Finite Group Databasebeta

← #10947981

Subgroups — #10947981

Order 768.

1252 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86313C211yesmaximal, characteristic
2C2×#4740C212yesmaximal, complemented
3C3×#2127C211yesmaximal, characteristic, complemented
4#71860C211yesmaximal, characteristic, complemented
5#70019C212yesmaximal, complemented
6#69159C212yesmaximal, complemented
7#69158C212yesmaximal, complemented
8#69152C212yesmaximal, complemented
9#69151C212yesmaximal, complemented
10#33358—31nomaximal, Sylow
11C2×D3×#41C411yescharacteristic, complemented
12C22×#237C2211yescharacteristic
13C3×#535C2211yescharacteristic
14#5825C411yescharacteristic, complemented
15#5805C2211yescharacteristic
16#5716C411yescharacteristic, complemented
17#5716C2211yescharacteristic
18#5715C411yescharacteristic, complemented
19#5715C2211yescharacteristic
20#5675C411yescharacteristic, complemented
21#5675C2211yescharacteristic
22#5668C411yescharacteristic, complemented
23#5668C2211yescharacteristic
24#5608—22no—
25#5606—22no—
26#5589—22no—
27#5586—22no—
28#5526C411yescharacteristic, complemented
29D3×#117C411yescharacteristic, complemented
30C2×#877—22no—
31C2×C3×#99C2212yescomplemented
32C2×#834C2212yescomplemented
33C2×#806—22no—
34#4740—22no—
35#4740—22no—
36#4740C2214yes—
37#4740C2214yescomplemented
38C3×#354C2212yescomplemented
39C3×#353C2212yescomplemented
40#4545C2212yescomplemented
41#4541C2212yescomplemented
42#4474C2214yescomplemented
43#4473C2214yescomplemented
44#3449—31no—
45C2×#410—32no—
46#2129—32no—
47#2128—32no—
48#2127—32no—
49#2127—31no—
50#2127D311yescharacteristic, complemented
51#2126—32no—
52#2126—32no—
53C2×C3×#41C2×C411yescharacteristic
54C22×C3×D4C2311yescharacteristic
55C22×#67D412yes—
56C23×Dic3D412yes—
57D3×#41—42no—
58D3×#41—42no—
59C2×#239C2×C411yescharacteristic, complemented
60C2×D3×Q8C2×C411yescharacteristic, complemented
61C2×#237—21no—
62C2×#237—42no—
63C2×#237—42no—
64C2×#237C2×C411yescharacteristic
65C2×#237C2×C411yescharacteristic
66C2×#237C2×C411yescharacteristic, complemented
67C2×#237D412yes—
68C2×#237D412yes—
69C2×#237C2311yescharacteristic
70C2×#237C2311yescharacteristic
71C2×D3×D4—21no—
72C2×D3×D4C2×C411yescharacteristic, complemented
73C2×#235—21no—
74C2×#235C2×C411yescharacteristic
75C22×C4×D3—21no—
76C22×C4×D3C2×C411yescharacteristic
77C22×Dic6C2×C411yescharacteristic
78C3×#127C2×C411yescharacteristic
79C3×#127C2311yescharacteristic
80C3×#124C2×C411yescharacteristic
81C3×#124C2311yescharacteristic
82C3×#121—22no—
83C3×#120—22no—
84C3×#117C2×C411yescharacteristic
85C2×C3×C22⋊C4—22no—
86#953—22no—
87C2×#217—22no—
