Finite Group Databasebeta

← #10944007

Subgroups — #10944007

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×#893C211yesmaximal, characteristic
2#82811C211yesmaximal, characteristic, complemented
3C2×#5046C211yesmaximal, characteristic
4C2×#4818C212yesmaximal, complemented
5C2×#4731C211yesmaximal, characteristic
6C3×#1872C211yesmaximal, characteristic, complemented
7#71559C211yesmaximal, characteristic, complemented
8#70276C212yesmaximal, complemented
9#69908C211yesmaximal, characteristic, complemented
10#68896C212yesmaximal, complemented
11#68892C212yesmaximal, complemented
12#32905—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C23×C4×D3—21no—
15C2×C3×C22⋊Q8C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C2×#944C2211yescharacteristic, complemented
18C22×#215C2212yes—
19C22×(C2×D3)⋊C4—21no—
20C2×#924C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22C22×#204C2212yes—
23C2×#893—21no—
24C2×#893—21no—
25C2×#893—28no—
26C2×#893C2212yes—
27C2×#893C2212yes—
28C2×C3×#103C2212yescomplemented
29C2×C3×C22⋊C8C2211yescharacteristic
30#5229C2211yescharacteristic, complemented
31C2×#830C2211yescharacteristic
32C2×#821C2212yes—
33C2×Dic3⋊C8C2212yescomplemented
34#5046—24no—
35#4976—21no—
36#4976—21no—
37#4975C2211yescharacteristic, complemented
38#4973C2211yescharacteristic, complemented
39C2×#810C2212yescomplemented
40C2×#809C2212yes—
41#4942—21no—
42#4942—21no—
43#4818—28no—
44#4731—24no—
45C3×#340C2212yescomplemented
46#4550C2212yescomplemented
47#4529C2212yescomplemented
48C22×C4×D4—31no—
49C2×#461—32no—
50C2×#450—31no—
51C2×#450—31no—
52#2857—31no—
53#1877—32no—
54#1877—32no—
55#1872—31no—
56#1872D311yescharacteristic, complemented
57#1802—32no—
58#1797—31no—
59C24×D3—21no—
60C22×C3×Q8C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4C2311yescharacteristic
63C23×Dic3D411yescharacteristic
64C23×Dic3D411yescharacteristic
65C22×D12—21no—
66C22×D12—22no—
67C22×D12—24no—
68C22×D12—24no—
69C22×D12D414yes—
70C22×D12C2312yes—
71C22×C4×D3—28no—
72C22×C4×D3—22no—
73C22×C4×D3—22no—
74C22×C4×D3—44no—
75C22×C4×D3—44no—
76C22×Dic6C2311yescharacteristic
77C22×C3×C8C2312yes—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—24no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C22⋊C4—21no—
86C2×#217—21no—
87#944—24no—
88C2×#215—24no—
89C2×#215—24no—
90C2×#215—24no—
91C2×(C2×D3)⋊C4—28no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—22no—
94C2×C4⋊Dic3C2311yescharacteristic
95#924—22no—
96#924—22no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3—24no—
103C2×C4×Dic3D411yescharacteristic
104C2×C4×Dic3D411yescharacteristic
105C2×C4×Dic3C2311yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2312yes—
108C2×#204—24no—
109C2×#204—24no—
110C2×#204—24no—
111#893—28no—
112#893—28no—
113#893—24no—
114#893—24no—
115#893—24no—
116#893D418yes—
117C3×#103—24no—
118C3×C22⋊C8—22no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123#830—22no—
124#821—24no—
125Dic3⋊C8—24no—
126#810—24no—
127#809—24no—
128C23×D4—31no—
129C24×C4—61no—
130C22×SD16—32no—
131C22×SD16—32no—
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⋊C4C2×D311yescharacteristic
141C22×C22⋊C4—61no—
142C22×C42—31no—
143#461—68no—
144#450—64no—
145#450—64no—
146C2×C4⋊C8—32no—
147C2×#103—32no—
148C2×#103C2×D312yescomplemented
149C2×#102—32no—
150C2×#102—32no—
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#340C2×D312yescomplemented
162#336—32no—
163C24×C3C2×D411yescharacteristic
164C23×D3—24no—
165C23×D3—24no—
166C23×D3—24no—
167C23×D3—22no—
168C23×D3—42no—
169C23×D3—42no—
170C23×D3—42no—
171C23×D3—42no—
172C2×C3×Q8—22no—
173C2×C3×Q8—21no—
174C2×C3×Q8—22no—
175C2×C3×Q8—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—24no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—42no—
