Finite Group Databasebeta

← C3×D3×#810 #13157616

Subgroups — C3×D3×#810 #13157616

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
2C32×D3×C4⋊C4C211yesmaximal, characteristic
3C2×C3×D3×C3⋊C8C211yesmaximal, characteristic
4C3×#63637C211yesmaximal, characteristic, complemented
5C32×#810C211yesmaximal, characteristic, complemented
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8C2×C3×#810—31nomaximal
9C3×D3×#103—31nomaximal
10D3×#810C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C32×C4×D3—21no—
13C2×C32×C4×D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3×#627—21no—
16C2×C3×D3×Dic3—21no—
17C3×D3×Dic6—22no—
18C3×D3×Dic6C414yescomplemented
19C33×C4⋊C4C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×C4⋊Dic3C2211yescharacteristic
22C32×Dic3⋊C4—21no—
23C2×C32×C3⋊C8C2211yescharacteristic
24C3×D3×C3⋊C8—22no—
25C2×C3×D3×Q8—31no—
26C22×C3×Dic6—31no—
27C2×D3×Dic6C2×C311yescharacteristic
28C2×C32×C4⋊C4—31no—
29C22×C3×C3⋊C8—31no—
30C2×C3×D3×C8—31no—
31C3×D3×C4⋊C4—21no—
32C3×D3×C4⋊C4D311yescharacteristic
33C3×D3×C4⋊C4C2×C311yescharacteristic
34C2×D3×C3⋊C8C2×C311yescharacteristic
35C32×#103—31no—
36#63637C2×C311yescharacteristic, complemented
37C3×#835—31no—
38C3×#816—31no—
39C3×#810—21no—
40C3×#810—31no—
41C3×#810—31no—
42C3×#810—31no—
43C3×#810—31no—
44C3×#810—61no—
45C3×#810D311yescharacteristic, direct product factor
46C3×#810C2×C311yescharacteristic, complemented
47#63567C2×C311yescharacteristic, complemented
48#63565C2×C311yescharacteristic, complemented
49C22×C32×D3D411yescharacteristic
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C3×#634—21no—
54C3×#634C2×C412yescomplemented
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C32×C4×D3—42no—
59C32×C4×D3C2×C412yes—
60C32×C4×D3D412yes—
61C32×Dic6—21no—
62C32×Dic6C2×C412yescomplemented
63C3×#627—24no—
64C3×D3×Dic3—24no—
65C3×C3⋊(C3⋊C8)—21no—
66C32×C3⋊C8—21no—
67C2×C3×#103—91no—
68C2×#810—31no—
69D3×#103—31no—
70C2×C32×Q8—31no—
71C22×C32×C4—31no—
72C22×C32×C4—61no—
73C2×#634C22×C311yescharacteristic
74C22×C3×Dic3—61no—
75C3×D3×Q8—34no—
76C3×D3×Q8—62no—
77C2×C3×C4×D3—21no—
78C2×C3×C4×D3—22no—
79C2×C3×C4×D3—21no—
80C2×C3×C4×D3—21no—
81C2×C3×C4×D3—61no—
82C2×C3×C4×D3C2×D311yescharacteristic
83C2×C3×C4×D3C22×C311yescharacteristic
84C2×C3×Dic6—21no—
85C2×C3×Dic6—31no—
86C2×C3×Dic6—31no—
87C2×C3×Dic6—34no—
88C2×C3×Dic6—62no—
89C2×C3×Dic6—61no—
90C2×C3×Dic6—61no—
91C2×C3×Dic6—61no—
92C2×C3×Dic6C2×D311yescharacteristic
93C2×C3×Dic6C22×C311yescharacteristic
94C2×#627—21no—
95C2×D3×Dic3—21no—
96D3×Dic6—22no—
97D3×Dic6C3×C414yescomplemented
98C2×C32×C8—31no—
99C32×C4⋊C4—21no—
100C32×C4⋊C4—21no—
101C32×C4⋊C4—21no—
102C32×C4⋊C4—31no—
103C32×C4⋊C4—41no—
104C32×C4⋊C4—61no—
105C32×C4⋊C4C2×D311yescharacteristic
106C32×C4⋊C4C2×D311yescharacteristic
107C32×C4⋊C4C22×C311yescharacteristic
108C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
109C3×C4⋊Dic3—21no—
110C3×C4⋊Dic3C2×D311yescharacteristic
111C3×C4⋊Dic3C22×C311yescharacteristic
112C3×Dic3⋊C4—22no—
113C3×Dic3⋊C4—21no—
114C3×Dic3⋊C4—21no—
115C2×C3×C3⋊C8—21no—
116C2×C3×C3⋊C8—31no—
117C2×C3×C3⋊C8—31no—
118C2×C3×C3⋊C8—62no—
119C2×C3×C3⋊C8—61no—
120C2×C3×C3⋊C8C2×D311yescharacteristic
