Finite Group Databasebeta

← #10828136

Subgroups — #10828136

Order 768.

1802 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5609C211yesmaximal, characteristic
2C22×#821C211yesmaximal, characteristic, complemented
3C2×#5034C211yesmaximal, characteristic, complemented
4#74028C211yesmaximal, characteristic, complemented
5C3×#1893C211yesmaximal, characteristic, complemented
6#71975C211yesmaximal, characteristic, complemented
7#70198C212yesmaximal, complemented
8#70177C212yesmaximal, complemented
9#69521C211yesmaximal, characteristic, complemented
10#68895C212yesmaximal, complemented
11#68823C212yesmaximal, complemented
12#24033—31nomaximal, Sylow
13C22×D3×D4—21no—
14C22×D3×D4C2211yescharacteristic
15C23×D12C2211yescharacteristic
16C23×C3×C8C2211yescharacteristic, complemented
17C2×C3×#121C2211yescharacteristic
18C2×#940C2211yescharacteristic
19C2×#937—21no—
20C2×C4×#67C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic, complemented
22#5609—24no—
23#5609—24no—
24C2×#895C2211yescharacteristic
25C2×#893C2211yescharacteristic
26C2×#884—21no—
27C2×#882—21no—
28C2×C3×#102C2212yescomplemented
29C3×#380C2211yescharacteristic, complemented
30C2×#832C2212yescomplemented
31C2×#830C2211yescharacteristic, complemented
32C2×#821—24no—
33C2×#821C2212yes—
34C2×#821C2212yescomplemented
35C2×(C2×D3)⋊C8C2212yescomplemented
36#5034—24no—
37#4987C2212yescomplemented
38#4984C2212yescomplemented
39C2×#809C2212yes—
40C2×#809C2212yescomplemented
41C2×#808C2212yescomplemented
42#4958—21no—
43#4950—21no—
44#4677—21no—
45#4659—21no—
46C3×#340C2212yescomplemented
47#4550C2212yescomplemented
48C2×D42—31no—
49C2×#481—31no—
50C22×#102—31no—
51#2487—31no—
52#2041—32no—
53#2034—32no—
54#1893D311yescharacteristic, complemented
55#1869—31no—
56#1864—32no—
57#1786—32no—
58#1786—31no—
59C24×D3—21no—
60C24×D3—21no—
61C22×C3×D4—21no—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C22×#67—21no—
65C22×#67—21no—
66C22×#67—21no—
67C22×#67D411yescharacteristic
68C23×Dic3—21no—
69C2×D3×D4—28no—
70C2×D3×D4—22no—
71C2×D3×D4—22no—
72C2×D3×D4—24no—
73C2×D3×D4—42no—
74C2×D3×D4—44no—
75C2×D3×D4—44no—
76C22×D12—24no—
77C22×D12—21no—
78C22×D12—21no—
79C22×D12—22no—
80C22×D12—42no—
81C22×D12D414yes—
82C22×D12C2311yescharacteristic
83C22×D12C2311yescharacteristic
84C22×C4×D3—21no—
85C22×C4×D3C2311yescharacteristic
86C22×C3×C8—22no—
87C22×C3×C8—42no—
88C22×C3×C8C2312yes—
89C3×#121—22no—
90C3×#121D414yes—
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×#217D411yescharacteristic
96#940—22no—
97#940—22no—
98#937—28no—
99C4×#67—22no—
100C4×#67—22no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—22no—
104C2×(C2×D3)⋊C4D411yescharacteristic
105C2×C4⋊Dic3—22no—
106C2×C4⋊Dic3—22no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—22no—
109C2×C4⋊Dic3C2312yes—
110C2×Dic3⋊C4D411yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2312yes—
113#895—22no—
114#895—22no—
115#893—22no—
116#893—22no—
117#884—28no—
118#882—28no—
119C3×#102—42no—
120C3×#95—21no—
121#832—24no—
122C3⋊#95—22no—
123C3⋊#95—22no—
124#830—22no—
125#821—24no—
126#821—24no—
127(C2×D3)⋊C8—42no—
128#809—24no—
129#809—42no—
130#808—24no—
131C23×D4—31no—
132C23×D4—31no—
133C23×D4—61no—
134C23×C8C2×D311yescharacteristic, complemented
135D42—64no—
136D42—64no—
137C2×#127—31no—
138C2×#121—31no—
139C2×#121—61no—
140C2×#121C2×D311yescharacteristic
141C2×#120—61no—
142C2×#120—61no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C22×C4⋊C4—31no—
146#481—64no—
147C2×#106—32no—
