Finite Group Databasebeta

← #10826268

Subgroups — #10826268

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×C3×SD16C211yesmaximal, characteristic, complemented
2C2×Q8×#67C211yesmaximal, characteristic, complemented
3#84053C211yesmaximal, characteristic
4C2×#5218C211yesmaximal, characteristic, complemented
5C2×#5170C212yesmaximal, complemented
6#72132C211yesmaximal, characteristic, complemented
7#72057C212yesmaximal, complemented
8#72054C212yesmaximal, complemented
9#72048C212yesmaximal, complemented
10#71975C211yesmaximal, characteristic, complemented
11#71972C211yesmaximal, characteristic, complemented
12#23855—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×D3×Q8C2211yescharacteristic, complemented
16C22×C3×SD16—24no—
17C22×C3×SD16—28no—
18C22×C3×SD16C2212yes—
19C22×C3×SD16C2212yes—
20C23×C3×C8C2211yescharacteristic
21C2×#953—21no—
22Q8×#67—24no—
23Q8×#67—24no—
24C2×#946—21no—
25C2×#946C2211yescharacteristic, complemented
26C2×Q8×Dic3C2211yescharacteristic, complemented
27C2×#944—21no—
28C2×#944C2211yescharacteristic, complemented
29C2×#938C2212yes—
30C22×#214C2212yes—
31C2×C4×#67—21no—
32C2×C4×#67C2211yescharacteristic
33C2×#924—21no—
34C2×#924C2211yescharacteristic
35#5247—21no—
36C2×#835C2212yescomplemented
37C2×#835C2212yescomplemented
38#5223—21no—
39#5222—21no—
40#5221C2212yes—
41#5218—24no—
42C2×#832C2212yes—
43#5212C2211yescharacteristic
44C2×#830C2211yescharacteristic
45#5170—28no—
46C2×(C2×D3)⋊C8C2212yescomplemented
47C2×#816C2212yescomplemented
48#4551C2212yescomplemented
49#4550C2212yescomplemented
50C23×SD16D311yescharacteristic, complemented
51C2×D4×Q8—31no—
52C2×#448—32no—
53C2×#450—31no—
54#2408—31no—
55#2013—32no—
56#2000—32no—
57#1892—31no—
58#1880—32no—
59#1869—31no—
60#1787—31no—
61C25×C3—21no—
62C22×C3×Q8—21no—
63C22×C3×Q8—22no—
64C22×C3×Q8—24no—
65C22×C3×Q8—42no—
66C22×C3×Q8D414yescomplemented
67C22×C3×Q8C2311yescharacteristic
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4—24no—
70C22×C3×D4—22no—
71C22×C3×D4—22no—
72C22×C3×D4—21no—
73C22×C3×D4—42no—
74C22×C3×D4C2312yes—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×#67—22no—
78C22×#67D411yescharacteristic
79C2×D3×Q8—22no—
80C2×D3×Q8—22no—
81C2×D3×Q8—24no—
82C2×D3×Q8—42no—
83C22×C4×D3—21no—
84C22×C4×D3C2311yescharacteristic
85C22×Dic6—21no—
86C22×Dic6C2311yescharacteristic
87C2×C3×SD16—232no—
88C2×C3×SD16—24no—
89C2×C3×SD16—24no—
90C2×C3×SD16—24no—
91C2×C3×SD16—24no—
92C2×C3×SD16—24no—
93C2×C3×SD16D418yescomplemented
94C22×C3×C8—22no—
95C22×C3×C8—24no—
96C22×C3×C8C2312yes—
97#953—44no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—22no—
101C2×#217—22no—
102C2×#217—22no—
103C2×#217D411yescharacteristic
104#946—28no—
105#946—22no—
106#946—22no—
107Q8×Dic3—22no—
108Q8×Dic3—22no—
109#944—28no—
110#944—22no—
111#944—22no—
112#938—24no—
113#938—42no—
114C2×#214—24no—
115C2×#214—24no—
116C2×#214—42no—
117C4×#67—28no—
118C4×#67—22no—
119C4×#67—22no—
120C2×(C2×D3)⋊C4—21no—
121C2×(C2×D3)⋊C4D411yescharacteristic
122C2×C4⋊Dic3—21no—
123C2×C4⋊Dic3C2312yes—
124C2×C4⋊Dic3C2311yescharacteristic
125#924—28no—
126#924—22no—
127#924D414yescomplemented
128C2×Dic3⋊C4—21no—
129C2×Dic3⋊C4—21no—
130C2×Dic3⋊C4—21no—
131C2×Dic3⋊C4—21no—
132C2×Dic3⋊C4D411yescharacteristic
133C2×C4×Dic3—21no—
