Finite Group Databasebeta

← C2×C3×#63983 #13175119

Subgroups — C2×C3×#63983 #13175119

Order 1728.

3490 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×D32C211yesmaximal, characteristic, complemented
2C22×C3×#638C211yesmaximal, characteristic, complemented
3C22×C32×#67C212yesmaximal, complemented
4C3×#63983C218yesmaximal, direct product factor
5C2×C3×#3722C211yesmaximal, characteristic, complemented
6C2×C3×(C2×D3)⋊Dic3C212yesmaximal, complemented
7C2×C3×#937—32nomaximal
8C2×#63983C311yesmaximal, characteristic, direct product factor
9C24×C33C2211yescharacteristic, complemented
10C23×C3×C3⋊D3C2211yescharacteristic
11C23×C32×D3C2212yes—
12C22×C3×D32—24no—
13C22×C3×D32—28no—
14C22×C3×D32C2214yescomplemented
15C2×C3×#638—24no—
16C2×C3×#638C2214yescomplemented
17C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
18C2×C32×#67—28no—
19C2×C32×#67C2218yescomplemented
20C22×C32×Dic3C2212yescomplemented
21C3×#3722C2214yescomplemented
22C3×(C2×D3)⋊Dic3C2218yescomplemented
23C24×C3×D3—32no—
24C23×D32C2×C311yescharacteristic, complemented
25C22×C32×D4—32no—
26C22×#638C2×C311yescharacteristic, complemented
27C22×C3×#67—22no—
28C22×C3×#67—32no—
29C22×C3×#67—61no—
30C22×C3×#67D312yescomplemented
31C22×C3×#67C2×C312yescomplemented
32C2×C3×#217—32no—
33C3×#937—316no—
34C2×C3×(C2×D3)⋊C4—32no—
35C2×C3×(C2×D3)⋊C4—32no—
36#63983C2×C318yesdirect product factor
37C2×#3722C2×C311yescharacteristic, complemented
38C2×(C2×D3)⋊Dic3C2×C312yescomplemented
39C23×C33—24no—
40C23×C33—22no—
41C23×C33C2311yescharacteristic
42C23×C33C2312yescomplemented
43C22×C3×C3⋊D3—24no—
44C22×C3×C3⋊D3D414yescomplemented
45C22×C3×C3⋊D3C2312yes—
46C22×C32×D3—24no—
47C22×C32×D3—24no—
48C22×C32×D3D418yescomplemented
49C22×C32×D3C2314yes—
50C2×C3×D32—232no—
51C2×C3×D32—28no—
52C2×C3×D32—216no—
53C3×#638—28no—
54C2×C3×C3⋊Dic3—22no—
55C2×C3×C3⋊Dic3C2312yescomplemented
56C32×#67—216no—
57C2×C32×Dic3—24no—
58C2×C32×Dic3C2314yescomplemented
59C2×C3×#120—91no—
60C2×#937—32no—
61C24×C32—21no—
62C24×C32—22no—
63C24×C32—32no—
64C24×C32—41no—
65C24×C32C2×D312yescomplemented
66C24×C32C22×C311yescharacteristic, complemented
67C23×C3⋊D3C22×C311yescharacteristic
68C23×C3×D3—22no—
69C23×C3×D3—32no—
70C23×C3×D3—38no—
71C23×C3×D3—68no—
72C23×C3×D3—68no—
73C23×C3×D3—68no—
74C23×C3×D3—61no—
75C23×C3×D3C2×D312yes—
76C23×C3×D3C22×C312yes—
77C22×D32—24no—
78C22×D32—28no—
79C22×D32C22×C314yescomplemented
80C2×C32×D4—38no—
81C2×C32×D4—68no—
82C22×C32×C4—32no—
83C2×#638—24no—
84C2×#638C22×C314yescomplemented
