Finite Group Databasebeta

← #10768701

Subgroups — #10768701

Order 768.

1508 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3537C211yesmaximal, characteristic, complemented
2#82927C211yesmaximal, characteristic, complemented
3#82653C211yesmaximal, characteristic, complemented
4C2×#5036C211yesmaximal, characteristic, complemented
5#71884C214yesmaximal, complemented
6#71845C214yesmaximal, complemented
7#71707C211yesmaximal, characteristic, complemented
8#71676C211yesmaximal, characteristic, complemented
9#71673C211yesmaximal, characteristic, complemented
10#18254—31nomaximal, Sylow
11C23×D12C2211yescharacteristic, complemented
12C3×#560—22no—
13C3×#553—21no—
14C3×#553—21no—
15C3×D42—22no—
16C3×D42—21no—
17C3×D42—21no—
18C3×#540—21no—
19C3×#538—21no—
20C3×#538C2212yescomplemented
21C3×#535—21no—
22C3×#534—21no—
23C3×#534C2212yescomplemented
24C3×#525C2211yescharacteristic
25C2×C3×#121C2211yescharacteristic
26C2×C3×#121C2211yescharacteristic
27C2×#949—21no—
28C2×#940—21no—
29C2×#937—22no—
30C22×#213—21no—
31C4×#217C2211yescharacteristic, complemented
32#5207C2211yescharacteristic, complemented
33C2×#830C2211yescharacteristic, complemented
34#5046—21no—
35#5046C2212yescomplemented
36#5036—22no—
37#5036—21no—
38#5036C2212yescomplemented
39#5036C2214yescomplemented
40#5001—21no—
41#4988—21no—
42C2×#809—21no—
43#4960C2214yescomplemented
44#4538C2212yescomplemented
45#4537C2212yescomplemented
46#4537C2214yescomplemented
47#4525C2214yescomplemented
48#3537D311yescharacteristic, complemented
49#3003—31no—
50C2×#447—31no—
51#2433—31no—
52#1600—34no—
53#1595—34no—
54#1507—31no—
55#1501—31no—
56#1471—31no—
57C24×D3—21no—
58C2×C3×#41D411yescharacteristic, complemented
59C22×C3×D4—22no—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4—22no—
63C22×C3×D4D412yes—
64C22×C3×D4D411yescharacteristic
65C22×C3×D4D411yescharacteristic, complemented
66C23×C3×C4C2311yescharacteristic
67C22×#67—24no—
68C22×D12—21no—
69C22×D12—21no—
70C22×D12—21no—
71C22×D12—21no—
72C22×D12—21no—
73C22×D12—44no—
74C22×D12C2312yes—
75C3×#127—22no—
76C3×#127—21no—
77C3×#126—22no—
78C3×#126—22no—
79C3×#125—21no—
80C3×#124—22no—
81C3×#123—22no—
82C3×#123—21no—
83C3×#123—22no—
84C3×#123—22no—
85C3×#123—22no—
86C3×C22⋊Q8D411yescharacteristic, complemented
87C3×C22⋊Q8C2311yescharacteristic
88C3×#121—22no—
89C3×#121—22no—
90C3×#121—22no—
91C3×#121—22no—
92C3×#121—21no—
93C3×#121—21no—
94C3×#121—22no—
95C3×#121—21no—
96C3×#121—21no—
97C3×#121—21no—
98C3×#121—21no—
99C3×#121—21no—
100C3×#121—21no—
101C3×#121—22no—
102C3×#121D412yescomplemented
103C3×#121D411yescharacteristic, complemented
104C3×#121C2312yes—
105C3×#121C2311yescharacteristic
106C3×#120—22no—
107C3×#120—22no—
108C3×#120—22no—
109C3×#120—21no—
110C3×#120—22no—
111C3×#120—22no—
112C3×#120—22no—
113C3×C4×D4—22no—
114C3×C4×D4—21no—
115C3×C4×D4—21no—
116C3×C4×D4—21no—
117C3×C4×D4—21no—
118C3×C4×D4—22no—
119C3×#117—21no—
120C3×#117D411yescharacteristic, complemented
121C2×C3×C4⋊C4D411yescharacteristic
122C2×C3×C4⋊C4D411yescharacteristic, complemented
123C2×C3×C22⋊C4—21no—
124C2×C3×C22⋊C4C2312yes—
125C2×#217—21no—
126#949—44no—
127C2×#215—22no—
128#940—44no—
129#937—48no—
130C2×#213—24no—
131C2×#213—22no—
132C2×#213—44no—
133C2×(C2×D3)⋊C4—24no—
134C2×C4×Dic3—21no—
135C2×#208—21no—
136C22×C3⋊C8—21no—
137C3⋊#95C2312yes—
138#830—21no—
139#830—21no—