134C2×C4×Dic3C2311yescharacteristic
135C22×C3⋊C8C2312yes—
136#835—24no—
137#835—42no—
138#832—24no—
139C3⋊#95—21no—
140#830—22no—
141(C2×D3)⋊C8—24no—
142#816—24no—
143C23×Q8—31no—
144C23×Q8C2×D311yescharacteristic
145C23×D4C2×D311yescharacteristic
146C22×SD16—24no—
147C22×SD16—28no—
148C22×SD16—32no—
149C22×SD16C2×D312yes—
150C22×SD16C2×D312yes—
151C23×C8C2×D311yescharacteristic
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×#121—32no—
161C2×#120—61no—
162C2×C4×Q8—31no—
163C2×C4×D4—31no—
164C2×C4×D4—61no—
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—61no—
174#392—61no—
175#390—32no—
176#380—31no—
177#379—61no—
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
187C23×D3C2×D411yescharacteristic
188C2×C3×Q8—24no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—22no—
192C2×C3×Q8—28no—
193C2×C3×Q8—22no—
194C2×C3×Q8—22no—
195C2×C3×Q8—44no—
196C2×C3×Q8—44no—
197C2×C3×Q8—48no—
198C2×C3×Q8C2×D412yes—
199C2×C3×Q8C2×D414yes—
200C2×C3×Q8C2×D412yes—
201C2×C3×D4—28no—
202C2×C3×D4—22no—
203C2×C3×D4—22no—
204C2×C3×D4—22no—
205C2×C3×D4—22no—
206C2×C3×D4—22no—
207C2×C3×D4—48no—
208C2×C3×D4—44no—
209C2×C3×D4—44no—
210C2×C3×D4—42no—
211C2×C3×D4C2×D414yes—
212C22×C3×C4—21no—
213C22×C3×C4—22no—
214C22×C3×C4—22no—
215C22×C3×C4—24no—
216C22×C3×C4—21no—
217C22×C3×C4—44no—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D412yes—
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2411yescharacteristic
222C2×#67—22no—
223C2×#67—22no—
224C2×#67—44no—
225C2×#67—44no—
226C2×#67—82no—
227C2×#67SD1614yescomplemented
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—22no—
231C22×Dic3C2×D411yescharacteristic
232C22×Dic3C2×D411yescharacteristic
233C22×Dic3C2×D411yescharacteristic
234D3×Q8—48no—
235D3×Q8—44no—
236D3×Q8—44no—
237C2×C4×D3—24no—
238C2×C4×D3—22no—
239C2×C4×D3—21no—
240C2×C4×D3—21no—
241C2×C4×D3—22no—
242C2×C4×D3—44no—
243C2×Dic6—22no—
244C2×Dic6—22no—
245C2×Dic6—24no—
246C2×Dic6—21no—
247C2×Dic6—44no—
248C2×Dic6C2×D412yes—
249C3×SD16—216no—
250C3×SD16—216no—
251C2×C3×C8—28no—
252C2×C3×C8—22no—
253C2×C3×C8—22no—
254C2×C3×C8C2×D414yes—
255#217—22no—
256#217—24no—
257#217—44no—
258#217—42no—
259#217—44no—
260#217—82no—
261#214—48no—
262(C2×D3)⋊C4—24no—
263(C2×D3)⋊C4—22no—
264C4⋊Dic3—24no—
265C4⋊Dic3—21no—
266C4⋊Dic3—22no—
267C4⋊Dic3—22no—
268C4⋊Dic3C2×D412yes—
269Dic3⋊C4—24no—
270Dic3⋊C4—24no—
271Dic3⋊C4—24no—
272Dic3⋊C4—24no—
273Dic3⋊C4—22no—
274C4×Dic3—24no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C2×C3⋊C8—22no—
278C2×C3⋊C8—22no—
279C2×C3⋊C8—22no—
280C25—21no—
281C22×Q8—21no—
282C22×Q8—22no—
283C22×Q8—24no—
284C22×Q8—31no—
285C22×Q8—42no—
286C22×Q8—61no—
287C22×Q8—62no—
288C22×Q8—62no—
289C22×Q8—64no—
290C22×Q8—122no—
291C22×Q8#6714yescomplemented
292C22×Q8C22×D311yescharacteristic
293C22×Q8C22×D311yescharacteristic
294C22×D4—24no—
295C22×D4—22no—
296C22×D4—21no—
297C22×D4—22no—
298C22×D4—31no—
299C22×D4—42no—
300C22×D4—62no—
301C22×D4C22×D312yes—
302C23×C4—21no—
303C23×C4—31no—
304C23×C4—61no—
305C23×C4C22×D311yescharacteristic
306C2×SD16—232no—
307C2×SD16—24no—
308C2×SD16—24no—
309C2×SD16—24no—
310C2×SD16—24no—
311C2×SD16—24no—
312C2×SD16—64no—
313C2×SD16—64no—
314C2×SD16—122no—
315C2×SD16#6718yescomplemented
316C22×C8—22no—
317C22×C8—24no—
318C22×C8—32no—
319C22×C8C22×D312yes—
