Finite Group Databasebeta

← C2×#24556 #6440549

Subgroups — C2×#24556 #6440549

Order 512.

3750 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic, complemented
2C22×#407C211yesmaximal, characteristic
3C2×#2848C211yesmaximal, characteristic
4C22×#380C211yesmaximal, characteristic
5#24556C2116yesmaximal, direct product factor
6C2×#2075C212yesmaximal, complemented
7C2×#2035C212yesmaximal, complemented
8C2×#1894C212yesmaximal, complemented
9C2×#1872C211yesmaximal, characteristic, complemented
10C2×#1853C211yesmaximal, characteristic, complemented
11C2×#1824C211yesmaximal, characteristic, complemented
12C2×#1809C212yesmaximal, complemented
13C25×C4C2211yescharacteristic
14C23×M16C2212yes—
15C22×C22⋊Q8C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4—21no—
21C2×#484—28no—
22C2×#484C2218yescomplemented
23C22×#107C2212yescomplemented
24C22×#106C2212yescomplemented
25C22×#103C2212yescomplemented
26C22×#103C2212yescomplemented
27C2×#407—28no—
28C2×#407C2218yes—
29C22×C22⋊C8C2212yes—
30C22×C22⋊C8C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32#2848—24no—
33#2848C2218yes—
34C2×#385—21no—
35C2×#385—21no—
36C2×#385—21no—
37C2×#380—24no—
38C2×#380—21no—
39C2×#380—21no—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2218yes—
46C2×#379—21no—
47C22×#95—21no—
48#2075C22116yescomplemented
49#2035C22116yescomplemented
50#1894C22116yescomplemented
51#1872C2218yescomplemented
52#1853C2218yescomplemented
53#1824C2218yescomplemented
54#1809C22116yescomplemented
55C2×#340C2212yescomplemented
56C2×#340C2212yescomplemented
57C2×#340C2212yescomplemented
58C2×#340C2212yescomplemented
59C26D411yescharacteristic
60C23×Q8C2311yescharacteristic
61C24×C4—24no—
62C24×C4—24no—
63C24×C4—28no—
64C24×C4—21no—
65C24×C4—21no—
66C24×C4—44no—
67C24×C4D414yes—
68C24×C4D411yescharacteristic
69C24×C4D411yescharacteristic
70C24×C4D411yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2314yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C22×M16—28no—
76C22×M16—28no—
77C22×M16—44no—
78C22×M16C2318yes—
79C23×C8C2312yes—
80C23×C8C2312yes—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—44no—
88C2×C22⋊Q8C2314yes—
89C2×C22⋊Q8C2314yes—
90C2×C22⋊Q8C2314yes—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4—28no—
109C22×C22⋊C4—22no—
110C22×C22⋊C4—22no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—48no—
118#484—232no—
119C2×#107—28no—
120C2×#107C2318yescomplemented
121C2×#106—44no—
122C2×#106C2318yescomplemented
123C2×#103—28no—
124C2×#103—44no—
125C2×#103C2318yescomplemented
126C2×#103C2318yescomplemented
127#407—232no—
128C2×C22⋊C8—24no—
129C2×C22⋊C8—28no—
130C2×C22⋊C8D418yes—
131C2×C22⋊C8C2318yes—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134#385—28no—
135#385—28no—
136#385—28no—
137#380—22no—
138#380—22no—
139#380—22no—
140#380—24no—
141#380—28no—
142#380—28no—
143#380—24no—
144#380C2314yes—
145#380C2318yes—
146#380C2314yes—
147#380C2318yes—
148#380C2314yes—
149#379—28no—
150C2×#95—28no—
151C2×#95—22no—
152C2×#95—22no—
153C2×#95—21no—
154C2×#95—21no—
155C2×#95—21no—
156#340C2318yescomplemented
157#340C2318yescomplemented
158#340C2318yescomplemented
159#340C2318yescomplemented
160C25—24no—
161C25—24no—
162C25—24no—
163C25—24no—
164C25—24no—
165C25—42no—
166C25—42no—
167C25C2×D411yescharacteristic
168C25C2×D411yescharacteristic
169C25C2×D414yes—
170C25C2×D411yescharacteristic
171C22×Q8—21no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8C2412yes—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—216no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—216no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—28no—
192C23×C4—28no—
193C23×C4—216no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—48no—
206C23×C4—48no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—432no—
211C23×C4—432no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4C2×D412yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D412yes—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D414yes—
226C23×C4C2×D412yes—
227C23×C4C2×D411yescharacteristic
228C23×C4#4114yes—
229C23×C4#4118yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C2×M16—216no—
235C2×M16—216no—
236C2×M16—48no—
237C2×M16—432no—
238C22×C8—24no—
239C22×C8—24no—
240C22×C8—24no—
241C22×C8—24no—
242C22×C8—24no—
243C22×C8C2×D418yes—
244C22×C8C2414yes—
245C22×C8C2414yes—
246C22⋊Q8—28no—
247C22⋊Q8—28no—
248C22⋊Q8—216no—
249C22⋊Q8—48no—
250C22⋊Q8—48no—
251C22⋊Q8—48no—
252C22⋊Q8—48no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—48no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4C2414yes—
282C2×C4⋊C4C2414yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—432no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4—44no—
303#107—216no—
304#106—48no—
305#103—216no—
306#103—48no—
307C22⋊C8—28no—
308C22⋊C8—216no—
309C22⋊C8C2×D4116yes—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—24no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—232no—
320C24—22no—
321C24—22no—
322C24—28no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—28no—
327C24—22no—
328C24—28no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—28no—
