Finite Group Databasebeta

← C2×C3×#63966 #13175080

Subgroups — C2×C3×#63966 #13175080

Order 1728.

2106 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#638C211yesmaximal, characteristic
2C23×C32×Dic3C211yesmaximal, characteristic, complemented
3C2×C32×#217C211yesmaximal, characteristic, complemented
4C3×#63966C218yesmaximal, direct product factor
5C2×C3×Dic3⋊Dic3C212yesmaximal, complemented
6C2×C3×#3722C212yesmaximal, complemented
7C2×C3×#929—31nomaximal
8C2×C3×#886—31nomaximal
9C2×#63966C311yesmaximal, characteristic, direct product factor
10C24×C33C2211yescharacteristic
11C23×C3×C3⋊D3C2211yescharacteristic
12C2×C3×#638—24no—
13C2×C3×#638C2214yes—
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C32×Dic3—24no—
16C22×C32×Dic3C2212yes—
17C22×C32×Dic3C2212yes—
18C22×C32×Dic3C2214yescomplemented
19C32×#217C2214yescomplemented
20C3×Dic3⋊Dic3C2218yescomplemented
21C3×#3722C2218yescomplemented
22C23×C32×C4—31no—
23C22×#638C2×C311yescharacteristic
24C22×C3×#67—31no—
25C22×C3×#67—31no—
26C22×C3×#67—61no—
27C23×C3×Dic3—21no—
28C23×C3×Dic3D311yescharacteristic, complemented
29C23×C3×Dic3C2×C311yescharacteristic, complemented
30C2×C32×C22⋊C4—31no—
31C2×C3×#217—21no—
32C2×C3×#217D311yescharacteristic, complemented
33C2×C3×#217C2×C311yescharacteristic, complemented
34C3×#929—38no—
35C2×C3×(C2×D3)⋊C4—32no—
36C2×C3×(C2×D3)⋊C4—32no—
37C2×C3×C4⋊Dic3—32no—
38C2×C3×Dic3⋊C4—32no—
39C3×#886—38no—
40#63966C2×C318yesdirect product factor
41C2×Dic3⋊Dic3C2×C312yescomplemented
42C2×#3722C2×C312yescomplemented
43C23×C33—24no—
44C23×C33D414yes—
45C23×C33C2312yes—
46C23×C33C2311yescharacteristic
47C22×C3×C3⋊D3—22no—
48C22×C3×C3⋊D3—24no—
49C22×C3×C3⋊D3C2312yes—
50C3×#638—28no—
51C2×C3×C3⋊Dic3—22no—
52C2×C3×C3⋊Dic3C2312yes—
53C2×C32×Dic3—24no—
54C2×C32×Dic3—28no—
55C2×C32×Dic3—24no—
56C2×C32×Dic3C2314yes—
57C2×C32×Dic3C2314yes—
58C2×C3×#123—91no—
59C2×#929—31no—
60C2×#886—31no—
61C24×C32—21no—
62C24×C32—21no—
63C24×C32—21no—
64C24×C32—41no—
65C24×C32C2×D311yescharacteristic
66C24×C32C2×D311yescharacteristic
67C24×C32C22×C311yescharacteristic
68C23×C3⋊D3C22×C311yescharacteristic
69C23×C3×D3—31no—
70C23×C3×D3—31no—
71C23×C3×D3—61no—
72C22×C32×C4—32no—
73C22×C32×C4—32no—
74C22×C32×C4—34no—
75C22×C32×C4—64no—
76C2×#638—24no—
77C2×#638C22×C314yes—
78C22×C3⋊Dic3C22×C311yescharacteristic
79C2×C3×#67—34no—
80C2×C3×#67—34no—
81C2×C3×#67—64no—
82C2×C3×#67—68no—
83C2×C3×#67—64no—
84C2×C3×#67—64no—
85C22×C3×Dic3—28no—
86C22×C3×Dic3—22no—
87C22×C3×Dic3—24no—
88C22×C3×Dic3—24no—
89C22×C3×Dic3—24no—
90C22×C3×Dic3—22no—
91C22×C3×Dic3—31no—
92C22×C3×Dic3—31no—
93C22×C3×Dic3—61no—
94C22×C3×Dic3C2×D312yes—
95C22×C3×Dic3C2×D312yes—
96C22×C3×Dic3C2×D314yescomplemented
97C22×C3×Dic3C22×C312yes—
98C22×C3×Dic3C22×C312yes—
99C22×C3×Dic3C22×C314yescomplemented
100C32×C22⋊C4—34no—
101C3×#217—24no—
102C3×#217C2×D314yescomplemented
