Finite Group Databasebeta

← C2×#24650 #6448937

Subgroups — C2×#24650 #6448937

Order 512.

3394 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#459C211yesmaximal, characteristic, complemented
2C22×#431C211yesmaximal, characteristic, complemented
3C2×#2856C211yesmaximal, characteristic, complemented
4C2×C4×#121C211yesmaximal, characteristic, complemented
5#24650C2116yesmaximal, direct product factor
6C2×#2033C212yesmaximal, complemented
7C2×#1999C212yesmaximal, complemented
8C2×#1982C211yesmaximal, characteristic, complemented
9C2×#1970C212yesmaximal, complemented
10C2×#1947C211yesmaximal, characteristic, complemented
11C2×#1934C212yesmaximal, complemented
12C2×#1802C211yesmaximal, characteristic, complemented
13C23×D8C2212yes—
14C22×#127C2211yescharacteristic
15C22×#121—21no—
16C22×#121—21no—
17C22×#121C2211yescharacteristic
18C22×#121C2212yes—
19C22×C4×D4—21no—
20C22×C4×D4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C2×#459—28no—
23C2×#459C2218yescomplemented
24C2×#431—28no—
25C2×#431C2218yescomplemented
26C22×#107C2212yes—
27C22×C4⋊C8C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C22×#102C2212yes—
30C22×#102C2212yes—
31C22×#102C2212yes—
32C22×#102C2212yes—
33C22×C22⋊C8C2212yes—
34C22×C4×C8C2212yes—
35#2856—24no—
36#2856C2218yescomplemented
37C4×#121—24no—
38C4×#121C2218yescomplemented
39C2×#390—21no—
40C2×#390—21no—
41C2×#390C2212yes—
42C2×#390C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#388—21no—
45C2×#386—21no—
46C2×#384C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4—21no—
50#2033C22116yescomplemented
51#1999C22116yescomplemented
52#1982C2218yescomplemented
53#1970C22116yescomplemented
54#1947C2218yescomplemented
55#1934C22116yescomplemented
56#1802C2218yescomplemented
57C2×#340C2212yes—
58C2×#340C2212yes—
59C23×D4—21no—
60C23×D4—22no—
61C23×D4—21no—
62C23×D4—21no—
63C23×D4C2312yes—
64C23×D4C2311yescharacteristic
65C24×C4—21no—
66C24×C4D411yescharacteristic
67C24×C4D411yescharacteristic
68C24×C4C2311yescharacteristic
69C22×D8—28no—
70C22×D8—28no—
71C22×D8—44no—
72C22×D8C2318yes—
73C23×C8C2312yes—
74C23×C8C2312yes—
75C2×#127—24no—
76C2×#127—22no—
77C2×#127—24no—
78C2×#127C2314yes—
79C2×#121—24no—
80C2×#121—22no—
81C2×#121—24no—
82C2×#121—28no—
83C2×#121—28no—
84C2×#121—24no—
85C2×#121—44no—
86C2×#121—44no—
87C2×#121—48no—
88C2×#121—48no—
89C2×#121C2314yes—
90C2×#121C2318yes—
91C2×C4×D4—28no—
92C2×C4×D4—22no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—48no—
96C2×C4×D4C2314yes—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—22no—
109C22×C22⋊C4—22no—
110C22×C22⋊C4—22no—
111C22×C22⋊C4—21no—
112C22×C42—24no—
113C22×C42—24no—
114C22×C42—21no—
115C22×C42D414yes—
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42C2314yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#459—232no—
122#431—232no—
123C2×#107—44no—
124C2×#107C2318yes—
125C2×C4⋊C8—24no—
126C2×C4⋊C8D418yescomplemented
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2314yes—
129C2×#102—28no—
130C2×#102—28no—
131C2×#102—44no—
132C2×#102—44no—
133C2×#102C2318yes—
134C2×#102C2318yes—
135C2×#102C2318yes—
136C2×#102C2318yes—
137C2×C22⋊C8—28no—
138C2×C22⋊C8C2318yes—
139C2×C4×C8—28no—
140C2×C4×C8C2318yes—
141#390—24no—
142#390—28no—
143#390—28no—
144#390—22no—
145#390C2318yes—
146#390C2314yes—
147#389—22no—
148#389C2314yes—
149#388—28no—
150#386—28no—
151#384—22no—
152#384C2314yes—
153#380—22no—
154#380C2314yes—
155C4×C4⋊C4—22no—
156C4×C4⋊C4C2314yes—
157C4×C22⋊C4—28no—
158C2×#95—21no—