88#944C2×C411yescharacteristic, complemented
89#944C2×C411yescharacteristic, complemented
90#940C2×C411yescharacteristic
91#940C2311yescharacteristic
92#939—22no—
93#939—22no—
94#938—22no—
95#938—22no—
96#938—22no—
97#937—22no—
98#937—22no—
99#936C2×C411yescharacteristic
100#936C2×C411yescharacteristic, complemented
101#936C2×C411yescharacteristic, complemented
102#936C2311yescharacteristic
103#935—22no—
104D4×Dic3—22no—
105D4×Dic3—22no—
106#926C2×C411yescharacteristic
107#924—22no—
108C2×Dic3⋊C4—42no—
109C2×C4×Dic3—42no—
110#897—22no—
111#892—21no—
112D3×C4⋊C4—21no—
113#886—22no—
114#884—22no—
115#884—22no—
116#883—22no—
117#883—22no—
118#67⋊C4—22no—
119#67⋊C4—22no—
120D3×C22⋊C4—21no—
121#878—22no—
122#878—22no—
123#877—21no—
124#877—24no—
125#877—24no—
126#877—42no—
127#877—42no—
128#875C2×C411yescharacteristic, complemented
129#875C2×C411yescharacteristic, complemented
130#872C2×C411yescharacteristic, complemented
131#872C2×C411yescharacteristic, complemented
132C42×D3C2×C411yescharacteristic, complemented
133C42×D3C2×C411yescharacteristic, complemented
134#869C2×C411yescharacteristic, complemented
135#869C2×C411yescharacteristic, complemented
136C3×#99—22no—
137C3×#99C2314yes—
138#834—22no—
139#834C2314yes—
140#806—24no—
141#806—24no—
142C22×#41—31no—
143C22×#41—31no—
144#537—62no—
145#536—31no—
146#536—31no—
147#536—31no—
148#536—31no—
149#535—31no—
150#535—31no—
151#535—31no—
152#535C2×D311yescharacteristic
153#534—62no—
154#532—31no—
155#532—31no—
156#532—31no—
157#532—31no—
158#525—62no—
159#525—62no—
160C2×#117—31no—
161C2×#117—62no—
162#410—34no—
163#410—34no—
164#410—62no—
165#410—62no—
166C2×#99—32no—
167C2×#99—62no—
168C2×#99C2×D312yescomplemented
169#354—32no—
170#354—34no—
171#354C2×D312yescomplemented
172#353—32no—
173#353—34no—
174#353C2×D312yescomplemented
175C24×C3C2×D412yes—
176C23×D3—21no—
177C23×D3C22⋊C411yescharacteristic
178C3×#41—42no—
179C2×C3×Q8C22×C411yescharacteristic
180C2×C3×D4—21no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—42no—
184C2×C3×D4C22×C411yescharacteristic
185C2×C3×D4C22×C411yescharacteristic
186C2×C3×D4C2×D412yes—
187C2×C3×D4C2411yescharacteristic
188C22×C3×C4—21no—
189C22×C3×C4—42no—
190C22×C3×C4C22×C411yescharacteristic
191C2×#67—21no—
192C2×#67—21no—
193C2×#67—22no—
194C2×#67—22no—
195C2×#67—21no—
196C2×#67—42no—
197C2×#67—42no—
198C2×#67C22⋊C412yes—
199C2×#67C22⋊C412yes—
200C2×#67C2×D412yes—
201C2×#67C2×D412yes—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—21no—
207C22×Dic3—42no—
208C22×Dic3C22⋊C412yes—
209C22×Dic3C22⋊C412yes—
210C22×Dic3C22⋊C412yes—
211C22×Dic3C22⋊C411yescharacteristic
