Finite Group Databasebeta

← #10770529

Subgroups — #10770529

Order 768.

1096 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3537C211yesmaximal, characteristic, complemented
2#82931C211yesmaximal, characteristic, complemented
3#82722C211yesmaximal, characteristic, complemented
4#82625C211yesmaximal, characteristic, complemented
5#71884C212yesmaximal, complemented
6#71882C212yesmaximal, complemented
7#71845C212yesmaximal, complemented
8#71843C212yesmaximal, complemented
9#71710C211yesmaximal, characteristic, complemented
10#71676C211yesmaximal, characteristic, complemented
11#71674C211yesmaximal, characteristic, complemented
12#18450—31nomaximal, Sylow
13C3×#560—22no—
14C3×#553—21no—
15C3×#553—21no—
16C3×D42—22no—
17C3×D42—21no—
18C3×D42—21no—
19C3×#540—21no—
20C3×#538—21no—
21C3×#538C2212yes—
22C3×#535—21no—
23C3×#534—21no—
24C3×#534C2212yes—
25C3×#525C2211yescharacteristic
26C2×C3×#121C2211yescharacteristic
27C2×C3×#121C2211yescharacteristic
28C2×#930C2211yescharacteristic, complemented
29#5238—21no—
30#5223—22no—
31#5223—21no—
32#5217—21no—
33#5211C2211yescharacteristic, complemented
34#5207C2211yescharacteristic, complemented
35C2×#830C2211yescharacteristic, complemented
36#5049—21no—
37#5046C2211yescharacteristic, complemented
38#5043C2211yescharacteristic, complemented
39#5039—21no—
40#5038—22no—
41#5036C2211yescharacteristic, complemented
42#5036C2212yescomplemented
43#5033C2211yescharacteristic, complemented
44#5033C2212yescomplemented
45#5003—21no—
46#4992—21no—
47#4967—21no—
48#4960C2212yescomplemented
49#4953C2212yescomplemented
50#4538C2212yescomplemented
51#4537C2212yescomplemented
52#4537C2212yescomplemented
53#4537C2212yescomplemented
54#4525C2212yescomplemented
55#4525C2212yescomplemented
56#3537D311yescharacteristic, complemented
57#3194—31no—
58#3186—31no—
59#2815—31no—
60#1612—32no—
61#1607—32no—
62#1600—32no—
63#1595—32no—
64#1516—31no—
65#1510—31no—
66#1471—31no—
67C2×C3×#41D411yescharacteristic, complemented
68C22×C3×D4—22no—
69C22×C3×D4—21no—
70C22×C3×D4—21no—
71C22×C3×D4—22no—
72C22×C3×D4D412yes—
73C22×C3×D4D411yescharacteristic
74C22×C3×D4D411yescharacteristic, complemented
75C23×C3×C4C2311yescharacteristic
76C22×#67—21no—
77C22×#67—22no—
78C22×D12C2311yescharacteristic
79C3×#127—22no—
80C3×#127—21no—
81C3×#126—22no—
82C3×#126—22no—
83C3×#125—21no—
84C3×#124—22no—
85C3×#123—22no—
86C3×#123—21no—
87C3×#123—22no—
88C3×#123—22no—
89C3×#123—22no—
90C3×C22⋊Q8D411yescharacteristic, complemented
91C3×C22⋊Q8C2311yescharacteristic
92C3×#121—22no—
93C3×#121—22no—
94C3×#121—22no—
95C3×#121—22no—
96C3×#121—21no—
97C3×#121—21no—
98C3×#121—22no—
99C3×#121—21no—
100C3×#121—21no—
101C3×#121—21no—
102C3×#121—21no—
103C3×#121—21no—
104C3×#121—21no—
105C3×#121—22no—
106C3×#121D412yescomplemented
107C3×#121D411yescharacteristic, complemented
108C3×#121C2312yes—
109C3×#121C2311yescharacteristic
110C3×#120—22no—
111C3×#120—22no—
112C3×#120—22no—
113C3×#120—21no—
114C3×#120—22no—
115C3×#120—22no—