159C2×#95—21no—
160C2×#95—21no—
161#340C2318yes—
162#340C2318yes—
163C25—21no—
164C25—22no—
165C25C2×D411yescharacteristic
166C22×D4—28no—
167C22×D4—24no—
168C22×D4—28no—
169C22×D4—22no—
170C22×D4—21no—
171C22×D4—24no—
172C22×D4—22no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—48no—
189C22×D4—48no—
190C22×D4—48no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4#4114yes—
196C22×D4C2414yes—
197C22×D4C2412yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—21no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—24no—
213C23×C4—42no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4D818yescomplemented
218C23×C4C2×D412yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D414yes—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D414yes—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C2×D8—216no—
229C2×D8—216no—
230C2×D8—48no—
231C2×D8—432no—
232C22×C8—24no—
233C22×C8—24no—
234C22×C8—24no—
235C22×C8—24no—
236C22×C8—24no—
237C22×C8C2×D418yes—
238C22×C8C2414yes—
239C22×C8C2414yes—
240#127—28no—
241#127—28no—
242#127—48no—
243#121—216no—
244#121—28no—
245#121—28no—
246#121—48no—
247#121—48no—
248#121—416no—
249#121—432no—
250#121—432no—
251C4×D4—28no—
252C4×D4—28no—
253C4×D4—432no—
254C4×D4—48no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4#4114yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—48no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—216no—
298C2×C42—28no—
299C2×C42—28no—
300C2×C42—24no—
301C2×C42—44no—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D414yes—
307C2×C42#4114yes—
308C2×C42C2412yes—
309C2×C42C2412yes—
310#107—48no—
311C4⋊C8—28no—
312C4⋊C8C2×D4116yescomplemented
313#102—216no—
314#102—216no—
315#102—48no—
316#102—48no—
317C22⋊C8—216no—
318C4×C8—216no—
319#95—24no—
320#95—24no—
321#95—24no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—24no—
329C24—22no—
330C24—24no—
331C24—42no—
332C24—42no—
333C24—42no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24—44no—
338C24—44no—
339C24—44no—
340C24—44no—
341C24—44no—
342C24—44no—
343C24—44no—
344C24C2×D814yes—
345C24C22×D412yes—
346C24C22×D411yescharacteristic
347C2×D4—216no—
348C2×D4—24no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—432no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—416no—
368C2×D4—416no—
369C2×D4—416no—
370C2×D4—42no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—42no—
376C2×D4—44no—
377C2×D4—48no—
378C2×D4—44no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—44no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—816no—
386C2×D4—832no—
387C2×D4—816no—
388C2×D4C2×#4114yes—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—232no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—44no—
418C22×C4—416no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—816no—
438C22×C4#12114yes—
439C22×C4#12114yes—
440C22×C4C2×D814yes—
441C22×C4C2×D814yes—
442C22×C4C2×D8116yescomplemented
443C22×C4#13514yes—
444C22×C4AGL(1,ℤ/8)14yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4C2511yescharacteristic
458D8—432no—
459C2×C8—216no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—48no—
466C2×C8C22×D418yes—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—48no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4C2×#4114yes—
482C22⋊C4—216no—
483C22⋊C4—416no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—416no—
488C22⋊C4—48no—
489C22⋊C4—416no—
490C42—24no—
491C42—28no—
492C42—28no—
493C42—22no—
494C42—22no—
495C42—48no—
496C42#12118yes—