140#830C2312yes—
141#830C2312yes—
142#809—24no—
143#809—22no—
144#809—22no—
145C23×D4—31no—
146C2×AGL(1,ℤ/8)—61no—
147C22×D8—61no—
148#560—22no—
149#553—21no—
150#553—21no—
151D42—22no—
152D42—21no—
153D42—21no—
154#540—21no—
155#538—21no—
156#538C2×D312yescomplemented
157#535—21no—
158#534—21no—
159#534C2×D312yescomplemented
160C2×#127—61no—
161#525C2×D311yescharacteristic
162C2×#121C2×D311yescharacteristic
163C2×#121C2×D311yescharacteristic
164C2×#120—62no—
165#450—32no—
166#450—61no—
167#447—32no—
168#447—34no—
169#447—62no—
170#447—61no—
171#418—61no—
172C2×#102—61no—
173#407—31no—
174C2×C22⋊C8—31no—
175#392—34no—
176#390—61no—
177C4×C22⋊C4—31no—
178#328—32no—
179#327—32no—
180#327—34no—
181#323—34no—
182C24×C3—22no—
183C24×C3—21no—
184C24×C3C2×D411yescharacteristic
185C24×C3C2×D412yes—
186C23×D3—22no—
187C23×D3—44no—
188C23×D3—41no—
189C23×D3—41no—
190C23×D3—44no—
191C23×D3—82no—
192C3×#41—42no—
193C2×C3×Q8C2×D411yescharacteristic
194C2×C3×D4—22no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4—22no—
200C2×C3×D4—22no—
201C2×C3×D4—22no—
202C2×C3×D4—22no—
203C2×C3×D4—21no—
204C2×C3×D4—22no—
205C2×C3×D4—22no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—42no—
210C2×C3×D4—42no—
211C2×C3×D4—44no—
212C2×C3×D4—82no—
213C2×C3×D4—82no—
214C2×C3×D4—84no—
215C2×C3×D4—84no—
216C2×C3×D4C2×D411yescharacteristic
217C2×C3×D4C2×D412yes—
218C22×C3×C4—22no—
219C22×C3×C4—22no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—42no—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D412yes—
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2411yescharacteristic
230C2×#67—48no—
231C2×#67—48no—
232C2×#67—84no—
233C22×Dic3—22no—
234C2×D12—24no—
235C2×D12—24no—
236C2×D12—22no—
237C2×D12—22no—
238C2×D12—24no—
239C2×D12—44no—
240C2×D12—44no—
241C2×D12—44no—
242C2×D12—42no—
243C2×D12—42no—
244C2×D12—42no—
245C2×D12—41no—
246C2×D12—41no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—22no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4C2×D412yes—
255C3×C4⋊C4C2×D411yescharacteristic
256C3×C4⋊C4C2×D411yescharacteristic
257C3×C22⋊C4—22no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—22no—
260C3×C22⋊C4—22no—
261C3×C22⋊C4—22no—
262C3×C22⋊C4—21no—
263C3×C22⋊C4—21no—
264C3×C22⋊C4—21no—
265C3×C22⋊C4—22no—
266C3×C22⋊C4—21no—
267C3×C22⋊C4—22no—
268C3×C22⋊C4—22no—
269C3×C22⋊C4—22no—
270C3×C22⋊C4—21no—
271C3×C42—22no—
272C3×C42—21no—
273#217—24no—
274#215—44no—
275#213—48no—
276#213—44no—
277(C2×D3)⋊C4—48no—
278C4×Dic3—42no—
279#208—42no—
280C2×C3⋊C8—22no—
281C2×C3⋊C8—22no—
282C2×C3⋊C8—42no—
283C25—61no—
284C2×#41#6711yescharacteristic, complemented
285C22×D4—22no—
286C22×D4—21no—
287C22×D4—21no—
288C22×D4—22no—
289C22×D4—32no—
290C22×D4—64no—
291C22×D4—61no—
292C22×D4—61no—
293C22×D4—61no—
294C22×D4—61no—
295C22×D4—61no—
296C22×D4—124no—
297C22×D4#6712yes—
298C22×D4#6711yescharacteristic
299C22×D4#6711yescharacteristic, complemented
300C23×C4C22×D311yescharacteristic
301AGL(1,ℤ/8)—124no—
302C2×SD16—62no—
303C2×D8—64no—
304C2×D8—62no—
305C2×D8—124no—
306C2×M16—61no—
307C22×C8—61no—
308#127—22no—
309#127—21no—
310#127—124no—
311#126—22no—
312#126—22no—
313#125—21no—
314#124—22no—
315#123—22no—
316#123—21no—
317#123—22no—
318#123—22no—
319#123—22no—
320C22⋊Q8#6711yescharacteristic, complemented
321C22⋊Q8C22×D311yescharacteristic
322#121—22no—
