Finite Group Databasebeta

← C3×#10591162 #13158341

Subgroups — C3×#10591162 #13158341

Order 1728.

1226 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×D12C211yesmaximal, characteristic, complemented
2C3×#64139C211yesmaximal, characteristic, complemented
3C2×C3×#3714C212yesmaximal, complemented
4C32×#830C211yesmaximal, characteristic, complemented
5C3×#63563C212yesmaximal, complemented
6C3×#5217—31nomaximal
7C3×#4732—31nomaximal
8#10591162C311yesmaximal, characteristic, direct product factor
9C23×C32×D3—21no—
10C22×C33×C4C2211yescharacteristic
11C2×C3×#638—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
13C2×C32×D12—21no—
14C2×C32×D12—24no—
15C2×C32×D12C2212yes—
16C3×#3752—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
18C2×C32×C3⋊C8C2212yescomplemented
19C3×#3714—24no—
20C22×C32×D4—31no—
21C22×C3×D12—21no—
22C22×C3×D12D311yescharacteristic, complemented
23C22×C3×D12C2×C311yescharacteristic, complemented
24#64139C2×C311yescharacteristic, complemented
25C2×C3×#213—32no—
26C3×#930—31no—
27C3×#930—31no—
28C3×#930—61no—
29C2×C3×D24—32no—
30C2×#3714C2×C312yescomplemented
31C32×C22⋊C8—31no—
32C3×#832—32no—
33C3×#830—21no—
34C3×#830D311yescharacteristic, complemented
35C3×#830C2×C311yescharacteristic, complemented
36C3×#821—32no—
37#63563C2×C312yescomplemented
38C23×C33D411yescharacteristic
39C22×C3×C3⋊D3—21no—
40C22×C32×D3—24no—
41C22×C32×D3—22no—
42C22×C32×D3—42no—
43C22×C32×D3—42no—
44C2×C33×C4—22no—
45C2×C33×C4D411yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C3×#638—42no—
48C2×C3×C3⋊Dic3—21no—
49C3×C4⋊(C3⋊D3)—22no—
50C32×D12—24no—
51C32×D12—22no—
52C32×D12D414yescomplemented
53C32×C3⋊C8—22no—
54C3×#447—91no—
55#5217—31no—
56#4732—31no—
57C24×C32—61no—
58C23×C3×D3—22no—
59C23×C3×D3—21no—
60C23×C3×D3—21no—
61C2×C32×D4—32no—
62C2×C32×D4—64no—
63C2×C32×D4—61no—
64C22×C32×C4—21no—
65C22×C32×C4—21no—
66C22×C32×C4—21no—
67C22×C32×C4—41no—
68C22×C32×C4C2×D311yescharacteristic
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C22×C311yescharacteristic
71C2×#638—21no—
72C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
73C2×C3×#67—62no—
74C2×C3×#67—61no—
75C2×C3×#67—61no—
76C2×C3×D12—22no—
77C2×C3×D12—28no—
78C2×C3×D12—22no—
79C2×C3×D12—21no—
80C2×C3×D12—21no—
81C2×C3×D12—24no—
82C2×C3×D12—24no—
83C2×C3×D12—31no—
84C2×C3×D12—31no—
85C2×C3×D12—61no—
86C2×C3×D12C2×D312yes—
87C2×C3×D12C22×C312yes—
88C2×C32×C8—32no—
89#3752—21no—
90C4⋊(C3⋊Dic3)C22×C311yescharacteristic, complemented
91C3×#213—64no—
92C3×(C2×D3)⋊C4—62no—
93C3×(C2×D3)⋊C4—61no—
94C3×(C2×D3)⋊C4—61no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3—61no—
98C2×C3×C3⋊C8—22no—
99C2×C3×C3⋊C8C2×D312yescomplemented
100C2×C3×C3⋊C8C22×C312yescomplemented
101C3×D24—64no—
102#3714—24no—
103C22×C33—21no—
104C22×C33—21no—
