Finite Group Databasebeta

← C2×C3×#63979 #13175109

Subgroups — C2×C3×#63979 #13175109

Order 1728.

2834 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#638C211yesmaximal, characteristic, complemented
2C22×C32×#67C212yesmaximal, complemented
3C22×C3×#626C212yesmaximal, complemented
4C22×C3×#625C211yesmaximal, characteristic, complemented
5C3×#63979C218yesmaximal, direct product factor
6C2×C3×Dic32C211yesmaximal, characteristic, complemented
7C2×C3×#940—32nomaximal
8C2×#63979C311yesmaximal, characteristic, direct product factor
9C24×C33C2211yescharacteristic
10C23×C3×C3⋊D3C2211yescharacteristic
11C23×C32×D3C2212yes—
12C2×C3×#638—24no—
13C2×C3×#638C2214yescomplemented
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C2×C32×#67—28no—
16C2×C32×#67C2218yescomplemented
17C22×C32×Dic3C2212yes—
18C2×C3×#626—28no—
19C2×C3×#626C2218yescomplemented
20C2×C3×#625—24no—
21C2×C3×#625C2214yescomplemented
22C3×Dic32C2214yescomplemented
23C22×C32×D4—32no—
24C22×#638C2×C311yescharacteristic, complemented
25C22×C3×#67—22no—
26C22×C3×#67—32no—
27C22×C3×#67—32no—
28C22×C3×#67—32no—
29C22×C3×#67—61no—
30C22×C3×#67D312yescomplemented
31C22×C3×#67C2×C312yescomplemented
32C22×C3×D12—32no—
33C22×#626C2×C312yescomplemented
34C22×#625C2×C311yescharacteristic, complemented
35C3×#940—316no—
36C2×C3×C4×Dic3—32no—
37#63979C2×C318yesdirect product factor
38C2×Dic32C2×C311yescharacteristic, complemented
39C23×C33—24no—
40C23×C33—22no—
41C23×C33C2312yes—
42C23×C33C2311yescharacteristic
43C22×C3×C3⋊D3—22no—
44C22×C3×C3⋊D3—24no—
45C22×C3×C3⋊D3C2312yes—
46C22×C32×D3—24no—
47C22×C32×D3—24no—
48C22×C32×D3—24no—
49C22×C32×D3C2314yes—
50C3×#638—28no—
51C2×C3×C3⋊Dic3D414yescomplemented
52C2×C3×C3⋊Dic3C2312yes—
53C32×#67—216no—
54C2×C32×Dic3D418yescomplemented
55C2×C32×Dic3C2314yes—
56C3×#626—216no—
57C3×#625—28no—
58C2×C3×#127—91no—
59C2×#940—32no—
60C24×C32—21no—
61C24×C32—22no—
62C24×C32—32no—
63C24×C32—41no—
64C24×C32C2×D312yes—
65C24×C32C22×C311yescharacteristic
66C23×C3⋊D3C22×C311yescharacteristic
67C23×C3×D3—22no—
68C23×C3×D3—32no—
69C23×C3×D3—61no—
70C23×C3×D3C2×D312yes—
71C23×C3×D3C22×C312yes—
72C2×C32×D4—38no—
73C2×C32×D4—68no—
74C22×C32×C4—32no—
75C2×#638—24no—
76C2×#638C22×C314yescomplemented
77C22×C3⋊Dic3C22×C311yescharacteristic
78C2×C3×#67—216no—
79C2×C3×#67—28no—
80C2×C3×#67—28no—
81C2×C3×#67—28no—
82C2×C3×#67—38no—
83C2×C3×#67—38no—
84C2×C3×#67—38no—
85C2×C3×#67—68no—
86C2×C3×#67—68no—
87C2×C3×#67—64no—
88C2×C3×#67—68no—
89C2×C3×#67—68no—
90C2×C3×#67C2×D318yescomplemented
91C2×C3×#67C22×C318yescomplemented
92C22×C3×Dic3—22no—
93C22×C3×Dic3—32no—
94C22×C3×Dic3—61no—
95C22×C3×Dic3C2×D312yes—
96C22×C3×Dic3C22×C312yes—
97C2×C3×D12—38no—
98C2×C3×D12—68no—
99C2×#626—28no—
100C2×#626C22×C318yescomplemented
101C2×#625—24no—