336C24—28no—
337C24—44no—
338C24—44no—
339C24—416no—
340C24—416no—
341C24—416no—
342C24—416no—
343C24—416no—
344C24—416no—
345C24#12114yes—
346C24#12314yes—
347C24AGL(1,ℤ/8)14yes—
348C24#13714yes—
349C24#13714yes—
350C24C22×D412yes—
351C24C22×D412yes—
352C24C22×D412yes—
353C24C22×D411yescharacteristic
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—24no—
357C2×Q8—24no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—48no—
362C2×Q8—48no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—216no—
372C22×C4—216no—
373C22×C4—216no—
374C22×C4—216no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—416no—
406C22×C4—464no—
407C22×C4—432no—
408C22×C4—432no—
409C22×C4—416no—
410C22×C4—416no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—48no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—816no—
438C22×C4—816no—
439C22×C4#12114yes—
440C22×C4#12118yes—
441C22×C4#12314yes—
442C22×C4#12318yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4118yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)14yes—
454C22×C4C2511yescharacteristic
455M16—432no—
456C2×C8—216no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—48no—
463C2×C8C22×D418yes—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—48no—
478C4⋊C4—42no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C22⋊C4—48no—
483C22⋊C4—416no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—416no—
489C22⋊C4—48no—
490C23—28no—
491C23—28no—
492C23—22no—
493C23—28no—
494C23—22no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—22no—
499C23—22no—
500C23—232no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—28no—
508C23—22no—
509C23—22no—
510C23—28no—
511C23—28no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—416no—
520C23—416no—
521C23—416no—
522C23—416no—
523C23—416no—
524C23—4128no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—416no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23#470116yes—
535C23C2×#12114yes—
536C23C2×#12112yes—
537C23C2×#12314yes—
538C23C2×#12312yes—
539C23#53412yes—
540C23C2×AGL(1,ℤ/8)14yes—
541C23C2×AGL(1,ℤ/8)12yes—
542C23C2×#13714yes—
543C23C2×#13714yes—
544C23C2×#13712yes—
545C23C2×#13712yes—
546C23#57512yes—
547C23C23×D411yescharacteristic
548Q8—44no—
549Q8—44no—
550Q8—84no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—216no—
558C2×C4—216no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—416no—
567C2×C4—416no—
568C2×C4—416no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—816no—
592C2×C4—816no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—88no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—88no—
599C2×C4C2×#12114yes—
600C2×C4C2×#12112yes—
601C2×C4C2×#12314yes—
602C2×C4C2×#12312yes—
603C2×C4#53412yes—
604C2×C4#54618yes—
605C2×C4#55118yes—
606C2×C4C23×D411yescharacteristic
607C2×C4C22×#4112yes—
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×H(5,𝔽2)12yes—
610C8—28no—
611C8—44no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—232no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—464no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—416no—
635C22—416no—
636C22#241214yes—
637C22C2×#47018yes—
638C22C2×#47014yes—
639C22#305514yes—
640C22#309214yes—
641C22#309514yes—
642C22#319314yes—
643C22#319614yes—
644C22C22×#12111yescharacteristic
645C22C22×#12311yescharacteristic
646C22C2×#53411yescharacteristic
647C22C22×AGL(1,ℤ/8)11yescharacteristic
648C22C22×#13711yescharacteristic
649C22C22×#13711yescharacteristic
650C22C2×#57511yescharacteristic
651C4—24no—
652C4—24no—
653C4—42no—
654C4—82no—
655C4—84no—
656C4—84no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4#241214yes—
661C4C2×#54612yes—
662C4C2×#55112yes—
663C2—28no—
664C2—44no—
665C2—44no—
666C2#2455618yesdirect product factor
667C2C2×#241211yescharacteristic
668C2C22×#47011yescharacteristic
669C2C2×#305511yescharacteristic
670C2C2×#309211yescharacteristic
671C2C2×#309511yescharacteristic
672C2C2×#319311yescharacteristic
673C2C2×#319611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC26D4
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×#484C2semidirect
5. #24556C2direct (N × H)
6. C2×#2075C2semidirect
7. C2×#2035C2semidirect
8. C2×#1894C2semidirect
9. C2×#1872C2semidirect
10. C2×#1853C2semidirect
11. C2×#1824C2semidirect
12. C2×#1809C2semidirect
22. C2×#484C22semidirect
23. C22×#107C22semidirect
24. C22×#106C22semidirect
25. C22×#103C22semidirect
26. C22×#103C22semidirect
48. #2075C22semidirect
49. #2035C22semidirect
50. #1894C22semidirect
51. #1872C22semidirect
52. #1853C22semidirect
53. #1824C22semidirect
54. #1809C22semidirect
55. C2×#340C22semidirect
56. C2×#340C22semidirect
57. C2×#340C22semidirect
58. C2×#340C22semidirect
120. C2×#107C23semidirect
122. C2×#106C23semidirect
125. C2×#103C23semidirect
126. C2×#103C23semidirect
156. #340C23semidirect
157. #340C23semidirect
158. #340C23semidirect
159. #340C23semidirect
666. C2#24556direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4842
C22×#4072
C2×#28482
C22×#3802
#245562
C2×#20752
C2×#20352
C2×#18942
C2×#18722
C2×#18532
C2×#18242
C2×#18092