212C22×Dic3C2×D412yes—
213C22×Dic3C2×D412yes—
214#239—41no—
215#239—41no—
216D3×Q8—42no—
217#237—42no—
218#237—41no—
219#237—41no—
220#237—42no—
221#237—42no—
222#237—42no—
223#237—42no—
224#237—42no—
225#237—41no—
226#237—41no—
227#237—41no—
228#237—41no—
229#237—42no—
230#237—42no—
231D3×D4—42no—
232D3×D4—44no—
233#235—44no—
234#235—42no—
235C2×D12—21no—
236C2×D12C22×C411yescharacteristic
237C2×C4×D3—22no—
238C2×C4×D3—22no—
239C2×C4×D3—21no—
240C2×C4×D3—21no—
241C2×C4×D3—21no—
242C2×C4×D3—21no—
243C2×C4×D3—21no—
244C2×C4×D3—42no—
245C2×C4×D3—42no—
246C2×C4×D3C22⋊C411yescharacteristic
247C2×C4×D3C22⋊C411yescharacteristic
248C2×C4×D3C22×C411yescharacteristic
249C2×C4×D3C22×C411yescharacteristic
250C2×Dic6—21no—
251C2×Dic6—42no—
252C2×Dic6C22⋊C412yes—
253C2×Dic6C22×C411yescharacteristic
254C2×Dic6C22×C411yescharacteristic
255C2×Dic6C22×C411yescharacteristic
256C3×C4⋊C4—21no—
257C3×C22⋊C4—21no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—22no—
260C3×C22⋊C4—24no—
261C3×C22⋊C4—42no—
262C3×C42C22×C411yescharacteristic
263C3×C42C22×C411yescharacteristic
264#217—21no—
265#217—22no—
266#217—22no—
267#217—22no—
268#217—22no—
269#217—24no—
270#217—42no—
271(C2×D3)⋊C4—21no—
272(C2×D3)⋊C4—21no—
273(C2×D3)⋊C4—22no—
274(C2×D3)⋊C4—22no—
275C4⋊Dic3—21no—
276C4⋊Dic3—22no—
277C4⋊Dic3—22no—
278Dic3⋊C4—21no—
279Dic3⋊C4—21no—
280Dic3⋊C4—22no—
281Dic3⋊C4—22no—
282Dic3⋊C4—44no—
283Dic3⋊C4—44no—
284C4×Dic3—21no—
285C4×Dic3—21no—
286C4×Dic3—44no—
287C4×Dic3—44no—
288C4×Dic3C22×C411yescharacteristic
289C4×Dic3C22×C411yescharacteristic
290C2×#41—31no—
291C2×#41—31no—
292C2×#41—32no—
293C2×#41—32no—
294C2×#41—31no—
295C2×#41—31no—
296C2×#41—31no—
297C2×#41—31no—
298C2×#41—31no—
299C2×#41—61no—
300C2×#41—61no—
301C2×#41—122no—
302C2×#41—122no—
303C2×#41—122no—
304C2×#41—122no—
305C2×#41C4×D311yescharacteristic
306C22×Q8—31no—
307C22×Q8—31no—
308C22×D4—32no—
309C22×D4—31no—
310C22×D4—61no—
311C22×D4C22×D311yescharacteristic
312C23×C4—32no—
313C23×C4—31no—
314C23×C4—61no—
315C4⋊Q8—31no—
316C4⋊Q8—31no—
317C4⋊Q8—31no—
318C4⋊Q8—31no—
319#127—31no—
320#127—31no—
321#127C4×D311yescharacteristic
322#127C22×D311yescharacteristic
323#124—31no—
324#124—31no—
325#124—31no—
326#124—31no—
327#124—31no—
328#124—31no—
329#124C4×D311yescharacteristic
330#124C22×D311yescharacteristic
331#123—62no—
332#123—62no—
333#123—62no—
334#123—62no—
335C22⋊Q8—62no—
336C22⋊Q8—62no—
337C22⋊Q8—62no—
338C22⋊Q8—62no—
339#121—22no—
340#121—62no—
341#121—62no—
342#121—62no—
343#121—62no—
344#121—62no—
345#121—62no—
346#121—62no—
347#120—22no—
348#120—62no—
349#120—62no—
350#120—62no—
351C4×D4—62no—