85C22×C3⋊Dic3C22×C311yescharacteristic, complemented
86C2×C3×#67—216no—
87C2×C3×#67—28no—
88C2×C3×#67—28no—
89C2×C3×#67—28no—
90C2×C3×#67—38no—
91C2×C3×#67—68no—
92C2×C3×#67—64no—
93C2×C3×#67—68no—
94C2×C3×#67C2×D318yescomplemented
95C2×C3×#67C22×C318yescomplemented
96C22×C3×Dic3—22no—
97C22×C3×Dic3—32no—
98C22×C3×Dic3—61no—
99C22×C3×Dic3C2×D312yescomplemented
100C22×C3×Dic3C22×C312yescomplemented
101C3×#217—38no—
102C3×(C2×D3)⋊C4—38no—
103C3×(C2×D3)⋊C4—38no—
104#3722C22×C314yescomplemented
105(C2×D3)⋊Dic3C22×C318yescomplemented
106C22×C33—24no—
107C22×C33—22no—
108C22×C33—44no—
109C22×C33C2×D412yes—
110C22×C33C2×D414yes—
111C22×C33C2411yescharacteristic
112C2×C3×C3⋊D3—28no—
113C2×C3×C3⋊D3—24no—
114C2×C3×C3⋊D3C2×D414yescomplemented
115C2×C32×D3—216no—
116C2×C32×D3—24no—
117C2×C32×D3—24no—
118C2×C32×D3C2×D418yescomplemented
119C3×D32—232no—
120C3×C3⋊Dic3—22no—
121C32×Dic3—24no—
122C25×C3—91no—
123C24×D3—32no—
124C22×C3×D4—32no—
125C22×C3×D4—32no—
126C22×C3×D4—62no—
127C22×C3×D4—91no—
128C22×#67—32no—
129C22×#67—61no—
130C22×#67C3×D312yescomplemented
131C3×#120—98no—
132C2×C3×C22⋊C4—91no—
133C2×C3×C22⋊C4—92no—
134C2×#217—32no—
135#937—316no—
136C2×(C2×D3)⋊C4—32no—
137C2×(C2×D3)⋊C4—32no—
138C23×C32—24no—
139C23×C32—22no—
140C23×C32—28no—
141C23×C32—24no—
142C23×C32—24no—
143C23×C32—24no—
144C23×C32—24no—
145C23×C32—21no—
146C23×C32—22no—
147C23×C32—22no—
148C23×C32—24no—
149C23×C32—38no—
150C23×C32—34no—
151C23×C32—41no—
152C23×C32—42no—
153C23×C32—44no—
154C23×C32—44no—
155C23×C32—44no—
156C23×C32—44no—
157C23×C32—48no—
158C23×C32—64no—
159C23×C32—64no—
160C23×C32C22×D312yes—
161C23×C32C22×D314yescomplemented
162C23×C32C23×C311yescharacteristic
163C23×C32C23×C312yescomplemented
164C22×C3⋊D3—24no—
165C22×C3⋊D3C3×D414yescomplemented
166C22×C3⋊D3C23×C312yes—
167C22×C3×D3—24no—
168C22×C3×D3—24no—
169C22×C3×D3—24no—
170C22×C3×D3—24no—
171C22×C3×D3—28no—
172C22×C3×D3—24no—
173C22×C3×D3—38no—
174C22×C3×D3—34no—
175C22×C3×D3—44no—
176C22×C3×D3—44no—
177C22×C3×D3—64no—
178C22×C3×D3—62no—
179C22×C3×D3—616no—
180C22×C3×D3—616no—
181C22×C3×D3—664no—
182C22×C3×D3—64no—
183C22×C3×D3—64no—
184C22×C3×D3—616no—
185C22×C3×D3—124no—
186C22×C3×D3#6718yescomplemented
187C22×C3×D3C3×D418yescomplemented
188C22×C3×D3C22×D314yes—
189C22×C3×D3C23×C314yes—
190C2×D32—232no—
191C2×D32—28no—
192C2×D32—216no—
193C32×D4—616no—
194C2×C32×C4—34no—
195C2×C32×C4—64no—
196#638—28no—
197C2×C3⋊Dic3—22no—
198C2×C3⋊Dic3C23×C312yescomplemented
199C3×#67—232no—
200C3×#67—216no—
201C3×#67—216no—
202C3×#67—616no—
203C3×#67—616no—
