Finite Group Databasebeta

← #10944034

Subgroups — #10944034

Order 768.

1894 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#893C211yesmaximal, characteristic, complemented
2#82698C211yesmaximal, characteristic, complemented
3C2×#5036C211yesmaximal, characteristic
4C2×#4787C212yesmaximal, complemented
5C2×#4731C211yesmaximal, characteristic, complemented
6C3×#1874C211yesmaximal, characteristic, complemented
7#71559C211yesmaximal, characteristic, complemented
8#70208C212yesmaximal, complemented
9#69908C211yesmaximal, characteristic, complemented
10#68896C212yesmaximal, complemented
11#68893C212yesmaximal, complemented
12#32908—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C23×C4×D3—21no—
15C2×C3×#121C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C2×#936C2211yescharacteristic, complemented
18C22×#213C2212yes—
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×#102C2212yescomplemented
29C2×C3×C22⋊C8C2211yescharacteristic
30#5221C2211yescharacteristic, complemented
31#5220—21no—
32#5220—21no—
33C2×#830C2211yescharacteristic
34C2×#821C2212yes—
35C2×Dic3⋊C8C2212yescomplemented
36#5036—24no—
37#4977C2211yescharacteristic, complemented
38#4973C2211yescharacteristic, complemented
39C2×#810C2212yescomplemented
40C2×#809C2212yes—
41#4945—21no—
42#4945—21no—
43#4787—28no—
44#4731—24no—
45C3×#340C2212yescomplemented
46#4550C2212yescomplemented
47#4529C2212yescomplemented
48C22×C4×D4—31no—
49C2×#463—32no—
50C2×#450—31no—
51C2×#447—31no—
52#2850—31no—
53#1879—32no—
54#1877—32no—
55#1874D311yescharacteristic, complemented
56#1872—31no—
57#1804—32no—
58#1797—31no—
59C24×D3—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4C2311yescharacteristic
64C23×Dic3D411yescharacteristic
65C23×Dic3D411yescharacteristic
66C22×D12—21no—
67C22×D12—22no—
68C22×D12—24no—
69C22×D12—24no—
70C22×D12D414yes—
71C22×D12C2312yes—
72C22×C4×D3—28no—
73C22×C4×D3—22no—
74C22×C4×D3—22no—
75C22×C4×D3—44no—
76C22×C4×D3—44no—
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2312yes—
79C3×#121—22no—
80C3×#121—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—24no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C22⋊C4—21no—
86C2×#217—21no—
87C2×#217—21no—
88C2×#217—21no—
89C2×#217—21no—
90C2×#217—21no—
91#936—24no—
92C2×#213—24no—
93C2×#213—24no—
94C2×#213—24no—
95C2×(C2×D3)⋊C4—28no—
96C2×(C2×D3)⋊C4—22no—
97C2×(C2×D3)⋊C4—22no—
98C2×C4⋊Dic3C2311yescharacteristic
99#924—22no—
100#924—22no—
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#893D418yescomplemented
117C3×#102—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×D8—32no—
132C2×#124—31no—
133C2×C22⋊Q8—31no—
134C2×#121C2×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#463—68no—
144#450—64no—
145#447—64no—
146C2×C4⋊C8—32no—
147C2×#103—32no—
148C2×#102—32no—
149C2×#102—32no—
150C2×#102C2×D312yescomplemented
151C2×C22⋊C8—31no—
152C2×C22⋊C8C2×D311yescharacteristic
153#391—31no—
154#390—31no—
155#388—61no—
156#388—61no—
157#386—61no—
158#386—61no—
159#382—31no—
160#340—32no—
161#340C2×D312yescomplemented
162#336—32no—
163C24×C3—21no—
164C24×C3C2×D411yescharacteristic
165C23×D3—24no—
166C23×D3—24no—
167C23×D3—24no—
168C23×D3—22no—
169C23×D3—42no—
170C23×D3—42no—
171C23×D3—42no—
172C23×D3—42no—
173C2×C3×D4—22no—
174C2×C3×D4—21no—
175C2×C3×D4—22no—
176C2×C3×D4—21no—
177C2×C3×D4—42no—
178C2×C3×D4—42no—
179C2×C3×D4—42no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—24no—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—42no—
186C22×C3×C4—42no—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D412yes—
189C22×C3×C4#4112yes—
190C22×C3×C4C2411yescharacteristic
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—41no—
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203C2×D12—24no—
204C2×D12—24no—
205C2×D12—216no—
206C2×D12—24no—
207C2×D12—22no—
208C2×D12—22no—
209C2×D12—22no—
210C2×D12—216no—
