Finite Group Databasebeta

← C2×#9081 #195919

Subgroups — C2×#9081 #195919

Order 512.

3134 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊C8C211yesmaximal, characteristic
2C22×#385C211yesmaximal, characteristic
3C2×#1779C211yesmaximal, characteristic
4C22×#324C212yesmaximal, complemented
5#9081C218yesmaximal, direct product factor
6C2×#1280C212yesmaximal, complemented
7C25×C4C2211yescharacteristic
8C24×C8—21no—
9C23×C4⋊C4C411yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C23×C22⋊C4C411yescharacteristic
12C23×C22⋊C4C411yescharacteristic
13C22×C22⋊C8—28no—
14C22×C22⋊C8—22no—
15C22×C22⋊C8—21no—
16C22×C22⋊C8—21no—
17C22×C22⋊C8—21no—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2214yes—
20C22×C22⋊C8C2212yes—
21C2×#385—21no—
22C2×#385—21no—
23C2×#385—21no—
24C2×#385—22no—
25C2×#385—21no—
26C2×#385—22no—
27C2×#385C414yes—
28C2×#385C2214yes—
29C22×#95C411yescharacteristic
30C22×#95C2211yescharacteristic
31#1779—22no—
32#1779C2214yes—
33C2×#324—28no—
34C2×#324C2218yescomplemented
35#1280C2218yescomplemented
36C26C2×C411yescharacteristic
37C24×C4—24no—
38C24×C4—24no—
39C24×C4—44no—
40C24×C4—44no—
41C24×C4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C412yes—
46C24×C4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4D412yes—
51C24×C4D411yescharacteristic
52C24×C4D412yes—
53C24×C4C2312yes—
54C24×C4C2311yescharacteristic
55C23×C8—28no—
56C23×C8—24no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—44no—
60C23×C8—44no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—44no—
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C414yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—24no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—44no—
82C22×C22⋊C4C818yescomplemented
83C22×C22⋊C4C2×C414yes—
84C22×C22⋊C4C2×C414yes—
85C2×C22⋊C8—28no—
86C2×C22⋊C8—216no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—216no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—44no—
97C2×C22⋊C8—44no—
98C2×C22⋊C8D418yes—
99C2×C22⋊C8C2314yes—
100C2×C22⋊C8C2314yes—
101#385—24no—
102#385—24no—
103#385—24no—
104#385—24no—
105#385—24no—
106#385—24no—
107#385—24no—
108#385—24no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95C2×C414yes—
117C2×#95C2×C412yes—
118C2×#95C2312yes—
119#324—216no—
120C25—24no—
121C25—24no—
122C25—24no—
123C25—24no—
124C25—42no—
125C25—42no—
126C25C22⋊C411yescharacteristic
127C25C22⋊C412yes—
128C25C22⋊C411yescharacteristic
129C25C2×C814yes—
130C25M1614yes—
131C25C22×C412yes—
132C25C22×C411yescharacteristic
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—28no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—28no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—42no—
170C23×C4—44no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—44no—
175C23×C4—42no—
176C23×C4—44no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—44no—
180C23×C4—432no—
181C23×C4—432no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4C22⋊C412yes—
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C412yes—
191C23×C4C22⋊C412yes—
192C23×C4C22⋊C412yes—
193C23×C4C22⋊C411yescharacteristic
194C23×C4C22⋊C412yes—
195C23×C4C2×C814yescomplemented
196C23×C4C2×C814yescomplemented
197C23×C4C22×C412yes—
198C23×C4C22×C412yes—
199C23×C4C22×C412yes—
200C23×C4C22×C412yes—
201C23×C4C22×C412yes—
202C23×C4C22×C411yescharacteristic
203C23×C4C2×D412yes—
204C23×C4C2×D412yes—
205C23×C4C2×D412yes—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D412yes—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D414yes—
212C23×C4#4114yes—
