Finite Group Databasebeta

← #10826271

Subgroups — #10826271

Order 768.

1930 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#214C211yesmaximal, characteristic, complemented
2C2×D4×Dic6C211yesmaximal, characteristic, complemented
3C3×#2408C211yesmaximal, characteristic
4C2×#5218C211yesmaximal, characteristic, complemented
5C2×#5040C212yesmaximal, complemented
6#71685C212yesmaximal, complemented
7#71610C211yesmaximal, characteristic, complemented
8#69661C212yesmaximal, complemented
9#69660C212yesmaximal, complemented
10#69525C211yesmaximal, characteristic, complemented
11#69514C211yesmaximal, characteristic, complemented
12#23855—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C23×Dic6C2211yescharacteristic, complemented
16C2×C3×#121C2212yes—
17C2×C3×#120—21no—
18C2×C3×C4×D4C2211yescharacteristic
19C2×D4×Dic3—21no—
20C22×#214—24no—
21C22×#214—28no—
22C22×#214C2212yes—
23C22×#214C2212yes—
24C22×#214C2212yes—
25C2×#924C2211yescharacteristic
26C2×#924C2211yescharacteristic, complemented
27C23×C3⋊C8C2211yescharacteristic
28D4×Dic6—24no—
29D4×Dic6—24no—
30C2×#888—21no—
31C2×#878—21no—
32C2×#878—21no—
33C2×#869C2211yescharacteristic, complemented
34C2×C4×Dic6C2211yescharacteristic, complemented
35C3×#394—21no—
36C3×#392—21no—
37C3×#390C2212yes—
38C3×#380C2211yescharacteristic
39C3×#379—21no—
40#5218—24no—
41C2×#832C2212yes—
42C2×#830C2211yescharacteristic
43C2×#830C2212yescomplemented
44#5040—28no—
45C2×#810C2212yescomplemented
46C2×#810C2212yescomplemented
47C2×#810C2212yescomplemented
48#4529C2212yescomplemented
49#4524C2212yescomplemented
50C23×SD16—31no—
51C2×D4×Q8—31no—
52C2×#448—32no—
53C2×#450—31no—
54#2408D311yescharacteristic
55#2013—32no—
56#2000—32no—
57#1892—31no—
58#1880—32no—
59#1869—31no—
60#1787—31no—
61C25×C3—21no—
62C22×C3×D4—24no—
63C22×C3×D4—22no—
64C22×C3×D4—22no—
65C22×C3×D4—22no—
66C22×C3×D4—21no—
67C22×C3×D4—42no—
68C22×C3×D4D411yescharacteristic
69C22×C3×D4C2312yes—
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C23×Dic3—21no—
73C23×Dic3—21no—
74C22×Dic6—24no—
75C22×Dic6—21no—
76C22×Dic6—22no—
77C22×Dic6—21no—
78C22×Dic6—22no—
79C22×Dic6—22no—
80C22×Dic6—24no—
81C22×Dic6—42no—
82C22×Dic6—42no—
83C22×Dic6D414yescomplemented
84C22×Dic6C2311yescharacteristic
85C22×Dic6C2311yescharacteristic
86C22×Dic6C2311yescharacteristic
87C3×#121—24no—
88C3×#121—42no—
89C3×#120—44no—
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C22⋊C4D411yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102C2×#217—21no—
103D4×Dic3—28no—
104C2×#214—232no—
105C2×#214—24no—
106C2×#214—24no—
107C2×#214—24no—
108C2×#214—24no—
109C2×#214—24no—
110C2×#214—24no—
111C2×#214—24no—
112C2×#214—42no—
113C2×#214D418yescomplemented
