Finite Group Databasebeta

← #10944005

Subgroups — #10944005

Order 768.

1868 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic3C211yesmaximal, characteristic
2C2×C3×#450C211yesmaximal, characteristic
3C2×#5218C211yesmaximal, characteristic
4C2×#5162C212yesmaximal, complemented
5#82811C211yesmaximal, characteristic, complemented
6#72047C212yesmaximal, complemented
7#71692C211yesmaximal, characteristic, complemented
8#70171C212yesmaximal, complemented
9#70153C212yesmaximal, complemented
10#69930C211yesmaximal, characteristic, complemented
11#69908C211yesmaximal, characteristic, complemented
12#32905—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C24×Dic3—21no—
15C22×C3×SD16C2212yes—
16C2×C3×C22⋊Q8C2211yescharacteristic
17C22×#217—21no—
18C2×#944C2211yescharacteristic, complemented
19C2×D4×Dic3—28no—
20C2×D4×Dic3—21no—
21C2×D4×Dic3—21no—
22C2×D4×Dic3C2212yes—
23C2×D4×Dic3C2212yes—
24C22×#214C2212yes—
25C22×C4⋊Dic3C2211yescharacteristic
26C2×#924C2211yescharacteristic
27C22×C4×Dic3C2211yescharacteristic
28C3×#450—24no—
29C2×C3×#102C2212yes—
30C2×C3×C22⋊C8C2211yescharacteristic
31#5229C2211yescharacteristic, complemented
32C2×#835C2212yescomplemented
33#5218—24no—
34C2×#832C2212yes—
35C2×#830C2211yescharacteristic
36#5162—28no—
37C2×#816C2212yescomplemented
38C2×Dic3⋊C8C2212yescomplemented
39#4976—21no—
40#4976—21no—
41#4975C2211yescharacteristic, complemented
42#4973C2211yescharacteristic, complemented
43#4942—21no—
44#4942—21no—
45#4551C2212yescomplemented
46#4550C2212yescomplemented
47#4529C2212yescomplemented
48C22×C4×D4—31no—
49C2×#461—32no—
50C2×#450—31no—
51C2×#450D311yescharacteristic
52#2857—31no—
53#1877—32no—
54#1877—32no—
55#1872—31no—
56#1872—31no—
57#1802—32no—
58#1797—31no—
59C25×C3—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—24no—
62C22×C3×D4—24no—
63C22×C3×D4—21no—
64C22×C3×D4—22no—
65C22×C3×D4D414yes—
66C22×C3×D4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C23×Dic3—28no—
69C23×Dic3—21no—
70C23×Dic3—22no—
71C23×Dic3—22no—
72C23×Dic3—44no—
73C23×Dic3—44no—
74C23×Dic3D411yescharacteristic
75C23×Dic3D411yescharacteristic
76C22×Dic6C2311yescharacteristic
77C2×C3×SD16—24no—
78C2×C3×SD16—24no—
79C2×C3×SD16—24no—
80C22×C3×C8C2312yes—
81C3×C22⋊Q8—22no—
82C3×C22⋊Q8—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—21no—
86C2×#217—28no—
87C2×#217—22no—
88C2×#217—22no—
89C2×#217—21no—
90#944—24no—
91D4×Dic3—28no—
92D4×Dic3—28no—
93D4×Dic3—24no—
94D4×Dic3—24no—
95D4×Dic3—24no—
96D4×Dic3D418yes—
97C2×#214—24no—
98C2×#214—24no—
99C2×#214—24no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—24no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2312yes—
104#924—22no—
105#924—22no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—24no—
112C2×C4×Dic3D411yescharacteristic
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2312yes—
117C3×#102—24no—
118C3×C22⋊C8—22no—
119#835—24no—
120#832—24no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125#830—22no—
126#816—24no—
127Dic3⋊C8—24no—
128C23×D4C2×D311yescharacteristic
129C24×C4—61no—
130C22×SD16—32no—
131C22×SD16C2×D312yes—
132C2×C4⋊Q8—31no—
133C2×C22⋊Q8—31no—
134C2×C22⋊Q8C2×D311yescharacteristic
135C2×C4×D4—32no—
136C2×C4×D4—32no—
137C2×C4×D4—61no—
138C2×C4×D4—61no—
139C2×C4×D4—68no—
140C22×C4⋊C4—31no—
141C22×C22⋊C4—61no—
142C22×C42—31no—
143#461—68no—
144#450—24no—
145#450—64no—
146C2×C4⋊C8—32no—
147C2×#103—32no—
148C2×#103—32no—
149C2×#102—32no—
150C2×#102C2×D312yes—
151C2×C22⋊C8—31no—
152C2×C22⋊C8C2×D311yescharacteristic
153#391—31no—
154#391—31no—
155#389—61no—
156#389—61no—
157#384—31no—
158#380—61no—
159#380—61no—
160#340—32no—
161#340—32no—
162#336—32no—
163C24×C3—24no—
164C24×C3—24no—
165C24×C3—24no—
166C24×C3—22no—
167C24×C3—42no—
168C24×C3—42no—
169C24×C3—42no—
170C24×C3—42no—
171C24×C3C2×D411yescharacteristic
172C2×C3×Q8—22no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
176C2×C3×D4—24no—
177C2×C3×D4—24no—
178C2×C3×D4—216no—
179C2×C3×D4—216no—
180C2×C3×D4—24no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4C2×D414yes—
185C2×C3×D4C2×D414yes—