323#121—21no—
324#121—21no—
325#121—22no—
326#121—22no—
327#121—22no—
328#121—22no—
329#121—21no—
330#121—21no—
331#121—21no—
332#121—21no—
333#121—21no—
334#121—21no—
335#121—22no—
336#121#6712yescomplemented
337#121#6711yescharacteristic, complemented
338#121C22×D312yes—
339#121C22×D311yescharacteristic
340#120—22no—
341#120—22no—
342#120—22no—
343#120—21no—
344#120—22no—
345#120—22no—
346#120—22no—
347#120—128no—
348C4×D4—22no—
349C4×D4—21no—
350C4×D4—21no—
351C4×D4—21no—
352C4×D4—21no—
353C4×D4—22no—
354#117—21no—
355#117#6711yescharacteristic, complemented
356C2×C4⋊C4#6711yescharacteristic
357C2×C4⋊C4#6711yescharacteristic, complemented
358C2×C22⋊C4—21no—
359C2×C22⋊C4—61no—
360C2×C22⋊C4—64no—
361C2×C22⋊C4C22×D312yes—
362C2×C42—61no—
363#102—64no—
364#102—62no—
365#102—62no—
366C22⋊C8—32no—
367C22⋊C8—32no—
368C22⋊C8—61no—
369C22⋊C8—61no—
370#95—32no—
371C23×C3—22no—
372C23×C3—22no—
373C23×C3—22no—
374C23×C3—22no—
375C23×C3—21no—
376C23×C3—24no—
377C23×C3—41no—
378C23×C3—42no—
379C23×C3—42no—
380C23×C3—42no—
381C23×C3—81no—
382C23×C3—81no—
383C23×C3—82no—
384C23×C3—82no—
385C23×C3—82no—
386C23×C3#12011yescharacteristic
387C23×C3#12011yescharacteristic
388C23×C3C22×D411yescharacteristic
389C22×D3—24no—
390C22×D3—44no—
391C22×D3—44no—
392C22×D3—44no—
393C22×D3—42no—
394C22×D3—42no—
395C22×D3—44no—
396C22×D3—816no—
397C22×D3—82no—
398C22×D3—82no—
399C22×D3—82no—
400C22×D3—82no—
401C3×Q8—41no—
402C3×D4—42no—
403C3×D4—41no—
404C3×D4—44no—
405C3×D4—42no—
406C3×D4—44no—
407C3×D4—42no—
408C3×D4—42no—
409C3×D4—42no—
410C3×D4—81no—
411C3×D4—82no—
412C3×D4—82no—
413C3×D4—81no—
414C3×D4—82no—
415C3×D4—82no—
416C3×D4—82no—
417C3×D4—82no—
418C3×D4—82no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—21no—
423C2×C3×C4—22no—
424C2×C3×C4—22no—
425C2×C3×C4—21no—
426C2×C3×C4—21no—
427C2×C3×C4—41no—
428C2×C3×C4—41no—
429C2×C3×C4—41no—
430C2×C3×C4—42no—
431C2×C3×C4—42no—
432C2×C3×C4—81no—
433C2×C3×C4—82no—
434C2×C3×C4—82no—
435C2×C3×C4#12011yescharacteristic
436C2×C3×C4#12011yescharacteristic
437C2×C3×C4C22×D411yescharacteristic
438C2×C3×C4C22×D411yescharacteristic
439#67—816no—
440C2×Dic3—24no—
441C2×Dic3—42no—
442C2×Dic3—42no—
443D12—24no—
444D12—24no—
445D12—42no—
446D12—42no—
447D12—44no—
448D12—44no—
449C3⋊C8—42no—
450C3⋊C8—42no—
451C24—22no—
452C24—21no—
453C24—62no—
454C24—124no—
455C24—124no—
456C24—121no—
457C24—121no—
458C24—242no—
459C24C2×#6711yescharacteristic
460C24C2×#6712yes—
461#41—42no—
462C2×Q8C2×#6711yescharacteristic
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—21no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—44no—
481C2×D4—64no—
482C2×D4—64no—
483C2×D4—64no—
484C2×D4—62no—
485C2×D4—62no—
486C2×D4—82no—
487C2×D4—82no—
488C2×D4—84no—
489C2×D4—84no—
490C2×D4—122no—
491C2×D4—122no—
492C2×D4—122no—
493C2×D4—124no—
494C2×D4—124no—
495C2×D4—124no—
496C2×D4—128no—
497C2×D4—128no—
498C2×D4—121no—
499C2×D4—121no—
500C2×D4—244no—
501C2×D4C2×#6711yescharacteristic
502C2×D4C2×#6712yes—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—42no—
508C22×C4—62no—
509C22×C4C2×#6711yescharacteristic
510C22×C4C2×#6711yescharacteristic
511C22×C4C2×#6711yescharacteristic
512C22×C4C2×#6712yes—
513C22×C4C2×#6711yescharacteristic
514C22×C4C2×#6711yescharacteristic
515C22×C4C23×D311yescharacteristic
516SD16—124no—
517D8—128no—
518D8—124no—
519M16—122no—
