Finite Group Databasebeta

← C2×#32981 #6629128

Subgroups — C2×#32981 #6629128

Order 512.

5260 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#117C211yesmaximal, characteristic, complemented
2C22×#482C212yesmaximal, complemented
3C2×#2847C211yesmaximal, characteristic
4#32981C2116yesmaximal, direct product factor
5C2×#1875C212yesmaximal, complemented
6C2×#1874C211yesmaximal, characteristic, complemented
7C2×#1874C211yesmaximal, characteristic, complemented
8C2×#1854C212yesmaximal, complemented
9C2×#1853C211yesmaximal, characteristic, complemented
10C2×#1852C211yesmaximal, characteristic, complemented
11C2×#1825C212yesmaximal, complemented
12C2×#1779C211yesmaximal, characteristic, complemented
13C25×C4C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C22×#120—21no—
18C22×#117—21no—
19C22×#117—21no—
20C22×#117—24no—
21C22×#117—216no—
22C22×#117C2212yes—
23C22×#117C2212yes—
24C22×#117C2218yescomplemented
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4—21no—
28C23×C42—21no—
29C2×#482—216no—
30C2×#482C22116yescomplemented
31C22×#107C2212yescomplemented
32C22×#106C2212yescomplemented
33C22×#102C2212yescomplemented
34C22×#102C2212yescomplemented
35C22×C22⋊C8C2212yes—
36C22×C22⋊C8C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38C22×C22⋊C8C2212yes—
39#2847—24no—
40#2847C2218yes—
41C2×#386—21no—
42C2×#386—21no—
43C2×#386—21no—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47#1875C22116yescomplemented
48#1874C2218yescomplemented
49#1874C2218yescomplemented
50#1854C22116yescomplemented
51#1853C2218yescomplemented
52#1852C2218yescomplemented
53#1825C22116yescomplemented
54#1779C2218yescomplemented
55C2×#340C2212yescomplemented
56C2×#340C2212yescomplemented
57C2×#340C2212yescomplemented
58C2×#340C2212yescomplemented
59C26D411yescharacteristic
60C23×D4—21no—
61C23×D4—21no—
62C23×D4—21no—
63C23×D4C2311yescharacteristic
64C24×C4—28no—
65C24×C4—28no—
66C24×C4—24no—
67C24×C4—24no—
68C24×C4—21no—
69C24×C4—21no—
70C24×C4D414yes—
71C24×C4D411yescharacteristic
72C24×C4D411yescharacteristic
73C24×C4D411yescharacteristic
74C24×C4C2314yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C23×C8C2312yes—
79C23×C8C2312yes—
80C2×#121—22no—
81C2×#121—22no—
82C2×#121—22no—
83C2×#121—28no—
84C2×#121—24no—
85C2×#121—24no—
86C2×#121—28no—
87C2×#121C2314yes—
88C2×#121C2314yes—
89C2×#121C2314yes—
90C2×#120—28no—
91C2×#120—48no—
92C2×#117—264no—
93C2×#117—216no—
94C2×#117—216no—
95C2×#117—28no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117—28no—
99C2×#117—28no—
100C2×#117—264no—
101C2×#117—28no—
102C2×#117—28no—
103C2×#117D4116yescomplemented
104C2×#117C2318yes—
105C2×#117C2318yes—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—28no—
111C22×C4⋊C4—28no—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—216no—
120C22×C22⋊C4—22no—
121C22×C22⋊C4—22no—
122C22×C22⋊C4—28no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C42—216no—
130C22×C42—22no—
131C22×C42—22no—
132C22×C42—28no—
133#482—264no—
134C2×#107—28no—
135C2×#107C2318yescomplemented
136C2×#106—28no—
137C2×#106C2318yescomplemented
138C2×#102—28no—
139C2×#102—28no—
140C2×#102C2318yescomplemented
141C2×#102C2318yescomplemented
142C2×C22⋊C8—28no—
143C2×C22⋊C8—24no—
144C2×C22⋊C8—24no—
145C2×C22⋊C8—28no—
146C2×C22⋊C8C2318yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2318yes—
150#386—28no—
151#386—28no—
152#386—28no—
153#380—22no—
154#380—22no—
155#380—22no—
156#380C2314yes—
157#380C2314yes—
158#380C2314yes—
159C2×#95—21no—
160C2×#95—21no—
161C2×#95—21no—
162#340C2318yescomplemented
163#340C2318yescomplemented
164#340C2318yescomplemented
165#340C2318yescomplemented
166C25—24no—
167C25—24no—
168C25—24no—
169C25—24no—
170C25—24no—
171C25—21no—
172C25—42no—
173C25—42no—
174C25C2×D411yescharacteristic
