Finite Group Databasebeta

← C2×#9113 #196357

Subgroups — C2×#9113 #196357

Order 512.

3452 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊C8C211yesmaximal, characteristic, complemented
2C2×#2847C211yesmaximal, characteristic
3C22×#327C212yesmaximal, complemented
4C2×C22⋊#95C211yesmaximal, characteristic, complemented
5#9113C218yesmaximal, direct product factor
6C2×#1294C212yesmaximal, complemented
7C25×C4C2211yescharacteristic
8C24×C8—21no—
9C22×#121C411yescharacteristic, complemented
10C22×#121C411yescharacteristic, complemented
11C22×#121C2211yescharacteristic
12C22×#120—21no—
13C23×C22⋊C4—21no—
14C22×C22⋊C8—28no—
15C22×C22⋊C8—21no—
16C22×C22⋊C8—22no—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2214yescomplemented
19#2847—22no—
20#2847—22no—
21#2847C414yescomplemented
22#2847C2214yes—
23C2×#386—21no—
24C2×#386—21no—
25C2×#386—21no—
26C2×#380C411yescharacteristic, complemented
27C2×#380C411yescharacteristic, complemented
28C2×#380C2211yescharacteristic
29C22×#95—21no—
30C22×#95C2212yes—
31C2×#327—28no—
32C2×#327C2218yescomplemented
33C22⋊#95—22no—
34C22⋊#95C2214yescomplemented
35#1294C2218yescomplemented
36C26D411yescharacteristic
37C23×D4—21no—
38C23×D4—21no—
39C23×D4—21no—
40C23×D4C2×C411yescharacteristic
41C24×C4—24no—
42C24×C4—24no—
43C24×C4—22no—
44C24×C4—44no—
45C24×C4—44no—
46C24×C4C2×C411yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C412yes—
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4D412yes—
52C24×C4D412yes—
53C24×C4D411yescharacteristic
54C24×C4C2312yes—
55C24×C4C2311yescharacteristic
56C23×C8—28no—
57C23×C8—22no—
58C23×C8—24no—
59C23×C8—44no—
60C23×C8—44no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—24no—
66C2×#121—24no—
67C2×#121—44no—
68C2×#121—44no—
69C2×#121C2×C412yes—
70C2×#121C2×C414yescomplemented
71C2×#121C2×C414yescomplemented
72C2×#121C2312yes—
73C2×#120—24no—
74C2×#120—24no—
75C2×#120—44no—
76C2×#120—44no—
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C22⋊C4—28no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—44no—
89C22×C22⋊C4—44no—
90C2×C22⋊C8—28no—
91C2×C22⋊C8—216no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—216no—
96C2×C22⋊C8D418yescomplemented
97C2×C22⋊C8C2314yes—
98#386—24no—
99#386—24no—
100#386—24no—
101#386—24no—
102#386—24no—
103#386—24no—
104#380—22no—
105#380—22no—
106#380—21no—
107#380—21no—
108#380C2×C412yes—
109#380C2×C414yescomplemented
110#380C2×C414yescomplemented
111#380C2312yes—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—24no—
116C2×#95—24no—
117C2×#95—22no—
118C2×#95—24no—
119C2×#95—44no—
120C2×#95C2314yes—
121#327—216no—
122C25—24no—
123C25—24no—
124C25—24no—
125C25—24no—
126C25—21no—
127C25—42no—
128C25—42no—
129C25C22⋊C411yescharacteristic
130C25C22⋊C412yes—
131C25C22⋊C411yescharacteristic
132C25D814yes—
133C25SD1614yes—
134C25C2×D412yes—
135C25C2×D411yescharacteristic
136C22×D4—22no—
137C22×D4—24no—
138C22×D4—24no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—24no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4—44no—
145C22×D4—44no—
146C22×D4—44no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4—44no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—44no—
154C22×D4C22×C412yes—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—28no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—28no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—28no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—24no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—44no—
185C23×C4—42no—
186C23×C4—44no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—44no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—44no—