520C2×C8—62no—
521C2×C8—62no—
522C2×C8—122no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4C2×#6712yes—
531C4⋊C4C2×#6711yescharacteristic
532C4⋊C4C2×#6711yescharacteristic
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—22no—
542C22⋊C4—21no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—21no—
547C22⋊C4—64no—
548C22⋊C4—128no—
549C42—22no—
550C42—21no—
551C42—122no—
552C22×C3—21no—
553C22×C3—21no—
554C22×C3—21no—
555C22×C3—21no—
556C22×C3—21no—
557C22×C3—21no—
558C22×C3—21no—
559C22×C3—44no—
560C22×C3—42no—
561C22×C3—42no—
562C22×C3—41no—
563C22×C3—42no—
564C22×C3—42no—
565C22×C3—41no—
566C22×C3—41no—
567C22×C3—42no—
568C22×C3—42no—
569C22×C3—84no—
570C22×C3—162no—
571C22×C3—164no—
572C22×C3#45312yes—
573C22×C3C2×#12011yescharacteristic
574C22×C3#57612yes—
575C2×D3—48no—
576C2×D3—44no—
577C2×D3—84no—
578C2×D3—88no—
579C2×D3—88no—
580C2×D3—82no—
581C2×D3—82no—
582C3×C4—21no—
583C3×C4—21no—
584C3×C4—21no—
585C3×C4—42no—
586C3×C4—41no—
587C3×C4—42no—
588C3×C4—41no—
589C3×C4—41no—
590C3×C4B(4,𝔽2)12yes—
591Dic3—44no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—21no—
597C23—24no—
598C23—41no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—64no—
603C23—81no—
604C23—81no—
605C23—82no—
606C23—82no—
607C23—82no—
608C23—124no—
609C23—124no—
610C23—124no—
611C23—122no—
612C23—124no—
613C23—122no—
614C23—242no—
615C23—242no—
616C23—242no—
617C23—2416no—
618C23—242no—
619C23#95311yescharacteristic
620C23#95311yescharacteristic
621C23C22×#6711yescharacteristic
622Q8—41no—
623D4—42no—
624D4—41no—
625D4—44no—
626D4—42no—
627D4—44no—
628D4—42no—
629D4—42no—
630D4—42no—
631D4—64no—
632D4—64no—
633D4—81no—
634D4—82no—
635D4—82no—
636D4—81no—
637D4—82no—
638D4—82no—
639D4—82no—
640D4—82no—
641D4—82no—
642D4—122no—
643D4—122no—
644D4—124no—
645D4—124no—
646D4—2416no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—21no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—64no—
661C2×C4—81no—
662C2×C4—82no—
663C2×C4—82no—
664C2×C4—122no—
665C2×C4—122no—
666C2×C4#95311yescharacteristic
667C2×C4#95311yescharacteristic
668C2×C4C22×#6711yescharacteristic
669C2×C4C22×#6711yescharacteristic
670C8—122no—
671C8—122no—
672C2×C3—22no—
673C2×C3—22no—
674C2×C3—41no—
675C2×C3—82no—
676C2×C3—82no—
677C2×C3—81no—
678C2×C3#300811yescharacteristic
679C2×C3C2×#45311yescharacteristic
680C2×C3C2×B(4,𝔽2)11yescharacteristic
681D3—84no—
682C22—21no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—21no—
687C22—21no—
688C22—21no—
689C22—44no—
690C22—42no—
691C22—42no—
692C22—41no—
693C22—42no—
694C22—42no—
695C22—41no—
696C22—41no—
697C22—42no—
698C22—42no—
699C22—84no—
700C22—128no—
701C22—124no—
702C22—162no—
703C22—164no—
704C22—244no—
705C22—248no—
706C22—248no—
707C22—242no—
708C22—242no—
709C22#524112yes—
710C22#583512yes—
711C22C2×#95311yescharacteristic
712C4—21no—
713C4—21no—
714C4—21no—
715C4—42no—
716C4—41no—
717C4—42no—
718C4—41no—
719C4—41no—
720C4—124no—
721C4#524412yes—
722C3#1825411yescharacteristic, complemented, Sylow
723C2—22no—
724C2—22no—
725C2—41no—
726C2—82no—
727C2—82no—
728C2—81no—
729C2—244no—
730C2#8410111yescharacteristic
731C2C2×#524111yescharacteristic
732C2C2×#524411yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3537C2
socleC22×C3C2×#120
hypercenter#3537D3
hypocenterC3#18254
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3537

