Finite Group Databasebeta

← #10830670

Subgroups — #10830670

Order 768.

1940 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
1C2×C3×D42C211yesmaximal, characteristic, complemented
2C2×#5217C212yesmaximal, complemented
3C2×#5072C211yesmaximal, characteristic, complemented
4#82415C211yesmaximal, characteristic, complemented
5C2×#5014C211yesmaximal, characteristic, complemented
6#72128C212yesmaximal, complemented
7#71765C211yesmaximal, characteristic, complemented
8#71684C212yesmaximal, complemented
9#71680C212yesmaximal, complemented
10#71615C211yesmaximal, characteristic, complemented
11#71605C211yesmaximal, characteristic, complemented
12#24248—31nomaximal, Sylow
13C23×C3×D4—21no—
14C23×C3×D4C2212yes—
15C3×D42—24no—
16C3×D42—24no—
17C2×C3×#127C2211yescharacteristic
18C2×C3×#121—21no—
19C2×C3×#121C2212yes—
20C2×C3×#120—22no—
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic
23C22×#213C2211yescharacteristic
24C22×#213C2212yescomplemented
25C2×#930C2212yescomplemented
26C2×C4×D12C2211yescharacteristic, complemented
27#5223—21no—
28#5217—28no—
29C2×#832C2212yes—
30C2×#832C2211yescharacteristic
31C2×#832C2212yescomplemented
32C2×#832C2212yescomplemented
33C2×#832C2212yescomplemented
34#5212C2212yescomplemented
35C2×#830C2212yescomplemented
36C2×#830C2212yescomplemented
37#5072—24no—
38#5014—24no—
39#4992—21no—
40C2×#809C2211yescharacteristic, complemented
41C2×#808C2211yescharacteristic, complemented
42#4956—22no—
43#4936C2211yescharacteristic, complemented
44C2×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
45C2×#803C2211yescharacteristic, complemented
46#4529C2212yescomplemented
47C2×D42D311yescharacteristic, complemented
48C2×#459—31no—
49C2×#447—32no—
50C2×#442—31no—
51#2535—31no—
52#2034—32no—
53#1999—32no—
54#1961—31no—
55#1955—31no—
56#1804—31no—
57#1786—32no—
58C25×C3—22no—
59C22×C3×D4—28no—
60C22×C3×D4—22no—
61C22×C3×D4—24no—
62C22×C3×D4—24no—
63C22×C3×D4—24no—
64C22×C3×D4—24no—
65C22×C3×D4—21no—
66C22×C3×D4—22no—
67C22×C3×D4—21no—
68C22×C3×D4—42no—
69C22×C3×D4—42no—
70C22×C3×D4—48no—
71C22×C3×D4D411yescharacteristic
72C22×C3×D4C2312yes—
73C22×C3×D4C2311yescharacteristic
74C23×C3×C4—21no—
75C23×C3×C4C2312yes—
76C22×#67—22no—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C3×#127—22no—
80C3×#127D414yescomplemented
81C3×#121—28no—
82C3×#121—24no—
83C3×#121—24no—
84C3×#120—216no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4D414yescomplemented
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4D412yes—
94C2×C3×C42C2311yescharacteristic
95C2×#217—22no—
96C2×#213—22no—
97C2×#213—22no—
98C2×#213—24no—
99C2×#213—24no—
100C2×#213—42no—
101C2×#213D414yescomplemented
102#930—24no—
103#930—42no—
104C2×(C2×D3)⋊C4—22no—
105C2×(C2×D3)⋊C4—21no—
106C2×C4⋊Dic3C2312yes—
107C2×C4⋊Dic3C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2312yes—
110C22×C3⋊C8C2311yescharacteristic
111C4×D12—22no—
112C4×D12—22no—
113#832—24no—
114#832—22no—
115#832—24no—
116#832—24no—
117#832—42no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—22no—
121#830—24no—
122#830—42no—
123#809—22no—
124#808—22no—
125C4⋊(C3⋊C8)—22no—
126#803—22no—
127C23×D4—21no—
128C23×D4C2×D312yes—
129C22×D8—31no—
130C22×D8—32no—
131D42—24no—
132D42—24no—
133C2×#127C2×D311yescharacteristic
134C2×#121—21no—
135C2×#121—32no—
136C2×#121C2×D312yes—
137C2×#120—22no—
138C2×C4×D4—31no—
139C2×C4×D4C2×D311yescharacteristic
140C2×C4×D4C2×D311yescharacteristic
141#459—64no—
142#447—68no—
143#442—64no—
144C2×#106—31no—
145C2×C4⋊C8—31no—
146C2×#102—32no—
147C2×#102—31no—
148C2×#102—32no—
149C2×#102—32no—
150C2×#102—31no—
151C2×#102—32no—
152C2×C22⋊C8—32no—
153C2×C22⋊C8—32no—
154C2×#97—31no—
155#397—61no—
156#392—61no—
157#386—62no—
158#382—31no—
159#380—32no—
160#340—32no—
161C24×C3—21no—
162C24×C3—28no—
163C24×C3—22no—
164C24×C3—44no—
165C24×C3—44no—
166C24×C3—44no—
167C24×C3—44no—
168C24×C3—82no—