193C23×C4—432no—
194C23×C4—432no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4C22⋊C412yes—
202C23×C4C22⋊C411yescharacteristic
203C23×C4C22⋊C412yes—
204C23×C4C22⋊C412yes—
205C23×C4C22⋊C412yes—
206C23×C4C22⋊C411yescharacteristic
207C23×C4C22⋊C412yes—
208C23×C4C22×C412yes—
209C23×C4C22×C412yes—
210C23×C4C22×C411yescharacteristic
211C23×C4C22×C411yescharacteristic
212C23×C4C2×D412yes—
213C23×C4C2×D412yes—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D412yes—
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D414yes—
219C23×C4#4114yes—
220C23×C4C2411yescharacteristic
221C22×C8—216no—
222C22×C8—28no—
223C22×C8—24no—
224C22×C8—432no—
225C22×C8—44no—
226C22×C8—44no—
227C22×C8—48no—
228C22×C8—48no—
229#121—28no—
230#121—24no—
231#121—24no—
232#121—48no—
233#121—48no—
234#121—48no—
235#121—42no—
236#121—44no—
237#121—42no—
238#121—48no—
239#120—416no—
240#120—48no—
241#120—48no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4C22×C412yes—
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C22×C412yes—
251C2×C22⋊C4—216no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—432no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C22⋊C8—216no—
272C22⋊C8—28no—
273C22⋊C8—24no—
274C22⋊C8C2×D418yescomplemented
275#95—22no—
276#95—24no—
277#95—24no—
278#95—22no—
279#95—24no—
280#95—24no—
281#95—48no—
282#95—44no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—24no—
292C24—24no—
293C24—216no—
294C24—216no—
295C24—24no—
296C24—22no—
297C24—24no—
298C24—22no—
299C24—24no—
300C24—22no—
301C24—24no—
302C24—22no—
303C24—24no—
304C24—24no—
305C24—22no—
306C24—22no—
307C24—216no—
308C24—216no—
309C24—22no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—48no—
314C24—48no—
315C24—42no—
316C24—416no—
317C24—416no—
318C24—416no—
319C24—416no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24#9914yes—
328C24#10214yes—
329C24#10218yes—
330C24#10214yes—
331C24#10414yes—
332C24C2×C22⋊C412yes—
333C24C2×C22⋊C411yescharacteristic
334C24C2×C22⋊C412yes—
335C24C2×C22⋊C411yescharacteristic
336C24#12114yes—
337C24C2×D814yes—
338C24C2×D812yes—
339C24C2×SD1614yes—
340C24C2×SD1612yes—
341C24#13514yes—
342C24C22×D411yescharacteristic
343C2×D4—24no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—44no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—44no—
358C2×D4—48no—
359C2×D4—44no—
360C2×D4—48no—
361C2×D4—42no—
362C2×D4—88no—
363C2×D4—816no—
364C2×D4—816no—
365C2×D4—88no—
366C2×D4—88no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—416no—
425C22×C4—416no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—432no—
430C22×C4—432no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—44no—
440C22×C4—416no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4#9914yes—
458C22×C4C2×C22⋊C412yes—
459C22×C4C2×C22⋊C411yescharacteristic
460C22×C4C2×C22⋊C411yescharacteristic
461C22×C4C2×C22⋊C412yes—
462C22×C4C4×D414yes—
463C22×C4C4×D414yes—
464C22×C4#12012yes—
465C22×C4#12012yes—
466C22×C4#12114yes—
467C22×C4#12312yes—
468C22×C4#12312yes—
469C22×C4C23×C411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C2×#4112yes—
473C2×C8—28no—
474C2×C8—432no—
475C2×C8—44no—
476C2×C8—44no—
477C2×C8—416no—
478C4⋊C4—24no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C22⋊C4—48no—
485C22⋊C4—432no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—48no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—44no—
494C22⋊C4—48no—
495C23—21no—
496C23—21no—
497C23—22no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—28no—
502C23—28no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—21no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—24no—