204C2×C3×Dic3—28no—
205C2×C3×Dic3—24no—
206C2×C3×Dic3—24no—
207C2×C3×Dic3—24no—
208C2×C3×Dic3—34no—
209C2×C3×Dic3—62no—
210C2×C3×Dic3—64no—
211C2×C3×Dic3—64no—
212C2×C3×Dic3C22×D314yescomplemented
213C2×C3×Dic3C23×C314yescomplemented
214C2×#120—91noSylow
215C2×C33—42no—
216C2×C33#12014yescomplemented
217C2×C33C22×D411yescharacteristic
218C2×C33C22×D412yes—
219C3×C3⋊D3—24no—
220C32×D3—28no—
221C24×C3—22no—
222C24×C3—21no—
223C24×C3—32no—
224C24×C3—32no—
225C24×C3—41no—
226C24×C3—62no—
227C24×C3—94no—
228C24×C3—91no—
229C24×C3—188no—
230C24×C3—184no—
231C24×C3D3211yescharacteristic, complemented
232C24×C3C2×C3×D312yescomplemented
233C23×D3—32no—
234C23×D3—38no—
235C23×D3—61no—
236C23×D3—68no—
237C23×D3—68no—
238C23×D3—68no—
239C23×D3C2×C3×D312yes—
240C2×C3×D4—38no—
241C2×C3×D4—38no—
242C2×C3×D4—68no—
243C2×C3×D4—68no—
244C2×C3×D4—68no—
245C2×C3×D4—616no—
246C2×C3×D4—94no—
247C2×C3×D4—184no—
248C22×C3×C4—32no—
249C22×C3×C4—32no—
250C22×C3×C4—62no—
251C22×C3×C4—91no—
252C2×#67—28no—
253C2×#67—38no—
254C2×#67—64no—
255C2×#67—68no—
256C2×#67—68no—
257C2×#67C2×C3×D318yescomplemented
258C22×Dic3—32no—
259C22×Dic3—61no—
260C22×Dic3C2×C3×D312yescomplemented
261C3×C22⋊C4—94no—
262C3×C22⋊C4—98no—
263#217—38no—
264(C2×D3)⋊C4—38no—
265(C2×D3)⋊C4—38no—
266C22×C32—24no—
267C22×C32—22no—
268C22×C32—28no—
269C22×C32—24no—
270C22×C32—24no—
271C22×C32—24no—
272C22×C32—24no—
273C22×C32—21no—
274C22×C32—22no—
275C22×C32—24no—
276C22×C32—22no—
277C22×C32—24no—
278C22×C32—24no—
279C22×C32—24no—
280C22×C32—38no—
281C22×C32—44no—
282C22×C32—42no—
283C22×C32—41no—
284C22×C32—44no—
285C22×C32—44no—
286C22×C32—44no—
287C22×C32—44no—
288C22×C32—48no—
289C22×C32—416no—
290C22×C32—48no—
291C22×C32—416no—
292C22×C32—48no—
293C22×C32—44no—
294C22×C32—616no—
295C22×C32—64no—
296C22×C32—64no—
297C22×C32D3×D414yes—
298C22×C32D3×D414yes—
299C22×C32C2×#6714yes—
300C22×C32C2×C3×D412yes—
301C22×C32C2×C3×D414yes—
302C22×C32C23×D312yes—
303C22×C32C24×C311yescharacteristic
304C2×C3⋊D3—28no—
305C2×C3⋊D3—24no—
306C2×C3⋊D3C2×C3×D414yescomplemented
307C2×C3×D3—216no—
308C2×C3×D3—216no—
309C2×C3×D3—24no—
310C2×C3×D3—24no—
311C2×C3×D3—24no—
312C2×C3×D3—24no—
313C2×C3×D3—28no—
314C2×C3×D3—38no—
315C2×C3×D3—416no—
316C2×C3×D3—44no—
317C2×C3×D3—44no—
318C2×C3×D3—64no—
319C2×C3×D3—664no—
320C2×C3×D3—616no—
321C2×C3×D3—64no—
322C2×C3×D3—64no—
323C2×C3×D3—124no—
324C2×C3×D3—128no—
325C2×C3×D3C2×#6718yescomplemented
326C2×C3×D3C2×C3×D418yescomplemented
327D32—232no—
328C32×C4—64no—
329C3⋊Dic3—22no—