175C25C2×D411yescharacteristic
176C25C2×D414yes—
177C25C2×D411yescharacteristic
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—21no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—24no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4C2412yes—
196C23×C4—264no—
197C23×C4—264no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—216no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—216no—
209C23×C4—216no—
210C23×C4—24no—
211C23×C4—216no—
212C23×C4—24no—
213C23×C4—216no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—28no—
223C23×C4—28no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D418yes—
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4#4118yes—
246C23×C4#4114yes—
247C23×C4#4114yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C22×C8—24no—
253C22×C8—24no—
254C22×C8—24no—
255C22×C8—24no—
256C22×C8—24no—
257C22×C8—24no—
258C22×C8C2414yes—
259C22×C8C2414yes—
260#121—216no—
261#121—28no—
262#121—28no—
263#121—216no—
264#121—48no—
265#121—48no—
266#121—48no—
267#120—432no—
268#117—232no—
269#117—264no—
270#117—264no—
271#117—232no—
272#117—216no—
273#117—216no—
274#117—216no—
275#117C2×D4132yescomplemented
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—216no—
279C2×C4⋊C4—216no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—216no—
287C2×C4⋊C4—216no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4C2×D418yes—
296C2×C4⋊C4C2×D418yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2414yes—
302C2×C22⋊C4—216no—
303C2×C22⋊C4—264no—
304C2×C22⋊C4—216no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C42—216no—
320C2×C42—264no—
321C2×C42—216no—
322C2×C42—28no—
323C2×C42—28no—
324#107—216no—
325#106—216no—
326#102—216no—
327#102—216no—
328C22⋊C8—216no—
329C22⋊C8—28no—
330C22⋊C8—28no—
331C22⋊C8—216no—
332#95—24no—
333#95—24no—
334#95—24no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—232no—
340C24—28no—
341C24—22no—
342C24—28no—
343C24—28no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—28no—
356C24—28no—
357C24—22no—
358C24—44no—
359C24—44no—
360C24—416no—
361C24—416no—
362C24—416no—
363C24—416no—
364C24—416no—
365C24—416no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24#12014yes—
374C24#12314yes—
375C24#12314yes—
376C24#13514yes—
377C24AGL(1,ℤ/8)14yes—
378C24C22×D412yes—
379C24C22×D412yes—
380C24C22×D412yes—
381C24C22×D411yescharacteristic
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—24no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—816no—
401C2×D4—816no—
402C2×D4—816no—
403C22×C4—232no—
404C22×C4—232no—
405C22×C4—232no—
406C22×C4—2128no—
407C22×C4—264no—
408C22×C4—264no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—216no—
431C22×C4—216no—
432C22×C4—216no—
433C22×C4—216no—
434C22×C4—216no—
435C22×C4—216no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—216no—
445C22×C4—216no—
446C22×C4—28no—
447C22×C4—28no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—432no—
454C22×C4—432no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4#12014yes—
473C22×C4#12018yes—
474C22×C4#12314yes—
475C22×C4#12314yes—
476C22×C4#12318yes—
477C22×C4#12318yes—
478C22×C4C22×D412yes—
479C22×C4C22×D418yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D414yes—
484C22×C4C2×#4118yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4114yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4114yes—
490C22×C4C2511yescharacteristic
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—48no—
498C2×C8—48no—
499C4⋊C4—216no—
500C4⋊C4—216no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4C22×D418yes—
513C4⋊C4C22×D418yes—
514C22⋊C4—232no—
515C22⋊C4—232no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C42—232no—
523C42—232no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—22no—
