Finite Group Databasebeta

← #10944032

Subgroups — #10944032

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×D4×Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#447C211yesmaximal, characteristic
3C2×#5218C211yesmaximal, characteristic, complemented
4C2×#5153C212yesmaximal, complemented
5#82698C211yesmaximal, characteristic, complemented
6#72037C212yesmaximal, complemented
7#71677C211yesmaximal, characteristic, complemented
8#70171C212yesmaximal, complemented
9#70154C212yesmaximal, complemented
10#69930C211yesmaximal, characteristic, complemented
11#69908C211yesmaximal, characteristic, complemented
12#32908—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C24×Dic3—21no—
15C22×C3×D8C2212yes—
16C2×C3×#121C2211yescharacteristic
17C22×#217—21no—
18C2×#936C2211yescharacteristic, 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×#447—24no—
29C2×C3×#102C2212yes—
30C2×C3×C22⋊C8C2211yescharacteristic
31#5221C2211yescharacteristic, complemented
32#5220—21no—
33#5220—21no—
34#5218—24no—
35C2×#832C2212yes—
36C2×#832C2212yescomplemented
37C2×#830C2211yescharacteristic
38#5153—28no—
39C2×#816C2212yescomplemented
40C2×Dic3⋊C8C2212yescomplemented
41#4977C2211yescharacteristic, complemented
42#4973C2211yescharacteristic, complemented
43#4945—21no—
44#4945—21no—
45#4551C2212yescomplemented
46#4550C2212yescomplemented
47#4529C2212yescomplemented
48C22×C4×D4—31no—
49C2×#463—32no—
50C2×#450—31no—
51C2×#447D311yescharacteristic
52#2850—31no—
53#1879—32no—
54#1877—32no—
55#1874—31no—
56#1872—31no—
57#1804—32no—
58#1797—31no—
59C25×C3—21no—
60C22×C3×D4—24no—
61C22×C3×D4—24no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4—22no—
65C22×C3×D4D414yes—
66C22×C3×D4C2311yescharacteristic
67C22×C3×D4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×Dic3—28no—
70C23×Dic3—21no—
71C23×Dic3—22no—
72C23×Dic3—22no—
73C23×Dic3—44no—
74C23×Dic3—44no—
75C23×Dic3D411yescharacteristic
76C23×Dic3D411yescharacteristic
77C22×Dic6C2311yescharacteristic
78C2×C3×D8—24no—
79C2×C3×D8—24no—
80C2×C3×D8—24no—
81C22×C3×C8C2312yes—
82C3×#121—22no—
83C3×#121—22no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—21no—
86C2×#217—28no—
87C2×#217—22no—
88C2×#217—22no—
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94#936—24no—
95D4×Dic3—28no—
96D4×Dic3—28no—
97D4×Dic3—24no—
98D4×Dic3—24no—
99D4×Dic3—24no—
100D4×Dic3D418yescomplemented
101C2×#214—24no—
102C2×#214—24no—
103C2×#214—24no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—24no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2312yes—
108#924—22no—
109#924—22no—
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#832—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×D8C2×D312yes—
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⋊C4—31no—
141C22×C22⋊C4—61no—
142C22×C42—31no—
143#463—68no—
144#450—64no—
145#447—24no—
146C2×C4⋊C8—32no—
147C2×#103—32no—
148C2×#102—32no—
149C2×#102—32no—
150C2×#102C2×D312yes—
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#340—32no—
162#336—32no—
163C24×C3—24no—
164C24×C3—24no—
165C24×C3—24no—
166C24×C3—22no—
167C24×C3—21no—
168C24×C3—42no—
169C24×C3—42no—
170C24×C3—42no—
171C24×C3—42no—
172C24×C3C2×D411yescharacteristic
173C2×C3×D4—24no—
174C2×C3×D4—24no—
175C2×C3×D4—216no—
176C2×C3×D4—216no—
177C2×C3×D4—24no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—21no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—21no—
184C2×C3×D4—22no—
185C2×C3×D4—42no—
186C2×C3×D4—42no—
187C2×C3×D4—42no—
188C2×C3×D4C2×D414yes—
189C2×C3×D4C2×D414yes—
190C2×C3×D4#4114yes—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—21no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D412yes—
196C22×C3×C4#4112yes—