211C2×D12C2×D414yes—
212C2×D12C2×D414yes—
213C2×D12#4114yes—
214C2×C4×D3—28no—
215C2×C4×D3—28no—
216C2×C4×D3—432no—
217C2×C4×D3—44no—
218C2×C4×D3—44no—
219C2×C4×D3—44no—
220C2×C4×D3—44no—
221C2×Dic6—22no—
222C2×Dic6—21no—
223C2×Dic6—22no—
224C2×Dic6—21no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4C2×D414yes—
233C3×C22⋊C4—42no—
234#217—42no—
235#217—42no—
236#217—42no—
237#217—42no—
238#217—42no—
239#213—48no—
240(C2×D3)⋊C4—28no—
241(C2×D3)⋊C4—28no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244Dic3⋊C4—42no—
245C4×Dic3—22no—
246C4×Dic3—22no—
247C4×Dic3—22no—
248C4×Dic3C2×D414yes—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252#204—48no—
253C25—61no—
254C22×Q8—31no—
255C22×D4—21no—
256C22×D4—34no—
257C22×D4—32no—
258C22×D4—64no—
259C22×D4—64no—
260C22×D4—62no—
261C22×D4—61no—
262C22×D4C22×D311yescharacteristic
263C23×C4—21no—
264C23×C4—31no—
265C23×C4—31no—
266C23×C4—62no—
267C23×C4—62no—
268C23×C4—68no—
269C23×C4—124no—
270C23×C4—124no—
271C23×C4C22×D311yescharacteristic
272C2×SD16—64no—
273C2×SD16—64no—
274C2×SD16—64no—
275C2×D8—64no—
276C2×D8—64no—
277C2×D8—64no—
278C22×C8—32no—
279C22×C8C22×D312yes—
280#124—64no—
281C22⋊Q8—62no—
282C22⋊Q8—62no—
283#121—22no—
284#121—22no—
285C4×D4—38no—
286C4×D4—64no—
287C4×D4—64no—
288C4×D4—68no—
289C4×D4—68no—
290C4×D4—64no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D312yes—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—62no—
299C2×C22⋊C4—62no—
300C2×C22⋊C4—68no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C42—31no—
307C2×C42—31no—
308C2×C42—31no—
309C2×C42—31no—
310C2×C42—64no—
311C4⋊C8—64no—
312#103—64no—
313#102—24no—
314#102—64no—
315#102—64no—
316C22⋊C8—22no—
317C22⋊C8—62no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3#12112yes—
334C23×C3#13512yes—
335C23×C3AGL(1,ℤ/8)12yes—
336C23×C3C22×D411yescharacteristic
337C22×D3—216no—
338C22×D3—24no—
339C22×D3—24no—
340C22×D3—24no—
341C22×D3—48no—
342C22×D3—48no—
343C22×D3—48no—
344C22×D3—48no—
345C22×D3—48no—
346C22×D3—42no—
347C22×D3—42no—
348C22×D3—42no—
349C22×D3—42no—
350C22×D3—48no—
351C3×D4—42no—
352C3×D4—42no—
353C3×D4—84no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—48no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4#12112yes—
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D412yes—
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4C2×#4112yes—
370C2×Dic3—28no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—44no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3#12012yes—
388C2×Dic3#12012yes—
389C2×Dic3AGL(1,ℤ/8)12yes—
390D12—216no—
391D12—24no—
392D12—24no—
393D12—24no—
394D12#12118yes—
395C4×D3—416no—
396C4×D3—416no—
397Dic6—42no—
398Dic6—42no—
399C3×C8—42no—
400C3⋊C8—42no—
401C24—21no—
402C24—64no—
403C24—64no—
404C24—64no—
405C24—62no—
406C24—122no—
407C24—122no—
408C24—122no—
409C24—122no—
410C24D3×D411yescharacteristic
411C2×Q8—61no—
412C2×Q8—62no—
413C2×Q8—61no—
414C2×Q8—62no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—34no—
420C2×D4—34no—
421C2×D4—34no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—616no—
426C2×D4—64no—
427C2×D4—64no—
428C2×D4—616no—
429C2×D4—64no—
430C2×D4—62no—
431C2×D4—62no—
432C2×D4—62no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—68no—
445C22×C4—68no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—124no—
454C22×C4—1232no—
455C22×C4—124no—
456C22×C4—124no—
457C22×C4—124no—
458C22×C4—121no—
459C22×C4#23512yes—
460C22×C4D3×D411yescharacteristic
461C22×C4D3×D412yes—
462C22×C4C23×D311yescharacteristic
463SD16—128no—
464D8—128no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—62no—
469C2×C8—62no—
470C2×C8—62no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—122no—
478C4⋊C4D3×D414yes—
479C22⋊C4—42no—
480C22⋊C4—68no—
481C22⋊C4—68no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C42—34no—
488C42—62no—
