Finite Group Databasebeta

← C2×C3×#63980 #13175110

Subgroups — C2×C3×#63980 #13175110

Order 1728.

2546 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C22×C32×#67C212yesmaximal, complemented
3C22×C3×#625C211yesmaximal, characteristic, complemented
4C3×#63980C218yesmaximal, direct product factor
5C2×C3×#3724C211yesmaximal, characteristic, complemented
6C2×C3×(C2×D3)⋊Dic3C212yesmaximal, complemented
7C2×C3×#939—32nomaximal
8C2×#63980C311yesmaximal, characteristic, direct product factor
9C24×C33C2211yescharacteristic
10C23×C32×D3C2212yes—
11C22×C3×C3⋊Dic3—24no—
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C22×C3×C3⋊Dic3C2214yescomplemented
15C2×C32×#67—28no—
16C2×C32×#67C2218yescomplemented
17C22×C32×Dic3C2212yes—
18C2×C3×#625—24no—
19C2×C3×#625C2214yescomplemented
20C3×#3724C2214yescomplemented
21C3×(C2×D3)⋊Dic3C2218yescomplemented
22C22×C32×D4—32no—
23C23×C3⋊Dic3C2×C311yescharacteristic, complemented
24C22×C3×#67—22no—
25C22×C3×#67—32no—
26C22×C3×#67D312yescomplemented
27C22×C3×#67C2×C312yescomplemented
28C23×C3×Dic3—32no—
29C23×C3×Dic3—61no—
30C22×#625C2×C311yescharacteristic, complemented
31C2×C3×#217—32no—
32C3×#939—316no—
33C2×C3×(C2×D3)⋊C4—32no—
34C2×C3×Dic3⋊C4—32no—
35#63980C2×C318yesdirect product factor
36C2×#3724C2×C311yescharacteristic, complemented
37C2×(C2×D3)⋊Dic3C2×C312yescomplemented
38C23×C33—22no—
39C23×C33—22no—
40C23×C33D414yescomplemented
41C23×C33C2312yes—
42C23×C33C2311yescharacteristic
43C22×C32×D3—24no—
44C22×C32×D3—24no—
45C22×C32×D3—24no—
46C22×C32×D3C2314yes—
47C2×C3×C3⋊Dic3—22no—
48C2×C3×C3⋊Dic3—28no—
49C2×C3×C3⋊Dic3D414yescomplemented
50C2×C3×C3⋊Dic3C2312yes—
51C2×C3×C3⋊Dic3C2312yes—
52C32×#67—216no—
53C2×C32×Dic3—24no—
54C2×C32×Dic3C2314yes—
55C3×#625—28no—
56C2×C3×#121—91no—
57C2×#939—32no—
58C24×C32—21no—
59C24×C32—22no—
60C24×C32—32no—
61C24×C32—41no—
62C24×C32C2×D312yes—
63C24×C32C22×C311yescharacteristic
64C23×C3×D3—22no—
65C23×C3×D3C2×D312yes—
66C23×C3×D3C22×C312yes—
67C2×C32×D4—38no—
68C2×C32×D4—68no—
69C22×C32×C4—32no—
70C22×C3⋊Dic3—24no—
71C22×C3⋊Dic3C22×C311yescharacteristic
72C22×C3⋊Dic3C22×C311yescharacteristic
73C22×C3⋊Dic3C22×C314yescomplemented
74C2×C3×#67—216no—
75C2×C3×#67—28no—
76C2×C3×#67—28no—
77C2×C3×#67—28no—
78C2×C3×#67—38no—
79C2×C3×#67—68no—
80C2×C3×#67C2×D318yescomplemented
81C2×C3×#67C22×C318yescomplemented
82C22×C3×Dic3—22no—
83C22×C3×Dic3—32no—
84C22×C3×Dic3—32no—
85C22×C3×Dic3—38no—
86C22×C3×Dic3—68no—
87C22×C3×Dic3—61no—
88C22×C3×Dic3—61no—
89C22×C3×Dic3—64no—
90C22×C3×Dic3—68no—
91C22×C3×Dic3C2×D312yes—
92C22×C3×Dic3C22×C312yes—
93C2×#625—24no—
94C2×#625C22×C314yescomplemented
95C3×#217—38no—
96C3×(C2×D3)⋊C4—38no—
97C3×Dic3⋊C4—38no—
98#3724C22×C314yescomplemented
99(C2×D3)⋊Dic3C22×C318yescomplemented
100C22×C33—28no—
101C22×C33—22no—
102C22×C33—22no—