320C4⋊Q8—62no—
321C4⋊Q8—68no—
322C4⋊Q8—62no—
323C22⋊Q8—34no—
324C22⋊Q8—62no—
325C22⋊Q8—68no—
326C22⋊Q8—68no—
327C22⋊Q8—62no—
328C22⋊Q8—62no—
329#121—64no—
330#121—122no—
331#120—124no—
332C4×Q8—62no—
333C4×Q8—62no—
334C4×D4—68no—
335C4×D4—62no—
336C4×D4—62no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—32no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4—61no—
345C2×C22⋊C4—31no—
346C2×C22⋊C4—31no—
347C2×C22⋊C4—61no—
348C2×C22⋊C4—61no—
349C2×C22⋊C4—62no—
350C2×C22⋊C4—62no—
351C2×C22⋊C4—61no—
352C2×C22⋊C4—62no—
353C2×C42—31no—
354C2×C42—61no—
355#103—64no—
356#103—64no—
357#103—122no—
358#102—64no—
359C22⋊C8—62no—
360C22⋊C8—64no—
361#95—61no—
362C23×C3—22no—
363C23×C3—22no—
364C23×C3—24no—
365C23×C3—48no—
366C23×C3—42no—
367C23×C3—42no—
368C23×C3—42no—
369C23×C3—42no—
370C23×C3—42no—
371C23×C3—42no—
372C23×C3—88no—
373C23×C3—88no—
374C23×C3—84no—
375C23×C3—84no—
376C23×C3#12012yes—
377C23×C3C2×SD1612yes—
378C23×C3#13712yes—
379C23×C3C22×D411yescharacteristic
380C22×D3—21no—
381C22×D3—22no—
382C22×D3—21no—
383C22×D3—22no—
384C22×D3C2×SD1612yescomplemented
385C3×Q8—22no—
386C3×Q8—24no—
387C3×Q8—22no—
388C3×Q8—22no—
389C3×Q8—48no—
390C3×Q8—42no—
391C3×Q8#12014yescomplemented
392C3×D4—24no—
393C3×D4—24no—
394C3×D4—48no—
395C3×D4—42no—
396C3×D4—42no—
397C2×C3×C4—22no—
398C2×C3×C4—24no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—24no—
403C2×C3×C4—22no—
404C2×C3×C4—48no—
405C2×C3×C4—42no—
406C2×C3×C4#12012yes—
407C2×C3×C4C22×D411yescharacteristic
408C2×C3×C4C22×D411yescharacteristic
409C2×C3×C4C22×D412yes—
410C2×C3×C4H(5,𝔽2)12yes—
411#67—28no—
412#67—88no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—22no—
417C2×Dic3—21no—
418C2×Dic3—22no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—42no—
422C2×Dic3—42no—
423C2×Dic3—42no—
424C2×Dic3—42no—
425C2×Dic3—42no—
426C2×Dic3C2×SD1612yescomplemented
427C4×D3—48no—
428C4×D3—41no—
429C4×D3—41no—
430C4×D3—42no—
431Dic6—24no—
432Dic6—48no—
433Dic6—41no—
434Dic6—41no—
435C3×C8—24no—
436C3⋊C8—42no—
437C24—22no—
438C24—31no—
439C24—44no—
440C24—42no—
441C24—42no—
442C24—42no—
443C24—82no—
444C24D3×D411yescharacteristic
445C2×Q8—24no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—22no—
449C2×Q8—28no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—32no—
453C2×Q8—44no—
454C2×Q8—44no—
455C2×Q8—48no—
456C2×Q8—64no—
457C2×Q8—61no—
458C2×Q8—62no—
459C2×Q8—62no—
460C2×Q8—124no—
461C2×Q8—128no—
462C2×Q8—124no—
463C2×Q8—124no—
464C2×Q8C2×#6712yes—
465C2×Q8C2×#6714yes—
466C2×Q8C2×#6712yes—
467C2×D4—28no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—34no—
474C2×D4—48no—
475C2×D4—44no—
476C2×D4—44no—
477C2×D4—42no—
478C2×D4—62no—
479C2×D4—62no—
480C2×D4—124no—
481C2×D4—124no—
482C2×D4—242no—
483C2×D4C2×#6714yes—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—21no—
489C22×C4—31no—
490C22×C4—31no—
491C22×C4—31no—
492C22×C4—44no—
493C22×C4—64no—
494C22×C4—61no—
495C22×C4—61no—
496C22×C4—61no—
497C22×C4—62no—
498C22×C4—62no—
499C22×C4—61no—
500C22×C4—62no—
501C22×C4—124no—
502C22×C4D3×D411yescharacteristic
503C22×C4D3×D412yes—
504C22×C4C2×#6712yes—
505C22×C4C23×D311yescharacteristic
506SD16—216no—
507SD16—216no—
508SD16—128no—
509C2×C8—28no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—62no—
513C2×C8—62no—