116C3×#120—22no—
117C3×C4×D4—22no—
118C3×C4×D4—21no—
119C3×C4×D4—21no—
120C3×C4×D4—21no—
121C3×C4×D4—21no—
122C3×C4×D4—22no—
123C3×#117—21no—
124C3×#117D411yescharacteristic, complemented
125C2×C3×C4⋊C4D411yescharacteristic
126C2×C3×C4⋊C4D411yescharacteristic, complemented
127C2×C3×C22⋊C4—21no—
128C2×C3×C22⋊C4C2312yes—
129C2×#217—22no—
130C2×#217—21no—
131C2×#215—21no—
132C2×#213—22no—
133C2×#213—21no—
134#930—21no—
135#930—21no—
136#930—21no—
137#930—21no—
138C2×(C2×D3)⋊C4—21no—
139C2×(C2×D3)⋊C4—22no—
140C2×C4⋊Dic3C2311yescharacteristic
141C2×#208—21no—
142C22×C3⋊C8—21no—
143#835—21no—
144#832—22no—
145#832—21no—
146C3⋊#95—22no—
147C3⋊#95—21no—
148C3⋊#95C2312yes—
149#830—21no—
150#830—21no—
151#830C2312yes—
152#830C2311yescharacteristic
153#830C2311yescharacteristic
154#809—22no—
155#809—21no—
156#809—21no—
157#808—21no—
158#807—22no—
159#807—21no—
160#560—22no—
161#553—21no—
162#553—21no—
163D42—22no—
164D42—21no—
165D42—21no—
166#540—21no—
167#538—21no—
168#538C2×D312yes—
169#535—21no—
170#534—21no—
171#534C2×D312yes—
172#525C2×D311yescharacteristic
173C2×#121—31no—
174C2×#121C2×D311yescharacteristic
175C2×#121C2×D311yescharacteristic
176#483—31no—
177#482—61no—
178#480—31no—
179#480—32no—
180#480—61no—
181#469—61no—
182#468—61no—
183#466—62no—
184#450—31no—
185#449—61no—
186#447—31no—
187#447—32no—
188#447—61no—
189#407—31no—
190C2×C22⋊C8—31no—
191#397—32no—
192#397—61no—
193#392—32no—
194#392—61no—
195#392—62no—
196#381—31no—
197#328—32no—
198#327—32no—
199#327—32no—
200#327—32no—
201#323—32no—
202#323—32no—
203C24×C3—22no—
204C24×C3—21no—
205C24×C3C2×D412yes—
206C24×C3C2×D411yescharacteristic
207C23×D3—21no—
208C3×#41—81no—
209C2×C3×Q8C2×D411yescharacteristic
210C2×C3×D4—22no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—22no—
214C2×C3×D4—22no—
215C2×C3×D4—22no—
216C2×C3×D4—22no—
217C2×C3×D4—22no—
218C2×C3×D4—22no—
219C2×C3×D4—21no—
220C2×C3×D4—22no—
221C2×C3×D4—22no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—81no—
226C2×C3×D4—82no—
227C2×C3×D4—81no—
228C2×C3×D4—82no—
229C2×C3×D4—82no—
230C2×C3×D4—82no—
231C2×C3×D4—82no—
232C2×C3×D4—84no—
233C2×C3×D4C2×D411yescharacteristic
234C2×C3×D4C2×D412yes—
235C22×C3×C4—22no—
236C22×C3×C4—22no—
237C22×C3×C4—21no—
238C22×C3×C4—21no—
239C22×C3×C4—81no—
240C22×C3×C4C2×D412yes—
241C22×C3×C4C2×D411yescharacteristic
242C22×C3×C4C2×D411yescharacteristic
243C22×C3×C4C2×D411yescharacteristic
244C22×C3×C4C2×D411yescharacteristic
245C22×C3×C4C2×D411yescharacteristic
246C22×C3×C4C2411yescharacteristic
247C2×#67—41no—
248C2×#67—41no—
249C2×#67—82no—
250C2×#67—82no—
251C2×#67—82no—
252C2×#67—82no—
253C22×Dic3—22no—
254C22×Dic3—21no—
255C2×D12—21no—
256C2×D12—21no—
257C2×D12—81no—
258C3×C4⋊C4—22no—
259C3×C4⋊C4—22no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4C2×D412yes—
266C3×C4⋊C4C2×D411yescharacteristic
267C3×C4⋊C4C2×D411yescharacteristic
268C3×C22⋊C4—22no—
269C3×C22⋊C4—22no—
270C3×C22⋊C4—22no—
271C3×C22⋊C4—22no—
272C3×C22⋊C4—22no—
273C3×C22⋊C4—21no—