103C22×C33C2×D412yes—
104C22×C33C2×D412yes—
105C22×C33C2×D414yescomplemented
106C22×C33#4112yes—
107C22×C33C2411yescharacteristic
108C2×C32×D3—24no—
109C2×C32×D3—24no—
110C2×C32×D3—24no—
111C2×C32×D3—48no—
112C3×C3⋊Dic3—22no—
113C3×C3⋊Dic3C2×D414yescomplemented
114C32×Dic3—24no—
115C22×C3×D4—32no—
116C22×C3×D4—32no—
117C22×C3×D4—62no—
118C22×C3×D4—91no—
119C23×C3×C4—91no—
120C22×#67—32no—
121C22×#67C3×D312yescomplemented
122C23×Dic3—32no—
123C23×Dic3—61no—
124C3×#121—98no—
125C2×C3×C4⋊C4—91no—
126C2×C3×C22⋊C4—92no—
127C2×#217—32no—
128#939—316no—
129C2×(C2×D3)⋊C4—32no—
130C2×Dic3⋊C4—32no—
131C23×C32—22no—
132C23×C32—22no—
133C23×C32—24no—
134C23×C32—24no—
135C23×C32—24no—
136C23×C32—24no—
137C23×C32—24no—
138C23×C32—28no—
139C23×C32—22no—
140C23×C32—21no—
141C23×C32—22no—
142C23×C32—24no—
143C23×C32—34no—
144C23×C32—44no—
145C23×C32—41no—
146C23×C32—42no—
147C23×C32—44no—
148C23×C32—44no—
149C23×C32—44no—
150C23×C32—44no—
151C23×C32—64no—
152C23×C32—64no—
153C23×C32—64no—
154C23×C32#6718yescomplemented
155C23×C32C3×D414yescomplemented
156C23×C32C22×D314yes—
157C23×C32C22×D312yes—
158C23×C32C23×C312yes—
159C23×C32C23×C311yescharacteristic
160C22×C3×D3—28no—
161C22×C3×D3—24no—
162C22×C3×D3—24no—
163C22×C3×D3—24no—
164C22×C3×D3—24no—
165C22×C3×D3—24no—
166C22×C3×D3—24no—
167C22×C3×D3—24no—
168C22×C3×D3—44no—
169C22×C3×D3—44no—
170C22×C3×D3C22×D314yes—
171C22×C3×D3C23×C314yes—
172C32×D4—616no—
173C2×C32×C4—34no—
174C2×C32×C4—64no—
175C2×C3⋊Dic3—22no—
176C2×C3⋊Dic3—28no—
177C2×C3⋊Dic3C3×D414yescomplemented
178C2×C3⋊Dic3C23×C312yes—
179C2×C3⋊Dic3C23×C312yes—
180C3×#67—232no—
181C3×#67—216no—
182C3×#67—216no—
183C3×#67—616no—
184C2×C3×Dic3—28no—
185C2×C3×Dic3—24no—
186C2×C3×Dic3—24no—
187C2×C3×Dic3—24no—
188C2×C3×Dic3—38no—
189C2×C3×Dic3—34no—
190C2×C3×Dic3—34no—
191C2×C3×Dic3—64no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—62no—
194C2×C3×Dic3—616no—
195C2×C3×Dic3—616no—
196C2×C3×Dic3—64no—
197C2×C3×Dic3—64no—
198C2×C3×Dic3C22×D314yes—
199C2×C3×Dic3C23×C314yes—
200#625—28no—
201C2×#121—91noSylow
202C2×C33—24no—
203C2×C33#12114yescomplemented
204C2×C33C22×D411yescharacteristic
205C2×C33C22×D411yescharacteristic
206C2×C33C2×#4111yescharacteristic
207C32×D3—44no—
208C24×C3—21no—
209C24×C3—22no—
210C24×C3—32no—
211C24×C3—32no—
212C24×C3—41no—
213C24×C3—62no—
214C24×C3D3211yescharacteristic
215C24×C3C2×C3×D312yes—
216C23×D3C2×C3×D312yes—
217C2×C3×D4—38no—
218C2×C3×D4—38no—
219C2×C3×D4—68no—
220C2×C3×D4—616no—
221C2×C3×D4—68no—
222C2×C3×D4—68no—
223C2×C3×D4—94no—
224C2×C3×D4—184no—
225C22×C3×C4—32no—
226C22×C3×C4—32no—
227C22×C3×C4—62no—
228C22×C3×C4—91no—
229C22×C3×C4—94no—
230C22×C3×C4—91no—
231C22×C3×C4—184no—
232C2×#67—28no—
233C2×#67—38no—
234C2×#67—68no—
235C2×#67C2×C3×D318yescomplemented
236C22×Dic3—32no—