529C23—22no—
530C23—28no—
531C23—28no—
532C23—22no—
533C23—22no—
534C23—232no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—28no—
544C23—28no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—416no—
554C23—416no—
555C23—416no—
556C23—416no—
557C23—416no—
558C23—4128no—
559C23—44no—
560C23—44no—
561C23—416no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23#482116yescomplemented
583C23C2×#12014yes—
584C23C2×#12012yes—
585C23C2×#12314yes—
586C23C2×#12314yes—
587C23C2×#12312yes—
588C23C2×#12312yes—
589C23C2×#13514yes—
590C23C2×#13512yes—
591C23C2×AGL(1,ℤ/8)14yes—
592C23C2×AGL(1,ℤ/8)12yes—
593C23#57512yes—
594C23#57512yes—
595C23C23×D411yescharacteristic
596D4—44no—
597D4—44no—
598D4—816no—
599D4—816no—
600D4—816no—
601D4—84no—
602C2×C4—232no—
603C2×C4—232no—
604C2×C4—232no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—216no—
615C2×C4—216no—
616C2×C4—216no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—28no—
625C2×C4—432no—
626C2×C4—432no—
627C2×C4—416no—
628C2×C4—416no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4C2×#12014yes—
646C2×C4C2×#12012yes—
647C2×C4C2×#12314yes—
648C2×C4C2×#12314yes—
649C2×C4C2×#12312yes—
650C2×C4C2×#12312yes—
651C2×C4#52518yes—
652C2×C4#52518yes—
653C2×C4C23×D412yes—
654C2×C4C22×#4111yescharacteristic
655C2×C4C22×#4111yescharacteristic
656C2×C4C22×#4112yes—
657C8—44no—
658C8—44no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—232no—
666C22—24no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—464no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—416no—
682C22—416no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—168no—
691C22#284914yes—
692C22#300214yes—
693C22#300614yes—
694C22C2×#48214yes—
695C22C2×#48218yescomplemented
696C22#309114yes—
697C22#309314yes—
698C22#309414yes—
699C22C22×#12011yescharacteristic
700C22C22×#12311yescharacteristic
701C22C22×#12311yescharacteristic
702C22C22×#13511yescharacteristic
703C22C22×AGL(1,ℤ/8)11yescharacteristic
704C22C2×#57511yescharacteristic
705C22C2×#57511yescharacteristic
706C4—24no—
707C4—24no—
708C4—24no—
709C4—48no—
710C4—48no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4#284914yes—
715C4C2×#52512yes—
716C4C2×#52512yes—
717C2—28no—
718C2—44no—
719C2—44no—
720C2—162no—
721C2#3298118yesdirect product factor
722C2C2×#284911yescharacteristic
723C2C2×#300211yescharacteristic
724C2C2×#300611yescharacteristic
725C2C22×#48211yescharacteristic
726C2C2×#309111yescharacteristic
727C2C2×#309311yescharacteristic
728C2C2×#309411yescharacteristic
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. C23×#117C2semidirect
2. C22×#482C2semidirect
4. #32981C2direct (N × H)
5. C2×#1875C2semidirect
6. C2×#1874C2semidirect
7. C2×#1874C2semidirect
8. C2×#1854C2semidirect
9. C2×#1853C2semidirect
10. C2×#1852C2semidirect
11. C2×#1825C2semidirect
12. C2×#1779C2semidirect
24. C22×#117C22semidirect
30. C2×#482C22semidirect
31. C22×#107C22semidirect
32. C22×#106C22semidirect
33. C22×#102C22semidirect
34. C22×#102C22semidirect
47. #1875C22semidirect
48. #1874C22semidirect
49. #1874C22semidirect
50. #1854C22semidirect
51. #1853C22semidirect
52. #1852C22semidirect
53. #1825C22semidirect
54. #1779C22semidirect
55. C2×#340C22semidirect
56. C2×#340C22semidirect
57. C2×#340C22semidirect
58. C2×#340C22semidirect
103. C2×#117D4semidirect
135. C2×#107C23semidirect
137. C2×#106C23semidirect
140. C2×#102C23semidirect
141. C2×#102C23semidirect
162. #340C23semidirect
163. #340C23semidirect
164. #340C23semidirect
165. #340C23semidirect
275. #117C2×D4semidirect
582. C23#482semidirect
695. C22C2×#482semidirect
721. C2#32981direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#1172
C22×#4822
C2×#28472
#329812
C2×#18752
C2×#18742
C2×#18542
C2×#18532
C2×#18522
C2×#18252
C2×#17792