102C2×#625C22×C314yescomplemented
103C3×C4×Dic3—38no—
104Dic32C22×C314yescomplemented
105C22×C33—24no—
106C22×C33—22no—
107C22×C33—44no—
108C22×C33C2×D412yes—
109C22×C33C2×D414yes—
110C22×C33C2411yescharacteristic
111C2×C3×C3⋊D3—22no—
112C2×C3×C3⋊D3—24no—
113C2×C3×C3⋊D3—44no—
114C2×C32×D3—24no—
115C2×C32×D3—24no—
116C2×C32×D3—24no—
117C2×C32×D3—48no—
118C3×C3⋊Dic3C2×D414yescomplemented
119C32×Dic3C2×D418yescomplemented
120C22×C3×D4—32no—
121C22×C3×D4—32no—
122C22×C3×D4—62no—
123C22×C3×D4—92no—
124C22×C3×D4—91no—
125C22×C3×D4—91no—
126C22×#67—32no—
127C22×#67—32no—
128C22×#67—32no—
129C22×#67—61no—
130C22×#67C3×D312yescomplemented
131C22×D12—32no—
132C3×#127—98no—
133C2×C3×C42—91no—
134#940—316no—
135C2×C4×Dic3—32no—
136C23×C32—24no—
137C23×C32—22no—
138C23×C32—28no—
139C23×C32—24no—
140C23×C32—24no—
141C23×C32—24no—
142C23×C32—24no—
143C23×C32—22no—
144C23×C32—21no—
145C23×C32—22no—
146C23×C32—24no—
147C23×C32—34no—
148C23×C32—41no—
149C23×C32—42no—
150C23×C32—44no—
151C23×C32—44no—
152C23×C32—44no—
153C23×C32—44no—
154C23×C32—48no—
155C23×C32—64no—
156C23×C32—64no—
157C23×C32—64no—
158C23×C32C22×D314yes—
159C23×C32C22×D312yes—
160C23×C32C23×C312yes—
161C23×C32C23×C311yescharacteristic
162C22×C3⋊D3—22no—
163C22×C3⋊D3—24no—
164C22×C3⋊D3C23×C312yes—
165C22×C3×D3—28no—
166C22×C3×D3—24no—
167C22×C3×D3—24no—
168C22×C3×D3—24no—
169C22×C3×D3—24no—
170C22×C3×D3—24no—
171C22×C3×D3—24no—
172C22×C3×D3—24no—
173C22×C3×D3—34no—
174C22×C3×D3—44no—
175C22×C3×D3—44no—
176C22×C3×D3—62no—
177C22×C3×D3—64no—
178C22×C3×D3—64no—
179C22×C3×D3—64no—
180C22×C3×D3—64no—
181C22×C3×D3—124no—
182C22×C3×D3C22×D314yes—
183C22×C3×D3C23×C314yes—
184C32×D4—616no—
185C2×C32×C4—38no—
186C2×C32×C4—34no—
187#638—28no—
188C2×C3⋊Dic3C3×D414yescomplemented
189C2×C3⋊Dic3C23×C312yes—
190C3×#67—232no—
191C3×#67—216no—
192C3×#67—216no—
193C3×#67—616no—
194C3×#67—616no—
195C3×#67—616no—
196C3×#67—616no—
197C2×C3×Dic3—28no—
198C2×C3×Dic3—24no—
199C2×C3×Dic3—38no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3—64no—
202C2×C3×Dic3—62no—
203C2×C3×Dic3#6718yescomplemented
204C2×C3×Dic3C3×D418yescomplemented
205C2×C3×Dic3C22×D314yes—
206C2×C3×Dic3C23×C314yes—
207C3×D12—616no—
208#626—216no—
209#625—28no—
210C2×#127—91noSylow
211C2×C33—42no—
212C2×C33#12714yescomplemented
213C2×C33C22×D411yescharacteristic
214C2×C33C22×D412yes—
215C3×C3⋊D3—42no—
216C32×D3—44no—
217C24×C3—22no—
218C24×C3—21no—
219C24×C3—32no—
220C24×C3—32no—
221C24×C3—41no—
222C24×C3—62no—
223C24×C3—91no—
224C24×C3D3211yescharacteristic
225C24×C3C2×C3×D312yes—
226C23×D3—32no—
227C23×D3—61no—
228C23×D3C2×C3×D312yes—
229C2×C3×D4—38no—
230C2×C3×D4—38no—
231C2×C3×D4—68no—
232C2×C3×D4—68no—
233C2×C3×D4—68no—
234C2×C3×D4—616no—
235C2×C3×D4—98no—
236C2×C3×D4—94no—
237C2×C3×D4—94no—