274C3×C22⋊C4—21no—
275C3×C22⋊C4—21no—
276C3×C22⋊C4—22no—
277C3×C22⋊C4—21no—
278C3×C22⋊C4—22no—
279C3×C22⋊C4—22no—
280C3×C22⋊C4—22no—
281C3×C22⋊C4—21no—
282C3×C42—22no—
283C3×C42—21no—
284#217—82no—
285#217—81no—
286#215—81no—
287#213—82no—
288#213—81no—
289(C2×D3)⋊C4—41no—
290(C2×D3)⋊C4—41no—
291(C2×D3)⋊C4—82no—
292C4⋊Dic3—21no—
293C4⋊Dic3—21no—
294#208—81no—
295C2×C3⋊C8—22no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—81no—
299C2×#41#6711yescharacteristic, complemented
300C22×D4—22no—
301C22×D4—21no—
302C22×D4—21no—
303C22×D4—22no—
304C22×D4—31no—
305C22×D4—61no—
306C22×D4—62no—
307C22×D4#6712yes—
308C22×D4#6711yescharacteristic
309C22×D4#6711yescharacteristic, complemented
310C23×C4C22×D311yescharacteristic
311C2×SD16—61no—
312C2×D8—62no—
313C2×D8—61no—
314C2×M16—61no—
315C22×C8—61no—
316#127—22no—
317#127—21no—
318#126—22no—
319#126—22no—
320#125—21no—
321#124—22no—
322#123—22no—
323#123—21no—
324#123—22no—
325#123—22no—
326#123—22no—
327C22⋊Q8#6711yescharacteristic, complemented
328C22⋊Q8C22×D311yescharacteristic
329#121—22no—
330#121—21no—
331#121—21no—
332#121—22no—
333#121—22no—
334#121—22no—
335#121—22no—
336#121—21no—
337#121—21no—
338#121—21no—
339#121—21no—
340#121—21no—
341#121—21no—
342#121—22no—
343#121—61no—
344#121—61no—
345#121—61no—
346#121—61no—
347#121#6712yescomplemented
348#121#6711yescharacteristic, complemented
349#121C22×D312yes—
350#121C22×D311yescharacteristic
351#120—22no—
352#120—22no—
353#120—22no—
354#120—21no—
355#120—22no—
356#120—22no—
357#120—22no—
358C4×D4—22no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4—22no—
364#117—21no—
365#117#6711yescharacteristic, complemented
366C2×C4⋊C4—31no—
367C2×C4⋊C4#6711yescharacteristic
368C2×C4⋊C4#6711yescharacteristic, complemented
369C2×C22⋊C4—21no—
370C2×C22⋊C4—61no—
371C2×C22⋊C4—61no—
372C2×C22⋊C4—62no—
373C2×C22⋊C4—62no—
374C2×C22⋊C4C22×D312yes—
375#107—62no—
376#107—61no—
377#106—61no—
378#103—61no—
379#102—62no—
380#102—61no—
381#102—62no—
382#102—61no—
383#102—61no—
384C22⋊C8—31no—
385C22⋊C8—31no—
386C22⋊C8—32no—
387C22⋊C8—61no—
388C22⋊C8—61no—
389#95—32no—
390#95—61no—
391#95—62no—
392C23×C3—22no—
393C23×C3—22no—
394C23×C3—22no—
395C23×C3—21no—
396C23×C3—22no—
397C23×C3—22no—
398C23×C3—41no—
399C23×C3—41no—
400C23×C3—42no—
401C23×C3—42no—
402C23×C3—42no—
403C23×C3—81no—
404C23×C3—81no—
405C23×C3—82no—
406C23×C3—82no—
407C23×C3—82no—
408C23×C3#12011yescharacteristic
409C23×C3#12011yescharacteristic
410C23×C3C22×D411yescharacteristic
411C22×D3—41no—
412C22×D3—42no—
413C22×D3—41no—
414C22×D3—41no—
415C22×D3—42no—
416C3×Q8—41no—
417C3×D4—42no—
418C3×D4—41no—
419C3×D4—42no—
420C3×D4—42no—
421C3×D4—44no—
422C3×D4—42no—
423C3×D4—42no—
424C3×D4—42no—
425C3×D4—42no—
426C3×D4—81no—
427C3×D4—82no—
428C3×D4—82no—
429C3×D4—81no—
430C3×D4—82no—
431C3×D4—82no—
432C3×D4—82no—
433C3×D4—82no—
434C3×D4—82no—
435C2×C3×C4—21no—
436C2×C3×C4—22no—
437C2×C3×C4—22no—
438C2×C3×C4—21no—
439C2×C3×C4—21no—
440C2×C3×C4—41no—
441C2×C3×C4—41no—