121C2×C3×C3⋊C8C22×C311yescharacteristic
122C3×D3×C8—62no—
123D3×C3⋊C8—22no—
124C22×C33C2×D411yescharacteristic
125C2×C32×D3C22⋊C412yes—
126C2×C32×D3SD1612yes—
127C2×C32×D3Q1612yes—
128C33×C4—41no—
129C33×C4C22×C411yescharacteristic
130C33×C4C2×D411yescharacteristic
131C3×C3⋊Dic3—22no—
132C32×Dic3—22no—
133C32×Dic3—41no—
134C32×Dic3C22⋊C412yes—
135C22×C3×Q8—91no—
136C2×D3×Q8—31no—
137C22×Dic6—31no—
138C22×C3×C8—91no—
139C2×C3×C4⋊C4—31no—
140C2×C3×C4⋊C4—31no—
141C2×C3×C4⋊C4—61no—
142C22×C3⋊C8—31no—
143C2×D3×C8—31no—
144D3×C4⋊C4C3×D311yescharacteristic
145C3×#103—31no—
146C3×#103—31no—
147C3×#103—61no—
148C3×#103—91no—
149C3×#103—91no—
150C3×#103—91no—
151#835—31no—
152#816—31no—
153#810—31no—
154#810—31no—
155#810—31no—
156#810—31no—
157#810—61no—
158#810C3×D311yescharacteristic, direct product factor
159C23×C32—31no—
160C22×C3×D3—21no—
161C22×C3×D3#6711yescharacteristic
162C22×C3×D3C3×D411yescharacteristic
163C32×Q8—32no—
164C32×Q8—61no—
165C2×C32×C4—21no—
166C2×C32×C4—22no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—31no—
173C2×C32×C4—32no—
174C2×C32×C4—32no—
175C2×C32×C4—41no—
176C2×C32×C4—41no—
177C2×C32×C4—41no—
178C2×C32×C4—42no—
179C2×C32×C4—61no—
180C2×C32×C4—61no—
181C2×C32×C4—122no—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C23×C311yescharacteristic
185C2×C3⋊Dic3—21no—
186#634—21no—
187#634C2×C3×C412yescomplemented
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—41no—
195C2×C3×Dic3—64no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—121no—
199C2×C3×Dic3#6711yescharacteristic
200C2×C3×Dic3C3×D411yescharacteristic
201C3×C4×D3—22no—
202C3×C4×D3—22no—
203C3×C4×D3—44no—
204C3×C4×D3—42no—
205C3×C4×D3—42no—
206C3×C4×D3—64no—
207C3×C4×D3C4×D312yes—
208C3×C4×D3D1212yes—
209C3×C4×D3C2×C3×C412yes—
210C3×C4×D3C3×D412yes—
211C3×Dic6—21no—
212C3×Dic6—21no—
213C3×Dic6—22no—
214C3×Dic6—32no—
215C3×Dic6—32no—
216C3×Dic6—41no—
217C3×Dic6—62no—
218C3×Dic6—64no—
219C3×Dic6—61no—
220C3×Dic6—61no—
221C3×Dic6—64no—
222C3×Dic6—121no—
223C3×Dic6C4×D312yescomplemented
224C3×Dic6C2×C3×C412yescomplemented
225#627—24no—
226D3×Dic3—24no—
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—91noSylow
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—61no—
251C22×C3×C4—61no—
252C22×C3×C4—62no—
253C22×C3×C4—181no—
254C22×Dic3—61no—
255D3×Q8—34no—
256D3×Q8—62no—
257C2×C4×D3—21no—
258C2×C4×D3—61no—
259C2×C4×D3C2×C3×D311yescharacteristic
260C2×Dic6—31no—
261C2×Dic6—31no—
262C2×Dic6—34no—
263C2×Dic6—61no—
264C2×Dic6—62no—
265C2×Dic6—61no—
266C2×Dic6—61no—
267C2×Dic6C2×C3×D311yescharacteristic
268C2×C3×C8—31no—
269C2×C3×C8—31no—
270C2×C3×C8—61no—
271C2×C3×C8—91no—
272C2×C3×C8—182no—
273C3×C4⋊C4—21no—
274C3×C4⋊C4—21no—
275C3×C4⋊C4—21no—
276C3×C4⋊C4—31no—
277C3×C4⋊C4—31no—
278C3×C4⋊C4—41no—
279C3×C4⋊C4—61no—
280C3×C4⋊C4—61no—
281C3×C4⋊C4—61no—
282C3×C4⋊C4—62no—
283C3×C4⋊C4D3211yescharacteristic
284C3×C4⋊C4C2×C3×D311yescharacteristic
285C3×C4⋊C4C2×C3×D311yescharacteristic
286C4⋊Dic3C2×C3×D311yescharacteristic
287Dic3⋊C4—21no—
288C2×C3⋊C8—31no—
289C2×C3⋊C8—31no—
290C2×C3⋊C8—61no—
291C2×C3⋊C8—62no—
292C2×C3⋊C8C2×C3×D311yescharacteristic
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×C32D3×D411yescharacteristic
