Finite Group Databasebeta

← #10948000

Subgroups — #10948000

Order 768.

1252 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86317C211yesmaximal, characteristic
2#78688C212yesmaximal, complemented
3C3×#2127C211yesmaximal, characteristic, complemented
4#71860C211yesmaximal, characteristic, complemented
5C2×#4473C212yesmaximal, complemented
6#69159C212yesmaximal, complemented
7D3×#354C212yesmaximal, complemented
8#69151C212yesmaximal, complemented
9#69150C212yesmaximal, complemented
10#33361—31nomaximal, Sylow
11C2×#1010C411yescharacteristic, complemented
12C2×#1004C2211yescharacteristic
13C3×#535C2211yescharacteristic
14#5832C411yescharacteristic, complemented
15C2×#936C2211yescharacteristic
16#5716C411yescharacteristic, complemented
17#5716C2211yescharacteristic
18#5715C411yescharacteristic, complemented
19#5715C2211yescharacteristic
20#5675C411yescharacteristic, complemented
21#5675C2211yescharacteristic
22D3×#124C411yescharacteristic, complemented
23D3×#124C2211yescharacteristic
24#5612—22no—
25#5598—22no—
26#5591—22no—
27#5591—22no—
28#5526C411yescharacteristic, complemented
29#5522C411yescharacteristic, complemented
30#5486—22no—
31C2×C3×#99C2212yes—
32C2×#834C2212yes—
33C2×#806—22no—
34D3×#99—22no—
35D3×#99C2214yes—
36#4740—22no—
37#4740C2214yes—
38C3×#354C2212yescomplemented
39C3×#353C2212yescomplemented
40#4545C2212yescomplemented
41#4541C2212yescomplemented
42#4473C2214yescomplemented
43#4473C2214yescomplemented
44#3500—31no—
45#2883—32no—
46#2129—32no—
47#2127—32no—
48#2127—31no—
49#2127D311yescharacteristic, complemented
50#2126—32no—
51C2×#354—32no—
52C2×#354—32no—
53C2×C3×#41C2×C411yescharacteristic, complemented
54C22×C3×D4C2311yescharacteristic
55C22×#67D412yes—
56C22×#67D412yes—
57#1010—44no—
58C2×D3×Q8C2×C411yescharacteristic, complemented
59C2×D3×Q8C2×C411yescharacteristic, complemented
60#1004—44no—
61C2×#237—21no—
62C2×#237—21no—
63C2×#237C2×C411yescharacteristic
64C2×#237C2×C411yescharacteristic, complemented
65C2×#237C2×C411yescharacteristic, complemented
66C2×#237D412yes—
67C2×#237C2311yescharacteristic
68C2×D3×D4C2×C411yescharacteristic
69C2×D3×D4D412yes—
70C2×D3×D4C2311yescharacteristic
71C2×#235—21no—
72C2×#235C2×C411yescharacteristic
73C2×#235C2×C411yescharacteristic
74C22×Dic6—21no—
75C22×Dic6C2×C411yescharacteristic, complemented
76C3×#127C2×C411yescharacteristic, complemented
77C3×#127C2311yescharacteristic
78C3×#124C2×C411yescharacteristic, complemented
79C3×#124C2311yescharacteristic
80C3×#121—22no—
81C3×#120—22no—
82C3×#117C2×C411yescharacteristic, complemented
83C2×C3×C22⋊C4—22no—
84C2×#217—22no—
85C2×#217—22no—
86#947C2×C411yescharacteristic, complemented
87#944C2×C411yescharacteristic, complemented
88#940C2×C411yescharacteristic, complemented
89#939—22no—
90#939—22no—
91#939—22no—
92#938—22no—
93#938—22no—
94#937—22no—
95#937—22no—
96#936—22no—
97#936C2×C411yescharacteristic, complemented
98#936C2×C411yescharacteristic, complemented
99#936C2×C411yescharacteristic, complemented
100#936C2311yescharacteristic
101#936C2311yescharacteristic
102D4×Dic3—22no—
103D4×Dic3—22no—
104C4×#67—22no—
105C4×#67—44no—
106C2×C4×Dic3C2×C411yescharacteristic, complemented
107#897—22no—
108#893—21no—
109#887—21no—
110#886—22no—
111#885—22no—
112#885—22no—
113#883—22no—
114#883—22no—
115#67⋊C4—21no—
116#67⋊C4—21no—
117#67⋊C4—44no—
118D3×C22⋊C4—22no—
119D3×C22⋊C4—22no—
120D3×C22⋊C4—24no—
121#878—22no—
122#878—22no—
123#877—24no—
124#875C2×C411yescharacteristic, complemented