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3—21no—
116C2×C4⋊Dic3C2311yescharacteristic
117#924—22no—
118#924—22no—
119#924—22no—
120#924D414yescomplemented
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4—21no—
124C2×C4×Dic3—21no—
125C22×C3⋊C8—22no—
126C22×C3⋊C8—24no—
127C22×C3⋊C8C2312yes—
128C22×C3⋊C8C2312yes—
129#888—28no—
130#878—28no—
131#878—28no—
132#869—22no—
133#869—22no—
134C4×Dic6—22no—
135C4×Dic6—22no—
136C3×#95—21no—
137#832—24no—
138#830—22no—
139#830—24no—
140#810—24no—
141#810—24no—
142#810—42no—
143C23×Q8—31no—
144C23×Q8—31no—
145C23×D4C2×D311yescharacteristic
146C22×SD16—32no—
147C22×SD16—32no—
148C22×SD16—32no—
149C22×SD16—68no—
150C22×SD16—64no—
151C23×C8—31no—
152D4×Q8—64no—
153D4×Q8—64no—
154C2×C4⋊Q8—31no—
155C2×C4⋊Q8—61no—
156C2×C22⋊Q8—31no—
157C2×C22⋊Q8—31no—
158C2×C22⋊Q8—61no—
159C2×C22⋊Q8—61no—
160C2×#121C2×D312yes—
161C2×#120—21no—
162C2×C4×Q8—31no—
163C2×C4×D4—61no—
164C2×C4×D4C2×D311yescharacteristic
165#450—64no—
166#448—68no—
167C2×#103—32no—
168C2×#103—32no—
169C2×#103—32no—
170C2×#102—32no—
171C2×C22⋊C8—31no—
172C2×C22⋊C8—32no—
173#394—21no—
174#392—21no—
175#390C2×D312yes—
176#380C2×D311yescharacteristic
177#379—21no—
178#340—32no—
179#336—32no—
180C24×C3—22no—
181C24×C3—44no—
182C24×C3—42no—
183C24×C3—42no—
184C24×C3—42no—
185C24×C3—82no—
186C24×C3C2×D411yescharacteristic
187C24×C3C2×D411yescharacteristic
188C2×C3×D4—22no—
189C2×C3×D4—22no—
190C2×C3×D4—28no—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—22no—
196C2×C3×D4—48no—
197C2×C3×D4—44no—
198C2×C3×D4—44no—
199C2×C3×D4—44no—
200C2×C3×D4—44no—
201C2×C3×D4—42no—
202C2×C3×D4—82no—
203C2×C3×D4SD1614yescomplemented
204C2×C3×D4C2×D414yes—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—22no—
211C22×C3×C4—22no—
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—24no—
221C22×Dic3—24no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—44no—
226C22×Dic3—44no—
227C2×Dic6—21no—
228C2×Dic6—22no—
229C2×Dic6—22no—
230C2×Dic6—22no—
231C2×Dic6—22no—
232C2×Dic6—22no—
233C2×Dic6—28no—
234C2×Dic6—24no—
235C2×Dic6—24no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—44no—
239C2×Dic6—48no—
240C2×Dic6—48no—
241C2×Dic6—44no—
242C2×Dic6—44no—
243C2×Dic6—44no—
244C2×Dic6—44no—
245C2×Dic6C2×D412yes—
246C2×Dic6C2×D412yes—
247C2×Dic6C2×D414yes—
248C2×Dic6C2×D412yes—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—22no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—44no—
255C3×C22⋊C4—42no—
256C3×C22⋊C4—44no—
257C3×C22⋊C4—82no—
258C3×C42—21no—
259C3×C42—21no—
260#217—24no—
261#217—24no—
262#214—216no—
263#214—216no—
264#214—48no—
265C4⋊Dic3—24no—
266C4⋊Dic3—24no—
267C4⋊Dic3—21no—
268C4⋊Dic3C2×D412yes—
269Dic3⋊C4—24no—
270Dic3⋊C4—24no—
271Dic3⋊C4—24no—
272C4×Dic3—24no—