169C24×C3—82no—
170C24×C3C2×D412yes—
171C23×D3—21no—
172C2×C3×D4—24no—
173C2×C3×D4—28no—
174C2×C3×D4—28no—
175C2×C3×D4—22no—
176C2×C3×D4—24no—
177C2×C3×D4—24no—
178C2×C3×D4—21no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—44no—
187C2×C3×D4—44no—
188C2×C3×D4—44no—
189C2×C3×D4—48no—
190C2×C3×D4—48no—
191C2×C3×D4—432no—
192C2×C3×D4—48no—
193C2×C3×D4—44no—
194C2×C3×D4—44no—
195C2×C3×D4—48no—
196C2×C3×D4—82no—
197C2×C3×D4C2×D412yes—
198C2×C3×D4C2×D414yes—
199C2×C3×D4C2×D412yes—
200C2×C3×D4#4112yes—
201C22×C3×C4—21no—
202C22×C3×C4—24no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—44no—
210C22×C3×C4C2×D412yes—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C2×#67—44no—
215C2×#67—44no—
216C2×#67—82no—
217C22×Dic3—22no—
218C22×Dic3—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12C2×D412yescomplemented
225C2×C4×D3—42no—
226C2×C4×D3—82no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4C2×D412yes—
230C3×C22⋊C4—28no—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—24no—
233C3×C42C2×D412yes—
234C3×C42C2×D412yes—
235#217—82no—
236#213—28no—
237#213—48no—
238(C2×D3)⋊C4—44no—
239(C2×D3)⋊C4—42no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—22no—
243C4⋊Dic3—22no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8C2×D412yescomplemented
253C25—22no—
254C22×D4—28no—
255C22×D4—22no—
256C22×D4—24no—
257C22×D4—24no—
258C22×D4—24no—
259C22×D4—24no—
260C22×D4—22no—
261C22×D4—21no—
262C22×D4—21no—
263C22×D4—31no—
264C22×D4—42no—
265C22×D4—42no—
266C22×D4—48no—
267C22×D4—62no—
268C22×D4#6711yescharacteristic
269C22×D4C22×D312yes—
270C22×D4C22×D311yescharacteristic
271C23×C4—21no—
272C23×C4—61no—
273C23×C4C22×D312yes—
274C2×D8—34no—
275C2×D8—62no—
276C2×D8—62no—
277C2×D8—64no—
278C2×D8—64no—
279C2×D8—122no—
280C22×C8—31no—
281C22×C8—32no—
282C22×C8—31no—
283#127—22no—
284#127#6714yescomplemented
285#121—28no—
286#121—24no—
287#121—24no—
288#121—64no—
289#121—122no—
290#120—216no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—62no—
295C4×D4—62no—
296C4×D4#6714yescomplemented
297C2×C4⋊C4—31no—
298C2×C4⋊C4—32no—
299C2×C4⋊C4#6711yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—62no—
305C2×C22⋊C4—62no—
306C2×C22⋊C4#6712yes—
307C2×C42C22×D311yescharacteristic
308#106—62no—
309C4⋊C8—62no—
310#102—64no—
311#102—64no—
312#102—62no—
313#102—64no—
314#102—62no—
315#102—122no—
316C22⋊C8—64no—
317C22⋊C8—122no—
318#97—62no—
319#95—62no—
320#95—61no—
321#95—61no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—24no—
332C23×C3—416no—
333C23×C3—416no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3—42no—
340C23×C3—42no—
341C23×C3—42no—
342C23×C3—88no—
343C23×C3—88no—
344C23×C3—88no—
345C23×C3—88no—
346C23×C3—88no—
347C23×C3AGL(1,ℤ/8)14yes—
348C23×C3C22×D411yescharacteristic
349C22×D3—22no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—42no—
353C22×D3—42no—
354C3×D4—28no—
355C3×D4—22no—
356C3×D4—22no—
357C3×D4—24no—
358C3×D4—22no—
359C3×D4—24no—
360C3×D4—48no—
361C3×D4—416no—
362C3×D4—416no—
363C3×D4—42no—
364C3×D4—42no—
365C3×D4—84no—
366C3×D4#12114yes—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—48no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—42no—
379C2×C3×C4#12012yes—
380C2×C3×C4#12012yes—
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4H(5,𝔽2)11yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387#67—88no—
388C2×Dic3—41no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393D12—22no—
394D12—22no—
395D12—41no—
396D12—41no—
397C4×D3—84no—
398C3⋊C8—22no—
399C3⋊C8—41no—
400C3⋊C8—42no—
401C24—21no—
402C24—28no—
403C24—22no—
404C24—44no—
405C24—44no—
406C24—44no—
407C24—44no—
408C24—61no—
409C24—82no—
410C24—82no—
411C24C2×#6712yes—
412C2×D4—24no—
413C2×D4—28no—
414C2×D4—28no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—32no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—48no—
431C2×D4—48no—
432C2×D4—432no—