518C23—28no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—216no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—216no—
531C23—28no—
532C23—28no—
533C23—28no—
534C23—28no—
535C23—22no—
536C23—48no—
537C23—44no—
538C23—44no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—464no—
544C23—464no—
545C23—48no—
546C23—48no—
547C23—48no—
548C23—48no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23#327116yescomplemented
568C23#38612yes—
569C23#38612yes—
570C23C2×#9914yes—
571C23C2×#9912yes—
572C23C2×#9912yes—
573C23C2×#10214yes—
574C23C2×#10212yes—
575C23C2×#10214yes—
576C23C2×#10212yes—
577C23#41612yes—
578C23#41612yes—
579C23C2×#10414yes—
580C23C2×#10412yes—
581C23#42112yes—
582C23#46518yes—
583C23#46618yes—
584C23C22×C22⋊C411yescharacteristic
585C23C2×#12112yes—
586C23C22×D811yescharacteristic
587C23C22×SD1611yescharacteristic
588C23C2×#13512yes—
589D4—84no—
590D4—84no—
591D4—816no—
592D4—88no—
593D4—816no—
594D4—84no—
595D4—84no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—216no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—48no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—48no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—48no—
631C2×C4—44no—
632C2×C4—48no—
633C2×C4—48no—
634C2×C4—48no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—832no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4#38612yes—
653C2×C4#38612yes—
654C2×C4C2×#9914yes—
655C2×C4C2×C4×D412yes—
656C2×C4C2×#12011yescharacteristic
657C2×C4C2×#12311yescharacteristic
658C8—48no—
659C22—21no—
660C22—21no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—24no—
665C22—216no—
666C22—232no—
667C22—216no—
668C22—21no—
669C22—21no—
670C22—22no—
671C22—24no—
672C22—22no—
673C22—24no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—22no—
681C22—22no—
682C22—24no—
683C22—24no—
684C22—24no—
685C22—416no—
686C22—432no—
687C22—416no—
688C22—42no—
689C22—416no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—168no—
701C22C2×#32714yes—
702C22C2×#32718yescomplemented
703C22#150114yes—
704C22#151614yes—
705C22#151914yes—
706C22#187114yes—
707C22#189314yes—
708C22#189314yes—
709C22#189814yes—
710C22C2×#38611yescharacteristic
711C22C22×#9911yescharacteristic
712C22C22×#9911yescharacteristic
713C22C22×#10211yescharacteristic
714C22C2×#41611yescharacteristic
715C22C22×#10411yescharacteristic
716C22C2×#42111yescharacteristic
717C22C2×#46512yes—
718C22C2×#46612yes—
719C4—21no—
720C4—21no—
721C4—22no—
722C4—24no—
723C4—22no—
724C4—22no—
725C4—42no—
726C4—88no—
727C4—82no—
728C4—82no—
729C4—82no—
730C2—22no—
731C2—24no—
732C2—28no—
733C2—22no—
734C2—44no—
735C2—162no—
736C2#911318yesdirect product factor
737C2C22×#32711yescharacteristic
738C2C2×#150111yescharacteristic
739C2C2×#151911yescharacteristic
740C2C2×#151611yescharacteristic
741C2C2×#187111yescharacteristic
742C2C2×#189311yescharacteristic
743C2C2×#189811yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2847C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊C8C2semidirect
3. C22×#327C2semidirect
4. C2×C22⋊#95C2semidirect
5. #9113C2direct (N × H)
6. C2×#1294C2semidirect
9. C22×#121C4semidirect
10. C22×#121C4semidirect
18. C22×C22⋊C8C22semidirect
21. #2847C4semidirect
26. C2×#380C4semidirect
27. C2×#380C4semidirect
32. C2×#327C22semidirect
34. C22⋊#95C22semidirect
35. #1294C22semidirect
70. C2×#121C2×C4semidirect
71. C2×#121C2×C4semidirect
96. C2×C22⋊C8D4semidirect
109. #380C2×C4semidirect
110. #380C2×C4semidirect
274. C22⋊C8C2×D4semidirect
567. C23#327semidirect
702. C22C2×#327semidirect
736. C2#9113direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊C82
C2×#28472
C22×#3272
C2×C22⋊#952
#91132
C2×#12942