148C2×#102—32no—
149C2×#102—32no—
150C2×#102—32no—
151C2×#102—32no—
152C2×#102—32no—
153C2×#102—64no—
154C2×#102C2×D312yescomplemented
155C2×C22⋊C8—31no—
156C2×C22⋊C8—32no—
157#400—61no—
158#399—61no—
159#389—32no—
160#386—61no—
161#385—61no—
162#380—32no—
163#380C2×D311yescharacteristic, complemented
164#340C2×D312yescomplemented
165#336—32no—
166C24×C3—21no—
167C24×C3C2×D411yescharacteristic
168C23×D3—21no—
169C23×D3—21no—
170C23×D3—24no—
171C23×D3—24no—
172C23×D3—44no—
173C23×D3—44no—
174C23×D3—42no—
175C23×D3—42no—
176C23×D3—42no—
177C23×D3—42no—
178C23×D3—82no—
179C23×D3—82no—
180C23×D3C2×D411yescharacteristic
181C2×C3×D4—24no—
182C2×C3×D4—21no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—42no—
186C2×C3×D4—42no—
187C2×C3×D4C2×D412yes—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2411yescharacteristic
196C2×#67—22no—
197C2×#67—22no—
198C2×#67—24no—
199C2×#67—24no—
200C2×#67—24no—
201C2×#67—44no—
202C2×#67—42no—
203C2×#67—42no—
204C2×#67—82no—
205C2×#67SD1614yes—
206C22×Dic3—22no—
207C22×Dic3—42no—
208C22×Dic3—82no—
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212D3×D4—48no—
213D3×D4—44no—
214D3×D4—432no—
215D3×D4—44no—
216C2×D12—21no—
217C2×D12—22no—
218C2×D12—22no—
219C2×D12—22no—
220C2×D12—28no—
221C2×D12—24no—
222C2×D12—24no—
223C2×D12—21no—
224C2×D12—44no—
225C2×D12—48no—
226C2×D12—44no—
227C2×D12—42no—
228C2×D12—42no—
229C2×D12C2×D412yes—
230C2×D12C2×D412yes—
231C2×D12#4114yes—
232C2×C4×D3—24no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—22no—
236C2×C4×D3—22no—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×C3×C8—22no—
240C2×C3×C8—22no—
241C2×C3×C8—22no—
242C2×C3×C8—44no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4C2×D412yes—
247C3×C22⋊C4—24no—
248C3×C22⋊C4—42no—
249#217—22no—
250(C2×D3)⋊C4—24no—
251(C2×D3)⋊C4—24no—
252(C2×D3)⋊C4—22no—
253(C2×D3)⋊C4—82no—
254C4⋊Dic3—22no—
255C4⋊Dic3—22no—
256C4⋊Dic3—42no—
257C4⋊Dic3—82no—
258Dic3⋊C4—22no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C2×C3⋊C8—22no—
262C2×C3⋊C8—22no—
263C2×C3⋊C8—22no—
264C25—61no—
265C25—61no—
266C22×D4—21no—
267C22×D4—31no—
268C22×D4—34no—
269C22×D4—31no—
270C22×D4—31no—
271C22×D4—64no—
272C22×D4—61no—
273C22×D4—61no—
274C22×D4—61no—
275C22×D4—61no—
276C22×D4—62no—
277C22×D4—64no—
278C22×D4—68no—
279C22×D4—62no—
280C22×D4—62no—
281C22×D4—61no—
282C22×D4—122no—
283C22×D4—122no—
284C22×D4—124no—
285C22×D4—124no—
286C22×D4C22×D311yescharacteristic
287C23×C4—31no—
288C23×C4—61no—
289C23×C4—61no—
290C23×C4C22×D311yescharacteristic
291C22×C8—22no—
292C22×C8—32no—
293C22×C8—42no—
294C22×C8C22×D312yes—
295#127—62no—
296#127—62no—
297#121—22no—
298#121—68no—
299#121—62no—
300#121—62no—
301#121D1214yes—
302#120—68no—
303#120—68no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—62no—
307C4×D4—62no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—32no—
310C2×C4⋊C4—62no—
311C2×C4⋊C4—62no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—62no—
314C2×C4⋊C4C22×D312yes—
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—31no—
319C2×C22⋊C4—31no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—62no—
323C2×C42—31no—
324#106—64no—
325#102—42no—
326#102—64no—
327#102—64no—
328#102—64no—
329#102—64no—
330#102—122no—
331C22⋊C8—62no—
332C22⋊C8—122no—
333#95—21no—
334#95—62no—
335#95—62no—
336C23×C3—22no—
337C23×C3—22no—
338C23×C3—22no—
339C23×C3—42no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3—42no—
343C23×C3#12012yes—
344C23×C3C2×SD1612yes—
345C23×C3AGL(1,ℤ/8)12yes—
