Finite Group Databasebeta

← C2×#24553 #6440545

Subgroups — C2×#24553 #6440545

Order 512.

4128 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#481C211yesmaximal, characteristic, complemented
2C23×C22⋊C8C211yesmaximal, characteristic, complemented
3C2×#2847C211yesmaximal, characteristic
4C22×#380C211yesmaximal, characteristic, complemented
5#24553C2116yesmaximal, direct product factor
6C2×#2077C212yesmaximal, complemented
7C2×#2034C212yesmaximal, complemented
8C2×#1893C212yesmaximal, complemented
9C2×#1874C211yesmaximal, characteristic, complemented
10C2×#1853C211yesmaximal, characteristic, complemented
11C2×#1824C211yesmaximal, characteristic, complemented
12C2×#1807C212yesmaximal, complemented
13C25×C4C2211yescharacteristic
14C24×C8C2212yes—
15C22×#121C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#120—21no—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4—21no—
22C2×#481—28no—
23C2×#481C2218yescomplemented
24C22×#106C2212yescomplemented
25C22×#106C2212yescomplemented
26C22×#102C2212yescomplemented
27C22×#102C2212yescomplemented
28C22×C22⋊C8—28no—
29C22×C22⋊C8C2212yes—
30C22×C22⋊C8C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×C22⋊C8C2218yescomplemented
33#2847—24no—
34#2847C2218yes—
35C2×#386—21no—
36C2×#386—21no—
37C2×#386—21no—
38C2×#380—21no—
39C2×#380—21no—
40C2×#380—24no—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#380C2218yescomplemented
47C22×#95—21no—
48#2077C22116yescomplemented
49#2034C22116yescomplemented
50#1893C22116yescomplemented
51#1874C2218yescomplemented
52#1853C2218yescomplemented
53#1824C2218yescomplemented
54#1807C22116yescomplemented
55C2×#340C2212yescomplemented
56C2×#340C2212yescomplemented
57C2×#340C2212yescomplemented
58C2×#340C2212yescomplemented
59C26D411yescharacteristic
60C23×D4—21no—
61C23×D4—21no—
62C23×D4—21no—
63C23×D4C2311yescharacteristic
64C24×C4—24no—
65C24×C4—24no—
66C24×C4—28no—
67C24×C4—21no—
68C24×C4—21no—
69C24×C4—44no—
70C24×C4D414yes—
71C24×C4D411yescharacteristic
72C24×C4D411yescharacteristic
73C24×C4D411yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2314yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C23×C8—28no—
79C23×C8—28no—
80C23×C8—44no—
81C23×C8C2312yes—
82C23×C8C2318yes—
83C23×C8C2312yes—
84C2×#121—22no—
85C2×#121—22no—
86C2×#121—22no—
87C2×#121—24no—
88C2×#121—24no—
89C2×#121—28no—
90C2×#121—44no—
91C2×#121C2314yes—
92C2×#121C2314yes—
93C2×#121C2314yes—
94C2×#120—28no—
95C2×#120—48no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—22no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—44no—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C22⋊C4—22no—
111C22×C22⋊C4—22no—
112C22×C22⋊C4—28no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—48no—
120#481—232no—
121C2×#106—28no—
122C2×#106—44no—
123C2×#106C2318yescomplemented
124C2×#106C2318yescomplemented
125C2×#102—28no—
126C2×#102—44no—
127C2×#102C2318yescomplemented
128C2×#102C2318yescomplemented
129C2×C22⋊C8—24no—
130C2×C22⋊C8—28no—
131C2×C22⋊C8—232no—
132C2×C22⋊C8D418yescomplemented
133C2×C22⋊C8C2318yes—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136#386—28no—
137#386—28no—
138#386—28no—
139#380—22no—
140#380—22no—
141#380—22no—
142#380—24no—
143#380—24no—
144#380—28no—
145#380—28no—
146#380C2314yes—
147#380C2318yes—
148#380C2314yes—
149#380C2318yes—
150#380C2314yes—
151C2×#95—22no—
152C2×#95—22no—
153C2×#95—28no—
154C2×#95—21no—
155C2×#95—21no—
156C2×#95—21no—
157#340C2318yescomplemented
158#340C2318yescomplemented
159#340C2318yescomplemented
160#340C2318yescomplemented
161C25—24no—
162C25—24no—
163C25—24no—
164C25—24no—
165C25—21no—
166C25—42no—
167C25—42no—
168C25SD1618yes—
169C25C2×D411yescharacteristic