103C3×#217C22×C314yescomplemented
104C3×(C2×D3)⋊C4—38no—
105C3×(C2×D3)⋊C4—38no—
106C3×C4⋊Dic3—38no—
107C3×Dic3⋊C4—38no—
108Dic3⋊Dic3C22×C318yescomplemented
109#3722C22×C318yescomplemented
110C22×C33—28no—
111C22×C33—24no—
112C22×C33C2×D414yes—
113C22×C33C2×D412yes—
114C22×C33#4114yes—
115C22×C33C2411yescharacteristic
116C2×C3×C3⋊D3—22no—
117C2×C3×C3⋊D3—24no—
118C2×C3×C3⋊D3—44no—
119C3×C3⋊Dic3—22no—
120C32×Dic3—24no—
121C32×Dic3—24no—
122C22×C3×D4—91no—
123C23×C3×C4—31no—
124C23×C3×C4—31no—
125C23×C3×C4—61no—
126C22×#67—31no—
127C22×#67—31no—
128C22×#67—61no—
129C23×Dic3C3×D311yescharacteristic, complemented
130C3×#123—98no—
131C2×C3×C4⋊C4—92no—
132C2×C3×C22⋊C4—31no—
133C2×C3×C22⋊C4—31no—
134C2×C3×C22⋊C4—61no—
135C2×C3×C22⋊C4—92no—
136C2×#217C3×D311yescharacteristic, complemented
137#929—38no—
138C2×(C2×D3)⋊C4—32no—
139C2×(C2×D3)⋊C4—32no—
140C2×C4⋊Dic3—32no—
141C2×Dic3⋊C4—32no—
142#886—38no—
143C23×C32—24no—
144C23×C32—28no—
145C23×C32—24no—
146C23×C32—24no—
147C23×C32—24no—
148C23×C32—24no—
149C23×C32—24no—
150C23×C32—22no—
151C23×C32—21no—
152C23×C32—22no—
153C23×C32—21no—
154C23×C32—21no—
155C23×C32—22no—
156C23×C32—44no—
157C23×C32—41no—
158C23×C32—42no—
159C23×C32—44no—
160C23×C32—44no—
161C23×C32—48no—
162C23×C32D1214yes—
163C23×C32#6714yes—
164C23×C32C3×D414yes—
165C23×C32C22×D312yes—
166C23×C32C22×D312yes—
167C23×C32C22×D311yescharacteristic
168C23×C32C22×D311yescharacteristic
169C23×C32C23×C312yes—
170C23×C32C23×C311yescharacteristic
171C22×C3⋊D3—22no—
172C22×C3⋊D3—24no—
173C22×C3⋊D3C23×C312yes—
174C22×C3×D3—32no—
175C22×C3×D3—32no—
176C22×C3×D3—62no—
177C22×C3×D3—64no—
178C22×C3×D3—62no—
179C22×C3×D3—64no—
180C22×C3×D3—64no—
181C22×C3×D3—62no—
182C22×C3×D3—124no—
183C2×C32×C4—34no—
184C2×C32×C4—34no—
185C2×C32×C4—68no—
186C2×C32×C4—64no—
187C2×C32×C4—64no—
188#638—28no—
189C2×C3⋊Dic3—22no—
190C2×C3⋊Dic3C23×C312yes—
191C3×#67—616no—
192C3×#67—68no—
193C3×#67—68no—
194C2×C3×Dic3—28no—
195C2×C3×Dic3—216no—
196C2×C3×Dic3—24no—
197C2×C3×Dic3—24no—
198C2×C3×Dic3—28no—
199C2×C3×Dic3—24no—
200C2×C3×Dic3—28no—
201C2×C3×Dic3—24no—
202C2×C3×Dic3—24no—
203C2×C3×Dic3—24no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—44no—
207C2×C3×Dic3—62no—
208C2×C3×Dic3—64no—
209C2×C3×Dic3—62no—
210C2×C3×Dic3—62no—
211C2×C3×Dic3C22×D314yes—
212C2×C3×Dic3C22×D314yes—
213C2×C3×Dic3C23×C314yes—
214C2×C3×Dic3C23×C314yes—
215C2×#123—91noSylow
216C2×C33—24no—
217C2×C33#12314yescomplemented
218C2×C33C22×D411yescharacteristic
219C2×C33C2×#4112yes—
220C3×C3⋊D3—42no—
221C24×C3—21no—
222C24×C3—21no—
223C24×C3—21no—
224C24×C3—41no—
225C24×C3—91no—
226C24×C3D3211yescharacteristic
227C24×C3C2×C3×D311yescharacteristic
228C24×C3C2×C3×D311yescharacteristic
229C23×D3—31no—
230C23×D3—31no—
231C23×D3—61no—
232C2×C3×D4—94no—
233C2×C3×D4—184no—
234C22×C3×C4—34no—
235C22×C3×C4—32no—
236C22×C3×C4—32no—