352C4×D4—62no—
353C4×D4—62no—
354C4×D4—62no—
355#117—31no—
356#117—31no—
357#117—31no—
358#117—61no—
359#117—61no—
360#117—64no—
361#117—64no—
362#117—122no—
363#117—122no—
364#117C4×D311yescharacteristic
365C2×C4⋊C4—61no—
366C2×C4⋊C4—122no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—61no—
369C2×C22⋊C4—62no—
370C2×C42—31no—
371C2×C42—31no—
372C2×C42—122no—
373#99—22no—
374#99—34no—
375#99—64no—
376#99—64no—
377#99—62no—
378#99C22×D314yes—
379C23×C3—22no—
380C23×C3—22no—
381C23×C3—21no—
382C23×C3—82no—
383C23×C3C2×C22⋊C412yes—
384C23×C3C2×C22⋊C411yescharacteristic
385C23×C3C22×D412yes—
386C22×D3—22no—
387C22×D3—21no—
388C22×D3—42no—
389C22×D3—82no—
390C22×D3—81no—
391C22×D3C2×C22⋊C411yescharacteristic
392C22×D3C2×C22⋊C411yescharacteristic
393C3×Q8—41no—
394C3×D4—41no—
395C3×D4—42no—
396C3×D4—41no—
397C3×D4—41no—
398C3×D4—42no—
399C3×D4—82no—
400C3×D4—82no—
401C2×C3×C4—22no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—42no—
406C2×C3×C4—44no—
407C2×C3×C4—84no—
408C2×C3×C4C2×C22⋊C411yescharacteristic
409C2×C3×C4C23×C411yescharacteristic
410#67—42no—
411#67—42no—
412#67—42no—
413#67—42no—
414#67—42no—
415#67—42no—
416#67—42no—
417#67—82no—
418#67—82no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—44no—
435C2×Dic3—84no—
436C2×Dic3#9912yes—
437C2×Dic3#9912yes—
438C2×Dic3C2×C22⋊C412yes—
439C2×Dic3C2×C22⋊C412yes—
440C2×Dic3C2×C22⋊C411yescharacteristic
441C2×Dic3C2×C22⋊C411yescharacteristic
442D12—41no—
443D12—42no—
444C4×D3—21no—
445C4×D3—21no—
446C4×D3—22no—
447C4×D3—22no—
448C4×D3—21no—
449C4×D3—21no—
450C4×D3—42no—
451C4×D3—42no—
452C4×D3—41no—
453C4×D3—41no—
454C4×D3—81no—
455C4×D3—81no—
456Dic6—42no—
457Dic6—41no—
458Dic6—41no—
459Dic6—42no—
460Dic6—41no—
461C24—31no—
462C24—61no—
463C24D3×D412yes—
464#41—42no—
465#41—122no—
466#41—122no—
467#41—122no—
468#41—124no—
469#41—122no—
470#41—122no—
471#41—122no—
472#41—122no—
473#41—121no—
474#41—122no—
475#41—121no—
476#41—122no—
477#41—121no—
478#41—121no—
479#41—121no—
480#41—121no—
481#41—121no—
482#41—121no—
483C2×Q8—32no—
484C2×Q8—31no—
485C2×Q8—31no—
486C2×Q8—31no—
487C2×Q8—61no—
488C2×Q8—122no—
489C2×Q8—122no—
490C2×Q8C2×C4×D311yescharacteristic
491C2×D4—21no—
492C2×D4—22no—
493C2×D4—22no—
494C2×D4—32no—
495C2×D4—32no—
496C2×D4—32no—
497C2×D4—32no—
498C2×D4—31no—
499C2×D4—42no—
500C2×D4—61no—
501C2×D4—61no—
502C2×D4—61no—
503C2×D4—61no—
504C2×D4—62no—
505C2×D4—62no—
506C2×D4—122no—
507C2×D4—122no—
508C2×D4—122no—
509C2×D4—124no—
510C2×D4C2×C4×D311yescharacteristic
511C2×D4C2×C4×D311yescharacteristic
512C2×D4D3×D412yes—
513C2×D4C23×D311yescharacteristic
514C22×C4—21no—