170C25C2×D414yes—
171C25C2×D411yescharacteristic
172C25C2×D411yescharacteristic
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—22no—
177C22×D4—21no—
178C22×D4—22no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4C2412yes—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—216no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—216no—
203C23×C4—24no—
204C23×C4—28no—
205C23×C4—22no—
206C23×C4—28no—
207C23×C4—216no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—48no—
219C23×C4—48no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—432no—
224C23×C4—432no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4C2×D412yes—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D412yes—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D414yes—
239C23×C4C2×D412yes—
240C23×C4C2×D411yescharacteristic
241C23×C4#4114yes—
242C23×C4#4118yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C22×C8—24no—
248C22×C8—24no—
249C22×C8—24no—
250C22×C8—216no—
251C22×C8—24no—
252C22×C8—24no—
253C22×C8—216no—
254C22×C8—48no—
255C22×C8—432no—
256C22×C8C2×D418yes—
257C22×C8C2414yes—
258C22×C8C2414yes—
259#121—28no—
260#121—28no—
261#121—216no—
262#121—48no—
263#121—48no—
264#121—48no—
265#121—48no—
266#120—432no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—48no—
289C2×C4⋊C4C2414yes—
290C2×C4⋊C4C2414yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4—432no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—48no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311#106—216no—
312#106—48no—
313#102—216no—
314#102—48no—
315C22⋊C8—28no—
316C22⋊C8—216no—
317C22⋊C8C2×D4116yescomplemented
318#95—28no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—24no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—28no—
332C24—22no—
333C24—28no—
334C24—22no—
335C24—22no—
336C24—28no—
337C24—22no—
338C24—28no—
339C24—22no—
340C24—22no—
341C24—232no—
342C24—232no—
343C24—22no—
344C24—44no—
345C24—416no—
346C24—44no—
347C24—416no—
348C24—416no—
349C24—416no—
350C24—416no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24#12114yes—
359C24#12314yes—
360C24C2×SD1614yes—
361C24C2×SD16116yes—
362C24C2×SD1614yes—
363C24C2×SD1614yes—
364C24#13514yes—
365C24AGL(1,ℤ/8)14yes—
366C24C22×D412yes—
367C24C22×D412yes—
368C24C22×D412yes—
369C24C22×D411yescharacteristic
370C2×D4—22no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—24no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—816no—
389C2×D4—816no—
390C2×D4—816no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—216no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—416no—
433C22×C4—464no—
434C22×C4—432no—
435C22×C4—432no—
436C22×C4—416no—
437C22×C4—416no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—42no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—48no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—816no—
462C22×C4—816no—
463C22×C4#12114yes—
464C22×C4#12118yes—
465C22×C4#12314yes—
466C22×C4#12318yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D414yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4118yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4114yes—
477C22×C4H(5,𝔽2)14yes—
478C22×C4C2511yescharacteristic
479C2×C8—216no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—432no—
486C2×C8—48no—
487C2×C8C22×D418yes—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C22⋊C4—48no—
504C22⋊C4—416no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—416no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—22no—
515C23—22no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—22no—
523C23—22no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—28no—
532C23—232no—
533C23—22no—
534C23—22no—