238C2×C3×D4—188no—
239C2×C3×D4—184no—
240C2×C3×D4—184no—
241C22×C3×C4—32no—
242C22×C3×C4—32no—
243C22×C3×C4—62no—
244C22×C3×C4—91no—
245C2×#67—28no—
246C2×#67—38no—
247C2×#67—38no—
248C2×#67—38no—
249C2×#67—64no—
250C2×#67—68no—
251C2×#67—68no—
252C2×#67—68no—
253C2×#67—68no—
254C2×#67C2×C3×D318yescomplemented
255C22×Dic3—32no—
256C22×Dic3—61no—
257C22×Dic3C2×C3×D312yes—
258C2×D12—38no—
259C2×D12—68no—
260C3×C42—94no—
261C4×Dic3—38no—
262C22×C32—24no—
263C22×C32—22no—
264C22×C32—28no—
265C22×C32—24no—
266C22×C32—24no—
267C22×C32—24no—
268C22×C32—24no—
269C22×C32—21no—
270C22×C32—22no—
271C22×C32—24no—
272C22×C32—22no—
273C22×C32—24no—
274C22×C32—24no—
275C22×C32—24no—
276C22×C32—44no—
277C22×C32—42no—
278C22×C32—41no—
279C22×C32—44no—
280C22×C32—44no—
281C22×C32—44no—
282C22×C32—44no—
283C22×C32—48no—
284C22×C32—416no—
285C22×C32—48no—
286C22×C32—416no—
287C22×C32—48no—
288C22×C32—44no—
289C22×C32—64no—
290C22×C32—64no—
291C22×C32—64no—
292C22×C32—128no—
293C22×C32D3×D414yes—
294C22×C32D3×D414yes—
295C22×C32C2×#6714yes—
296C22×C32C2×C3×D412yes—
297C22×C32C2×C3×D414yes—
298C22×C32C23×D312yes—
299C22×C32C24×C311yescharacteristic
300C2×C3⋊D3—22no—
301C2×C3⋊D3—24no—
302C2×C3⋊D3—44no—
303C2×C3×D3—28no—
304C2×C3×D3—24no—
305C2×C3×D3—24no—
306C2×C3×D3—24no—
307C2×C3×D3—24no—
308C2×C3×D3—24no—
309C2×C3×D3—24no—
310C2×C3×D3—44no—
311C2×C3×D3—44no—
312C2×C3×D3—416no—
313C2×C3×D3—48no—
314C2×C3×D3—48no—
315C2×C3×D3—64no—
316C2×C3×D3—64no—
317C2×C3×D3—64no—
318C2×C3×D3—64no—
319C2×C3×D3—124no—
320C2×C3×D3—128no—
321C2×C3×D3—128no—
322C32×C4—38no—
323C3⋊Dic3C2×C3×D414yescomplemented
324C3×Dic3—28no—
325C3×Dic3—38no—
326C3×Dic3—64no—
327C3×Dic3C2×#6718yescomplemented
328C3×Dic3C2×C3×D418yescomplemented
329C22×D4—32no—
330C22×D4—92no—
331C22×D4—91no—
332C22×D4—91no—
333#127—98no—
334C2×C42—91no—
335C33C2×#12711yescharacteristic, complemented, Sylow
336C23×C3—24no—
337C23×C3—28no—
338C23×C3—24no—
339C23×C3—22no—
340C23×C3—24no—
341C23×C3—24no—
342C23×C3—24no—
343C23×C3—22no—
344C23×C3—21no—
345C23×C3—24no—
346C23×C3—22no—
347C23×C3—34no—
348C23×C3—34no—
349C23×C3—44no—
350C23×C3—44no—
351C23×C3—44no—
352C23×C3—48no—
353C23×C3—44no—
354C23×C3—42no—
355C23×C3—41no—
356C23×C3—64no—
357C23×C3—68no—
358C23×C3—64no—
359C23×C3—64no—
360C23×C3—64no—
361C23×C3—64no—
362C23×C3—64no—
363C23×C3—64no—
364C23×C3—92no—
365C23×C3—124no—
366C23×C3—124no—
367C23×C3—184no—
368C23×C3—182no—
369C23×C3C2×D3212yes—
370C23×C3C2×D3211yescharacteristic
371C23×C3C22×C3×D314yes—
372C23×C3C22×C3×D312yes—
373C22×D3—24no—
374C22×D3—24no—
375C22×D3—24no—
376C22×D3—34no—
377C22×D3—62no—
378C22×D3—64no—
379C22×D3—64no—
380C22×D3—64no—
381C22×D3—64no—
382C22×D3—124no—
383C22×D3C22×C3×D314yes—
384C3×D4—616no—
385C3×D4—616no—
386C3×D4—632no—