125#875C2×C411yescharacteristic, complemented
126#872C2×C411yescharacteristic, complemented
127#872C2×C411yescharacteristic, complemented
128C42×D3C2×C411yescharacteristic, complemented
129C42×D3C2×C411yescharacteristic, complemented
130#869C2×C411yescharacteristic, complemented
131#869C2×C411yescharacteristic, complemented
132C3×#99—22no—
133C3×#99C2314yes—
134#834—22no—
135#834C2314yes—
136#806—24no—
137#806—24no—
138C2×2−1+4—31no—
139C2×H(5,𝔽2)—31no—
140#548—62no—
141#546—62no—
142#546—62no—
143#546—62no—
144#536—31no—
145#536—31no—
146#536—31no—
147#535—31no—
148#535—31no—
149#535C2×D311yescharacteristic
150#532—31no—
151#532—31no—
152#532—31no—
153C2×#124—31no—
154C2×#124—31no—
155C2×#124—31no—
156#518—62no—
157C4×#41—31no—
158#410—34no—
159#410—62no—
160C2×#99—32no—
161C2×#99—34no—
162C2×#99—62no—
163C2×#99—62no—
164C2×#99C2×D312yes—
165#354—32no—
166#354—34no—
167#354—34no—
168#354C2×D312yescomplemented
169#353—32no—
170#353C2×D312yescomplemented
171C24×C3C2×D412yes—
172C23×D3C22⋊C412yes—
173C23×D3C2×D412yes—
174C3×#41—42no—
175C2×C3×Q8C22×C411yescharacteristic
176C2×C3×D4—21no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—42no—
180C2×C3×D4C22×C411yescharacteristic
181C2×C3×D4C22×C411yescharacteristic
182C2×C3×D4C2×D412yes—
183C2×C3×D4C2411yescharacteristic
184C22×C3×C4—21no—
185C22×C3×C4—42no—
186C22×C3×C4C22×C411yescharacteristic
187C2×#67—24no—
188C2×#67—24no—
189C2×#67—21no—
190C2×#67—21no—
191C2×#67—22no—
192C2×#67—42no—
193C2×#67—42no—
194C2×#67C22⋊C411yescharacteristic
195C2×#67C22⋊C412yes—
196C2×#67C22⋊C412yes—
197C2×#67C22⋊C412yes—
198C2×#67C22⋊C411yescharacteristic
199C2×#67C2×D412yes—
200C2×#67C2×D412yes—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—42no—
206C22×Dic3C22⋊C412yes—
207C22×Dic3C2×D412yes—
208D3×Q8—42no—
209D3×Q8—42no—
210#237—44no—
211#237—42no—
212#237—42no—
213#237—42no—
214#237—42no—
215#237—44no—
216#237—44no—
217D3×D4—42no—
218D3×D4—42no—
219D3×D4—44no—
220#235—24no—
221#235—44no—
222#235—42no—
223C2×D12C22⋊C411yescharacteristic
224C2×D12C22×C411yescharacteristic
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×C4×D3—44no—
230C2×C4×D3C22⋊C412yes—
231C2×C4×D3C22×C411yescharacteristic
232C2×C4×D3C22×C411yescharacteristic
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—44no—
237C2×Dic6—42no—
238C2×Dic6C22⋊C411yescharacteristic
239C2×Dic6C22×C411yescharacteristic
240C2×Dic6C22×C411yescharacteristic
241C2×Dic6C22×C411yescharacteristic
242C3×C4⋊C4—21no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—21no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—42no—
248C3×C42C22×C411yescharacteristic
249C3×C42C22×C411yescharacteristic
250#217—22no—
251#217—22no—
252#217—22no—
253#217—22no—
254#217—22no—
255#217—24no—
256#217—42no—
257(C2×D3)⋊C4—21no—
258(C2×D3)⋊C4—21no—
259(C2×D3)⋊C4—22no—
260(C2×D3)⋊C4—22no—
261(C2×D3)⋊C4—22no—
262(C2×D3)⋊C4—44no—
263C4⋊Dic3—22no—
264C4⋊Dic3—22no—
265Dic3⋊C4—21no—
266Dic3⋊C4—21no—
267Dic3⋊C4—22no—
268Dic3⋊C4—44no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—44no—
273C4×Dic3C22×C411yescharacteristic
274C4×Dic3C22×C411yescharacteristic
2752−1+4—124no—
276H(5,𝔽2)—124no—
277C2×#41—31no—
278C2×#41—32no—
279C2×#41—31no—
280C2×#41—31no—
281C2×#41—31no—
282C2×#41—31no—
283C2×#41—31no—
284C2×#41—61no—
285C2×#41—61no—
286C2×#41—61no—
287C2×#41C4×D311yescharacteristic, complemented
288C22×Q8—31no—
289C22×Q8—31no—
290C22×Q8—31no—
291C22×Q8—61no—
292C22×D4—32no—
293C22×D4—32no—