237C22×Dic3—32no—
238C22×Dic3—38no—
239C22×Dic3—61no—
240C22×Dic3—61no—
241C22×Dic3—64no—
242C22×Dic3—68no—
243C22×Dic3—68no—
244C22×Dic3C2×C3×D312yes—
245C3×C4⋊C4—94no—
246C3×C22⋊C4—98no—
247#217—38no—
248(C2×D3)⋊C4—38no—
249Dic3⋊C4—38no—
250C22×C32—28no—
251C22×C32—22no—
252C22×C32—22no—
253C22×C32—24no—
254C22×C32—24no—
255C22×C32—216no—
256C22×C32—216no—
257C22×C32—24no—
258C22×C32—24no—
259C22×C32—21no—
260C22×C32—22no—
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—24no—
264C22×C32—22no—
265C22×C32—24no—
266C22×C32—24no—
267C22×C32—24no—
268C22×C32—28no—
269C22×C32—42no—
270C22×C32—42no—
271C22×C32—44no—
272C22×C32—42no—
273C22×C32—41no—
274C22×C32—416no—
275C22×C32—44no—
276C22×C32—44no—
277C22×C32—44no—
278C22×C32—416no—
279C22×C32—44no—
280C22×C32—64no—
281C22×C32—64no—
282C22×C32—64no—
283C22×C32—128no—
284C22×C32D3×D414yes—
285C22×C32#23714yes—
286C22×C32C2×#6714yes—
287C22×C32C2×#6718yescomplemented
288C22×C32C2×C3×D412yes—
289C22×C32C2×C3×D412yes—
290C22×C32C2×C3×D414yescomplemented
291C22×C32C3×#4112yes—
292C22×C32C23×D312yes—
293C22×C32C24×C311yescharacteristic
294C2×C3×D3—28no—
295C2×C3×D3—24no—
296C2×C3×D3—24no—
297C2×C3×D3—24no—
298C2×C3×D3—24no—
299C2×C3×D3—24no—
300C2×C3×D3—24no—
301C2×C3×D3—44no—
302C2×C3×D3—44no—
303C2×C3×D3—416no—
304C2×C3×D3—48no—
305C2×C3×D3—48no—
306C32×C4—64no—
307C3⋊Dic3—22no—
308C3⋊Dic3C2×C3×D414yescomplemented
309C3×Dic3—28no—
310C3×Dic3—24no—
311C3×Dic3—24no—
312C3×Dic3—38no—
313C3×Dic3—64no—
314C3×Dic3—64no—
315C3×Dic3—64no—
316C22×D4—32no—
317C22×D4—91no—
318C23×C4—91no—
319#121—98no—
320C2×C4⋊C4—91no—
321C2×C22⋊C4—92no—
322C33C2×#12111yescharacteristic, complemented, Sylow
323C23×C3—24no—
324C23×C3—24no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—24no—
328C23×C3—24no—
329C23×C3—24no—
330C23×C3—28no—
331C23×C3—22no—
332C23×C3—22no—
333C23×C3—21no—
334C23×C3—24no—
335C23×C3—34no—
336C23×C3—34no—
337C23×C3—44no—
338C23×C3—44no—
339C23×C3—44no—
340C23×C3—44no—
341C23×C3—44no—
342C23×C3—42no—
343C23×C3—41no—
344C23×C3—64no—
345C23×C3—68no—
346C23×C3—64no—
347C23×C3—64no—
348C23×C3—64no—
349C23×C3—64no—
350C23×C3—64no—
351C23×C3—64no—
352C23×C3—124no—
353C23×C3—124no—
354C23×C3#62514yescomplemented
355C23×C3C3×#6718yescomplemented
356C23×C3C2×D3212yes—
357C23×C3C2×D3211yescharacteristic
358C23×C3C22×C3×D314yes—
359C23×C3C22×C3×D312yes—
360C22×D3—24no—
361C22×D3—24no—
362C22×D3—24no—
363C22×D3C22×C3×D314yes—
364C3×D4—632no—
365C3×D4—616no—
366C3×D4—616no—
367C3×D4—188no—
368C2×C3×C4—34no—
369C2×C3×C4—34no—
370C2×C3×C4—64no—
371C2×C3×C4—68no—
372C2×C3×C4—64no—
373C2×C3×C4—64no—
374C2×C3×C4—92no—
375C2×C3×C4—94no—
376C2×C3×C4—92no—
377C2×C3×C4—188no—
378C2×C3×C4—182no—
379#67—216no—
380#67—616no—
381C2×Dic3—24no—
382C2×Dic3—34no—
383C2×Dic3—38no—
384C2×Dic3—34no—