186C2×C3×D4#4114yes—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D412yes—
193C22×C3×C4#4112yes—
194C22×C3×C4C2411yescharacteristic
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—24no—
198C22×Dic3—28no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—28no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—41no—
206C22×Dic3—44no—
207C22×Dic3—432no—
208C22×Dic3—42no—
209C22×Dic3—42no—
210C22×Dic3—44no—
211C22×Dic3—44no—
212C22×Dic3—44no—
213C22×Dic3SD1614yescomplemented
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C2×Dic6—22no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—22no—
222C3×SD16—48no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—42no—
229C3×C22⋊C4—42no—
230#217—28no—
231#217—28no—
232#217—42no—
233#214—48no—
234C4⋊Dic3—24no—
235C4⋊Dic3—22no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3C2×D414yes—
239Dic3⋊C4—42no—
240Dic3⋊C4—42no—
241Dic3⋊C4—42no—
242Dic3⋊C4—42no—
243Dic3⋊C4—42no—
244C4×Dic3—22no—
245C4×Dic3—22no—
246C4×Dic3—22no—
247C4×Dic3C2×D414yes—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C25—21no—
252C22×Q8—31no—
253C22×Q8C22×D311yescharacteristic
254C22×D4—24no—
255C22×D4—24no—
256C22×D4—21no—
257C22×D4—22no—
258C22×D4#6714yes—
259C22×D4C22×D312yes—
260C23×C4—31no—
261C23×C4—31no—
262C23×C4—62no—
263C23×C4—62no—
264C23×C4—68no—
265C23×C4—61no—
266C23×C4—124no—
267C23×C4—124no—
268C23×C4C22×D311yescharacteristic
269C2×SD16—24no—
270C2×SD16—24no—
271C2×SD16—24no—
272C2×SD16—64no—
273C2×SD16—64no—
274C2×SD16—64no—
275C22×C8—32no—
276C22×C8C22×D312yes—
277C4⋊Q8—64no—
278C22⋊Q8—22no—
279C22⋊Q8—22no—
280C22⋊Q8—62no—
281C22⋊Q8—62no—
282C4×D4—38no—
283C4×D4—64no—
284C4×D4—64no—
285C4×D4—68no—
286C4×D4—68no—
287C4×D4—64no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—32no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—64no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—62no—
301C2×C22⋊C4—62no—
302C2×C22⋊C4—68no—
303C2×C22⋊C4—61no—
304C2×C42—31no—
305C2×C42—31no—
306C2×C42—31no—
307C2×C42—31no—
308C2×C42—64no—
309C4⋊C8—64no—
310#103—64no—
311#103—64no—
312#102—24no—
313#102—64no—
314C22⋊C8—22no—
315C22⋊C8—62no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—216no—
325C23×C3—24no—
326C23×C3—24no—
327C23×C3—24no—
328C23×C3—48no—
329C23×C3—48no—
330C23×C3—48no—
331C23×C3—48no—
332C23×C3—48no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—48no—
338C23×C3#12112yes—
339C23×C3C2×SD1612yes—
340C23×C3#13712yes—
341C23×C3C22×D411yescharacteristic
342C3×Q8—42no—
343C3×Q8—42no—
344C3×D4—216no—
345C3×D4—24no—
346C3×D4—24no—
347C3×D4—24no—
348C3×D4#12118yes—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4#12112yes—
360C2×C3×C4C22×D412yes—
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C2×#4112yes—
363C2×C3×C4C2×#4111yescharacteristic
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—28no—
369C2×Dic3—21no—
370C2×Dic3—24no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—44no—
374C2×Dic3—416no—
375C2×Dic3—48no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—416no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3#12012yes—
386C2×Dic3#12012yes—
387C2×Dic3C2×SD1612yes—
388C2×Dic3C2×SD1612yes—
389C2×Dic3AGL(1,ℤ/8)12yes—
390Dic6—42no—
391Dic6—42no—
392C3×C8—42no—
393C3⋊C8—42no—
394C24—24no—
395C24—24no—
396C24—24no—
397C24—22no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24D3×D411yescharacteristic
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—62no—
408C2×Q8—61no—
409C2×Q8—62no—
410C2×Q8—61no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—216no—
414C2×D4—216no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4D3×D414yes—
420C2×D4#23714yes—
421C2×D4C2×#6714yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—31no—
427C22×C4—34no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—68no—
435C22×C4—64no—
436C22×C4—68no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—124no—
442C22×C4—124no—
443C22×C4—124no—
444C22×C4—121no—
445C22×C4—1232no—
446C22×C4—124no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4D3×D411yescharacteristic
450C22×C4#23712yes—
451C22×C4C2×#6712yes—
452C22×C4C23×D311yescharacteristic