302C22×C32C2×#6711yescharacteristic
303C22×C32C2×C3×D411yescharacteristic
304C2×C3×D3—22no—
305C2×C3×D3—22no—
306C2×C3×D3—22no—
307C2×C3×D3(C2×D3)⋊C412yes—
308C2×C3×D3#21412yes—
309C2×C3×D3#21612yes—
310C2×C3×D3C3×C22⋊C412yes—
311C2×C3×D3C3×SD1612yes—
312C2×C3×D3C3×Q1612yes—
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—41no—
323C32×C4—42no—
324C32×C4—41no—
325C32×C4—41no—
326C32×C4—62no—
327C32×C4—81no—
328C32×C4—81no—
329C32×C4—82no—
330C32×C4—121no—
331C32×C4C2×C4×D311yescharacteristic
332C32×C4C2×C4×D311yescharacteristic
333C32×C4C2×D1211yescharacteristic
334C32×C4D3×D411yescharacteristic
335C32×C4C22×C3×C411yescharacteristic
336C32×C4C2×C3×D411yescharacteristic
337C3⋊Dic3—22no—
338C3×Dic3—22no—
339C3×Dic3—22no—
340C3×Dic3—22no—
341C3×Dic3—42no—
342C3×Dic3—42no—
343C3×Dic3—41no—
344C3×Dic3—41no—
345C3×Dic3—62no—
346C3×Dic3—62no—
347C3×Dic3—122no—
348C3×Dic3(C2×D3)⋊C412yes—
349C3×Dic3C3×C22⋊C412yes—
350C22×Q8—91no—
351C22×C8—91no—
352C2×C4⋊C4—31no—
353#103—31no—
354#103—91no—
355#103—91no—
356#103—91no—
357C33C2×#10311yescharacteristic, complemented, Sylow
358C23×C3—31no—
359C23×C3—31no—
360C23×C3—61no—
361C22×D3C3×#6711yescharacteristic
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—22no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—31no—
379C2×C3×C4—31no—
380C2×C3×C4—32no—
381C2×C3×C4—32no—
382C2×C3×C4—32no—
383C2×C3×C4—32no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—42no—
387C2×C3×C4—41no—
388C2×C3×C4—62no—
389C2×C3×C4—62no—
390C2×C3×C4—61no—
391C2×C3×C4—61no—
392C2×C3×C4—61no—
393C2×C3×C4—62no—
394C2×C3×C4—61no—
395C2×C3×C4—61no—
396C2×C3×C4—121no—
397C2×C3×C4—122no—
398C2×C3×C4—122no—
399C2×C3×C4—124no—
400C2×C3×C4—184no—
401C2×C3×C4—181no—
402C2×C3×C4C2×D3211yescharacteristic
403C2×C3×C4C22×C3×D311yescharacteristic
404C2×C3×C4C22×C3×D311yescharacteristic
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—64no—
408C2×Dic3—61no—
409C2×Dic3—61no—
410C2×Dic3—121no—
411C2×Dic3C3×#6711yescharacteristic
412C4×D3—42no—
413C4×D3—64no—
414C4×D3C3×C4×D312yes—
415C4×D3C3×D1212yes—
416Dic6—21no—
417Dic6—32no—
418Dic6—32no—
419Dic6—62no—
420Dic6—64no—
421Dic6—61no—
422Dic6—61no—
423Dic6—64no—
424Dic6—121no—
425Dic6C3×C4×D312yescomplemented
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×C32D3×SD1611yescharacteristic
449C2×C32D3×Q1611yescharacteristic
450C2×C32C2×(C2×D3)⋊C411yescharacteristic
451C2×C32C2×#21411yescharacteristic
452C2×C32C2×#21611yescharacteristic
453C2×C32C2×C3×C22⋊C411yescharacteristic
454C2×C32C2×C3×SD1611yescharacteristic
455C2×C32C2×C3×Q1611yescharacteristic
456C3×D3—24no—
457C3×D3#81014yesdirect product factor
458C3×D3C3×#10314yescomplemented
459C2×Q8—31no—
460C2×Q8—94no—
461C2×Q8—91no—
462C2×Q8—181no—
463C2×Q8—182no—
464C22×C4—31no—
465C22×C4—61no—
466C22×C4—181no—
467C2×C8—31no—
468C2×C8—91no—
469C2×C8—182no—
470C4⋊C4—31no—
471C4⋊C4—61no—
472C4⋊C4C3×D3211yescharacteristic
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×#6711yescharacteristic
487C22×C3C3×D3×D411yescharacteristic
488C22×C3C2×C3×#6711yescharacteristic
489C2×D3C3×(C2×D3)⋊C412yes—
490C2×D3C3×#21412yes—
491C2×D3C3×#21612yes—