385C2×Dic3—64no—
386C2×Dic3—62no—
387C2×Dic3—62no—
388C2×Dic3—616no—
389C2×Dic3—616no—
390C2×Dic3—64no—
391C2×Dic3—64no—
392C2×Dic3C22×C3×D314yes—
393C2×C32—24no—
394C2×C32—28no—
395C2×C32—28no—
396C2×C32—28no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—22no—
401C2×C32—22no—
402C2×C32—22no—
403C2×C32—24no—
404C2×C32—44no—
405C2×C32—41no—
406C2×C32—41no—
407C2×C32—41no—
408C2×C32—48no—
409C2×C32—48no—
410C2×C32—124no—
411C2×C32#93918yescomplemented
412C2×C32C3×#12114yescomplemented
413C2×C32C2×D3×D412yes—
414C2×C32C2×#23712yes—
415C2×C32C22×#6712yes—
416C2×C32C22×C3×D411yescharacteristic
417C2×C32C22×C3×D411yescharacteristic
418C2×C32C2×C3×#4111yescharacteristic
419C3×D3—48no—
420C3×D3—44no—
421C3×D3—44no—
422C24—32no—
423C24C3×D3211yescharacteristic
424C2×D4—38no—
425C2×D4—68no—
426C2×D4—94no—
427C2×D4—184no—
428C22×C4—32no—
429C22×C4—91no—
430C22×C4—91no—
431C22×C4—94no—
432C22×C4—184no—
433C4⋊C4—94no—
434C22⋊C4—98no—
435C22×C3—24no—
436C22×C3—24no—
437C22×C3—216no—
438C22×C3—216no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—24no—
442C22×C3—24no—
443C22×C3—28no—
444C22×C3—21no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—24no—
448C22×C3—24no—
449C22×C3—22no—
450C22×C3—24no—
451C22×C3—28no—
452C22×C3—22no—
453C22×C3—24no—
454C22×C3—44no—
455C22×C3—44no—
456C22×C3—416no—
457C22×C3—44no—
458C22×C3—44no—
459C22×C3—416no—
460C22×C3—41no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—44no—
464C22×C3—42no—
465C22×C3—68no—
466C22×C3—64no—
467C22×C3—64no—
468C22×C3—64no—
469C22×C3—64no—
470C22×C3—64no—
471C22×C3—64no—
472C22×C3—124no—
473C22×C3—124no—
474C22×C3—1216no—
475C22×C3—128no—
476C22×C3—128no—
477C22×C3#380512yes—
478C22×C3C2×#62512yes—
479C22×C3C2×#62514yescomplemented
480C22×C3#381112yes—
481C22×C3C3×D3×D414yes—
482C22×C3C3×#23714yes—
483C22×C3C2×C3×#6714yes—
484C22×C3C2×C3×#6718yescomplemented
485C22×C3C22×D3211yescharacteristic
486C22×C3C23×C3×D312yes—
487C2×D3—24no—
488C2×D3—24no—
489C2×D3—24no—
490C2×D3—48no—
491C3×C4—68no—
492C3×C4—64no—
493C3×C4—64no—
494C3×C4—94no—
495C3×C4—182no—
496Dic3—24no—
497Dic3—38no—
498Dic3—64no—
499Dic3—64no—
500Dic3—64no—
501C32—22no—
502C32—21no—
503C32—41no—
504C32C2×#93912yescomplemented
505C32C2×C3×#12111yescharacteristic, complemented
506C23—22no—
507C23—22no—
508C23—34no—
509C23—64no—
510C23—64no—
511C23—64no—
512C23C3×#62514yescomplemented
513C23C2×C3×D3212yes—
514C23C2×C3×D3211yescharacteristic
515D4—616no—
516D4—188no—
517C2×C4—34no—
518C2×C4—64no—
519C2×C4—92no—
520C2×C4—94no—
521C2×C4—92no—
522C2×C4—188no—
523C2×C4—182no—
524C2×C3—28no—
525C2×C3—28no—
526C2×C3—24no—
527C2×C3—21no—
528C2×C3—22no—
529C2×C3—22no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—22no—
533C2×C3—24no—
534C2×C3—28no—
535C2×C3—48no—
536C2×C3—48no—
537C2×C3—41no—