237C22×C3×C4—32no—
238C22×C3×C4—32no—
239C22×C3×C4—34no—
240C22×C3×C4—62no—
241C22×C3×C4—64no—
242C22×C3×C4—62no—
243C22×C3×C4—68no—
244C22×C3×C4—64no—
245C22×C3×C4—64no—
246C22×C3×C4—91no—
247C2×#67—34no—
248C2×#67—34no—
249C2×#67—64no—
250C2×#67—68no—
251C2×#67—64no—
252C2×#67—64no—
253C22×Dic3—24no—
254C22×Dic3—31no—
255C22×Dic3—31no—
256C22×Dic3—61no—
257C22×Dic3C2×C3×D312yes—
258C22×Dic3C2×C3×D312yes—
259C22×Dic3C2×C3×D314yescomplemented
260C3×C4⋊C4—98no—
261C3×C22⋊C4—34no—
262C3×C22⋊C4—34no—
263C3×C22⋊C4—64no—
264C3×C22⋊C4—98no—
265#217C2×C3×D314yescomplemented
266(C2×D3)⋊C4—38no—
267(C2×D3)⋊C4—38no—
268C4⋊Dic3—38no—
269Dic3⋊C4—38no—
270C22×C32—28no—
271C22×C32—24no—
272C22×C32—216no—
273C22×C32—28no—
274C22×C32—28no—
275C22×C32—28no—
276C22×C32—24no—
277C22×C32—24no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—24no—
282C22×C32—22no—
283C22×C32—24no—
284C22×C32—22no—
285C22×C32—24no—
286C22×C32—24no—
287C22×C32—24no—
288C22×C32—24no—
289C22×C32—22no—
290C22×C32—42no—
291C22×C32—44no—
292C22×C32—44no—
293C22×C32—41no—
294C22×C32—48no—
295C22×C32—44no—
296C22×C32—44no—
297C22×C32—48no—
298C22×C32—416no—
299C22×C32—48no—
300C22×C32C2×D1214yes—
301C22×C32C2×D1212yes—
302C22×C32#23514yes—
303C22×C32#23714yes—
304C22×C32C2×#6714yes—
305C22×C32C2×#6712yes—
306C22×C32C2×C3×D414yes—
307C22×C32C2×C3×D412yes—
308C22×C32C3×#4114yes—
309C22×C32C23×D311yescharacteristic
310C22×C32C23×D311yescharacteristic
311C22×C32C24×C311yescharacteristic
312C2×C3⋊D3—22no—
313C2×C3⋊D3—24no—
314C2×C3⋊D3—44no—
315C2×C3×D3—64no—
316C2×C3×D3—62no—
317C2×C3×D3—64no—
318C2×C3×D3—64no—
319C2×C3×D3—62no—
320C2×C3×D3—124no—
321C2×C3×D3—128no—
322C2×C3×D3—124no—
323C2×C3×D3—124no—
324C32×C4—64no—
325C32×C4—64no—
326C3⋊Dic3—22no—
327C3×Dic3—28no—
328C3×Dic3—24no—
329C3×Dic3—24no—
330C3×Dic3—24no—
331C3×Dic3—24no—
332C3×Dic3—44no—
333C3×Dic3—64no—
334C3×Dic3—62no—
335C3×Dic3—62no—
336C22×D4—91no—
337C23×C4—31no—
338#123—98no—
339C2×C4⋊C4—92no—
340C2×C22⋊C4—31no—
341C2×C22⋊C4—92no—
342C33C2×#12311yescharacteristic, complemented, Sylow
343C23×C3—28no—
344C23×C3—24no—
345C23×C3—24no—
346C23×C3—24no—
347C23×C3—24no—
348C23×C3—24no—
349C23×C3—24no—
350C23×C3—22no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—44no—
357C23×C3—44no—
358C23×C3—48no—
359C23×C3—44no—
360C23×C3—42no—
361C23×C3—41no—
362C23×C3—92no—
363C23×C3—184no—
364C23×C3—182no—
365C23×C3#62614yes—
366C23×C3C3×D1214yes—
367C23×C3C3×#6714yes—
368C23×C3C2×D3212yes—
369C23×C3C2×D3211yescharacteristic
370C23×C3C22×C3×D312yes—
371C23×C3C22×C3×D312yes—
372C23×C3C22×C3×D311yescharacteristic
373C23×C3C22×C3×D311yescharacteristic
374C22×D3—32no—
375C22×D3—32no—
376C22×D3—62no—
377C22×D3—64no—
378C22×D3—62no—
379C22×D3—64no—
380C22×D3—64no—
381C22×D3—62no—
382C22×D3—124no—
383C3×D4—188no—
384C2×C3×C4—34no—
385C2×C3×C4—34no—
386C2×C3×C4—34no—