515C22×C4—31no—
516C22×C4—32no—
517C22×C4—32no—
518C22×C4—32no—
519C22×C4—32no—
520C22×C4—32no—
521C22×C4—31no—
522C22×C4—31no—
523C22×C4—31no—
524C22×C4—31no—
525C22×C4—42no—
526C22×C4—62no—
527C22×C4—62no—
528C22×C4—61no—
529C22×C4—61no—
530C22×C4—61no—
531C22×C4—61no—
532C22×C4—62no—
533C22×C4—62no—
534C22×C4—62no—
535C22×C4—62no—
536C22×C4—61no—
537C22×C4—61no—
538C22×C4—122no—
539C22×C4—122no—
540C22×C4—122no—
541C22×C4C2×C4×D311yescharacteristic
542C4⋊C4—21no—
543C4⋊C4—61no—
544C4⋊C4—61no—
545C4⋊C4—62no—
546C4⋊C4—62no—
547C4⋊C4—62no—
548C4⋊C4—62no—
549C4⋊C4—61no—
550C4⋊C4—124no—
551C4⋊C4—124no—
552C22⋊C4—22no—
553C22⋊C4—21no—
554C22⋊C4—22no—
555C22⋊C4—24no—
556C22⋊C4—42no—
557C22⋊C4—62no—
558C22⋊C4—62no—
559C22⋊C4—61no—
560C22⋊C4—61no—
561C22⋊C4—62no—
562C22⋊C4—62no—
563C22⋊C4—62no—
564C22⋊C4—61no—
565C22⋊C4—62no—
566C22⋊C4—64no—
567C22⋊C4—122no—
568C42—31no—
569C42—31no—
570C42—61no—
571C42—61no—
572C42—124no—
573C42—124no—
574C42C2×C4×D311yescharacteristic
575C42C2×C4×D311yescharacteristic
576C22×C3—22no—
577C22×C3—22no—
578C22×C3—41no—
579C22×C3—42no—
580C22×C3—42no—
581C22×C3—44no—
582C22×C3—42no—
583C22×C3—41no—
584C22×C3—82no—
585C22×C3C2×#9912yes—
586C22×C3C22×C22⋊C411yescharacteristic
587C2×D3—21no—
588C2×D3—21no—
589C2×D3—41no—
590C2×D3—41no—
591C2×D3—42no—
592C2×D3—41no—
593C2×D3—82no—
594C2×D3—84no—
595C3×C4—21no—
596C3×C4—21no—
597C3×C4—41no—
598C3×C4—41no—
599C3×C4—41no—
600C3×C4—84no—
601Dic3—22no—
602Dic3—22no—
603Dic3—41no—
604Dic3—41no—
605Dic3—41no—
606Dic3—84no—
607Dic3C2×#9912yes—
608Dic3C2×#9911yescharacteristic
609Dic3C2×#9911yescharacteristic
610C23—22no—
611C23—22no—
612C23—21no—
613C23—31no—
614C23—31no—
615C23—62no—
616C23—61no—
617C23—82no—
618C23—122no—
619C23—241no—
620C23—242no—
621C23D3×C22⋊C412yes—
622C23D3×C22⋊C411yescharacteristic
623C23C2×D3×D412yes—
624Q8—41no—
625Q8—122no—
626Q8—121no—
627Q8—121no—
628Q8—121no—
629Q8—122no—
630D4—41no—
631D4—42no—
632D4—41no—
633D4—41no—
634D4—42no—
635D4—82no—
636D4—82no—
637D4—121no—
638D4—122no—
639D4—122no—
640D4—122no—
641D4—122no—
642D4—122no—
643D4—122no—
644D4—122no—
645D4—122no—
646D4—242no—
647D4—242no—
648C2×C4—22no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—31no—
653C2×C4—31no—
654C2×C4—32no—
655C2×C4—32no—
656C2×C4—32no—
657C2×C4—32no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—61no—
661C2×C4—61no—
662C2×C4—62no—
663C2×C4—62no—
664C2×C4—61no—
665C2×C4—61no—
666C2×C4—62no—
667C2×C4—62no—
668C2×C4—62no—
669C2×C4—62no—
670C2×C4—62no—
671C2×C4—62no—
672C2×C4—62no—
673C2×C4—62no—
674C2×C4—62no—
675C2×C4—62no—
676C2×C4—61no—
677C2×C4—61no—