330C3×Dic3—28no—
331C3×Dic3—24no—
332C3×Dic3—24no—
333C3×Dic3—64no—
334C3×Dic3—64no—
335C25—91no—
336C22×D4—32no—
337C22×D4—91no—
338#120—98no—
339C2×C22⋊C4—91no—
340C2×C22⋊C4—92no—
341C33C2×#12011yescharacteristic, complemented, Sylow
342C23×C3—24no—
343C23×C3—28no—
344C23×C3—24no—
345C23×C3—22no—
346C23×C3—24no—
347C23×C3—24no—
348C23×C3—24no—
349C23×C3—22no—
350C23×C3—21no—
351C23×C3—24no—
352C23×C3—22no—
353C23×C3—34no—
354C23×C3—34no—
355C23×C3—38no—
356C23×C3—38no—
357C23×C3—44no—
358C23×C3—44no—
359C23×C3—44no—
360C23×C3—48no—
361C23×C3—44no—
362C23×C3—41no—
363C23×C3—42no—
364C23×C3—64no—
365C23×C3—68no—
366C23×C3—64no—
367C23×C3—64no—
368C23×C3—64no—
369C23×C3—64no—
370C23×C3—92no—
371C23×C3—94no—
372C23×C3—124no—
373C23×C3—124no—
374C23×C3—1816no—
375C23×C3—188no—
376C23×C3—1832no—
377C23×C3—184no—
378C23×C3C2×D3211yescharacteristic
379C23×C3C2×D3212yescomplemented
380C23×C3C22×C3×D312yes—
381C23×C3C22×C3×D314yescomplemented
382C22×D3—24no—
383C22×D3—24no—
384C22×D3—34no—
385C22×D3—38no—
386C22×D3—64no—
387C22×D3—62no—
388C22×D3—616no—
389C22×D3—616no—
390C22×D3—664no—
391C22×D3—64no—
392C22×D3—64no—
393C22×D3—616no—
394C22×D3—124no—
395C22×D3C3×#6718yescomplemented
396C22×D3C22×C3×D314yes—
397C3×D4—616no—
398C3×D4—616no—
399C3×D4—632no—
400C3×D4—188no—
401C2×C3×C4—34no—
402C2×C3×C4—34no—
403C2×C3×C4—64no—
404C2×C3×C4—68no—
405C2×C3×C4—64no—
406C2×C3×C4—64no—
407C2×C3×C4—92no—
408C2×C3×C4—182no—
409#67—216no—
410#67—616no—
411#67—616no—
412C2×Dic3—24no—
413C2×Dic3—34no—
414C2×Dic3—62no—
415C2×Dic3—64no—
416C2×Dic3—64no—
417C2×Dic3C22×C3×D314yescomplemented
418C2×C32—28no—
419C2×C32—22no—
420C2×C32—21no—
421C2×C32—22no—
422C2×C32—22no—
423C2×C32—22no—
424C2×C32—24no—
425C2×C32—44no—
426C2×C32—42no—
427C2×C32—41no—
428C2×C32—48no—
429C2×C32—44no—
430C2×C32—48no—
431C2×C32—44no—
432C2×C32—42no—
433C2×C32—68no—
434C2×C32#93718yescomplemented
435C2×C32C3×#12014yescomplemented
436C2×C32C2×D3×D412yes—
437C2×C32C2×D3×D412yes—
438C2×C32C22×#6712yes—
439C2×C32C22×C3×D411yescharacteristic
440C2×C32C22×C3×D412yes—
441C3⋊D3—24no—
442C3×D3—28no—
443C3×D3—28no—
444C3×D3—48no—
445C3×D3—68no—
446C3×D3—124no—
447C24—32no—
448C24—94no—
449C24—91no—
450C24—188no—
451C24—184no—
452C24C3×D3211yescharacteristic, complemented
453C2×D4—38no—
454C2×D4—68no—
455C2×D4—94no—
456C2×D4—184no—
457C22×C4—32no—
458C22×C4—91no—
459C22⋊C4—94no—
460C22⋊C4—98no—
461C22×C3—24no—
462C22×C3—28no—
463C22×C3—24no—
464C22×C3—22no—
465C22×C3—24no—
466C22×C3—24no—
467C22×C3—24no—
468C22×C3—21no—