387C2×C3×C4—34no—
388C2×C3×C4—64no—
389C2×C3×C4—64no—
390C2×C3×C4—616no—
391C2×C3×C4—68no—
392C2×C3×C4—64no—
393C2×C3×C4—68no—
394C2×C3×C4—64no—
395C2×C3×C4—68no—
396C2×C3×C4—64no—
397C2×C3×C4—64no—
398C2×C3×C4—92no—
399C2×C3×C4—124no—
400C2×C3×C4—182no—
401#67—616no—
402#67—68no—
403#67—68no—
404C2×Dic3—24no—
405C2×Dic3—24no—
406C2×Dic3—28no—
407C2×Dic3—32no—
408C2×Dic3—32no—
409C2×Dic3—62no—
410C2×Dic3—64no—
411C2×Dic3—62no—
412C2×Dic3—62no—
413C2×Dic3C22×C3×D314yes—
414C2×Dic3C22×C3×D314yes—
415C2×C32—24no—
416C2×C32—28no—
417C2×C32—24no—
418C2×C32—24no—
419C2×C32—24no—
420C2×C32—24no—
421C2×C32—22no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—22no—
425C2×C32—22no—
426C2×C32—21no—
427C2×C32—24no—
428C2×C32—44no—
429C2×C32—41no—
430C2×C32—42no—
431C2×C32—44no—
432C2×C32—44no—
433C2×C32—48no—
434C2×C32#88614yescomplemented
435C2×C32#92914yescomplemented
436C2×C32C3×#12314yescomplemented
437C2×C32C22×D1211yescharacteristic
438C2×C32C2×#23512yes—
439C2×C32C2×#23712yes—
440C2×C32C22×#6711yescharacteristic
441C2×C32C22×C3×D411yescharacteristic
442C2×C32C2×C3×#4112yes—
443C3⋊D3—42no—
444C3×D3—124no—
445C3×D3—122no—
446C3×D3—122no—
447C24—91no—
448C24C3×D3211yescharacteristic
449C2×D4—94no—
450C2×D4—184no—
451C22×C4—32no—
452C22×C4—32no—
453C22×C4—34no—
454C22×C4—64no—
455C22×C4—91no—
456C4⋊C4—98no—
457C22⋊C4—34no—
458C22⋊C4—98no—
459C22×C3—28no—
460C22×C3—216no—
461C22×C3—28no—
462C22×C3—28no—
463C22×C3—24no—
464C22×C3—24no—
465C22×C3—24no—
466C22×C3—28no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—24no—
473C22×C3—22no—
474C22×C3—24no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—24no—
478C22×C3—24no—
479C22×C3—44no—
480C22×C3—44no—
481C22×C3—48no—
482C22×C3—48no—
483C22×C3—48no—
484C22×C3—416no—
485C22×C3—41no—
486C22×C3—44no—
487C22×C3—44no—
488C22×C3—42no—
489C22×C3—184no—
490C22×C3—182no—
491C22×C3—364no—
492C22×C3#380414yes—
493C22×C3C2×#62614yes—
494C22×C3C2×#62612yes—
495C22×C3C2×C3×D1214yes—
496C22×C3C2×C3×D1212yes—
497C22×C3C3×#23514yes—
498C22×C3C3×#23714yes—
499C22×C3C2×C3×#6714yes—
500C22×C3C2×C3×#6712yes—
501C22×C3C22×D3211yescharacteristic
502C22×C3C23×C3×D311yescharacteristic
503C22×C3C23×C3×D311yescharacteristic
504C2×D3—64no—
505C2×D3—62no—
506C2×D3—64no—
507C2×D3—64no—
508C2×D3—62no—
509C2×D3—124no—
510C2×D3—128no—
511C2×D3—124no—
512C2×D3—124no—
513C3×C4—68no—
514C3×C4—64no—
515C3×C4—64no—
516C3×C4—64no—
517C3×C4—64no—
518C3×C4—124no—
519C3×C4—182no—
520Dic3—24no—
521Dic3—24no—
522Dic3—64no—
523Dic3—62no—
524Dic3—62no—
525C32—21no—
526C32—21no—
527C32—21no—
528C32—41no—
529C32C2×#88611yescharacteristic, complemented
530C32C2×#92911yescharacteristic, complemented
531C32C2×C3×#12311yescharacteristic, complemented
532C23—24no—
533C23—92no—
534C23—184no—
535C23—182no—
536C23C3×#62614yes—
537C23C2×C3×D3212yes—
538C23C2×C3×D3211yescharacteristic