Complements

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

NH ≅ G/Ntype
1. C3×#3537C2semidirect
2. #82927C2semidirect
3. #82653C2semidirect
4. C2×#5036C2semidirect
5. #71884C2semidirect
6. #71845C2semidirect
7. #71707C2semidirect
8. #71676C2semidirect
9. #71673C2semidirect
11. C23×D12C22semidirect
20. C3×#538C22semidirect
23. C3×#534C22semidirect
31. C4×#217C22semidirect
32. #5207C22semidirect
33. C2×#830C22semidirect
35. #5046C22semidirect
38. #5036C22semidirect
39. #5036C22semidirect
43. #4960C22semidirect
44. #4538C22semidirect
45. #4537C22semidirect
46. #4537C22semidirect
47. #4525C22semidirect
48. #3537D3semidirect
58. C2×C3×#41D4semidirect
65. C22×C3×D4D4semidirect
86. C3×C22⋊Q8D4semidirect
102. C3×#121D4semidirect
103. C3×#121D4semidirect
120. C3×#117D4semidirect
122. C2×C3×C4⋊C4D4semidirect
156. #538C2×D3semidirect
159. #534C2×D3semidirect
284. C2×#41#67semidirect
299. C22×D4#67semidirect
320. C22⋊Q8#67semidirect
336. #121#67semidirect
337. #121#67semidirect
355. #117#67semidirect
357. C2×C4⋊C4#67semidirect
722. C3#18254semidirect

Maximal subgroups

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

subgroupindex
C3×#35372
#829272
#826532
C2×#50362
#718842
#718452
#717072
#716762
#716732
#182543