514C2×C8—62no—
515C2×C8C2×#6714yes—
516C4⋊C4—32no—
517C4⋊C4—64no—
518C4⋊C4—64no—
519C4⋊C4—64no—
520C4⋊C4—64no—
521C4⋊C4—62no—
522C4⋊C4—64no—
523C4⋊C4—61no—
524C4⋊C4—62no—
525C4⋊C4—62no—
526C22⋊C4—64no—
527C22⋊C4—62no—
528C22⋊C4—64no—
529C22⋊C4—62no—
530C22⋊C4—124no—
531C22⋊C4—124no—
532C22⋊C4—122no—
533C22⋊C4—242no—
534C42—64no—
535C42—61no—
536C42—61no—
537C22×C3—22no—
538C22×C3—22no—
539C22×C3—24no—
540C22×C3—24no—
541C22×C3—44no—
542C22×C3—44no—
543C22×C3—44no—
544C22×C3—816no—
545C22×C3—82no—
546C22×C3—82no—
547C22×C3—82no—
548C22×C3—82no—
549C22×C3#44814yescomplemented
550C22×C3C2×#12011yescharacteristic
551C22×C3D4212yes—
552C22×C3C22×SD1611yescharacteristic
553C22×C3C2×#13711yescharacteristic
554C22×C3#57712yes—
555C2×D3—22no—
556C2×D3—24no—
557C2×D3—41no—
558C2×D3—41no—
559C2×D3—42no—
560C3×C4—24no—
561C3×C4—22no—
562C3×C4—42no—
563C3×C4C2×#12011yescharacteristic
564C3×C4#53411yescharacteristic
565C3×C4D4212yes—
566Dic3—22no—
567Dic3—42no—
568Dic3—41no—
569Dic3—41no—
570Dic3—42no—
571Dic3—82no—
572C23—22no—
573C23—22no—
574C23—24no—
575C23—32no—
576C23—48no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—61no—
584C23—62no—
585C23—61no—
586C23—62no—
587C23—88no—
588C23—88no—
589C23—84no—
590C23—84no—
591C23D3×SD1612yes—
592C23#91512yes—
593C23#93712yes—
594C23C2×D3×D411yescharacteristic
595Q8—24no—
596Q8—22no—
597Q8—22no—
598Q8—22no—
599Q8—48no—
600Q8—42no—
601Q8—64no—
602Q8—128no—
603Q8—121no—
604Q8—121no—
605Q8#95314yescomplemented
606D4—24no—
607D4—24no—
608D4—48no—
609D4—42no—
610D4—42no—
611D4—68no—
612D4—248no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—32no—
621C2×C4—48no—
622C2×C4—42no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—62no—
626C2×C4—62no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—128no—
632C2×C4—122no—
633C2×C4—122no—
634C2×C4—122no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—122no—
638C2×C4—122no—
639C2×C4—122no—
640C2×C4#93712yes—
641C2×C4C2×D3×D411yescharacteristic
642C2×C4#100412yes—
643C2×C4C22×#6712yes—
644C2×C4C22×#6711yescharacteristic
645C8—24no—
646C8—122no—
647C2×C3—24no—
648C2×C3—84no—
649C2×C3#240311yescharacteristic
650C2×C3C2×#44811yescharacteristic
651C2×C3#301011yescharacteristic
652C2×C3#326612yes—
653C2×C3D4×SD1612yes—
654D3—42no—
655C22—22no—
656C22—22no—
657C22—24no—
658C22—24no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—62no—
663C22—64no—
664C22—816no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—121no—
670C22—121no—
671C22—122no—
672C22#517014yescomplemented
673C22C2×D3×SD1611yescharacteristic
674C22C2×#91511yescharacteristic
675C22#576212yes—
676C22C2×#93711yescharacteristic
677C22D4×#6712yes—
678C4—22no—
679C4—24no—
680C4—42no—
681C4—62no—
682C4—121no—
683C4—122no—
684C4—121no—
685C4—122no—
686C4—242no—
687C4D4×#6712yes—
688C4#580411yescharacteristic
689C4C2×#95311yescharacteristic
690C3#2385511yescharacteristic, complemented, Sylow
691C2—24no—
692C2—84no—
693C2—122no—
694C2C2×#517011yescharacteristic
695C2#8386011yescharacteristic
696C2SD16×#6712yes—
697C2#8386612yes—
698C2#8405211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC23×C3×SD16C2
socleC23×C3C22×D4
prime radical#84053C2
hypercenterC23×SD16D3
hypocenterC3#23855
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×SD16