294C22×D4—31no—
295C22×D4—32no—
296C22×D4—31no—
297C22×D4C22×D311yescharacteristic
298C4⋊Q8—31no—
299C4⋊Q8—31no—
300C4⋊Q8—31no—
301#127—31no—
302#127C4×D311yescharacteristic, complemented
303#127C22×D311yescharacteristic
304#124—31no—
305#124—31no—
306#124—31no—
307#124—31no—
308#124—31no—
309#124—31no—
310#124—31no—
311#124—31no—
312#124—62no—
313#124—62no—
314#124—62no—
315#124C4×D311yescharacteristic, complemented
316#124C22×D311yescharacteristic
317#123—62no—
318#123—62no—
319#123—62no—
320C22⋊Q8—62no—
321C22⋊Q8—62no—
322C22⋊Q8—62no—
323#121—22no—
324#121—62no—
325#121—62no—
326#121—62no—
327#121—62no—
328#121—62no—
329#120—22no—
330#120—62no—
331#120—62no—
332C4×Q8—61no—
333C4×D4—61no—
334C4×D4—61no—
335C4×D4—61no—
336C4×D4—62no—
337C4×D4—62no—
338C4×D4—62no—
339C4×D4—124no—
340C4×D4—124no—
341#117—31no—
342#117—31no—
343#117—64no—
344#117C4×D311yescharacteristic, complemented
345C2×C22⋊C4—22no—
346C2×C22⋊C4—62no—
347C2×C22⋊C4—62no—
348C2×C22⋊C4—64no—
349C2×C22⋊C4—62no—
350C2×C22⋊C4—62no—
351C2×C42—31no—
352C2×C42—31no—
353C2×C42—31no—
354#99—22no—
355#99—34no—
356#99—64no—
357#99—64no—
358#99—62no—
359#99C22×D314yes—
360C23×C3—22no—
361C23×C3—22no—
362C23×C3—21no—
363C23×C3—82no—
364C23×C3C2×C22⋊C412yes—
365C23×C3C2×C22⋊C411yescharacteristic
366C23×C3C22×D412yes—
367C22×D3—24no—
368C22×D3—42no—
369C22×D3—82no—
370C22×D3—82no—
371C22×D3C2×C22⋊C412yes—
372C22×D3C2×C22⋊C411yescharacteristic
373C22×D3C2×C22⋊C411yescharacteristic
374C3×Q8—41no—
375C3×D4—41no—
376C3×D4—42no—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—42no—
380C3×D4—82no—
381C3×D4—82no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—42no—
387C2×C3×C4—44no—
388C2×C3×C4—162no—
389C2×C3×C4C2×C22⋊C411yescharacteristic
390C2×C3×C4C23×C411yescharacteristic
391#67—22no—
392#67—22no—
393#67—22no—
394#67—24no—
395#67—24no—
396#67—42no—
397#67—42no—
398#67—42no—
399#67—42no—
400#67—82no—
401#67#9914yes—
402C2×Dic3—24no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—42no—
408C2×Dic3—42no—
409C2×Dic3—42no—
410C2×Dic3—42no—
411C2×Dic3—44no—
412C2×Dic3—81no—
413C2×Dic3—162no—
414C2×Dic3C2×C22⋊C411yescharacteristic
415C2×Dic3C2×C22⋊C412yes—
416C2×Dic3C2×C22⋊C411yescharacteristic
417D12—22no—
418D12—41no—
419C4×D3—22no—
420C4×D3—22no—
421C4×D3—42no—
422C4×D3—42no—
423C4×D3—42no—
424C4×D3—81no—
425C4×D3—81no—
426C4×D3—164no—
427Dic6—22no—
428Dic6—41no—
429Dic6—42no—
430Dic6—42no—
431Dic6—42no—
432Dic6—41no—
433Dic6—41no—
434C24—32no—
435C24—32no—
436C24D3×D412yes—
437#41—42no—
438#41—64no—
439#41—122no—
440#41—124no—
441#41—122no—
442#41—122no—
443#41—124no—
444#41—122no—
445#41—122no—
446#41—124no—
447#41—124no—
448C2×Q8—31no—
449C2×Q8—31no—
450C2×Q8—31no—
451C2×Q8—31no—
452C2×Q8—61no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—122no—
456C2×Q8—124no—
457C2×Q8—122no—
458C2×Q8—122no—
459C2×Q8C2×C4×D311yescharacteristic
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—31no—
464C2×D4—32no—
465C2×D4—32no—
466C2×D4—31no—
467C2×D4—32no—
468C2×D4—32no—
469C2×D4—32no—
470C2×D4—31no—
471C2×D4—31no—
472C2×D4—42no—
473C2×D4—64no—
474C2×D4—64no—
475C2×D4—61no—
476C2×D4—61no—
477C2×D4—62no—
478C2×D4—122no—
479C2×D4—122no—
480C2×D4—124no—
481C2×D4—122no—
482C2×D4—122no—
483C2×D4C2×C4×D311yescharacteristic
484C2×D4C2×C4×D311yescharacteristic
485C2×D4D3×D412yes—
486C2×D4C23×D311yescharacteristic
487C22×C4—21no—
488C22×C4—32no—
489C22×C4—32no—
490C22×C4—31no—
491C22×C4—31no—
492C22×C4—32no—
493C22×C4—42no—