273C2×C3⋊C8—28no—
274C2×C3⋊C8—22no—
275C2×C3⋊C8—22no—
276C2×C3⋊C8—22no—
277C2×C3⋊C8—22no—
278C2×C3⋊C8—22no—
279C2×C3⋊C8C2×D414yes—
280C25—21no—
281C22×Q8—34no—
282C22×Q8—31no—
283C22×Q8—31no—
284C22×Q8—31no—
285C22×Q8—62no—
286C22×Q8—62no—
287C22×Q8—64no—
288C22×Q8—64no—
289C22×Q8—62no—
290C22×Q8—61no—
291C22×Q8—61no—
292C22×Q8—122no—
293C22×Q8—122no—
294C22×D4—24no—
295C22×D4—22no—
296C22×D4—22no—
297C22×D4—22no—
298C22×D4—21no—
299C22×D4—42no—
300C22×D4#6711yescharacteristic
301C22×D4C22×D312yes—
302C23×C4—61no—
303C23×C4—61no—
304C23×C4C22×D311yescharacteristic
305C23×C4C22×D311yescharacteristic
306C2×SD16—38no—
307C2×SD16—64no—
308C2×SD16—64no—
309C2×SD16—64no—
310C2×SD16—64no—
311C2×SD16—632no—
312C2×SD16—64no—
313C2×SD16—64no—
314C2×SD16—64no—
315C2×SD16—122no—
316C22×C8—32no—
317C22×C8—32no—
318C22×C8—64no—
319C22×C8—62no—
320C4⋊Q8—62no—
321C4⋊Q8—62no—
322C4⋊Q8—68no—
323C22⋊Q8—34no—
324C22⋊Q8—68no—
325C22⋊Q8—68no—
326C22⋊Q8—62no—
327C22⋊Q8—62no—
328C22⋊Q8—62no—
329#121—24no—
330#121—42no—
331#120—44no—
332C4×Q8—62no—
333C4×Q8—62no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—68no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4#6711yescharacteristic
344C2×C4⋊C4C22×D312yes—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—61no—
351C2×C22⋊C4#6711yescharacteristic
352C2×C22⋊C4#6711yescharacteristic
353C2×C42—61no—
354C2×C42C22×D311yescharacteristic
355#103—64no—
356#103—64no—
357#103—122no—
358#102—64no—
359C22⋊C8—64no—
360C22⋊C8—62no—
361#95—21no—
362C23×C3—22no—
363C23×C3—22no—
364C23×C3—21no—
365C23×C3—22no—
366C23×C3—21no—
367C23×C3—22no—
368C23×C3—24no—
369C23×C3—48no—
370C23×C3—42no—
371C23×C3—42no—
372C23×C3—42no—
373C23×C3—42no—
374C23×C3—42no—
375C23×C3—42no—
376C23×C3—88no—
377C23×C3—88no—
378C23×C3—84no—
379C23×C3—84no—
380C23×C3#12012yes—
381C23×C3C2×SD1612yes—
382C23×C3C2×SD1612yescomplemented
383C23×C3#13712yes—
384C23×C3C22×D411yescharacteristic
385C3×D4—28no—
386C3×D4—24no—
387C3×D4—24no—
388C3×D4—48no—
389C3×D4—42no—
390C3×D4—42no—
391C3×D4—88no—
392C2×C3×C4—21no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—24no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—42no—
405C2×C3×C4—42no—
406C2×C3×C4—42no—
407C2×C3×C4—42no—
408C2×C3×C4#12012yes—
409C2×C3×C4C2×SD1612yescomplemented
410C2×C3×C4C22×D411yescharacteristic
411C2×C3×C4C22×D412yes—
412C2×C3×C4C22×D411yescharacteristic
413C2×C3×C4H(5,𝔽2)12yes—
414C2×Dic3—22no—
415C2×Dic3—24no—
416C2×Dic3—22no—
417C2×Dic3—22no—
418C2×Dic3—22no—
419C2×Dic3—48no—
420C2×Dic3—48no—
421C2×Dic3—42no—
422C2×Dic3—42no—
423C2×Dic3—42no—
424Dic6—22no—
425Dic6—24no—
426Dic6—22no—
427Dic6—22no—
428Dic6—24no—
429Dic6—48no—
430Dic6—48no—
431Dic6—41no—
432Dic6—41no—
433Dic6—42no—