346C23×C3C22×D411yescharacteristic
347C22×D3—21no—
348C22×D3—22no—
349C22×D3—22no—
350C22×D3—24no—
351C22×D3—21no—
352C22×D3—22no—
353C22×D3—22no—
354C22×D3—48no—
355C22×D3—48no—
356C22×D3—48no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—42no—
360C22×D3—42no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—42no—
364C22×D3—48no—
365C22×D3—88no—
366C22×D3—84no—
367C22×D3—84no—
368C22×D3—88no—
369C22×D3—88no—
370C22×D3—88no—
371C22×D3C2×SD1612yes—
372C3×D4—24no—
373C3×D4—48no—
374C3×D4—41no—
375C3×D4—41no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—24no—
380C2×C3×C4—22no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—42no—
384C2×C3×C4—42no—
385C2×C3×C4—42no—
386C2×C3×C4#12012yes—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C2×#4112yes—
390C2×C3×C4H(5,𝔽2)12yes—
391#67—28no—
392#67—48no—
393#67—48no—
394#67—84no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—42no—
402C2×Dic3—42no—
403C2×Dic3—42no—
404C2×Dic3—84no—
405C2×Dic3C2×SD1612yes—
406D12—22no—
407D12—22no—
408D12—22no—
409D12—24no—
410D12—48no—
411D12—48no—
412D12—42no—
413D12#12314yes—
414C4×D3—48no—
415C4×D3—41no—
416C4×D3—41no—
417C4×D3—42no—
418C3×C8—42no—
419C3⋊C8—42no—
420C24—21no—
421C24—31no—
422C24—64no—
423C24—61no—
424C24—61no—
425C24—64no—
426C24—124no—
427C24—122no—
428C24—124no—
429C24—122no—
430C24—122no—
431C24—122no—
432C24—242no—
433C24—242no—
434C24D3×D411yescharacteristic
435C2×D4—24no—
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—34no—
440C2×D4—32no—
441C2×D4—32no—
442C2×D4—34no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—68no—
446C2×D4—64no—
447C2×D4—62no—
448C2×D4—62no—
449C2×D4—64no—
450C2×D4—64no—
451C2×D4—64no—
452C2×D4—64no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—62no—
456C2×D4—62no—
457C2×D4—62no—
458C2×D4—124no—
459C2×D4—128no—
460C2×D4—124no—
461C2×D4—122no—
462C2×D4—122no—
463C2×D4—122no—
464C2×D4—124no—
465C2×D4—122no—
466C2×D4—128no—
467C2×D4—124no—
468C2×D4—1232no—
469C2×D4—124no—
470C2×D4—242no—
471C2×D4C2×D1212yes—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—64no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—62no—
483C22×C4—62no—
484C22×C4—62no—
485C22×C4—122no—
486C22×C4—122no—
487C22×C4—122no—
488C22×C4—242no—
489C22×C4C2×D1212yes—
490C22×C4D3×D411yescharacteristic
491C22×C4D3×D412yes—
492C22×C4C23×D311yescharacteristic
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—44no—
497C2×C8—62no—
498C2×C8—62no—
499C2×C8—62no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—62no—
504C4⋊C4—62no—
505C4⋊C4—62no—
506C4⋊C4—122no—
507C4⋊C4—242no—
508C4⋊C4C2×D1212yes—
509C22⋊C4—24no—
510C22⋊C4—42no—
511C22⋊C4—64no—
512C22⋊C4—64no—
513C22⋊C4—62no—
514C22⋊C4—62no—
515C22⋊C4—242no—
516C42—61no—
517C42—61no—
518C22×C3—22no—
519C22×C3—22no—
520C22×C3—24no—
521C22×C3—24no—
522C22×C3—42no—
523C22×C3—42no—
524C22×C3—42no—
525C22×C3—82no—
526C22×C3—82no—
527C22×C3#45014yes—
528C22×C3C2×#12011yescharacteristic
529C22×C3#54612yes—
530C22×C3C22×SD1611yescharacteristic
531C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
532C22×C3#57612yes—
533C2×D3—24no—
534C2×D3—22no—
535C2×D3—24no—
536C2×D3—41no—
537C2×D3—42no—
538C2×D3—41no—
539C2×D3—42no—
540C2×D3—44no—
541C2×D3—42no—
542C2×D3—44no—
543C2×D3—42no—
544C2×D3—42no—
545C2×D3—42no—
546C2×D3—816no—
547C2×D3—816no—
548C2×D3—82no—
549C2×D3—82no—
550C3×C4—22no—
551C3×C4—42no—
552C3×C4—82no—
553C3×C4C2×#12311yescharacteristic
554C3×C4#53411yescharacteristic
555C3×C4#54612yes—