469C22×C3—22no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—24no—
473C22×C3—22no—
474C22×C3—24no—
475C22×C3—38no—
476C22×C3—38no—
477C22×C3—44no—
478C22×C3—44no—
479C22×C3—44no—
480C22×C3—48no—
481C22×C3—44no—
482C22×C3—48no—
483C22×C3—416no—
484C22×C3—416no—
485C22×C3—48no—
486C22×C3—44no—
487C22×C3—41no—
488C22×C3—44no—
489C22×C3—42no—
490C22×C3—68no—
491C22×C3—616no—
492C22×C3—64no—
493C22×C3—64no—
494C22×C3—64no—
495C22×C3—64no—
496C22×C3—616no—
497C22×C3—94no—
498C22×C3—124no—
499C22×C3—124no—
500C22×C3—1216no—
501C22×C3—1832no—
502C22×C3—184no—
503C22×C3—188no—
504C22×C3D3×#6714yes—
505C22×C3#381112yes—
506C22×C3C3×D3×D414yes—
507C22×C3C3×D3×D414yes—
508C22×C3C2×C3×#6714yes—
509C22×C3C22×D3211yescharacteristic
510C22×C3C23×C3×D312yes—
511C2×D3—216no—
512C2×D3—24no—
513C2×D3—24no—
514C2×D3—38no—
515C2×D3—64no—
516C2×D3—664no—
517C2×D3—616no—
518C2×D3—64no—
519C2×D3—64no—
520C2×D3—124no—
521C2×D3—128no—
522C2×D3C2×C3×#6718yescomplemented
523C3×C4—68no—
524C3×C4—64no—
525C3×C4—64no—
526C3×C4—182no—
527Dic3—24no—
528Dic3—64no—
529Dic3—64no—
530C32—22no—
531C32—21no—
532C32—41no—
533C32C2×#93712yescomplemented
534C32C2×C3×#12011yescharacteristic, complemented
535C23—24no—
536C23—22no—
537C23—34no—
538C23—38no—
539C23—64no—
540C23—64no—
541C23—92no—
542C23—94no—
543C23—1816no—
544C23—188no—
545C23—1832no—
546C23—184no—
547C23C2×C3×D3211yescharacteristic
548C23C2×C3×D3212yescomplemented
549D4—616no—
550D4—188no—
551C2×C4—34no—
552C2×C4—64no—
553C2×C4—92no—
554C2×C4—182no—
555C2×C3—22no—
556C2×C3—22no—
557C2×C3—22no—
558C2×C3—21no—
559C2×C3—22no—
560C2×C3—24no—
561C2×C3—28no—
562C2×C3—44no—
563C2×C3—48no—
564C2×C3—48no—
565C2×C3—44no—
566C2×C3—42no—
567C2×C3—42no—
568C2×C3—41no—
569C2×C3—44no—
570C2×C3—68no—
571C2×C3—68no—
572C2×C3—128no—
573C2×C3—184no—
574C2×C3#6398314yesdirect product factor
575C2×C3C3×#93718yescomplemented
576C2×C3C2×D3×#6712yes—
577C2×C3C2×#381111yescharacteristic
578C2×C3C2×C3×D3×D412yes—
579C2×C3C2×C3×D3×D412yes—
580C2×C3C22×C3×#6712yes—
581D3—28no—
582D3—68no—
583D3—124no—
584C22—24no—
585C22—22no—
586C22—38no—
587C22—44no—
588C22—64no—
589C22—64no—
590C22—616no—
591C22—94no—
592C22—1832no—
593C22—184no—
594C22—188no—
595C22C3×D3×#6714yes—
596C22C3×#381112yes—
597C22C22×C3×D3211yescharacteristic
598C4—64no—
599C4—182no—
600C3—21no—
601C3—22no—
602C3—41no—
603C3C2×#6398311yescharacteristic, direct product factor
604C3C2×C3×#93712yescomplemented
605C2—42no—
606C2—68no—
607C2—184no—
608C2C3×#6398314yesdirect product factor
609C2C2×C3×D3×#6712yes—
