21no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—41no—
278C22×C32—42no—
279C22×C32—42no—
280C22×C32—44no—
281C22×C32—41no—
282C22×C32—41no—
283C22×C32—42no—
284C22×C32—42no—
285C22×C32—41no—
286C22×C32—41no—
287C22×C32—41no—
288C22×C32—41no—
289C22×C32—81no—
290C22×C32—81no—
291C22×C32—81no—
292C22×C32—81no—
293C22×C32—82no—
294C22×C32—82no—
295C22×C32D3×D411yescharacteristic
296C22×C32C2×#6711yescharacteristic
297C22×C32C2×C3×D411yescharacteristic
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—32no—
305C32×C4—41no—
306C32×C4—41no—
307C32×C4—61no—
308C32×C4—62no—
309C32×C4—121no—
310C32×C4C2×C4×D311yescharacteristic
311C32×C4#23511yescharacteristic
312C32×C4#23711yescharacteristic
313C32×C4C22×Dic311yescharacteristic
314C32×C4C22×C3×C411yescharacteristic
315C32×C4C3×#4111yescharacteristic
316C3⋊Dic3—41no—
317C3×Dic3—22no—
318C3×Dic3—21no—
319C3×Dic3—22no—
320C3×Dic3—21no—
321C3×Dic3—22no—
322C3×Dic3—42no—
323C3×Dic3—41no—
324C3×Dic3—42no—
325C3×Dic3—41no—
326C3×Dic3—41no—
327C3×Dic3—121no—
328C3×Dic3—121no—
329C3×Dic3—241no—
330C3×Dic3#21412yesdirect product factor
331C3×Dic3C3×SD1612yescomplemented
332C2×SD16—31no—
333C4×Q8—91no—
334C4×D4—31no—
335#106—91no—
336#103—91no—
337#102—91no—
338C4×C8—91no—
339C33C4×SD1611yescharacteristic, complemented, Sylow
340C23×C3—22no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—42no—
347C3×Q8—32no—
348C3×Q8—32no—
349C3×Q8—62no—
350C3×Q8—61no—
351C3×Q8—61no—
352C3×Q8—121no—
353C3×D4—22no—
354C3×D4—21no—
355C3×D4—21no—
356C3×D4—21no—
357C3×D4—22no—
358C3×D4—22no—
359C3×D4—22no—
360C3×D4—41no—
361C3×D4—41no—
362C3×D4—42no—
363C3×D4—42no—
364C3×D4D3×Dic312yes—
365C3×D4C3×C4×D312yes—
366C3×D4C2×C3×Dic312yes—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—31no—
371C2×C3×C4—31no—
372C2×C3×C4—31no—
373C2×C3×C4—31no—
374C2×C3×C4—41no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—62no—
380C2×C3×C4—61no—
381C2×C3×C4—61no—
382C2×C3×C4—121no—
383C2×C3×C4—122no—
384C2×C3×C4—122no—
385C2×C3×C4—124no—
386C2×C3×C4—181no—
387C2×C3×C4C2×D3211yescharacteristic
388C2×C3×C4C22×C3×D311yescharacteristic
389C2×C3×C4C22×C3×D311yescharacteristic
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—42no—
393C2×Dic3—61no—
394C2×Dic3—61no—
395C2×Dic3—121no—
396C2×Dic3C3×#6711yescharacteristic
397C2×Dic3C3×#6711yescharacteristic
398Dic6—21no—
399Dic6C2×C3×Dic312yescomplemented
400C3×C8—32no—
401C3×C8—32no—
402C3×C8—62no—
403C3×C8—181no—
404C3⋊C8—61no—
405C3⋊C8—61no—
406C3⋊C8—121no—
407C3⋊C8C2×C3×Dic312yescomplemented
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—41no—
418C2×C32—41no—
419C2×C32—41no—
420C2×C32—42no—
421C2×C32—44no—
422C2×C32—42no—
423C2×C32—42no—
424C2×C32—82no—
425C2×C32—82no—
426C2×C32—84no—
427C2×C32D3×SD1611yescharacteristic
428C2×C32#91611yescharacteristic
429C2×C32C4×#6711yescharacteristic
430C2×C32C2×#21411yescharacteristic
431C2×C32D4×Dic311yescharacteristic
432C2×C32#95011yescharacteristic
433C2×C32C3×C4×D411yescharacteristic
434C2×C32C2×C3×SD1611yescharacteristic
435C2×C32C3×#13511yescharacteristic
436C2×Q8—31no—
437C2×D4C3×D3211yescharacteristic
438C22×C4—61no—
439SD16—34no—
440C2×C8—31no—
441C2×C8—91no—
442C4⋊C4—31no—
443C4⋊C4—91no—
444C4⋊C4—181no—
445C22⋊C4—61no—
446C42—31no—
447C42—181no—
448C22×C3—24no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—44no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3—41no—
466C22×C3—41no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3—82no—
472C22×C3—82no—
473C22×C3D3×#6711yescharacteristic