Complements

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

NH ≅ G/Ntype
1. C23×C3×SD16C2semidirect
2. C2×Q8×#67C2semidirect
4. C2×#5218C2semidirect
5. C2×#5170C2semidirect
6. #72132C2semidirect
7. #72057C2semidirect
8. #72054C2semidirect
9. #72048C2semidirect
10. #71975C2semidirect
11. #71972C2semidirect
15. C22×D3×Q8C22semidirect
25. C2×#946C22semidirect
26. C2×Q8×Dic3C22semidirect
28. C2×#944C22semidirect
36. C2×#835C22semidirect
37. C2×#835C22semidirect
46. C2×(C2×D3)⋊C8C22semidirect
47. C2×#816C22semidirect
48. #4551C22semidirect
49. #4550C22semidirect
50. C23×SD16D3semidirect
66. C22×C3×Q8D4semidirect
93. C2×C3×SD16D4semidirect
127. #924D4semidirect
227. C2×#67SD16semidirect
291. C22×Q8#67semidirect
315. C2×SD16#67semidirect
384. C22×D3C2×SD16semidirect
391. C3×Q8#120semidirect
426. C2×Dic3C2×SD16semidirect
549. C22×C3#448semidirect
605. Q8#953semidirect
672. C22#5170semidirect
690. C3#23855semidirect

Maximal subgroups

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

subgroupindex
C23×C3×SD162
C2×Q8×#672
#840532
C2×#52182
C2×#51702
#721322
#720572
#720542
#720482
#719752
#719722
#238553