Finite Group Databasebeta

← #10826935

Subgroups — #10826935

Order 768.

2252 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#121C211yesmaximal, characteristic
2C22×#924C211yesmaximal, characteristic, complemented
3C2×#5218C212yesmaximal, complemented
4C2×#5040C212yesmaximal, complemented
5#72126C211yesmaximal, characteristic, complemented
6#71685C214yesmaximal, complemented
7#71677C212yesmaximal, complemented
8#71559C212yesmaximal, complemented
9#23919—31nomaximal, Sylow
10C23×C3×D4—21no—
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C23×Dic6C2211yescharacteristic, complemented
14C2×C3×#121—21no—
15C2×C3×#121—21no—
16C2×C3×#121—28no—
17C2×C3×#121C2212yes—
18C2×C3×#121C2212yes—
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4—21no—
21C22×#217—21no—
22C22×#214C2214yescomplemented
23C22×C4⋊Dic3C2211yescharacteristic, complemented
24C2×#924—21no—
25C2×#924—21no—
26C2×#924—28no—
27C2×#924C2212yes—
28C2×#924C2212yescomplemented
29C22×Dic3⋊C4—21no—
30#5218—28no—
31C2×#832C2214yescomplemented
32C2×#830C2212yes—
33C2×#830C2212yescomplemented
34C2×#830C2212yescomplemented
35#5040—28no—
36C2×#810C2214yescomplemented
37#4529C2214yescomplemented
38C22×C22⋊Q8—31no—
39C22×#121D311yescharacteristic
40C2×#448—32no—
41C2×#450—32no—
42#2000—34no—
43#1874—32no—
44#1872—32no—
45#1779—31no—
46C25×C3—21no—
47C25×C3D411yescharacteristic
48C22×C3×D4—28no—
49C22×C3×D4—22no—
50C22×C3×D4—22no—
51C22×C3×D4—24no—
52C22×C3×D4—21no—
53C22×C3×D4—24no—
54C22×C3×D4—42no—
55C22×C3×D4—44no—
56C22×C3×D4—44no—
57C22×C3×D4C2312yes—
58C23×C3×C4—21no—
59C23×C3×C4—24no—
60C23×C3×C4—24no—
61C23×C3×C4—42no—
62C23×C3×C4D411yescharacteristic
63C23×C3×C4D411yescharacteristic
64C23×C3×C4D411yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×Dic3—21no—
69C23×Dic3—21no—
70C22×Dic6—21no—
71C22×Dic6—24no—
72C22×Dic6—24no—
73C22×Dic6—42no—
74C22×Dic6C2312yes—
75C3×#121—28no—
76C3×#121—28no—
77C3×#121—24no—
78C3×#121—24no—
79C3×#121—24no—
80C3×#121D418yes—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—24no—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C22⋊C4—28no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×#217—28no—
88C2×#217—22no—
89C2×#217—22no—
90C2×#214—28no—
91C2×#214—28no—
92C2×#214—44no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—24no—
95C2×C4⋊Dic3C2312yes—
96#924—28no—
97#924—28no—
98#924—24no—
99#924—24no—
100#924—24no—
101#924D418yescomplemented
102C2×Dic3⋊C4—28no—
103C2×Dic3⋊C4—22no—
104C2×Dic3⋊C4—22no—
105C22×C3⋊C8C2314yes—
106#832—28no—
107#830—24no—
108#830—24no—
109#830—42no—
110#810—28no—
111C23×Q8—31no—
112C23×D4—21no—
113C23×D4C2×D311yescharacteristic
114C24×C4C2×D311yescharacteristic
115C22×SD16—34no—
116C2×C22⋊Q8—32no—
117C2×C22⋊Q8—32no—
118C2×C22⋊Q8—68no—
119C2×C22⋊Q8—61no—
120C2×C22⋊Q8—61no—
121C2×#121—21no—
122C2×#121—21no—
123C2×#121—28no—
124C2×#121C2×D312yes—
125C2×#121C2×D312yes—
126C22×C4⋊C4—31no—
127C22×C4⋊C4—61no—
128C22×C4⋊C4C2×D311yescharacteristic
129C22×C22⋊C4—21no—
130C22×C22⋊C4—61no—
131#450—68no—
132#448—68no—
133C2×#103—34no—
134C2×#102—34no—
135C2×C22⋊C8—32no—
136C2×C22⋊C8—32no—
137C2×C22⋊C8—32no—
138#340—34no—
139C24×C3—22no—
140C24×C3—22no—
141C24×C3—22no—
142C24×C3—22no—
143C24×C3—24no—
144C24×C3—22no—
145C24×C3—41no—
146C24×C3—41no—
147C24×C3—42no—
148C24×C3—42no—
149C24×C3—42no—
150C24×C3—42no—
151C24×C3—42no—
152C24×C3—42no—
153C24×C3SD1614yescomplemented
154C24×C3C2×D411yescharacteristic
155C24×C3C2×D411yescharacteristic
156C24×C3C2×D411yescharacteristic
157C2×C3×D4—24no—
158C2×C3×D4—28no—
159C2×C3×D4—28no—
160C2×C3×D4—24no—
161C2×C3×D4—22no—
162C2×C3×D4—24no—
163C2×C3×D4—416no—
164C2×C3×D4—42no—
165C2×C3×D4—432no—
166C2×C3×D4—44no—
167C2×C3×D4—44no—
168C2×C3×D4—44no—
169C2×C3×D4—44no—
170C2×C3×D4—42no—
171C2×C3×D4C2×D414yes—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—24no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—41no—
188C22×C3×C4—41no—
189C22×C3×C4—41no—
190C22×C3×C4—416no—
191C22×C3×C4—42no—
192C22×C3×C4—42no—