105C22×C33D812yescomplemented
106C22×C33C2×D411yescharacteristic
107C2×C3×C3⋊D3—41no—
108C2×C3×C3⋊D3—41no—
109C2×C3×C3⋊D3—41no—
110C2×C32×D3—24no—
111C2×C32×D3—48no—
112C2×C32×D3—42no—
113C2×C32×D3—42no—
114C33×C4—21no—
115C33×C4C2×D412yes—
116C3×C3⋊Dic3—41no—
117C22×C3×D4—31no—
118C22×C3×D4—31no—
119C22×C3×D4—61no—
120C22×D12C3×D311yescharacteristic, complemented
121C2×C3×D8—92no—
122C3×#121—91no—
123C2×#213—32no—
124#930—31no—
125#930—31no—
126#930—61no—
127C2×D24—32no—
128C3×#102—92no—
129C3×C22⋊C8—31no—
130C3×C22⋊C8—31no—
131C3×C22⋊C8—61no—
132#832—32no—
133#830C3×D311yescharacteristic, complemented
134#821—32no—
135C23×C32—21no—
136C23×C32—21no—
137C23×C32—21no—
138C23×C32—41no—
139C23×C32—64no—
140C23×C32—62no—
141C23×C32—122no—
142C23×C32—122no—
143C23×C32D1211yescharacteristic
144C23×C32#6711yescharacteristic
145C23×C32C3×D411yescharacteristic
146C22×C3⋊D3—21no—
147C22×C3×D3—28no—
148C22×C3×D3—24no—
149C22×C3×D3—24no—
150C22×C3×D3—24no—
151C22×C3×D3—22no—
152C22×C3×D3—22no—
153C22×C3×D3—44no—
154C22×C3×D3—42no—
155C22×C3×D3—42no—
156C22×C3×D3—44no—
157C22×C3×D3—42no—
158C22×C3×D3—42no—
159C22×C3×D3—62no—
160C22×C3×D3—61no—
161C22×C3×D3—61no—
162C32×D4—34no—
163C32×D4—64no—
164C32×D4—62no—
165C2×C32×C4—22no—
166C2×C32×C4—24no—
167C2×C32×C4—22no—
168C2×C32×C4—22no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—41no—
176C2×C32×C4—41no—
177C2×C32×C4—42no—
178C2×C32×C4—42no—
179C2×C32×C4—44no—
180C2×C32×C4D1211yescharacteristic
181C2×C32×C4#6711yescharacteristic
182C2×C32×C4C3×D411yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C23×C311yescharacteristic
186#638—42no—
187C2×C3⋊Dic3—21no—
188C4⋊(C3⋊D3)—22no—
189C3×#67—124no—
190C3×#67—122no—
191C3×#67—122no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C3×D12—28no—
196C3×D12—24no—
197C3×D12—24no—
198C3×D12—24no—
199C3×D12—22no—
200C3×D12—22no—
201C3×D12—24no—
202C3×D12—62no—
203C3×D12—62no—
204C3×D12—122no—
205C3×D12#6714yescomplemented
206C3×D12C3×D414yescomplemented
207C32×C8—62no—
208C3×C3⋊C8—22no—
209C3×C3⋊C8—22no—
210C3×C3⋊C8—42no—
211#447—91noSylow
212C2×C33—22no—
213C2×C33#12011yescharacteristic
214C2×C33C2×D811yescharacteristic
215C2×C33AGL(1,ℤ/8)11yescharacteristic
216C3×C3⋊D3—81no—
217C32×D3—44no—
218C24×C3—62no—
219C24×C3—61no—
220C24×C3—61no—
221C23×D3—21no—
222C2×C3×D4—32no—
223C2×C3×D4—32no—
224C2×C3×D4—62no—
225C2×C3×D4—64no—
226C2×C3×D4—64no—
227C2×C3×D4—68no—
228C2×C3×D4—62no—
229C2×C3×D4—61no—
230C2×C3×D4—61no—
231C2×C3×D4—91no—
232C2×C3×D4—181no—
233C22×C3×C4—21no—
234C22×C3×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—41no—
237C22×C3×C4D3211yescharacteristic
238C22×C3×C4C2×C3×D311yescharacteristic
239C22×C3×C4C2×C3×D311yescharacteristic
240C2×#67—62no—
241C2×#67—61no—
242C2×#67—61no—
243C2×D12—21no—