434Dic6#12014yescomplemented
435C3⋊C8—24no—
436C3⋊C8—42no—
437C24—22no—
438C24—44no—
439C24—42no—
440C24—42no—
441C24—42no—
442C24—82no—
443C24C2×#6711yescharacteristic
444C24C2×#6711yescharacteristic
445C2×Q8—34no—
446C2×Q8—32no—
447C2×Q8—32no—
448C2×Q8—32no—
449C2×Q8—64no—
450C2×Q8—68no—
451C2×Q8—64no—
452C2×Q8—61no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—62no—
456C2×Q8—62no—
457C2×Q8—62no—
458C2×Q8—62no—
459C2×Q8—62no—
460C2×Q8—124no—
461C2×Q8—124no—
462C2×Q8—124no—
463C2×Q8—128no—
464C2×Q8—124no—
465C2×Q8—128no—
466C2×Q8—124no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—28no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—48no—
476C2×D4—44no—
477C2×D4—44no—
478C2×D4—44no—
479C2×D4—44no—
480C2×D4—42no—
481C2×D4—82no—
482C2×D4#21414yescomplemented
483C2×D4D3×D414yes—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—61no—
492C22×C4—64no—
493C22×C4—64no—
494C22×C4—61no—
495C22×C4—61no—
496C22×C4—61no—
497C22×C4—124no—
498C22×C4—124no—
499C22×C4D3×D412yes—
500C22×C4D3×D412yes—
501C22×C4C2×#6711yescharacteristic
502C22×C4C2×#6711yescharacteristic
503C22×C4C2×#6711yescharacteristic
504C22×C4C2×#6711yescharacteristic
505C22×C4C23×D311yescharacteristic
506SD16—616no—
507SD16—616no—
508SD16—128no—
509C2×C8—34no—
510C2×C8—68no—
511C2×C8—62no—
512C2×C8—62no—
513C2×C8—62no—
514C2×C8—62no—
515C2×C8—62no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—32no—
520C4⋊C4—64no—
521C4⋊C4—64no—
522C4⋊C4—64no—
523C4⋊C4—64no—
524C4⋊C4—64no—
525C4⋊C4—61no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—64no—
532C22⋊C4—64no—
533C22⋊C4—82no—
534C42—21no—
535C42—21no—
536C42—64no—
537C22×C3—22no—
538C22×C3—22no—
539C22×C3—24no—
540C22×C3—24no—
541C22×C3—22no—
542C22×C3—24no—
543C22×C3—41no—
544C22×C3—41no—
545C22×C3—42no—
546C22×C3—44no—
547C22×C3—44no—
548C22×C3—44no—
549C22×C3—816no—
550C22×C3—82no—
551C22×C3—82no—
552C22×C3—82no—
553C22×C3—82no—
554C22×C3#44814yescomplemented
555C22×C3C2×#12011yescharacteristic
556C22×C3D4212yes—
557C22×C3C22×SD1611yescharacteristic
558C22×C3C2×#13711yescharacteristic
559C22×C3#57712yes—
560C3×C4—22no—
561C3×C4—22no—
562C3×C4—41no—
563C3×C4—41no—
564C3×C4—82no—
565C3×C4C2×#12011yescharacteristic
566C3×C4#53411yescharacteristic
567C3×C4D4212yes—
568Dic3—24no—
569Dic3—42no—
570Dic3—42no—
571Dic3—42no—
572C23—22no—
573C23—22no—
574C23—21no—
575C23—22no—
576C23—21no—
577C23—22no—
578C23—24no—
579C23—48no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—88no—
587C23—88no—
588C23—84no—
589C23—84no—
590C23C2×#21412yes—
591C23C2×#21412yescomplemented
592C23#93712yes—
593C23#95112yes—
594C23C22×#6711yescharacteristic
595Q8—34no—
596Q8—62no—
597Q8—64no—
598Q8—62no—
599Q8—62no—
600Q8—64no—
601Q8—128no—