497C42C22×D414yes—
498C42C2×#4114yes—
499C23—28no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—28no—
504C23—28no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—48no—
514C23—48no—
515C23—42no—
516C23—48no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23—88no—
534C23—88no—
535C23—88no—
536C23—88no—
537C23—88no—
538C23—88no—
539C23—88no—
540C23C2×#12112yes—
541C23#52512yes—
542C23#53512yes—
543C23C22×D814yes—
544C23C22×D812yes—
545C23C2×#13512yes—
546C23C2×AGL(1,ℤ/8)12yes—
547C23#57512yes—
548C23C23×D411yescharacteristic
549D4—24no—
550D4—24no—
551D4—432no—
552D4—48no—
553D4—44no—
554D4—42no—
555D4—44no—
556D4—44no—
557D4—42no—
558D4—44no—
559D4—816no—
560D4—832no—
561D4—88no—
562D4—816no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—232no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—416no—
578C2×C4—48no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—84no—
597C2×C4—88no—
598C2×C4—84no—
599C2×C4—816no—
600C2×C4#466116yes—
601C2×C4C2×#12114yes—
602C2×C4C2×#12112yes—
603C2×C4C2×#12114yes—
604C2×C4#52512yes—
605C2×C4#53512yes—
606C2×C4#54614yes—
607C2×C4#55014yes—
608C2×C4#55314yes—
609C2×C4#55514yes—
610C2×C4C22×D814yes—
611C2×C4C22×D814yes—
612C2×C4C2×#13514yes—
613C2×C4C2×AGL(1,ℤ/8)14yes—
614C2×C4C23×D411yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C2×C4C2×H(5,𝔽2)11yescharacteristic
620C8—28no—
621C8—44no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—42no—
626C22—416no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—82no—
634C22—82no—
635C22—816no—
636C22—82no—
637C22—82no—
638C22—84no—
639C22—84no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22—84no—
644C22—84no—
645C22—1616no—
646C22#241414yes—
647C22C2×#46614yes—
648C22#305514yes—
649C22#310014yes—
650C22#310614yes—
651C22#320514yes—
652C22#321214yes—
653C22C22×#12111yescharacteristic
654C22C2×#52511yescharacteristic
655C22C2×#53511yescharacteristic
656C22C23×D811yescharacteristic
657C22C22×#13511yescharacteristic
658C22C22×AGL(1,ℤ/8)11yescharacteristic
659C22C2×#57511yescharacteristic
660C4—24no—
661C4—24no—
662C4—24no—
663C4—44no—
664C4—42no—
665C4—82no—
666C4—82no—
667C4—84no—
668C4#241414yes—
669C4C2×#46618yes—
670C4C2×#54611yescharacteristic
671C4C2×#55011yescharacteristic
672C4C2×#55311yescharacteristic
673C4C2×#55511yescharacteristic
674C2—44no—
675C2—84no—
676C2—164no—
677C2#2465018yesdirect product factor
678C2C2×#241411yescharacteristic
679C2C22×#46611yescharacteristic
680C2C2×#305511yescharacteristic
681C2C2×#310011yescharacteristic
682C2C2×#310611yescharacteristic
683C2C2×#320511yescharacteristic
684C2C2×#321211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#459C2semidirect
2. C22×#431C2semidirect
3. C2×#2856C2semidirect
4. C2×C4×#121C2semidirect
5. #24650C2direct (N × H)
6. C2×#2033C2semidirect
7. C2×#1999C2semidirect
8. C2×#1982C2semidirect
9. C2×#1970C2semidirect
10. C2×#1947C2semidirect
11. C2×#1934C2semidirect
12. C2×#1802C2semidirect
23. C2×#459C22semidirect
25. C2×#431C22semidirect
36. #2856C22semidirect
38. C4×#121C22semidirect
50. #2033C22semidirect
51. #1999C22semidirect
52. #1982C22semidirect
53. #1970C22semidirect
54. #1947C22semidirect
55. #1934C22semidirect
56. #1802C22semidirect
126. C2×C4⋊C8D4semidirect
217. C23×C4D8semidirect
312. C4⋊C8C2×D4semidirect
442. C22×C4C2×D8semidirect
677. C2#24650direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4592
C22×#4312
C2×#28562
C2×C4×#1212
#246502
C2×#20332
C2×#19992
C2×#19822
C2×#19702
C2×#19472
C2×#19342
C2×#18022