387C3×D4—1816no—
388C3×D4—188no—
389C3×D4—188no—
390C2×C3×C4—34no—
391C2×C3×C4—34no—
392C2×C3×C4—38no—
393C2×C3×C4—38no—
394C2×C3×C4—64no—
395C2×C3×C4—68no—
396C2×C3×C4—92no—
397C2×C3×C4—94no—
398#67—216no—
399#67—616no—
400#67—616no—
401#67—616no—
402#67—616no—
403C2×Dic3—34no—
404C2×Dic3—38no—
405C2×Dic3—64no—
406C2×Dic3—62no—
407C2×Dic3C3×#6718yescomplemented
408C2×Dic3C22×C3×D314yes—
409D12—616no—
410C2×C32—28no—
411C2×C32—22no—
412C2×C32—21no—
413C2×C32—22no—
414C2×C32—22no—
415C2×C32—22no—
416C2×C32—24no—
417C2×C32—44no—
418C2×C32—42no—
419C2×C32—41no—
420C2×C32—48no—
421C2×C32—44no—
422C2×C32—48no—
423C2×C32—44no—
424C2×C32—42no—
425C2×C32—124no—
426C2×C32#94018yescomplemented
427C2×C32C3×#12714yescomplemented
428C2×C32C2×D3×D412yes—
429C2×C32C2×D3×D412yes—
430C2×C32C22×#6712yes—
431C2×C32C22×C3×D411yescharacteristic
432C2×C32C22×C3×D412yes—
433C3⋊D3—42no—
434C3×D3—48no—
435C3×D3—44no—
436C3×D3—44no—
437C3×D3—124no—
438C3×D3—124no—
439C24—32no—
440C24—91no—
441C24C3×D3211yescharacteristic
442C2×D4—38no—
443C2×D4—68no—
444C2×D4—98no—
445C2×D4—94no—
446C2×D4—94no—
447C2×D4—188no—
448C2×D4—184no—
449C2×D4—184no—
450C22×C4—32no—
451C22×C4—91no—
452C42—94no—
453C22×C3—24no—
454C22×C3—28no—
455C22×C3—24no—
456C22×C3—22no—
457C22×C3—24no—
458C22×C3—24no—
459C22×C3—24no—
460C22×C3—21no—
461C22×C3—22no—
462C22×C3—24no—
463C22×C3—24no—
464C22×C3—24no—
465C22×C3—22no—
466C22×C3—24no—
467C22×C3—44no—
468C22×C3—44no—
469C22×C3—44no—
470C22×C3—48no—
471C22×C3—44no—
472C22×C3—48no—
473C22×C3—416no—
474C22×C3—416no—
475C22×C3—48no—
476C22×C3—44no—
477C22×C3—41no—
478C22×C3—44no—
479C22×C3—42no—
480C22×C3—68no—
481C22×C3—64no—
482C22×C3—64no—
483C22×C3—64no—
484C22×C3—64no—
485C22×C3—64no—
486C22×C3—64no—
487C22×C3—124no—
488C22×C3—124no—
489C22×C3—1216no—
490C22×C3—128no—
491C22×C3—128no—
492C22×C3—184no—
493C22×C3—182no—
494C22×C3—364no—
495C22×C3D3×#6714yes—
496C22×C3#381112yes—
497C22×C3C3×D3×D414yes—
498C22×C3C3×D3×D414yes—
499C22×C3C2×C3×#6714yes—
500C22×C3C22×D3211yescharacteristic
501C22×C3C23×C3×D312yes—
502C2×D3—24no—
503C2×D3—24no—
504C2×D3—24no—
505C2×D3—48no—
506C2×D3—64no—
507C2×D3—64no—
508C2×D3—64no—
509C2×D3—64no—
510C2×D3—124no—
511C2×D3—128no—
512C2×D3—128no—
513C3×C4—38no—
514C3×C4—38no—
515C3×C4—68no—
516C3×C4—94no—
517Dic3—38no—
518Dic3—64no—
519Dic3C2×C3×#6718yescomplemented
520C32—22no—
521C32—21no—
522C32—41no—
523C32C2×#94012yescomplemented
524C32C2×C3×#12711yescharacteristic, complemented
525C23—24no—
526C23—22no—
527C23—34no—
528C23—64no—
529C23—64no—
530C23—64no—
531C23—92no—
532C23—184no—
533C23—182no—
534C23C2×C3×D3212yes—
535C23C2×C3×D3211yescharacteristic
536D4—616no—
537D4—1816no—
538D4—188no—
539D4—188no—
540C2×C4—34no—
541C2×C4—38no—
542C2×C4—92no—
543C2×C4—94no—
544C2×C3—22no—