193C22×C3×C4C2×D414yes—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D414yes—
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—24no—
200C22×Dic3—24no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—42no—
204C22×Dic3—42no—
205C22×Dic3—42no—
206C22×Dic3—42no—
207C2×Dic6—24no—
208C2×Dic6—24no—
209C2×Dic6—24no—
210C2×Dic6—22no—
211C2×Dic6—416no—
212C2×Dic6—42no—
213C2×Dic6—42no—
214C2×Dic6C2×D414yescomplemented
215C3×C4⋊C4—24no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4C2×D414yes—
218C3×C22⋊C4—28no—
219C3×C22⋊C4—28no—
220#217—28no—
221#217—28no—
222#214—416no—
223C4⋊Dic3—24no—
224C4⋊Dic3—22no—
225C4⋊Dic3C2×D414yescomplemented
226Dic3⋊C4—28no—
227Dic3⋊C4—28no—
228C2×C3⋊C8—24no—
229C2×C3⋊C8—24no—
230C2×C3⋊C8—24no—
231C25—21no—
232C25#6711yescharacteristic
233C22×Q8—32no—
234C22×Q8—64no—
235C22×Q8—61no—
236C22×Q8—64no—
237C22×Q8—122no—
238C22×D4—28no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—24no—
242C22×D4—24no—
243C22×D4—21no—
244C22×D4—42no—
245C22×D4—44no—
246C22×D4—44no—
247C22×D4C22×D312yes—
248C23×C4—21no—
249C23×C4—24no—
250C23×C4—24no—
251C23×C4—42no—
252C23×C4—61no—
253C23×C4—61no—
254C23×C4#6711yescharacteristic
255C23×C4#6711yescharacteristic
256C23×C4#6711yescharacteristic
257C23×C4C22×D311yescharacteristic
258C23×C4C22×D311yescharacteristic
259C23×C4C22×D311yescharacteristic
260C2×SD16—68no—
261C2×SD16—68no—
262C2×SD16—124no—
263C22×C8—34no—
264C22⋊Q8—38no—
265C22⋊Q8—64no—
266C22⋊Q8—64no—
267C22⋊Q8—68no—
268C22⋊Q8—68no—
269C22⋊Q8—64no—
270#121—28no—
271#121—28no—
272#121—24no—
273#121—24no—
274#121—24no—
275#121#6718yes—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—32no—
279C2×C4⋊C4—64no—
280C2×C4⋊C4—68no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4C22×D312yes—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—68no—
289C2×C22⋊C4—62no—
290C2×C22⋊C4—62no—
291#103—68no—
292#102—68no—
293C22⋊C8—64no—
294C22⋊C8—64no—
295C22⋊C8—122no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—28no—
311C23×C3—28no—
312C23×C3—24no—
313C23×C3—44no—
314C23×C3—44no—
315C23×C3—44no—
316C23×C3—44no—
317C23×C3—44no—
318C23×C3—48no—
319C23×C3—48no—
320C23×C3—48no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—84no—
328C23×C3—84no—
329C23×C3—84no—
330C23×C3—84no—
331C23×C3#12012yes—
332C23×C3#12012yes—
333C23×C3C2×SD1612yes—
334C23×C3C2×SD1612yes—
335C23×C3C2×SD1612yes—
336C23×C3AGL(1,ℤ/8)12yes—
337C23×C3#13712yes—
338C23×C3C22×D411yescharacteristic
339C3×D4—28no—
340C3×D4—48no—
341C3×D4—416no—
342C3×D4—416no—
343C3×D4—44no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—28no—
349C2×C3×C4—28no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—24no—
353C2×C3×C4—21no—
354C2×C3×C4—44no—
355C2×C3×C4—44no—
356C2×C3×C4—44no—
357C2×C3×C4—48no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—44no—
361C2×C3×C4#12012yes—
362C2×C3×C4#12012yes—
363C2×C3×C4C22×D412yes—
364C2×C3×C4C22×D412yes—
365C2×C3×C4H(5,𝔽2)12yes—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—48no—
369C2×Dic3—48no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374Dic6—28no—
375Dic6—48no—
376Dic6—44no—
377C3⋊C8—44no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—24no—
383C24—22no—
384C24—41no—
385C24—41no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24#21414yescomplemented
393C24C2×#6711yescharacteristic
394C24C2×#6711yescharacteristic
395C24C2×#6711yescharacteristic
396C2×Q8—34no—
397C2×Q8—64no—
398C2×Q8—62no—
399C2×Q8—64no—
400C2×Q8—64no—
401C2×Q8—1216no—
402C2×Q8—122no—
403C2×Q8—122no—
404C2×D4—24no—
405C2×D4—28no—
406C2×D4—28no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—416no—
411C2×D4—42no—
412C2×D4—432no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4C2×#6714yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—416no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—64no—
441C22×C4—64no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4D3×D414yes—
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6714yes—