535C23—28no—
536C23—22no—
537C23—232no—
538C23—28no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—416no—
543C23—416no—
544C23—416no—
545C23—416no—
546C23—416no—
547C23—4128no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23#465116yes—
567C23#481116yescomplemented
568C23C2×#12114yes—
569C23C2×#12112yes—
570C23C2×#12314yes—
571C23C2×#12312yes—
572C23#53412yes—
573C23C22×SD1614yes—
574C23C22×SD1614yes—
575C23C22×SD1614yes—
576C23C22×SD1612yes—
577C23C2×#13514yes—
578C23C2×#13512yes—
579C23C2×AGL(1,ℤ/8)14yes—
580C23C2×AGL(1,ℤ/8)12yes—
581C23#57512yes—
582C23C23×D411yescharacteristic
583D4—44no—
584D4—44no—
585D4—816no—
586D4—816no—
587D4—816no—
588D4—84no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—216no—
596C2×C4—216no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—416no—
605C2×C4—416no—
606C2×C4—416no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—816no—
627C2×C4—816no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—88no—
631C2×C4—84no—
632C2×C4—88no—
633C2×C4C2×#12114yes—
634C2×C4C2×#12112yes—
635C2×C4C2×#12314yes—
636C2×C4C2×#12312yes—
637C2×C4#53412yes—
638C2×C4#54618yes—
639C2×C4#55118yes—
640C2×C4C23×D411yescharacteristic
641C2×C4C22×#4112yes—
642C2×C4C22×#4111yescharacteristic
643C2×C4C2×H(5,𝔽2)12yes—
644C8—28no—
645C8—44no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—232no—
659C22—232no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—464no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—416no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—168no—
677C22#241214yes—
678C22C2×#46518yes—
679C22C2×#46514yes—
680C22#305514yes—
681C22C2×#48114yes—
682C22C2×#48118yescomplemented
683C22#309114yes—
684C22#318714yes—
685C22#318814yes—
686C22C22×#12111yescharacteristic
687C22C22×#12311yescharacteristic
688C22C2×#53411yescharacteristic
689C22C23×SD1611yescharacteristic
690C22C22×#13511yescharacteristic
691C22C22×AGL(1,ℤ/8)11yescharacteristic
692C22C2×#57511yescharacteristic
693C4—24no—
694C4—24no—
695C4—42no—
696C4—82no—
697C4—84no—
698C4—82no—
699C4—84no—
700C4—82no—
701C4#241214yes—
702C4C2×#54612yes—
703C4C2×#55112yes—
704C2—28no—
705C2—28no—
706C2—44no—
707C2—162no—
708C2#2455318yesdirect product factor
709C2C2×#241211yescharacteristic
710C2C22×#46511yescharacteristic
711C2C2×#305511yescharacteristic
712C2C22×#48111yescharacteristic
713C2C2×#309111yescharacteristic
714C2C2×#318711yescharacteristic
715C2C2×#318811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#481C2semidirect
2. C23×C22⋊C8C2semidirect
4. C22×#380C2semidirect
5. #24553C2direct (N × H)
6. C2×#2077C2semidirect
7. C2×#2034C2semidirect
8. C2×#1893C2semidirect
9. C2×#1874C2semidirect
10. C2×#1853C2semidirect
11. C2×#1824C2semidirect
12. C2×#1807C2semidirect
23. C2×#481C22semidirect
24. C22×#106C22semidirect
25. C22×#106C22semidirect
26. C22×#102C22semidirect
27. C22×#102C22semidirect
32. C22×C22⋊C8C22semidirect
46. C2×#380C22semidirect
48. #2077C22semidirect
49. #2034C22semidirect
50. #1893C22semidirect
51. #1874C22semidirect
52. #1853C22semidirect
53. #1824C22semidirect
54. #1807C22semidirect
55. C2×#340C22semidirect
56. C2×#340C22semidirect
57. C2×#340C22semidirect
58. C2×#340C22semidirect
123. C2×#106C23semidirect
124. C2×#106C23semidirect
127. C2×#102C23semidirect
128. C2×#102C23semidirect
132. C2×C22⋊C8D4semidirect
157. #340C23semidirect
158. #340C23semidirect
159. #340C23semidirect
160. #340C23semidirect
317. C22⋊C8C2×D4semidirect
567. C23#481semidirect
682. C22C2×#481semidirect
708. C2#24553direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4812
C23×C22⋊C82
C2×#28472
C22×#3802
#245532
C2×#20772
C2×#20342
C2×#18932
C2×#18742
C2×#18532
C2×#18242
C2×#18072