442C2×C3×C4—41no—
443C2×C3×C4—41no—
444C2×C3×C4—41no—
445C2×C3×C4—41no—
446C2×C3×C4—42no—
447C2×C3×C4—42no—
448C2×C3×C4—81no—
449C2×C3×C4—82no—
450C2×C3×C4—82no—
451C2×C3×C4#12011yescharacteristic
452C2×C3×C4#12011yescharacteristic
453C2×C3×C4C22×D411yescharacteristic
454C2×C3×C4C22×D411yescharacteristic
455#67—82no—
456#67—82no—
457#67—88no—
458C2×Dic3—42no—
459C2×Dic3—42no—
460C2×Dic3—41no—
461C2×Dic3—42no—
462C2×Dic3—42no—
463D12—42no—
464D12—41no—
465D12—41no—
466C3⋊C8—42no—
467C3⋊C8—41no—
468C3⋊C8—41no—
469C24—22no—
470C24—21no—
471C24—61no—
472C24C2×#6712yes—
473C24C2×#6711yescharacteristic
474#41—81no—
475C2×Q8C2×#6711yescharacteristic
476C2×D4—22no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—61no—
492C2×D4—61no—
493C2×D4—81no—
494C2×D4—82no—
495C2×D4—81no—
496C2×D4—82no—
497C2×D4—82no—
498C2×D4—82no—
499C2×D4—82no—
500C2×D4—84no—
501C2×D4—121no—
502C2×D4—121no—
503C2×D4—241no—
504C2×D4—242no—
505C2×D4—242no—
506C2×D4—242no—
507C2×D4—242no—
508C2×D4C2×#6711yescharacteristic
509C2×D4C2×#6712yes—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—62no—
515C22×C4—61no—
516C22×C4—81no—
517C22×C4C2×#6712yes—
518C22×C4C2×#6711yescharacteristic
519C22×C4C2×#6711yescharacteristic
520C22×C4C2×#6711yescharacteristic
521C22×C4C2×#6711yescharacteristic
522C22×C4C2×#6711yescharacteristic
523C22×C4C23×D311yescharacteristic
524SD16—241no—
525D8—242no—
526D8—241no—
527M16—241no—
528C2×C8—62no—
529C2×C8—61no—
530C2×C8—61no—
531C2×C8—241no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—61no—
540C4⋊C4—61no—
541C4⋊C4C2×#6711yescharacteristic
542C4⋊C4C2×#6712yes—
543C4⋊C4C2×#6711yescharacteristic
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—21no—
552C22⋊C4—22no—
553C22⋊C4—21no—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—21no—
558C22⋊C4—121no—
559C22⋊C4—121no—
560C22⋊C4—242no—
561C22⋊C4—242no—
562C22⋊C4—241no—
563C42—22no—
564C42—21no—
565C22×C3—21no—
566C22×C3—21no—
567C22×C3—21no—
568C22×C3—21no—
569C22×C3—21no—
570C22×C3—21no—
571C22×C3—21no—
572C22×C3—21no—
573C22×C3—42no—
574C22×C3—42no—
575C22×C3—41no—
576C22×C3—42no—
577C22×C3—42no—
578C22×C3—41no—
579C22×C3—41no—
580C22×C3—42no—
581C22×C3—42no—
582C22×C3—82no—
583C22×C3—82no—
584C22×C3—82no—
585C22×C3—162no—
586C22×C3—164no—
587C22×C3C2×#12011yescharacteristic
588C22×C3#57612yes—
589C2×D3—82no—
590C2×D3—82no—
591C2×D3—81no—
592C2×D3—81no—
593C2×D3—81no—
594C3×C4—21no—
595C3×C4—21no—
596C3×C4—21no—
597C3×C4—21no—
598C3×C4—42no—
599C3×C4—41no—
600C3×C4—42no—
601C3×C4—41no—
602C3×C4—41no—
603Dic3—82no—
604Dic3—81no—
605C23—22no—
606C23—22no—
607C23—22no—
608C23—21no—
609C23—22no—
610C23—22no—
611C23—41no—
612C23—41no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—81no—
617C23—81no—
618C23—82no—
619C23—82no—
620C23—82no—
621C23—121no—
622C23—121no—
623C23—122no—
624C23—122no—
625C23—121no—
626C23#95311yescharacteristic
627C23#95311yescharacteristic
628C23C22×#6711yescharacteristic
629Q8—41no—
630D4—42no—
631D4—41no—
632D4—42no—
633D4—42no—