453C22×C4C23×D311yescharacteristic
454SD16—1216no—
455C2×C8—64no—
456C2×C8—64no—
457C2×C8—64no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—34no—
461C4⋊C4—68no—
462C4⋊C4—64no—
463C4⋊C4—68no—
464C4⋊C4—62no—
465C4⋊C4C2×#6714yes—
466C22⋊C4—28no—
467C22⋊C4—28no—
468C22⋊C4—68no—
469C22⋊C4—68no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—416no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—44no—
491C22×C3—44no—
492C22×C3—44no—
493C22×C3—816no—
494C22×C3—82no—
495C22×C3—82no—
496C22×C3—82no—
497C22×C3—82no—
498C22×C3#44814yescomplemented
499C22×C3#45014yescomplemented
500C22×C3C2×#12011yescharacteristic
501C22×C3C2×#12011yescharacteristic
502C22×C3#53411yescharacteristic
503C22×C3C22×SD1611yescharacteristic
504C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
505C22×C3C2×#13711yescharacteristic
506C22×C3#57511yescharacteristic
507C3×C4—22no—
508C3×C4—22no—
509C3×C4—41no—
510C3×C4—44no—
511C3×C4D4214yes—
512Dic3—44no—
513Dic3—44no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—28no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—28no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—24no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—48no—
537C23—48no—
538C23—48no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23#93212yes—
550C23C2×#21412yes—
551C23C2×#21412yes—
552C23C2×#21412yes—
553C23#93712yes—
554C23#95112yes—
555C23#95312yes—
556C23C22×#6711yescharacteristic
557Q8—68no—
558Q8—128no—
559Q8—124no—
560D4—28no—
561D4—48no—
562D4—416no—
563D4—416no—
564D4—44no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—24no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—64no—
583C2×C4—64no—
584C2×C4—128no—
585C2×C4—128no—
586C2×C4—122no—
587C2×C4—122no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4#93712yes—
591C2×C4#95312yes—
592C2×C4C2×D3×D412yes—
593C2×C4#100412yes—
594C2×C4C22×#6712yes—
595C8—124no—
596C2×C3—24no—
597C2×C3—24no—
598C2×C3—42no—
599C2×C3—84no—
600C2×C3#240311yescharacteristic
601C2×C3C2×#45011yescharacteristic
602C2×C3C2×#44811yescharacteristic
603C2×C3#300611yescharacteristic
604C2×C3#300711yescharacteristic
605C2×C3#318711yescharacteristic
606C2×C3#319311yescharacteristic
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—416no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—816no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22#504014yescomplemented
636C22#521814yescomplemented
637C22C2×#93211yescharacteristic
638C22C22×#21411yescharacteristic
639C22C2×#93711yescharacteristic
640C22#580411yescharacteristic
641C22#582211yescharacteristic
642C22C2×#95111yescharacteristic
643C22C2×#95311yescharacteristic
644C4—22no—
645C4—22no—
646C4—44no—
647C4—41no—
648C4—124no—
649C4—124no—
650C4D4×#6714yes—
651C3#2391911yescharacteristic, complemented, Sylow
652C2—24no—
653C2—24no—
654C2—42no—
655C2—84no—
656C2#8262411yescharacteristic
657C2C2×#504011yescharacteristic
658C2#8263011yescharacteristic
659C2#8263111yescharacteristic
660C2C2×#521811yescharacteristic
661C2#8405211yescharacteristic
662C2#8408411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC22×C3×#121C2
socleC23×C3C22×D4
hypercenterC22×#121D3
hypocenterC3#23919
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
2. C22×#924C2semidirect
3. C2×#5218C2semidirect
4. C2×#5040C2semidirect
5. #72126C2semidirect
6. #71685C2semidirect
7. #71677C2semidirect
8. #71559C2semidirect
13. C23×Dic6C22semidirect
22. C22×#214C22semidirect
23. C22×C4⋊Dic3C22semidirect
28. C2×#924C22semidirect
31. C2×#832C22semidirect
33. C2×#830C22semidirect
34. C2×#830C22semidirect
36. C2×#810C22semidirect
37. #4529C22semidirect
101. #924D4semidirect
153. C24×C3SD16semidirect
214. C2×Dic6C2×D4semidirect
225. C4⋊Dic3C2×D4semidirect
392. C24#214semidirect
498. C22×C3#448semidirect
499. C22×C3#450semidirect
635. C22#5040semidirect
636. C22#5218semidirect
651. C3#23919semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C22×#9242
C2×#52182
C2×#50402
#721262
#716852
#716772
#715592
#239193