538C2×C3—41no—
539C2×C3—41no—
540C2×C3—44no—
541C2×C3—128no—
542C2×C3—124no—
543C2×C3—124no—
544C2×C3#6398014yesdirect product factor
545C2×C3C3×#93918yescomplemented
546C2×C3C2×#380511yescharacteristic
547C2×C3C22×#62511yescharacteristic
548C2×C3C2×#381111yescharacteristic
549C2×C3C2×C3×D3×D412yes—
550C2×C3C2×C3×#23712yes—
551C2×C3C22×C3×#6712yes—
552D3—44no—
553C22—22no—
554C22—22no—
555C22—28no—
556C22—64no—
557C22—64no—
558C22—64no—
559C22—128no—
560C22C3×#380512yes—
561C22C2×C3×#62512yes—
562C22C2×C3×#62514yescomplemented
563C22C3×#381112yes—
564C22C22×C3×D3211yescharacteristic
565C4—64no—
566C4—94no—
567C4—182no—
568C3—21no—
569C3—22no—
570C3—41no—
571C3C2×#6398011yescharacteristic, direct product factor
572C3C2×C3×#93912yescomplemented
573C2—24no—
574C2—124no—
575C2C3×#6398014yesdirect product factor
576C2C2×C3×#380511yescharacteristic
577C2C22×C3×#62511yescharacteristic
578C2C2×C3×#381111yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC24×C33C22
socleC23×C33C23
hypercenterC24×C3D32
hypocenterC32C2×C3×#121
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×C3⋊Dic3C2semidirect
2. C22×C32×#67C2semidirect
3. C22×C3×#625C2semidirect
4. C3×#63980C2direct (N × H)
5. C2×C3×#3724C2semidirect
6. C2×C3×(C2×D3)⋊Dic3C2semidirect
8. C2×#63980C3direct (N × H)
14. C22×C3×C3⋊Dic3C22semidirect
16. C2×C32×#67C22semidirect
19. C2×C3×#625C22semidirect
20. C3×#3724C22semidirect
21. C3×(C2×D3)⋊Dic3C22semidirect
23. C23×C3⋊Dic3C2×C3semidirect
26. C22×C3×#67D3semidirect
27. C22×C3×#67C2×C3semidirect
30. C22×#625C2×C3semidirect
35. #63980C2×C3direct (N × H)
36. C2×#3724C2×C3semidirect
37. C2×(C2×D3)⋊Dic3C2×C3semidirect
40. C23×C33D4semidirect
49. C2×C3×C3⋊Dic3D4semidirect
73. C22×C3⋊Dic3C22×C3semidirect
80. C2×C3×#67C2×D3semidirect
81. C2×C3×#67C22×C3semidirect
94. C2×#625C22×C3semidirect
98. #3724C22×C3semidirect
99. (C2×D3)⋊Dic3C22×C3semidirect
105. C22×C33C2×D4semidirect
113. C3×C3⋊Dic3C2×D4semidirect
121. C22×#67C3×D3semidirect
154. C23×C32#67semidirect
155. C23×C32C3×D4semidirect
177. C2×C3⋊Dic3C3×D4semidirect
203. C2×C33#121semidirect
235. C2×#67C2×C3×D3semidirect
287. C22×C32C2×#67semidirect
290. C22×C32C2×C3×D4semidirect
308. C3⋊Dic3C2×C3×D4semidirect
322. C33C2×#121semidirect
354. C23×C3#625semidirect
355. C23×C3C3×#67semidirect
411. C2×C32#939semidirect
412. C2×C32C3×#121semidirect
479. C22×C3C2×#625semidirect
484. C22×C3C2×C3×#67semidirect
504. C32C2×#939semidirect
505. C32C2×C3×#121semidirect
512. C23C3×#625semidirect
544. C2×C3#63980direct (N × H)
545. C2×C3C3×#939semidirect
562. C22C2×C3×#625semidirect
571. C3C2×#63980direct (N × H)
572. C3C2×C3×#939semidirect
575. C2C3×#63980direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C22×C32×#672
C22×C3×#6252
C3×#639802
C2×C3×#37242
C2×C3×(C2×D3)⋊Dic32
C2×C3×#9393
C2×#639803