474C22×C3C3×D3×D411yescharacteristic
475C22×C3C2×C3×#6711yescharacteristic
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—32no—
483C3×C4—32no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—62no—
487C3×C4—61no—
488C3×C4—62no—
489C3×C4—61no—
490C3×C4—62no—
491C3×C4—121no—
492C3×C4—122no—
493C3×C4—122no—
494C3×C4—121no—
495C3×C4—121no—
496C3×C4—361no—
497C3×C4C2×D3×Dic311yescharacteristic
498C3×C4#380411yescharacteristic
499C3×C4C2×C3×C4×D311yescharacteristic
500C3×C4C3×#23511yescharacteristic
501C3×C4C3×#23711yescharacteristic
502C3×C4C22×C3×Dic311yescharacteristic
503Dic3—22no—
504Dic3—21no—
505Dic3—41no—
506Dic3—121no—
507Dic3—121no—
508Dic3—241no—
509Dic3C3×#21412yesdirect product factor
510C32—21no—
511C32—21no—
512C32—21no—
513C32—41no—
514C32C4×#21411yescharacteristic, complemented
515C32Dic3×SD1611yescharacteristic, complemented
516C32C3×C4×SD1611yescharacteristic, complemented
517C23—21no—
518Q8—32no—
519Q8—61no—
520D4—21no—
521D4C3×D3×Dic312yes—
522C2×C4—31no—
523C2×C4—31no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—122no—
527C2×C4—181no—
528C2×C4C2×C3×D3211yescharacteristic
529C8—32no—
530C8—181no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—44no—
541C2×C3—42no—
542C2×C3—42no—
543C2×C3—42no—
544C2×C3—41no—
545C2×C3—41no—
546C2×C3—41no—
547C2×C3—82no—
548C2×C3—82no—
549C2×C3—84no—
550C2×C3D3×#21411yescharacteristic
551C2×C3#6393511yescharacteristic
552C2×C3Dic3×#6711yescharacteristic
553C2×C3C3×D3×SD1611yescharacteristic
554C2×C3C3×#91611yescharacteristic
555C2×C3C3×C4×#6711yescharacteristic
556C2×C3C2×C3×#21411yescharacteristic
557C2×C3C3×D4×Dic311yescharacteristic
558C2×C3C3×#95011yescharacteristic
559C22—22no—
560C22—41no—
561C22—41no—
562C22C3×D3×#6711yescharacteristic
563C4—32no—
564C4—62no—
565C4—61no—
566C4—121no—
567C4—361no—
568C4C2×C3×D3×Dic311yescharacteristic
569C4C3×#380411yescharacteristic
570C3—21no—
571C3—21no—
572C3—21no—
573C3—41no—
574C3Dic3×#21411yescharacteristic, direct product factor
575C3C3×C4×#21411yescharacteristic, complemented
576C3C3×Dic3×SD1611yescharacteristic, complemented
577C2—42no—
578C2C3×D3×#21411yescharacteristic
579C2C3×#6393511yescharacteristic
580C2C3×Dic3×#6711yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×D4D32
hypocenterC32C3×C4×SD16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
3. C3×Dic3×Dic6C2semidirect
5. C3×#63573C2semidirect
6. C3×#63565C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
10. Dic3×#214C3direct (N × H)
17. C32×#214C4semidirect
33. Dic3×Dic6C2×C3semidirect
44. #63573C2×C3semidirect
45. #63565C2×C3semidirect
46. Dic3×C3⋊C8C2×C3semidirect
58. C32×Dic6C2×C4semidirect
60. C32×C3⋊C8C2×C4semidirect
90. C3×#214Dic3direct (N × H)
91. C3×#214C3×C4semidirect
123. C32×Dic3SD16semidirect
206. C3×Dic6C2×Dic3semidirect
207. C3×Dic6C2×C3×C4semidirect
214. C3×C3⋊C8C2×Dic3semidirect
215. C3×C3⋊C8C2×C3×C4semidirect
256. #214C3×Dic3direct (N × H)
330. C3×Dic3#214direct (N × H)
331. C3×Dic3C3×SD16semidirect
339. C33C4×SD16semidirect
399. Dic6C2×C3×Dic3semidirect
407. C3⋊C8C2×C3×Dic3semidirect
509. Dic3C3×#214direct (N × H)
514. C32C4×#214semidirect
515. C32Dic3×SD16semidirect
516. C32C3×C4×SD16semidirect
574. C3Dic3×#214direct (N × H)
575. C3C3×C4×#214semidirect
576. C3C3×Dic3×SD16semidirect

Maximal subgroups

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

subgroupindex
C32×D4×Dic32
C2×C32×#2142
C3×Dic3×Dic62
C3×#636592
C3×#635732
C3×#635652
C3×Dic3×C3⋊C82
C3×Dic3×SD163
C3×C4×#2143
Dic3×#2143