556Dic3—22no—
557Dic3—41no—
558Dic3—41no—
559Dic3—82no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—32no—
564C23—41no—
565C23—41no—
566C23—42no—
567C23—42no—
568C23—62no—
569C23—61no—
570C23—61no—
571C23—62no—
572C23—62no—
573C23—64no—
574C23—62no—
575C23—128no—
576C23—128no—
577C23—128no—
578C23—121no—
579C23—121no—
580C23—122no—
581C23—121no—
582C23—121no—
583C23—122no—
584C23—128no—
585C23—122no—
586C23—244no—
587C23—248no—
588C23—248no—
589C23—244no—
590C23—248no—
591C23—248no—
592C23#88212yes—
593C23#91112yes—
594C23D3×SD1612yes—
595C23C2×D3×D411yescharacteristic
596D4—24no—
597D4—34no—
598D4—48no—
599D4—41no—
600D4—41no—
601D4—68no—
602D4—62no—
603D4—64no—
604D4—62no—
605D4—62no—
606D4—128no—
607D4—128no—
608D4—128no—
609D4—128no—
610D4—122no—
611D4—244no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—32no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—62no—
628C2×C4—62no—
629C2×C4—128no—
630C2×C4—121no—
631C2×C4—122no—
632C2×C4—121no—
633C2×C4—122no—
634C2×C4—122no—
635C2×C4—122no—
636C2×C4—244no—
637C2×C4#88212yes—
638C2×C4C22×D1211yescharacteristic
639C2×C4C2×D3×D411yescharacteristic
640C2×C4#100412yes—
641C2×C4D3×#4112yes—
642C8—42no—
643C8—122no—
644C2×C3—24no—
645C2×C3—82no—
646C2×C3#240511yescharacteristic
647C2×C3C2×#45011yescharacteristic
648C2×C3#300811yescharacteristic
649C2×C3#329512yes—
650C2×C3#331812yes—
651D3—42no—
652D3—44no—
653D3—82no—
654C22—24no—
655C22—24no—
656C22—22no—
657C22—22no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—64no—
662C22—62no—
663C22—64no—
664C22—82no—
665C22—82no—
666C22—124no—
667C22—122no—
668C22—124no—
669C22—122no—
670C22—121no—
671C22—121no—
672C22—122no—
673C22—122no—
674C22—122no—
675C22—122no—
676C22—2416no—
677C22—2416no—
678C22—242no—
679C22—242no—
680C22#477614yes—
681C22C2×#88211yescharacteristic
682C22#561012yes—
683C22C2×#91111yescharacteristic
684C22C2×D3×SD1611yescharacteristic
685C22#576912yes—
686C4—22no—
687C4—42no—
688C4—62no—
689C4—82no—
690C4—121no—
691C4—121no—
692C4—242no—
693C4#555412yes—
694C4#558911yescharacteristic
695C4D3×#12311yescharacteristic
696C3#2403311yescharacteristic, complemented, Sylow
697C2—24no—
698C2—82no—
699C2—122no—
700C2—124no—
701C2—242no—
702C2#7468611yescharacteristic
703C2C2×#477611yescharacteristic
704C2#7883411yescharacteristic
705C2#8068012yes—
706C2#8073212yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1893C2
socleC23×C3C22×D4
prime radicalC2×#5609C2
hypercenter#1893D3
hypocenterC3#24033
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
2. C22×#821C2semidirect
3. C2×#5034C2semidirect
4. #74028C2semidirect
5. C3×#1893C2semidirect
6. #71975C2semidirect
7. #70198C2semidirect
8. #70177C2semidirect
9. #69521C2semidirect
10. #68895C2semidirect
11. #68823C2semidirect
16. C23×C3×C8C22semidirect
21. C22×C4⋊Dic3C22semidirect
28. C2×C3×#102C22semidirect
29. C3×#380C22semidirect
30. C2×#832C22semidirect
31. C2×#830C22semidirect
34. C2×#821C22semidirect
35. C2×(C2×D3)⋊C8C22semidirect
37. #4987C22semidirect
38. #4984C22semidirect
40. C2×#809C22semidirect
41. C2×#808C22semidirect
46. C3×#340C22semidirect
47. #4550C22semidirect
54. #1893D3semidirect
134. C23×C8C2×D3semidirect
154. C2×#102C2×D3semidirect
163. #380C2×D3semidirect
164. #340C2×D3semidirect
696. C3#24033semidirect

Maximal subgroups

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

subgroupindex
C2×#56092
C22×#8212
C2×#50342
#740282
C3×#18932
#719752
#701982
#701772
#695212
#688952
#688232
#240333