610C2C2×C3×#381111yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC24×C33C22
socleC23×C33C23
hypercenterC24×C3D32
hypocenterC32C2×C3×#120
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
1. C23×C3×D32C2semidirect
2. C22×C3×#638C2semidirect
3. C22×C32×#67C2semidirect
4. C3×#63983C2direct (N × H)
5. C2×C3×#3722C2semidirect
6. C2×C3×(C2×D3)⋊Dic3C2semidirect
8. C2×#63983C3direct (N × H)
9. C24×C33C22semidirect
14. C22×C3×D32C22semidirect
16. C2×C3×#638C22semidirect
17. C22×C3×C3⋊Dic3C22semidirect
19. C2×C32×#67C22semidirect
20. C22×C32×Dic3C22semidirect
21. C3×#3722C22semidirect
22. C3×(C2×D3)⋊Dic3C22semidirect
24. C23×D32C2×C3semidirect
26. C22×#638C2×C3semidirect
30. C22×C3×#67D3semidirect
31. C22×C3×#67C2×C3semidirect
36. #63983C2×C3direct (N × H)
37. C2×#3722C2×C3semidirect
38. C2×(C2×D3)⋊Dic3C2×C3semidirect
42. C23×C33C23semidirect
44. C22×C3×C3⋊D3D4semidirect
48. C22×C32×D3D4semidirect
55. C2×C3×C3⋊Dic3C23semidirect
58. C2×C32×Dic3C23semidirect
65. C24×C32C2×D3semidirect
66. C24×C32C22×C3semidirect
79. C22×D32C22×C3semidirect
84. C2×#638C22×C3semidirect
85. C22×C3⋊Dic3C22×C3semidirect
94. C2×C3×#67C2×D3semidirect
95. C2×C3×#67C22×C3semidirect
99. C22×C3×Dic3C2×D3semidirect
100. C22×C3×Dic3C22×C3semidirect
104. #3722C22×C3semidirect
105. (C2×D3)⋊Dic3C22×C3semidirect
114. C2×C3×C3⋊D3C2×D4semidirect
118. C2×C32×D3C2×D4semidirect
130. C22×#67C3×D3semidirect
161. C23×C32C22×D3semidirect
163. C23×C32C23×C3semidirect
165. C22×C3⋊D3C3×D4semidirect
186. C22×C3×D3#67semidirect
187. C22×C3×D3C3×D4semidirect
198. C2×C3⋊Dic3C23×C3semidirect
212. C2×C3×Dic3C22×D3semidirect
213. C2×C3×Dic3C23×C3semidirect
216. C2×C33#120semidirect
231. C24×C3D32semidirect
232. C24×C3C2×C3×D3semidirect
257. C2×#67C2×C3×D3semidirect
260. C22×Dic3C2×C3×D3semidirect
306. C2×C3⋊D3C2×C3×D4semidirect
325. C2×C3×D3C2×#67semidirect
326. C2×C3×D3C2×C3×D4semidirect
341. C33C2×#120semidirect
379. C23×C3C2×D32semidirect
381. C23×C3C22×C3×D3semidirect
395. C22×D3C3×#67semidirect
417. C2×Dic3C22×C3×D3semidirect
434. C2×C32#937semidirect
435. C2×C32C3×#120semidirect
452. C24C3×D32semidirect
522. C2×D3C2×C3×#67semidirect
533. C32C2×#937semidirect
534. C32C2×C3×#120semidirect
548. C23C2×C3×D32semidirect
574. C2×C3#63983direct (N × H)
575. C2×C3C3×#937semidirect
603. C3C2×#63983direct (N × H)
604. C3C2×C3×#937semidirect
608. C2C3×#63983direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×D322
C22×C3×#6382
C22×C32×#672
C3×#639832
C2×C3×#37222
C2×C3×(C2×D3)⋊Dic32
C2×C3×#9373
C2×#639833