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—42no—
501C3×C4—41no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—62no—
507C3×C4—62no—
508C3×C4—62no—
509C3×C4—81no—
510C3×C4—81no—
511C3×C4—82no—
512C3×C4—122no—
513C3×C4—122no—
514C3×C4—121no—
515C3×C4—121no—
516C3×C4—182no—
517C3×C4C4×D3211yescharacteristic
518C3×C4D3×D1211yescharacteristic
519C3×C4C2×C3×C4×D311yescharacteristic
520C3×C4C2×C3×C4×D311yescharacteristic
521C3×C4C2×C3×D1211yescharacteristic
522C3×C4C3×D3×D411yescharacteristic
523Dic3—22no—
524Dic3—41no—
525Dic3—62no—
526Dic3—62no—
527Dic3—122no—
528Dic3C3×(C2×D3)⋊C412yes—
529C32—21no—
530C32—21no—
531C32—21no—
532C32—41no—
533C32D3×#10311yescharacteristic, complemented
534C32C2×#81011yescharacteristic, complemented
535C32C2×C3×#10311yescharacteristic, complemented
536C23—31no—
537Q8—32no—
538Q8—61no—
539Q8—92no—
540Q8—184no—
541Q8—181no—
542C2×C4—21no—
543C2×C4—31no—
544C2×C4—32no—
545C2×C4—32no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—122no—
549C2×C4—184no—
550C2×C4—181no—
551C2×C4C2×C3×D3211yescharacteristic
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×(C2×D3)⋊C411yescharacteristic
570C2×C3D3×#21411yescharacteristic
571C2×C3D3×#21611yescharacteristic
572C2×C3C3×D3×C22⋊C411yescharacteristic
573C2×C3C3×D3×SD1611yescharacteristic
574C2×C3C3×D3×Q1611yescharacteristic
575C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
576C2×C3C2×C3×#21411yescharacteristic
577C2×C3C2×C3×#21611yescharacteristic
578D3C3×#81014yesdirect product factor
579C22—32no—
580C22—32no—
581C22—32no—
582C22C3×D3×#6711yescharacteristic
583C4—32no—
584C4—41no—
585C4—62no—
586C4—121no—
587C4—182no—
588C4C3×C4×D3211yescharacteristic
589C4C3×D3×D1211yescharacteristic
590C3—21no—
591C3—21no—
592C3—21no—
593C3—41no—
594C3D3×#81011yescharacteristic, direct product factor
595C3C3×D3×#10311yescharacteristic, complemented
596C3C2×C3×#81011yescharacteristic, complemented
597C2—34no—
598C2C3×D3×(C2×D3)⋊C411yescharacteristic
599C2C3×D3×#21411yescharacteristic
600C2C3×D3×#21611yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
cosocleC2×C32×C4C23×C3
hypercenterC3×C4⋊C4D32
hypocenterC32C2×C3×#103
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C3×#63637C2semidirect
5. C32×#810C2semidirect
6. C3×#63567C2semidirect
7. C3×#63565C2semidirect
10. D3×#810C3direct (N × H)
18. C3×D3×Dic6C4semidirect
36. #63637C2×C3semidirect
45. C3×#810D3direct (N × H)
46. C3×#810C2×C3semidirect
47. #63567C2×C3semidirect
48. #63565C2×C3semidirect
54. C3×#634C2×C4semidirect
62. C32×Dic6C2×C4semidirect
97. D3×Dic6C3×C4semidirect
158. #810C3×D3direct (N × H)
187. #634C2×C3×C4semidirect
223. C3×Dic6C4×D3semidirect
224. C3×Dic6C2×C3×C4semidirect
239. C32×D3#103semidirect
357. C33C2×#103semidirect
425. Dic6C3×C4×D3semidirect
457. C3×D3#810direct (N × H)
458. C3×D3C3×#103semidirect
533. C32D3×#103semidirect
534. C32C2×#810semidirect
535. C32C2×C3×#103semidirect
578. D3C3×#810direct (N × H)
594. C3D3×#810direct (N × H)
595. C3C3×D3×#103semidirect
596. C3C2×C3×#810semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×Dic62
C32×D3×C4⋊C42
C2×C3×D3×C3⋊C82
C3×#636372
C32×#8102
C3×#635672
C3×#635652
C2×C3×#8103
C3×D3×#1033
D3×#8103