678C2×C4—61no—
679C2×C4—84no—
680C2×C4—122no—
681C2×C4—121no—
682C2×C4—121no—
683C2×C4—121no—
684C2×C4—121no—
685C2×C4—122no—
686C2×C4—124no—
687C2×C4—241no—
688C2×C4—241no—
689C2×C4—244no—
690C2×C4D3×C22⋊C411yescharacteristic
691C2×C4C22×C4×D311yescharacteristic
692C2×C3—21no—
693C2×C3—22no—
694C2×C3—42no—
695C2×C3—42no—
696C2×C3—81no—
697C2×C3C22×#9911yescharacteristic
698D3—41no—
699D3—41no—
700D3—81no—
701C22—22no—
702C22—22no—
703C22—41no—
704C22—42no—
705C22—42no—
706C22—44no—
707C22—42no—
708C22—41no—
709C22—61no—
710C22—61no—
711C22—82no—
712C22—121no—
713C22—121no—
714C22—122no—
715C22—121no—
716C22—242no—
717C22—244no—
718C22D3×#9912yes—
719C22C2×D3×C22⋊C411yescharacteristic
720C4—21no—
721C4—21no—
722C4—31no—
723C4—31no—
724C4—32no—
725C4—41no—
726C4—41no—
727C4—41no—
728C4—62no—
729C4—62no—
730C4—84no—
731C4—121no—
732C4—121no—
733C4—121no—
734C4—244no—
735C3#3335811yescharacteristic, complemented, Sylow
736C2—21no—
737C2—22no—
738C2—42no—
739C2—42no—
740C2—81no—
741C2—121no—
742C2—121no—
743C2—241no—
744C2C2×D3×#9911yescharacteristic
subgroupG/H
centerC2C2×D3×#99
commutatorC2×C3×C4C23×C4
FrattiniC2×D4C23×D3
FittingC3×#2127C2
socleC2×C3C22×#99
prime radical#86313C2
cosocleC2×C3×D4C24
hypercenter#2127D3
hypocenterC3#33358
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. C2×#4740C2semidirect
3. C3×#2127C2semidirect
4. #71860C2semidirect
5. #70019C2semidirect
6. #69159C2semidirect
7. #69158C2semidirect
8. #69152C2semidirect
9. #69151C2semidirect
11. C2×D3×#41C4semidirect
14. #5825C4semidirect
16. #5716C4semidirect
18. #5715C4semidirect
20. #5675C4semidirect
22. #5668C4semidirect
28. #5526C4semidirect
29. D3×#117C4semidirect
31. C2×C3×#99C22semidirect
32. C2×#834C22semidirect
37. #4740C22semidirect
38. C3×#354C22semidirect
39. C3×#353C22semidirect
40. #4545C22semidirect
41. #4541C22semidirect
42. #4474C22semidirect
43. #4473C22semidirect
50. #2127D3semidirect
59. C2×#239C2×C4semidirect
60. C2×D3×Q8C2×C4semidirect
66. C2×#237C2×C4semidirect
72. C2×D3×D4C2×C4semidirect
88. #944C2×C4semidirect
89. #944C2×C4semidirect
100. #936C2×C4semidirect
101. #936C2×C4semidirect
128. #875C2×C4semidirect
129. #875C2×C4semidirect
130. #872C2×C4semidirect
131. #872C2×C4semidirect
132. C42×D3C2×C4semidirect
133. C42×D3C2×C4semidirect
134. #869C2×C4semidirect
135. #869C2×C4semidirect
168. C2×#99C2×D3semidirect
171. #354C2×D3semidirect
174. #353C2×D3semidirect
735. C3#33358semidirect

Maximal subgroups

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

subgroupindex
#863132
C2×#47402
C3×#21272
#718602
#700192
#691592
#691582
#691522
#691512
#333583