494C22×C4—61no—
495C22×C4—61no—
496C22×C4—61no—
497C22×C4—61no—
498C22×C4—61no—
499C22×C4—61no—
500C22×C4—61no—
501C22×C4—61no—
502C22×C4—124no—
503C22×C4—122no—
504C22×C4C2×C4×D311yescharacteristic
505C4⋊C4—21no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—62no—
509C4⋊C4—62no—
510C4⋊C4—62no—
511C4⋊C4—124no—
512C22⋊C4—22no—
513C22⋊C4—21no—
514C22⋊C4—22no—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—62no—
518C22⋊C4—62no—
519C22⋊C4—62no—
520C22⋊C4—61no—
521C22⋊C4—61no—
522C22⋊C4—62no—
523C22⋊C4—62no—
524C22⋊C4—62no—
525C22⋊C4—62no—
526C22⋊C4—62no—
527C22⋊C4—64no—
528C22⋊C4—124no—
529C22⋊C4—122no—
530C42—31no—
531C42—31no—
532C42—61no—
533C42—61no—
534C42—61no—
535C42—124no—
536C42C2×C4×D311yescharacteristic
537C42C2×C4×D311yescharacteristic
538C22×C3—22no—
539C22×C3—22no—
540C22×C3—41no—
541C22×C3—42no—
542C22×C3—42no—
543C22×C3—42no—
544C22×C3—41no—
545C22×C3—82no—
546C22×C3—82no—
547C22×C3C2×#9912yes—
548C22×C3C22×C22⋊C411yescharacteristic
549C2×D3—21no—
550C2×D3—22no—
551C2×D3—41no—
552C2×D3—41no—
553C2×D3—42no—
554C2×D3—84no—
555C2×D3—162no—
556C2×D3C2×#9912yes—
557C3×C4—21no—
558C3×C4—21no—
559C3×C4—41no—
560C3×C4—41no—
561C3×C4—41no—
562C3×C4—84no—
563Dic3—21no—
564Dic3—22no—
565Dic3—41no—
566Dic3—41no—
567Dic3—41no—
568Dic3—41no—
569Dic3—84no—
570Dic3C2×#9912yes—
571C23—22no—
572C23—22no—
573C23—21no—
574C23—31no—
575C23—32no—
576C23—31no—
577C23—64no—
578C23—82no—
579C23—122no—
580C23—242no—
581C23—242no—
582C23D3×C22⋊C412yes—
583C23D3×C22⋊C411yescharacteristic
584C23C2×D3×D412yes—
585Q8—41no—
586Q8—62no—
587Q8—121no—
588Q8—121no—
589Q8—121no—
590Q8—122no—
591Q8—122no—
592Q8—122no—
593D4—34no—
594D4—41no—
595D4—42no—
596D4—41no—
597D4—41no—
598D4—42no—
599D4—62no—
600D4—64no—
601D4—64no—
602D4—62no—
603D4—62no—
604D4—62no—
605D4—82no—
606D4—82no—
607D4—121no—
608D4—122no—
609D4—122no—
610D4—122no—
611D4—122no—
612D4—242no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—31no—
618C2×C4—31no—
619C2×C4—32no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—62no—
623C2×C4—62no—
624C2×C4—64no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—122no—
630C2×C4—122no—
631C2×C4—122no—
632C2×C4—122no—
633C2×C4—122no—
634C2×C4—122no—
635C2×C4—122no—
636C2×C4—124no—
637C2×C4—162no—
638C2×C4—241no—
639C2×C4—241no—
640C2×C4—241no—
641C2×C4—484no—
642C2×C4—482no—
643C2×C4D3×C22⋊C411yescharacteristic
644C2×C4C22×C4×D311yescharacteristic
645C2×C3—21no—
646C2×C3—22no—
647C2×C3—42no—
648C2×C3—42no—
649C2×C3—81no—
650C2×C3C22×#9911yescharacteristic
651D3—22no—
652D3—42no—
653D3—41no—
654C22—22no—
655C22—22no—
656C22—32no—
657C22—41no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—41no—
662C22—61no—
663C22—62no—
664C22—82no—
665C22—82no—
666C22—121no—
667C22—121no—
668C22—122no—
669C22—244no—
670C22—482no—
671C22D3×#9912yes—
672C22C2×D3×C22⋊C411yescharacteristic
673C4—21no—
674C4—21no—
675C4—32no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—61no—
680C4—62no—
681C4—84no—
682C4—121no—
683C4—121no—
684C4—121no—
685C4—121no—
686C4—244no—
687C3#3336111yescharacteristic, complemented, Sylow
688C2—21no—
689C2—22no—
690C2—42no—
691C2—42no—
692C2—62no—
693C2—81no—
694C2—122no—
695C2—121no—
696C2C2×D3×#9911yescharacteristic
subgroupG/H
centerC2C2×D3×#99
commutatorC2×C3×C4C23×C4
FrattiniC2×D4C23×D3
FittingC3×#2127C2
socleC2×C3C22×#99
prime radical#86317C2
cosocleC2×C3×D4C24
hypercenter#2127D3
hypocenterC3#33361
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ #2127