183C22×C3×C4—42no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D412yes—
186C22×C3×C4#4112yes—
187C22×C3×C4C2411yescharacteristic
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—41no—
195C22×Dic3SD1614yescomplemented
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C2×D12—24no—
201C2×D12—24no—
202C2×D12—216no—
203C2×D12—24no—
204C2×D12—22no—
205C2×D12—22no—
206C2×D12—22no—
207C2×D12—216no—
208C2×D12C2×D414yes—
209C2×D12C2×D414yes—
210C2×D12#4114yes—
211C2×C4×D3—28no—
212C2×C4×D3—28no—
213C2×C4×D3—44no—
214C2×C4×D3—432no—
215C2×C4×D3—44no—
216C2×C4×D3—44no—
217C2×C4×D3—44no—
218C2×Dic6—22no—
219C2×Dic6—21no—
220C2×Dic6—22no—
221C2×Dic6—21no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4C2×D414yes—
231C3×C22⋊C4—42no—
232#217—42no—
233#215—48no—
234(C2×D3)⋊C4—28no—
235(C2×D3)⋊C4—28no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238Dic3⋊C4—42no—
239Dic3⋊C4—42no—
240Dic3⋊C4—42no—
241Dic3⋊C4—42no—
242Dic3⋊C4—42no—
243C4×Dic3—22no—
244C4×Dic3—22no—
245C4×Dic3—22no—
246C4×Dic3C2×D414yes—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—22no—
250#204—48no—
251C25—61no—
252C22×Q8—31no—
253C22×Q8C22×D311yescharacteristic
254C22×D4—34no—
255C22×D4—32no—
256C22×D4—64no—
257C22×D4—64no—
258C22×D4—62no—
259C22×D4—61no—
260C23×C4—21no—
261C23×C4—31no—
262C23×C4—31no—
263C23×C4—62no—
264C23×C4—62no—
265C23×C4—68no—
266C23×C4—124no—
267C23×C4—124no—
268C23×C4C22×D311yescharacteristic
269C2×SD16—64no—
270C2×SD16—64no—
271C2×SD16—64no—
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—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C4⋊C4C22×D312yes—
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—24no—
311#103—64no—
312#102—64no—
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#12112yes—
325C23×C3C2×SD1612yes—
326C23×C3#13712yes—
327C23×C3C22×D411yescharacteristic
328C22×D3—216no—
329C22×D3—24no—
330C22×D3—24no—
331C22×D3—24no—
332C22×D3—48no—
333C22×D3—48no—
334C22×D3—48no—
335C22×D3—48no—
336C22×D3—48no—
337C22×D3—42no—
338C22×D3—42no—
339C22×D3—42no—
340C22×D3—42no—
341C22×D3—48no—
342C3×Q8—42no—
343C3×Q8—42no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—48no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4#12112yes—
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C22×D412yes—
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4112yes—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—28no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—44no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3#12012yes—
379C2×Dic3#12012yes—
380C2×Dic3C2×SD1612yes—
381C2×Dic3C2×SD1612yes—
382C2×Dic3AGL(1,ℤ/8)12yes—
383D12—216no—
384D12—24no—
385D12—24no—
386D12—24no—
387D12#12118yes—
388C4×D3—416no—
389C4×D3—416no—
390Dic6—42no—
391Dic6—42no—
392C3×C8—42no—
393C3⋊C8—42no—
394C24—64no—
395C24—64no—
396C24—64no—
397C24—62no—
398C24—122no—
399C24—122no—
400C24—122no—
401C24—122no—
402C24D3×D411yescharacteristic
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—61no—
408C2×Q8—62no—
409C2×Q8—61no—
410C2×Q8—62no—
411C2×D4—34no—
412C2×D4—34no—
413C2×D4—34no—
414C2×D4—616no—
415C2×D4—64no—
416C2×D4—64no—
417C2×D4—616no—
418C2×D4—64no—
419C2×D4—62no—
420C2×D4—62no—
421C2×D4—62no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—31no—
429C22×C4—34no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—68no—
436C22×C4—68no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—124no—
444C22×C4—124no—
445C22×C4—124no—
446C22×C4—1232no—
447C22×C4—124no—
448C22×C4—121no—
449C22×C4#23512yes—
450C22×C4D3×D411yescharacteristic
451C22×C4D3×D412yes—
452C22×C4C23×D311yescharacteristic
453SD16—128no—
454SD16—128no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—62no—
459C2×C8—62no—
460C2×C8—62no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—122no—
469C4⋊C4—122no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4—122no—