244C2×D12—24no—
245C2×D12—31no—
246C2×D12—31no—
247C2×D12—61no—
248C2×D12C2×C3×D312yes—
249C3×D8—184no—
250C2×C3×C8—32no—
251C2×C3×C8—32no—
252C2×C3×C8—62no—
253C3×C4⋊C4—91no—
254C3×C22⋊C4—181no—
255#213—64no—
256(C2×D3)⋊C4—62no—
257(C2×D3)⋊C4—61no—
258(C2×D3)⋊C4—61no—
259C4⋊Dic3—31no—
260C4⋊Dic3—31no—
261C4⋊Dic3—61no—
262C2×C3⋊C8C2×C3×D312yescomplemented
263D24—64no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32—22no—
267C22×C32—22no—
268C22×C32—21no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—22no—
273C22×C32—22no—
274C22×C32—22no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—42no—
279C22×C32—41no—
280C22×C32—41no—
281C22×C32—41no—
282C22×C32—41no—
283C22×C32—41no—
284C22×C32—42no—
285C22×C32—42no—
286C22×C32—64no—
287C22×C32—128no—
288C22×C32—122no—
289C22×C32—122no—
290C22×C32D2412yescomplemented
291C22×C32#21312yescomplemented
292C22×C32C3×D812yescomplemented
293C22×C32C2×D1211yescharacteristic
294C22×C32C2×#6711yescharacteristic
295C22×C32C2×C3×D411yescharacteristic
296C2×C3⋊D3—41no—
297C2×C3⋊D3—41no—
298C2×C3⋊D3—41no—
299C2×C3×D3—28no—
300C2×C3×D3—24no—
301C2×C3×D3—24no—
302C2×C3×D3—416no—
303C2×C3×D3—48no—
304C2×C3×D3—48no—
305C2×C3×D3—44no—
306C2×C3×D3—44no—
307C2×C3×D3—42no—
308C2×C3×D3—42no—
309C2×C3×D3—42no—
310C2×C3×D3—42no—
311C2×C3×D3—122no—
312C2×C3×D3—122no—
313C2×C3×D3—122no—
314C2×C3×D3—121no—
315C2×C3×D3—121no—
316C2×C3×D3—121no—
317C2×C3×D3—121no—
318C2×C3×D3—121no—
319C2×C3×D3—121no—
320C32×C4—21no—
321C32×C4—22no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—22no—
325C32×C4—22no—
326C32×C4—22no—
327C32×C4—42no—
328C32×C4—41no—
329C32×C4—41no—
330C32×C4—42no—
331C32×C4D3×D412yes—
332C32×C4C2×#6712yes—
333C32×C4C2×C3×D412yes—
334C3⋊Dic3—41no—
335C3×Dic3—122no—
336C3×Dic3—121no—
337C3×Dic3—121no—
338C22×D4—31no—
339C2×D8—92no—
340#121—91no—
341#102—92no—
342C22⋊C8—31no—
343C33#44711yescharacteristic, complemented, Sylow
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—41no—
348C23×C3—68no—
349C23×C3—64no—
350C23×C3—64no—
351C23×C3—64no—
352C23×C3—62no—
353C23×C3—62no—
354C23×C3—124no—
355C23×C3—122no—
356C23×C3—122no—
357C23×C3—124no—
358C23×C3—122no—
359C23×C3—122no—
360C23×C3—181no—
361C23×C3#62611yescharacteristic
362C23×C3C3×D1211yescharacteristic
363C23×C3C3×#6711yescharacteristic
364C22×D3—24no—
365C22×D3—22no—
366C22×D3—42no—
367C22×D3—42no—
368C22×D3—62no—
369C22×D3—61no—
370C22×D3—61no—
371C3×D4—34no—
372C3×D4—34no—
373C3×D4—64no—
374C3×D4—68no—
375C3×D4—64no—
376C3×D4—64no—
377C3×D4—64no—
378C3×D4—62no—
379C3×D4—62no—
380C3×D4—182no—
381C3×D4—362no—
382C2×C3×C4—24no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—42no—
393C2×C3×C4—42no—
394C2×C3×C4—44no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—181no—