Complements

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

NH ≅ G/Ntype
2. #78688C2semidirect
3. C3×#2127C2semidirect
4. #71860C2semidirect
5. C2×#4473C2semidirect
6. #69159C2semidirect
7. D3×#354C2semidirect
8. #69151C2semidirect
9. #69150C2semidirect
11. C2×#1010C4semidirect
14. #5832C4semidirect
16. #5716C4semidirect
18. #5715C4semidirect
20. #5675C4semidirect
22. D3×#124C4semidirect
28. #5526C4semidirect
29. #5522C4semidirect
38. C3×#354C22semidirect
39. C3×#353C22semidirect
40. #4545C22semidirect
41. #4541C22semidirect
42. #4473C22semidirect
43. #4473C22semidirect
49. #2127D3semidirect
53. C2×C3×#41C2×C4semidirect
58. C2×D3×Q8C2×C4semidirect
59. C2×D3×Q8C2×C4semidirect
64. C2×#237C2×C4semidirect
65. C2×#237C2×C4semidirect
75. C22×Dic6C2×C4semidirect
76. C3×#127C2×C4semidirect
78. C3×#124C2×C4semidirect
82. C3×#117C2×C4semidirect
86. #947C2×C4semidirect
87. #944C2×C4semidirect
88. #940C2×C4semidirect
97. #936C2×C4semidirect
98. #936C2×C4semidirect
99. #936C2×C4semidirect
106. C2×C4×Dic3C2×C4semidirect
124. #875C2×C4semidirect
125. #875C2×C4semidirect
126. #872C2×C4semidirect
127. #872C2×C4semidirect
128. C42×D3C2×C4semidirect
129. C42×D3C2×C4semidirect
130. #869C2×C4semidirect
131. #869C2×C4semidirect
168. #354C2×D3semidirect
170. #353C2×D3semidirect
287. C2×#41C4×D3semidirect
302. #127C4×D3semidirect
315. #124C4×D3semidirect
344. #117C4×D3semidirect
687. C3#33361semidirect

Maximal subgroups

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

subgroupindex
#863172
#786882
C3×#21272
#718602
C2×#44732
#691592
D3×#3542
#691512
#691502
#333613