473C4⋊C4D3×D414yes—
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#46114yescomplemented
489C22×C3C2×#12011yescharacteristic
490C22×C3C2×#12111yescharacteristic
491C22×C3#52511yescharacteristic
492C22×C3C22×SD1611yescharacteristic
493C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
494C22×C3C2×#13711yescharacteristic
495C22×C3#57511yescharacteristic
496C2×D3—28no—
497C2×D3—432no—
498C2×D3—44no—
499C2×D3—44no—
500C2×D3—44no—
501C2×D3—44no—
502C2×D3—44no—
503C2×D3—44no—
504C3×C4—22no—
505C3×C4—44no—
506C3×C4—81no—
507C3×C4—81no—
508C3×C4C2×#12112yes—
509C3×C4#52512yes—
510Dic3—22no—
511Dic3—22no—
512Dic3—41no—
513Dic3—81no—
514Dic3—81no—
515Dic3#45014yescomplemented
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—616no—
521C23—64no—
522C23—64no—
523C23—64no—
524C23—128no—
525C23—128no—
526C23—128no—
527C23—128no—
528C23—128no—
529C23—122no—
530C23—122no—
531C23—122no—
532C23—122no—
533C23—128no—
534C23#88412yes—
535C23D3×SD1612yes—
536C23#91812yes—
537C23C2×D3×D411yescharacteristic
538Q8—42no—
539Q8—42no—
540Q8—122no—
541Q8—122no—
542D4—38no—
543D4—616no—
544D4—64no—
545D4—64no—
546D4—64no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—32no—
553C2×C4—32no—
554C2×C4—32no—
555C2×C4—32no—
556C2×C4—32no—
557C2×C4—48no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—68no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—1216no—
575C2×C4—1216no—
576C2×C4—124no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4#88412yes—
584C2×C4C2×#23511yescharacteristic
585C2×C4C2×D3×D412yes—
586C2×C4C2×D3×D411yescharacteristic
587C2×C4D3×#4112yes—
588C8—42no—
589C8—122no—
590C2×C3—24no—
591C2×C3#284711yescharacteristic
592C2×C3C2×#45011yescharacteristic
593C2×C3#300711yescharacteristic
594C2×C3C2×#46111yescharacteristic
595C2×C3#303611yescharacteristic
596C2×C3#310111yescharacteristic
597C2×C3#311011yescharacteristic
598D3—48no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—68no—
606C22—1232no—
607C22—124no—
608C22—124no—
609C22—124no—
610C22—124no—
611C22—124no—
612C22—124no—
613C22#481814yescomplemented
614C22C2×#88411yescharacteristic
615C22D3×#12011yescharacteristic
616C22#565611yescharacteristic
617C22C2×D3×SD1611yescharacteristic
618C22C2×#91811yescharacteristic
619C22D3×AGL(1,ℤ/8)11yescharacteristic
620C22#577711yescharacteristic
621C4—22no—
622C4—34no—
623C4—44no—
624C4—62no—
625C4—62no—
626C4—81no—
627C4—81no—
628C4—121no—
629C4—241no—
630C4—241no—
631C4#554712yes—
632C4D3×#12112yes—
633C3#3290511yescharacteristic, complemented, Sylow
634C2—24no—
635C2—128no—
636C2#7493511yescharacteristic
637C2C2×#481811yescharacteristic
638C2#7891011yescharacteristic
639C2D3×#45011yescharacteristic
640C2#8018411yescharacteristic
641C2#8141211yescharacteristic
642C2#8145411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1872C2
socleC23×C3C22×D4
prime radicalC23×D12C22
hypercenter#1872D3
hypocenterC3#32905
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
2. #82811C2semidirect
4. C2×#4818C2semidirect
6. C3×#1872C2semidirect
7. #71559C2semidirect
8. #70276C2semidirect
9. #69908C2semidirect
10. #68896C2semidirect
11. #68892C2semidirect
17. C2×#944C22semidirect
28. C2×C3×#103C22semidirect
30. #5229C22semidirect
33. C2×Dic3⋊C8C22semidirect
37. #4975C22semidirect
38. #4973C22semidirect
39. C2×#810C22semidirect
45. C3×#340C22semidirect
46. #4550C22semidirect
47. #4529C22semidirect
56. #1872D3semidirect
148. C2×#103C2×D3semidirect
161. #340C2×D3semidirect
195. C22×Dic3SD16semidirect
488. C22×C3#461semidirect
515. Dic3#450semidirect
613. C22#4818semidirect
633. C3#32905semidirect

Maximal subgroups

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

subgroupindex
C22×#8932
#828112
C2×#50462
C2×#48182
C2×#47312
C3×#18722
#715592
#702762
#699082
#688962
#688922
#329053