398C2×C3×C4#62611yescharacteristic
399C2×C3×C4C3×D1211yescharacteristic
400C2×C3×C4C3×#6711yescharacteristic
401C2×C3×C4C2×D3211yescharacteristic
402C2×C3×C4C22×C3×D311yescharacteristic
403C2×C3×C4C22×C3×D311yescharacteristic
404#67—124no—
405#67—122no—
406#67—122no—
407C2×Dic3—62no—
408C2×Dic3—61no—
409C2×Dic3—61no—
410D12—24no—
411D12—22no—
412D12—62no—
413D12—62no—
414D12—122no—
415D12C3×#6714yescomplemented
416C3×C8—62no—
417C3×C8—62no—
418C3×C8—122no—
419C3⋊C8—22no—
420C2×C32—22no—
421C2×C32—24no—
422C2×C32—22no—
423C2×C32—22no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—41no—
434C2×C32—41no—
435C2×C32—41no—
436C2×C32—42no—
437C2×C32—42no—
438C2×C32—44no—
439C2×C32—124no—
440C2×C32#88211yescharacteristic
441C2×C32C2×D2411yescharacteristic
442C2×C32#90811yescharacteristic
443C2×C32C2×#21311yescharacteristic
444C2×C32#93211yescharacteristic
445C2×C32#95311yescharacteristic
446C2×C32C3×#12011yescharacteristic
447C2×C32C2×C3×D811yescharacteristic
448C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
449C3⋊D3—81no—
450C3×D3—48no—
451C3×D3—44no—
452C3×D3—44no—
453C3×D3—242no—
454C3×D3—241no—
455C3×D3—241no—
456C24—61no—
457C2×D4—32no—
458C2×D4—64no—
459C2×D4—61no—
460C2×D4—91no—
461C2×D4—181no—
462C22×C4C3×D3211yescharacteristic
463D8—184no—
464C2×C8—32no—
465C4⋊C4—91no—
466C22⋊C4—181no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—41no—
488C22×C3—42no—
489C22×C3—68no—
490C22×C3—64no—
491C22×C3—64no—
492C22×C3—1216no—
493C22×C3—128no—
494C22×C3—128no—
495C22×C3—124no—
496C22×C3—124no—
497C22×C3—122no—
498C22×C3—122no—
499C22×C3—122no—
500C22×C3—122no—
501C22×C3—361no—
502C22×C3—361no—
503C22×C3—361no—
504C22×C3#371412yescomplemented
505C22×C3C3×D2412yescomplemented
506C22×C3C3×#21312yescomplemented
507C22×C3C2×#62611yescharacteristic
508C22×C3C2×C3×D1211yescharacteristic
509C22×C3C2×C3×#6711yescharacteristic
510C2×D3—24no—
511C2×D3—48no—
512C2×D3—42no—
513C2×D3—42no—
514C2×D3—122no—
515C2×D3—122no—
516C2×D3—122no—
517C2×D3—121no—
518C2×D3—121no—
519C2×D3—121no—
520C2×D3—121no—
521C2×D3—121no—
522C2×D3—121no—
523C3×C4—22no—
524C3×C4—21no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—22no—
528C3×C4—22no—
529C3×C4—22no—
530C3×C4—41no—
531C3×C4—41no—
532C3×C4—42no—
533C3×C4—42no—
534C3×C4—361no—
535C3×C4D3×#6712yes—
536C3×C4C3×D3×D412yes—
537C3×C4C2×C3×#6712yes—
538Dic3—122no—
539Dic3—121no—
540Dic3—121no—
541C32—21no—
542C32—21no—
543C32—21no—
544C32—41no—
545C32#473211yescharacteristic, complemented
546C32#521711yescharacteristic, complemented
547C32C3×#44711yescharacteristic, complemented
548C23—64no—
549C23—62no—
550C23—122no—
551C23—122no—
552C23—181no—
553C23C3×#62611yescharacteristic
554D4—34no—
555D4—64no—
556D4—62no—
557D4—182no—
558D4—362no—
559C2×C4—22no—
560C2×C4—181no—