197C22×C3×C4C2411yescharacteristic
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—24no—
202C22×Dic3—28no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—28no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—41no—
210C22×Dic3—432no—
211C22×Dic3—44no—
212C22×Dic3—44no—
213C22×Dic3—44no—
214C22×Dic3—42no—
215C22×Dic3—42no—
216C22×Dic3—44no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C2×Dic6—22no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—22no—
225C3×D8—48no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C22⋊C4—42no—
232#217—28no—
233#217—28no—
234#217—42no—
235#217—42no—
236#217—42no—
237#217—42no—
238#217—42no—
239#214—48no—
240C4⋊Dic3—24no—
241C4⋊Dic3—22no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3C2×D414yes—
245Dic3⋊C4—42no—
246C4×Dic3—22no—
247C4×Dic3—22no—
248C4×Dic3—22no—
249C4×Dic3C2×D414yes—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C25—21no—
254C22×Q8—31no—
255C22×D4—24no—
256C22×D4—24no—
257C22×D4—21no—
258C22×D4—21no—
259C22×D4—22no—
260C22×D4#6714yes—
261C22×D4C22×D311yescharacteristic
262C22×D4C22×D312yes—
263C23×C4—31no—
264C23×C4—31no—
265C23×C4—62no—
266C23×C4—62no—
267C23×C4—68no—
268C23×C4—61no—
269C23×C4—124no—
270C23×C4—124no—
271C23×C4C22×D311yescharacteristic
272C2×SD16—64no—
273C2×SD16—64no—
274C2×SD16—64no—
275C2×D8—24no—
276C2×D8—24no—
277C2×D8—24no—
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—32no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—64no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4C22×D311yescharacteristic
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—216no—
327C23×C3—24no—
328C23×C3—24no—
329C23×C3—24no—
330C23×C3—48no—
331C23×C3—48no—
332C23×C3—48no—
333C23×C3—48no—
334C23×C3—48no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3—48no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3—41no—
343C23×C3—41no—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3#12112yes—
348C23×C3#13512yes—
349C23×C3AGL(1,ℤ/8)12yes—
350C23×C3C22×D411yescharacteristic
351C3×D4—216no—
352C3×D4—24no—
353C3×D4—24no—
354C3×D4—24no—
355C3×D4—42no—
356C3×D4—42no—
357C3×D4—84no—
358C3×D4#12118yes—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4#12112yes—
367C2×C3×C4C22×D412yes—
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C2×#4112yes—
370C2×C3×C4C2×#4111yescharacteristic
371C2×Dic3—28no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—24no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—44no—
383C2×Dic3—416no—
384C2×Dic3—416no—
385C2×Dic3—48no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3#12012yes—
395C2×Dic3#12012yes—
396C2×Dic3AGL(1,ℤ/8)12yes—
397Dic6—42no—
398Dic6—42no—
399C3×C8—42no—
400C3⋊C8—42no—
401C24—24no—
402C24—24no—
403C24—24no—
404C24—22no—
405C24—21no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24D3×D411yescharacteristic
411C2×Q8—62no—
412C2×Q8—61no—
413C2×Q8—62no—
414C2×Q8—61no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—216no—
418C2×D4—216no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4D3×D414yes—
431C2×D4#23714yes—
432C2×D4C2×#6714yes—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—68no—
444C22×C4—62no—
445C22×C4—64no—
446C22×C4—68no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—124no—
452C22×C4—121no—
453C22×C4—1232no—
454C22×C4—124no—
455C22×C4—124no—
456C22×C4—124no—
457C22×C4—122no—
458C22×C4—122no—
459C22×C4D3×D411yescharacteristic
460C22×C4#23712yes—
461C22×C4C2×#6712yes—
462C22×C4C23×D311yescharacteristic
463SD16—128no—
464D8—48no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—62no—
469C2×C8—62no—
470C2×C8—62no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—34no—
474C4⋊C4—64no—