634D4—44no—
635D4—42no—
636D4—42no—
637D4—42no—
638D4—42no—
639D4—81no—
640D4—82no—
641D4—82no—
642D4—81no—
643D4—82no—
644D4—82no—
645D4—82no—
646D4—82no—
647D4—82no—
648D4—122no—
649D4—121no—
650D4—121no—
651D4—242no—
652D4—242no—
653D4—248no—
654C2×C4—21no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—81no—
668C2×C4—82no—
669C2×C4—82no—
670C2×C4—122no—
671C2×C4—122no—
672C2×C4—122no—
673C2×C4—122no—
674C2×C4—121no—
675C2×C4#95311yescharacteristic
676C2×C4#95311yescharacteristic
677C2×C4C22×#6711yescharacteristic
678C2×C4C22×#6711yescharacteristic
679C8—122no—
680C8—121no—
681C8—121no—
682C2×C3—22no—
683C2×C3—41no—
684C2×C3—41no—
685C2×C3—82no—
686C2×C3—82no—
687C2×C3—81no—
688C2×C3#300811yescharacteristic
689C2×C3#301911yescharacteristic
690C2×C3#302611yescharacteristic
691D3—161no—
692C22—21no—
693C22—21no—
694C22—21no—
695C22—21no—
696C22—21no—
697C22—21no—
698C22—21no—
699C22—21no—
700C22—42no—
701C22—42no—
702C22—41no—
703C22—42no—
704C22—42no—
705C22—41no—
706C22—41no—
707C22—42no—
708C22—42no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—162no—
713C22—164no—
714C22—242no—
715C22—242no—
716C22—241no—
717C22—241no—
718C22—241no—
719C22#583512yes—
720C22C2×#95311yescharacteristic
721C4—21no—
722C4—21no—
723C4—21no—
724C4—21no—
725C4—42no—
726C4—41no—
727C4—42no—
728C4—41no—
729C4—41no—
730C4—242no—
731C4—241no—
732C3#1845011yescharacteristic, complemented, Sylow
733C2—22no—
734C2—41no—
735C2—41no—
736C2—82no—
737C2—82no—
738C2—81no—
739C2—481no—
740C2#8410111yescharacteristic
741C2#8410411yescharacteristic
742C2#8411011yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3537C2
socleC22×C3C2×#120
hypercenter#3537D3
hypocenterC3#18450
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. #82931C2semidirect
3. #82722C2semidirect
4. #82625C2semidirect
5. #71884C2semidirect
6. #71882C2semidirect
7. #71845C2semidirect
8. #71843C2semidirect
9. #71710C2semidirect
10. #71676C2semidirect
11. #71674C2semidirect
28. C2×#930C22semidirect
33. #5211C22semidirect
34. #5207C22semidirect
35. C2×#830C22semidirect
37. #5046C22semidirect
38. #5043C22semidirect
41. #5036C22semidirect
42. #5036C22semidirect
43. #5033C22semidirect
44. #5033C22semidirect
48. #4960C22semidirect
49. #4953C22semidirect
50. #4538C22semidirect
51. #4537C22semidirect
52. #4537C22semidirect
53. #4537C22semidirect
54. #4525C22semidirect
55. #4525C22semidirect
56. #3537D3semidirect
67. C2×C3×#41D4semidirect
74. C22×C3×D4D4semidirect
90. C3×C22⋊Q8D4semidirect
106. C3×#121D4semidirect
107. C3×#121D4semidirect
124. C3×#117D4semidirect
126. C2×C3×C4⋊C4D4semidirect
299. C2×#41#67semidirect
309. C22×D4#67semidirect
327. C22⋊Q8#67semidirect
347. #121#67semidirect
348. #121#67semidirect
365. #117#67semidirect
368. C2×C4⋊C4#67semidirect
732. C3#18450semidirect

Maximal subgroups

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

subgroupindex
C3×#35372
#829312
#827222
#826252
#718842
#718822
#718452
#718432
#717102
#716762
#716742
#184503