561C2×C4C3×#62611yescharacteristic
562C2×C4C2×C3×D3211yescharacteristic
563C8—62no—
564C2×C3—24no—
565C2×C3—22no—
566C2×C3—22no—
567C2×C3—22no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—42no—
578C2×C3—42no—
579C2×C3—44no—
580C2×C3—41no—
581C2×C3—41no—
582C2×C3—41no—
583C2×C3—128no—
584C2×C3—124no—
585C2×C3—124no—
586C2×C3—721no—
587C2×C3C2×#371411yescharacteristic
588C2×C3#6383011yescharacteristic
589C2×C3#6397711yescharacteristic
590C2×C3C3×#88211yescharacteristic
591C2×C3C2×C3×D2411yescharacteristic
592C2×C3C3×#90811yescharacteristic
593C2×C3C2×C3×#21311yescharacteristic
594C2×C3C3×#93211yescharacteristic
595C2×C3C3×#95311yescharacteristic
596D3—44no—
597D3—242no—
598D3—241no—
599D3—241no—
600C22—21no—
601C22—21no—
602C22—64no—
603C22—128no—
604C22—122no—
605C22—122no—
606C22—361no—
607C22—361no—
608C22—361no—
609C22C3×#371412yescomplemented
610C22C2×C3×#62611yescharacteristic
611C4—21no—
612C4—361no—
613C4C3×D3×#6712yes—
614C3—21no—
615C3—21no—
616C3—21no—
617C3—41no—
618C3#1059116211yescharacteristic, direct product factor
619C3C3×#473211yescharacteristic, complemented
620C3C3×#521711yescharacteristic, complemented
621C2—22no—
622C2—124no—
623C2—721no—
624C2C2×C3×#371411yescharacteristic
625C2C3×#6383011yescharacteristic
626C2C3×#6397711yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
hypercenterC22×C3×C4D32
hypocenterC32C3×#447
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C32×D12C2semidirect
2. C3×#64139C2semidirect
3. C2×C3×#3714C2semidirect
4. C32×#830C2semidirect
5. C3×#63563C2semidirect
8. #10591162C3direct (N × H)
12. C2×C3×C4⋊(C3⋊D3)C22semidirect
17. C3×C4⋊(C3⋊Dic3)C22semidirect
18. C2×C32×C3⋊C8C22semidirect
22. C22×C3×D12D3semidirect
23. C22×C3×D12C2×C3semidirect
24. #64139C2×C3semidirect
30. C2×#3714C2×C3semidirect
34. C3×#830D3semidirect
35. C3×#830C2×C3semidirect
37. #63563C2×C3semidirect
52. C32×D12D4semidirect
72. C2×C4⋊(C3⋊D3)C22×C3semidirect
90. C4⋊(C3⋊Dic3)C22×C3semidirect
99. C2×C3×C3⋊C8C2×D3semidirect
100. C2×C3×C3⋊C8C22×C3semidirect
105. C22×C33D8semidirect
120. C22×D12C3×D3semidirect
133. #830C3×D3semidirect
205. C3×D12#67semidirect
206. C3×D12C3×D4semidirect
262. C2×C3⋊C8C2×C3×D3semidirect
290. C22×C32D24semidirect
291. C22×C32#213semidirect
292. C22×C32C3×D8semidirect
343. C33#447semidirect
415. D12C3×#67semidirect
504. C22×C3#3714semidirect
505. C22×C3C3×D24semidirect
506. C22×C3C3×#213semidirect
545. C32#4732semidirect
546. C32#5217semidirect
547. C32C3×#447semidirect
609. C22C3×#3714semidirect
618. C3#10591162direct (N × H)
619. C3C3×#4732semidirect
620. C3C3×#5217semidirect

Maximal subgroups

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

subgroupindex
C22×C32×D122
C3×#641392
C2×C3×#37142
C32×#8302
C3×#635632
C3×#52173
C3×#47323
#105911623