475C4⋊C4—62no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—122no—
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—28no—
498C22×C3—432no—
499C22×C3—44no—
500C22×C3—44no—
501C22×C3—44no—
502C22×C3—44no—
503C22×C3—44no—
504C22×C3—44no—
505C22×C3—81no—
506C22×C3—81no—
507C22×C3—81no—
508C22×C3—81no—
509C22×C3—81no—
510C22×C3—81no—
511C22×C3—81no—
512C22×C3#46314yescomplemented
513C22×C3C2×#12011yescharacteristic
514C22×C3C2×#12111yescharacteristic
515C22×C3#52511yescharacteristic
516C22×C3C2×#13511yescharacteristic
517C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
518C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
519C22×C3#57511yescharacteristic
520C3×C4—22no—
521C3×C4—81no—
522C3×C4C2×#12112yes—
523C3×C4#52512yes—
524Dic3—22no—
525Dic3—22no—
526Dic3—22no—
527Dic3—41no—
528Dic3—44no—
529Dic3—81no—
530Dic3—81no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—216no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—48no—
544C23—48no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23#91112yes—
557C23#91212yes—
558C23#93912yes—
559C23C2×D3×D411yescharacteristic
560Q8—122no—
561Q8—122no—
562D4—216no—
563D4—24no—
564D4—24no—
565D4—24no—
566D4—42no—
567D4—42no—
568D4—84no—
569D4#93918yes—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—32no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—68no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—64no—
591C2×C4—124no—
592C2×C4—1216no—
593C2×C4—1216no—
594C2×C4—128no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4#93912yes—
604C2×C4C2×D3×D412yes—
605C2×C4C2×#23712yes—
606C2×C4C2×#23711yescharacteristic
607C2×C4C22×#6711yescharacteristic
608C8—42no—
609C8—122no—
610C2×C3—24no—
611C2×C3—48no—
612C2×C3—161no—
613C2×C3#284711yescharacteristic
614C2×C3#300211yescharacteristic
615C2×C3#300311yescharacteristic
616C2×C3C2×#46311yescharacteristic
617C2×C3#303611yescharacteristic
618C2×C3#310511yescharacteristic
619C2×C3#311411yescharacteristic
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—28no—
627C22—432no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22#515314yescomplemented
642C22#558611yescharacteristic
643C22D3×#12011yescharacteristic
644C22C2×#91111yescharacteristic
645C22C2×#91211yescharacteristic
646C22D3×AGL(1,ℤ/8)11yescharacteristic
647C22#577311yescharacteristic
648C22C2×#93911yescharacteristic
649C4—22no—
650C4—62no—
651C4—62no—
652C4—62no—
653C4—81no—
654C4—121no—
655C4—124no—
656C4—241no—
657C4—241no—
658C4#558612yes—
659C4C2×#93912yes—
660C3#3290811yescharacteristic, complemented, Sylow
661C2—24no—
662C2—48no—
663C2—161no—
664C2#7998911yescharacteristic
665C2#7999111yescharacteristic
666C2#8000111yescharacteristic
667C2#8000211yescharacteristic
668C2#8260111yescharacteristic
669C2C2×#515311yescharacteristic
670C2#8393111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#447C2
socleC23×C3C22×D4
hypercenterC2×#447D3
hypocenterC3#32908
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
1. C22×D4×Dic3C2semidirect
3. C2×#5218C2semidirect
4. C2×#5153C2semidirect
5. #82698C2semidirect
6. #72037C2semidirect
7. #71677C2semidirect
8. #70171C2semidirect
9. #70154C2semidirect
10. #69930C2semidirect
11. #69908C2semidirect
18. C2×#936C22semidirect
31. #5221C22semidirect
36. C2×#832C22semidirect
39. C2×#816C22semidirect
40. C2×Dic3⋊C8C22semidirect
41. #4977C22semidirect
42. #4973C22semidirect
45. #4551C22semidirect
46. #4550C22semidirect
47. #4529C22semidirect
100. D4×Dic3D4semidirect
512. C22×C3#463semidirect
641. C22#5153semidirect
660. C3#32908semidirect

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×#4472
C2×#52182
C2×#51532
#826982
#720372
#716772
#701712
#701542
#699302
#699082
#329083