545C2×C3—22no—
546C2×C3—22no—
547C2×C3—21no—
548C2×C3—22no—
549C2×C3—24no—
550C2×C3—28no—
551C2×C3—44no—
552C2×C3—48no—
553C2×C3—48no—
554C2×C3—44no—
555C2×C3—42no—
556C2×C3—42no—
557C2×C3—41no—
558C2×C3—44no—
559C2×C3—128no—
560C2×C3—124no—
561C2×C3—124no—
562C2×C3—362no—
563C2×C3#6397914yesdirect product factor
564C2×C3C3×#94018yescomplemented
565C2×C3C2×D3×#6712yes—
566C2×C3C2×#381111yescharacteristic
567C2×C3C2×C3×D3×D412yes—
568C2×C3C2×C3×D3×D412yes—
569C2×C3C22×C3×#6712yes—
570D3—44no—
571D3—124no—
572D3—124no—
573C22—24no—
574C22—22no—
575C22—44no—
576C22—64no—
577C22—64no—
578C22—64no—
579C22—128no—
580C22—184no—
581C22—182no—
582C22—364no—
583C22C3×D3×#6714yes—
584C22C3×#381112yes—
585C22C22×C3×D3211yescharacteristic
586C4—38no—
587C4—94no—
588C3—21no—
589C3—22no—
590C3—41no—
591C3C2×#6397911yescharacteristic, direct product factor
592C3C2×C3×#94012yescomplemented
593C2—42no—
594C2—124no—
595C2—362no—
596C2C3×#6397914yesdirect product factor
597C2C2×C3×D3×#6712yes—
598C2C2×C3×#381111yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC24×C33C22
socleC23×C33C23
hypercenterC24×C3D32
hypocenterC32C2×C3×#127
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×#638C2semidirect
2. C22×C32×#67C2semidirect
3. C22×C3×#626C2semidirect
4. C22×C3×#625C2semidirect
5. C3×#63979C2direct (N × H)
6. C2×C3×Dic32C2semidirect
8. C2×#63979C3direct (N × H)
13. C2×C3×#638C22semidirect
16. C2×C32×#67C22semidirect
19. C2×C3×#626C22semidirect
21. C2×C3×#625C22semidirect
22. C3×Dic32C22semidirect
24. C22×#638C2×C3semidirect
30. C22×C3×#67D3semidirect
31. C22×C3×#67C2×C3semidirect
33. C22×#626C2×C3semidirect
34. C22×#625C2×C3semidirect
37. #63979C2×C3direct (N × H)
38. C2×Dic32C2×C3semidirect
51. C2×C3×C3⋊Dic3D4semidirect
54. C2×C32×Dic3D4semidirect
76. C2×#638C22×C3semidirect
90. C2×C3×#67C2×D3semidirect
91. C2×C3×#67C22×C3semidirect
100. C2×#626C22×C3semidirect
102. C2×#625C22×C3semidirect
104. Dic32C22×C3semidirect
118. C3×C3⋊Dic3C2×D4semidirect
119. C32×Dic3C2×D4semidirect
130. C22×#67C3×D3semidirect
188. C2×C3⋊Dic3C3×D4semidirect
203. C2×C3×Dic3#67semidirect
204. C2×C3×Dic3C3×D4semidirect
212. C2×C33#127semidirect
254. C2×#67C2×C3×D3semidirect
323. C3⋊Dic3C2×C3×D4semidirect
327. C3×Dic3C2×#67semidirect
328. C3×Dic3C2×C3×D4semidirect
335. C33C2×#127semidirect
407. C2×Dic3C3×#67semidirect
426. C2×C32#940semidirect
427. C2×C32C3×#127semidirect
519. Dic3C2×C3×#67semidirect
523. C32C2×#940semidirect
524. C32C2×C3×#127semidirect
563. C2×C3#63979direct (N × H)
564. C2×C3C3×#940semidirect
591. C3C2×#63979direct (N × H)
592. C3C2×C3×#940semidirect
596. C2C3×#63979direct (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
C22×C32×#672
C22×C3×#6262
C22×C3×#6252
C3×#639792
C2×C3×Dic322
C2×C3×#9403
C2×#639793