602Q8—128no—
603Q8—121no—
604Q8—121no—
605Q8—122no—
606D4—28no—
607D4—24no—
608D4—24no—
609D4—48no—
610D4—42no—
611D4—42no—
612D4—88no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—62no—
630C2×C4—64no—
631C2×C4—62no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—128no—
635C2×C4—128no—
636C2×C4—122no—
637C2×C4—122no—
638C2×C4—122no—
639C2×C4C2×#21412yescomplemented
640C2×C4#93712yes—
641C2×C4C2×D3×D411yescharacteristic
642C2×C4C2×D3×D412yes—
643C2×C4C2×D3×D411yescharacteristic
644C2×C4#100412yes—
645C8—64no—
646C8—122no—
647C2×C3—24no—
648C2×C3—42no—
649C2×C3—84no—
650C2×C3#240311yescharacteristic
651C2×C3C2×#44811yescharacteristic
652C2×C3#301011yescharacteristic
653C2×C3#326612yes—
654C2×C3D4×SD1612yes—
655C22—24no—
656C22—24no—
657C22—22no—
658C22—22no—
659C22—24no—
660C22—22no—
661C22—42no—
662C22—41no—
663C22—41no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—816no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22#504014yescomplemented
673C22C22×#21411yescharacteristic
674C22C2×#93711yescharacteristic
675C22D4×#6712yes—
676C22C2×#95111yescharacteristic
677C22#583612yes—
678C4—22no—
679C4—22no—
680C4—41no—
681C4—41no—
682C4—64no—
683C4—82no—
684C4—122no—
685C4—122no—
686C4—122no—
687C4D3×#12011yescharacteristic
688C4#558911yescharacteristic
689C4#559012yes—
690C3#2385511yescharacteristic, complemented, Sylow
691C2—24no—
692C2—42no—
693C2—84no—
694C2#8260411yescharacteristic
695C2C2×#504011yescharacteristic
696C2#8339011yescharacteristic
697C2D4×#21412yes—
698C2#8350612yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2408C2
socleC23×C3C22×D4
hypercenter#2408D3
hypocenterC3#23855
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. C23×#214C2semidirect
2. C2×D4×Dic6C2semidirect
4. C2×#5218C2semidirect
5. C2×#5040C2semidirect
6. #71685C2semidirect
7. #71610C2semidirect
8. #69661C2semidirect
9. #69660C2semidirect
10. #69525C2semidirect
11. #69514C2semidirect
15. C23×Dic6C22semidirect
26. C2×#924C22semidirect
33. C2×#869C22semidirect
34. C2×C4×Dic6C22semidirect
43. C2×#830C22semidirect
45. C2×#810C22semidirect
46. C2×#810C22semidirect
47. C2×#810C22semidirect
48. #4529C22semidirect
49. #4524C22semidirect
83. C22×Dic6D4semidirect
113. C2×#214D4semidirect
120. #924D4semidirect
203. C2×C3×D4SD16semidirect
382. C23×C3C2×SD16semidirect
409. C2×C3×C4C2×SD16semidirect
434. Dic6#120semidirect
482. C2×D4#214semidirect
554. C22×C3#448semidirect
591. C23C2×#214semidirect
639. C2×C4C2×#214semidirect
672. C22#5040semidirect
690. C3#23855semidirect

Maximal subgroups

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

subgroupindex
C23×#2142
C2×D4×Dic62
C3×#24082
C2×#52182
C2×#50402
#716852
#716102
#696612
#696602
#695252
#695142
#238553