453SD16—48no—
454SD16—128no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—62no—
459C2×C8—62no—
460C2×C8—62no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—34no—
464C4⋊C4—42no—
465C4⋊C4—64no—
466C4⋊C4—62no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—122no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4—122no—
473C4⋊C4—122no—
474C22⋊C4—42no—
475C22⋊C4—68no—
476C22⋊C4—68no—
477C22⋊C4—122no—
478C42—34no—
479C42—62no—
480C42—62no—
481C42—62no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—28no—
489C22×C3—432no—
490C22×C3—44no—
491C22×C3—44no—
492C22×C3—44no—
493C22×C3—44no—
494C22×C3—44no—
495C22×C3—44no—
496C22×C3#46114yescomplemented
497C22×C3C2×#12011yescharacteristic
498C22×C3C2×#12111yescharacteristic
499C22×C3#52511yescharacteristic
500C22×C3C22×SD1611yescharacteristic
501C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
502C22×C3C2×#13711yescharacteristic
503C22×C3#57511yescharacteristic
504C3×C4—22no—
505C3×C4—81no—
506C3×C4—81no—
507C3×C4C2×#12112yes—
508C3×C4#52512yes—
509Dic3—22no—
510Dic3—22no—
511Dic3—44no—
512Dic3—41no—
513Dic3—81no—
514Dic3—81no—
515Dic3#45014yescomplemented
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—216no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—48no—
525C23—48no—
526C23—48no—
527C23—48no—
528C23—48no—
529C23—48no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23D3×SD1612yes—
535C23#91512yes—
536C23#93912yes—
537C23C2×D3×D411yescharacteristic
538Q8—42no—
539Q8—42no—
540Q8—122no—
541Q8—122no—
542D4—216no—
543D4—24no—
544D4—24no—
545D4—24no—
546D4#93918yes—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—32no—
552C2×C4—32no—
553C2×C4—32no—
554C2×C4—32no—
555C2×C4—32no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—68no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—64no—
571C2×C4—1216no—
572C2×C4—124no—
573C2×C4—1216no—
574C2×C4—128no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4#93912yes—
584C2×C4C2×D3×D412yes—
585C2×C4C2×#23712yes—
586C2×C4C2×#23711yescharacteristic
587C2×C4C22×#6711yescharacteristic
588C8—42no—
589C8—122no—
590C2×C3—24no—
591C2×C3—48no—
592C2×C3#284711yescharacteristic
593C2×C3C2×#45011yescharacteristic
594C2×C3#300711yescharacteristic
595C2×C3C2×#46111yescharacteristic
596C2×C3#303611yescharacteristic
597C2×C3#310111yescharacteristic
598C2×C3#311011yescharacteristic
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—28no—
606C22—432no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22#516214yescomplemented
614C22#558611yescharacteristic
615C22D3×#12011yescharacteristic
616C22C2×D3×SD1611yescharacteristic
617C22C2×#91511yescharacteristic
618C22D3×AGL(1,ℤ/8)11yescharacteristic
619C22#577311yescharacteristic
620C22C2×#93911yescharacteristic
621C4—22no—
622C4—34no—
623C4—62no—
624C4—62no—
625C4—81no—
626C4—81no—
627C4—124no—
628C4—121no—
629C4—241no—
630C4—241no—
631C4#558612yes—
632C4C2×#93912yes—
633C3#3290511yescharacteristic, complemented, Sylow
634C2—24no—
635C2—48no—
636C2#8013211yescharacteristic
637C2#8013411yescharacteristic
638C2D3×#45011yescharacteristic
639C2#8014711yescharacteristic
640C2#8260111yescharacteristic
641C2C2×#516211yescharacteristic
642C2#8393111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#450C2
socleC23×C3C22×D4
prime radicalC23×C3×D4C22
hypercenterC2×#450D3
hypocenterC3#32905
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
4. C2×#5162C2semidirect
5. #82811C2semidirect
6. #72047C2semidirect
7. #71692C2semidirect
8. #70171C2semidirect
9. #70153C2semidirect
10. #69930C2semidirect
11. #69908C2semidirect
18. C2×#944C22semidirect
31. #5229C22semidirect
32. C2×#835C22semidirect
37. C2×#816C22semidirect
38. C2×Dic3⋊C8C22semidirect
41. #4975C22semidirect
42. #4973C22semidirect
45. #4551C22semidirect
46. #4550C22semidirect
47. #4529C22semidirect
213. C22×Dic3SD16semidirect
496. C22×C3#461semidirect
515. Dic3#450semidirect
613. C22#5162semidirect
633. C3#32905semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic32
C2×C3×#4502
C2×#52182
C2×#51622
#828112
#720472
#716922
#701712
#701532
#699302
#699082
#329053