433C2×D4—48no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—48no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—82no—
443C2×D4—124no—
444C2×D4—124no—
445C2×D4—242no—
446C2×D4D3×D412yes—
447C2×D4#23712yes—
448C2×D4C2×#6714yes—
449C2×D4C2×#6712yes—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—44no—
459C22×C4—62no—
460C22×C4—61no—
461C22×C4—122no—
462C22×C4—242no—
463C22×C4C2×#6712yes—
464C22×C4C2×#6711yescharacteristic
465C22×C4C2×#6711yescharacteristic
466C22×C4C23×D311yescharacteristic
467D8—68no—
468D8—128no—
469C2×C8—32no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—62no—
473C2×C8—61no—
474C2×C8—62no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—62no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—62no—
481C4⋊C4—62no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4C2×#6712yes—
485C22⋊C4—28no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—124no—
489C22⋊C4—122no—
490C22⋊C4—242no—
491C42C2×#6712yes—
492C42C2×#6712yes—
493C22×C3—24no—
494C22×C3—24no—
495C22×C3—24no—
496C22×C3—24no—
497C22×C3—42no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—44no—
503C22×C3—42no—
504C22×C3—44no—
505C22×C3—44no—
506C22×C3—44no—
507C22×C3—44no—
508C22×C3—832no—
509C22×C3—82no—
510C22×C3—82no—
511C22×C3—82no—
512C22×C3—82no—
513C22×C3C2×#12011yescharacteristic
514C22×C3D4211yescharacteristic
515C22×C3#54611yescharacteristic
516C22×C3C2×AGL(1,ℤ/8)12yes—
517C22×C3#57612yes—
518C2×D3—42no—
519C2×D3—44no—
520C2×D3—81no—
521C2×D3—81no—
522C2×D3—82no—
523C3×C4—22no—
524C3×C4—22no—
525C3×C4—42no—
526C3×C4—42no—
527C3×C4C2×#12111yescharacteristic
528C3×C4#53511yescharacteristic
529C3×C4D4211yescharacteristic
530C3×C4#54611yescharacteristic
531Dic3—81no—
532Dic3—82no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—416no—
544C23—416no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—62no—
554C23—88no—
555C23—88no—
556C23—88no—
557C23—88no—
558C23—88no—
559C23—122no—
560C23—121no—
561C23—122no—
562C23—121no—
563C23#93214yes—
564C23C22×#6711yescharacteristic
565D4—28no—
566D4—22no—
567D4—24no—
568D4—22no—
569D4—22no—
570D4—24no—
571D4—48no—
572D4—416no—
573D4—416no—
574D4—42no—
575D4—42no—
576D4—62no—
577D4—62no—
578D4—84no—
579D4—121no—
580D4—121no—
581D4—248no—
582D4#93814yes—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—48no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4—121no—
600C2×C4—244no—
601C2×C4#95312yes—
602C2×C4#95312yes—
603C2×C4C2×D3×D411yescharacteristic
604C2×C4C2×#23711yescharacteristic
605C2×C4#100411yescharacteristic
606C2×C4#100411yescharacteristic
607C2×C4C22×#6711yescharacteristic
608C2×C4C22×#6711yescharacteristic
609C8—62no—
610C8—122no—
611C8—121no—
612C2×C3—44no—
613C2×C3—44no—
614C2×C3—84no—
615C2×C3#240811yescharacteristic
616C2×C3#300311yescharacteristic
617C2×C3#300811yescharacteristic
618C2×C3#326412yes—
619C2×C3#331812yes—
620D3—82no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—42no—
626C22—44no—
627C22—42no—
628C22—44no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—832no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—124no—
642C22—122no—
643C22—242no—
644C22—241no—
645C22—241no—
646C22C2×#93212yes—
647C22D4×#6711yescharacteristic
648C22#580611yescharacteristic
649C22#583512yes—
650C22C2×#95311yescharacteristic
651C4—22no—
652C4—22no—
653C4—42no—
654C4—42no—
655C4—242no—
656C4—241no—
657C4C2×#93811yescharacteristic
658C4D4×#6711yescharacteristic
659C4#580511yescharacteristic
660C4#580611yescharacteristic
661C3#2424811yescharacteristic, complemented, Sylow
662C2—44no—
663C2—44no—
664C2—84no—
665C2—242no—
666C2#8348512yes—
667C2#8348812yes—
668C2#8403811yescharacteristic
669C2#8405311yescharacteristic
670C2#8410111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×D42C2
socleC23×C3C22×D4
hypercenterC2×D42D3
hypocenterC3#24248
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D42