489C42—62no—
490C42—62no—
491C22×C3—22no—
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3—81no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3#46314yescomplemented
505C22×C3C2×#12011yescharacteristic
506C22×C3C2×#12111yescharacteristic
507C22×C3#52511yescharacteristic
508C22×C3C2×#13511yescharacteristic
509C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
510C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
511C22×C3#57511yescharacteristic
512C2×D3—28no—
513C2×D3—432no—
514C2×D3—44no—
515C2×D3—44no—
516C2×D3—44no—
517C2×D3—44no—
518C2×D3—44no—
519C2×D3—44no—
520C3×C4—22no—
521C3×C4—44no—
522C3×C4—81no—
523C3×C4C2×#12112yes—
524C3×C4#52512yes—
525Dic3—22no—
526Dic3—22no—
527Dic3—22no—
528Dic3—41no—
529Dic3—81no—
530Dic3—81no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—616no—
543C23—64no—
544C23—64no—
545C23—64no—
546C23—128no—
547C23—128no—
548C23—128no—
549C23—128no—
550C23—128no—
551C23—122no—
552C23—122no—
553C23—122no—
554C23—122no—
555C23—128no—
556C23#88412yes—
557C23#91112yes—
558C23#91612yes—
559C23C2×D3×D411yescharacteristic
560Q8—122no—
561Q8—122no—
562D4—38no—
563D4—42no—
564D4—42no—
565D4—616no—
566D4—64no—
567D4—64no—
568D4—64no—
569D4—84no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—32no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—68no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—1216no—
595C2×C4—1216no—
596C2×C4—124no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4#88412yes—
604C2×C4C2×#23511yescharacteristic
605C2×C4C2×D3×D412yes—
606C2×C4C2×D3×D411yescharacteristic
607C2×C4D3×#4112yes—
608C8—42no—
609C8—122no—
610C2×C3—24no—
611C2×C3—161no—
612C2×C3#284711yescharacteristic
613C2×C3#300211yescharacteristic
614C2×C3#300311yescharacteristic
615C2×C3C2×#46311yescharacteristic
616C2×C3#303611yescharacteristic
617C2×C3#310511yescharacteristic
618C2×C3#311411yescharacteristic
619D3—48no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—68no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—1232no—
635C22—124no—
636C22—124no—
637C22—124no—
638C22—124no—
639C22—124no—
640C22—124no—
641C22#478714yescomplemented
642C22C2×#88411yescharacteristic
643C22D3×#12011yescharacteristic
644C22#565611yescharacteristic
645C22C2×#91111yescharacteristic
646C22C2×#91611yescharacteristic
647C22D3×AGL(1,ℤ/8)11yescharacteristic
648C22#577711yescharacteristic
649C4—22no—
650C4—44no—
651C4—62no—
652C4—62no—
653C4—62no—
654C4—81no—
655C4—121no—
656C4—241no—
657C4—241no—
658C4#554712yes—
659C4D3×#12112yes—
660C3#3290811yescharacteristic, complemented, Sylow
661C2—24no—
662C2—128no—
663C2—161no—
664C2#7493511yescharacteristic
665C2C2×#478711yescharacteristic
666C2#7891011yescharacteristic
667C2#8000211yescharacteristic
668C2#8003811yescharacteristic
669C2#8127111yescharacteristic
670C2#8130911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1874C2
socleC23×C3C22×D4
prime radicalC2×#5036C2
hypercenter#1874D3
hypocenterC3#32908
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
1. C22×#893C2semidirect
2. #82698C2semidirect
4. C2×#4787C2semidirect
5. C2×#4731C2semidirect
6. C3×#1874C2semidirect
7. #71559C2semidirect
8. #70208C2semidirect
9. #69908C2semidirect
10. #68896C2semidirect
11. #68893C2semidirect
17. C2×#936C22semidirect
28. C2×C3×#102C22semidirect
30. #5221C22semidirect
35. C2×Dic3⋊C8C22semidirect
37. #4977C22semidirect
38. #4973C22semidirect
39. C2×#810C22semidirect
45. C3×#340C22semidirect
46. #4550C22semidirect
47. #4529C22semidirect
55. #1874D3semidirect
116. #893D4semidirect
150. C2×#102C2×D3semidirect
161. #340C2×D3semidirect
504. C22×C3#463semidirect
641. C22#4787semidirect
660. C3#32908semidirect

Maximal subgroups

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

subgroupindex
C22×#8932
#826982
C2×#50362
C2×#47872
C2×#47312
C3×#18742
#715592
#702082
#699082
#688962
#688932
#329083