539D4—188no—
540C2×C4—34no—
541C2×C4—34no—
542C2×C4—64no—
543C2×C4—68no—
544C2×C4—64no—
545C2×C4—92no—
546C2×C4—182no—
547C2×C3—28no—
548C2×C3—24no—
549C2×C3—24no—
550C2×C3—24no—
551C2×C3—22no—
552C2×C3—22no—
553C2×C3—22no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—24no—
558C2×C3—24no—
559C2×C3—24no—
560C2×C3—44no—
561C2×C3—44no—
562C2×C3—48no—
563C2×C3—42no—
564C2×C3—41no—
565C2×C3—44no—
566C2×C3—362no—
567C2×C3#6396614yesdirect product factor
568C2×C3C3×#88614yescomplemented
569C2×C3C3×#92914yescomplemented
570C2×C3C2×#380412yes—
571C2×C3C22×#62611yescharacteristic
572C2×C3C22×C3×D1211yescharacteristic
573C2×C3C2×C3×#23512yes—
574C2×C3C2×C3×#23712yes—
575C2×C3C22×C3×#6711yescharacteristic
576D3—124no—
577D3—122no—
578D3—122no—
579C22—24no—
580C22—28no—
581C22—184no—
582C22—182no—
583C22—364no—
584C22C3×#380414yes—
585C22C2×C3×#62614yes—
586C22C2×C3×#62612yes—
587C22C22×C3×D3211yescharacteristic
588C4—64no—
589C4—64no—
590C4—182no—
591C3—21no—
592C3—21no—
593C3—21no—
594C3—41no—
595C3C2×#6396611yescharacteristic, direct product factor
596C3C2×C3×#88611yescharacteristic, complemented
597C3C2×C3×#92911yescharacteristic, complemented
598C2—24no—
599C2—362no—
600C2C3×#6396614yesdirect product factor
601C2C2×C3×#380412yes—
602C2C22×C3×#62611yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC24×C33C22
socleC23×C33C23
prime radicalC22×C3×#638C2
hypercenterC24×C3D32
hypocenterC32C2×C3×#123
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
2. C23×C32×Dic3C2semidirect
3. C2×C32×#217C2semidirect
4. C3×#63966C2direct (N × H)
5. C2×C3×Dic3⋊Dic3C2semidirect
6. C2×C3×#3722C2semidirect
9. C2×#63966C3direct (N × H)
18. C22×C32×Dic3C22semidirect
19. C32×#217C22semidirect
20. C3×Dic3⋊Dic3C22semidirect
21. C3×#3722C22semidirect
28. C23×C3×Dic3D3semidirect
29. C23×C3×Dic3C2×C3semidirect
32. C2×C3×#217D3semidirect
33. C2×C3×#217C2×C3semidirect
40. #63966C2×C3direct (N × H)
41. C2×Dic3⋊Dic3C2×C3semidirect
42. C2×#3722C2×C3semidirect
96. C22×C3×Dic3C2×D3semidirect
99. C22×C3×Dic3C22×C3semidirect
102. C3×#217C2×D3semidirect
103. C3×#217C22×C3semidirect
108. Dic3⋊Dic3C22×C3semidirect
109. #3722C22×C3semidirect
129. C23×Dic3C3×D3semidirect
136. C2×#217C3×D3semidirect
217. C2×C33#123semidirect
259. C22×Dic3C2×C3×D3semidirect
265. #217C2×C3×D3semidirect
342. C33C2×#123semidirect
434. C2×C32#886semidirect
435. C2×C32#929semidirect
436. C2×C32C3×#123semidirect
529. C32C2×#886semidirect
530. C32C2×#929semidirect
531. C32C2×C3×#123semidirect
567. C2×C3#63966direct (N × H)
568. C2×C3C3×#886semidirect
569. C2×C3C3×#929semidirect
595. C3C2×#63966direct (N × H)
596. C3C2×C3×#886semidirect
597. C3C2×C3×#929semidirect
600. C2C3×#63966direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C23×C32×Dic32
C2×C32×#2172
C3×#639662
C2×C3×Dic3⋊Dic32
C2×C3×#37222
C2×C3×#9293
C2×C3×#8863
C2×#639663