Complements

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

NH ≅ G/Ntype
1. C2×C3×D42C2semidirect
2. C2×#5217C2semidirect
3. C2×#5072C2semidirect
4. #82415C2semidirect
5. C2×#5014C2semidirect
6. #72128C2semidirect
7. #71765C2semidirect
8. #71684C2semidirect
9. #71680C2semidirect
10. #71615C2semidirect
11. #71605C2semidirect
24. C22×#213C22semidirect
25. C2×#930C22semidirect
26. C2×C4×D12C22semidirect
31. C2×#832C22semidirect
32. C2×#832C22semidirect
33. C2×#832C22semidirect
34. #5212C22semidirect
35. C2×#830C22semidirect
36. C2×#830C22semidirect
40. C2×#809C22semidirect
41. C2×#808C22semidirect
43. #4936C22semidirect
44. C2×C4⋊(C3⋊C8)C22semidirect
45. C2×#803C22semidirect
46. #4529C22semidirect
47. C2×D42D3semidirect
80. C3×#127D4semidirect
88. C3×C4×D4D4semidirect
101. C2×#213D4semidirect
224. C2×D12C2×D4semidirect
252. C2×C3⋊C8C2×D4semidirect
284. #127#67semidirect
296. C4×D4#67semidirect
661. C3#24248semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D422
C2×#52172
C2×#50722
#824152
C2×#50142
#721282
#717652
#716842
#716802
#716152
#716052
#242483