213C23×C4C2411yescharacteristic
214C22×C8—216no—
215C22×C8—28no—
216C22×C8—24no—
217C22×C8—24no—
218C22×C8—432no—
219C22×C8—48no—
220C22×C8—44no—
221C22×C8—44no—
222C22×C8—48no—
223C22×C8—44no—
224C22×C8—44no—
225C22×C8—44no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—48no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—216no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—48no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4C2×C8116yescomplemented
258C22⋊C8—216no—
259C22⋊C8—28no—
260C22⋊C8—24no—
261C22⋊C8—24no—
262C22⋊C8—24no—
263C22⋊C8—48no—
264C22⋊C8—48no—
265C22⋊C8—44no—
266C22⋊C8C2×D418yes—
267#95—24no—
268#95—22no—
269#95—22no—
270#95—24no—
271#95—22no—
272#95—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—24no—
282C24—24no—
283C24—216no—
284C24—216no—
285C24—24no—
286C24—22no—
287C24—24no—
288C24—22no—
289C24—24no—
290C24—22no—
291C24—24no—
292C24—22no—
293C24—24no—
294C24—22no—
295C24—24no—
296C24—22no—
297C24—216no—
298C24—216no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24—48no—
303C24—48no—
304C24—42no—
305C24—416no—
306C24—416no—
307C24—416no—
308C24—416no—
309C24C22⋊C814yes—
310C24C22⋊C818yes—
311C24C22⋊C814yes—
312C24#9914yes—
313C24#10114yes—
314C24C2×C22⋊C412yes—
315C24C2×C22⋊C411yescharacteristic
316C24C2×C22⋊C412yes—
317C24C2×C22⋊C411yescharacteristic
318C24C4×D414yes—
319C24C22×C814yes—
320C24C22×C812yes—
321C24C2×M1614yes—
322C24C2×M1612yes—
323C24#13114yes—
324C24C23×C411yescharacteristic
325C22×C4—216no—
326C22×C4—216no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—216no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—416no—
394C22×C4—416no—
395C22×C4—416no—
396C22×C4—416no—
397C22×C4—416no—
398C22×C4—432no—
399C22×C4—432no—
400C22×C4—416no—
401C22×C4—416no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—48no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—48no—
414C22×C4—42no—
415C22×C4—48no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—88no—
434C22×C4—88no—
435C22×C4#9914yes—
436C22×C4C2×C22⋊C412yes—
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C412yes—
440C22×C4C4×D414yes—
441C22×C4#12012yes—
442C22×C4#12012yes—
443C22×C4#12114yes—
444C22×C4#12114yes—
445C22×C4#12312yes—
446C22×C4#12312yes—
447C22×C4C22×C814yescomplemented
448C22×C4C22×C814yescomplemented
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C2×#4112yes—
453C2×C8—28no—
454C2×C8—432no—
455C2×C8—44no—
456C2×C8—44no—
457C2×C8—416no—
458C2×C8—44no—
459C2×C8—44no—
460C2×C8—44no—
461C2×C8—88no—
462C4⋊C4—48no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C22⋊C4—216no—
470C22⋊C4—216no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C23—22no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—22no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—28no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—216no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—216no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—22no—
514C23—48no—
515C23—44no—
516C23—44no—
517C23—48no—
518C23—48no—
519C23—464no—
520C23—464no—
521C23—48no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—48no—
526C23—48no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23#324116yescomplemented
532C23#38612yes—
533C23#38612yes—
534C23C2×C22⋊C814yes—
535C23C2×C22⋊C812yes—
536C23C2×C22⋊C814yes—
537C23C2×C22⋊C812yes—
538C23#40812yes—
539C23#40812yes—
540C23C2×#9914yes—
541C23C2×#9912yes—
542C23C2×#9912yes—
543C23C2×#10114yes—
544C23C2×#10112yes—
545C23#41312yes—
546C23C8×D418yes—
547C23#43518yes—
548C23C22×C22⋊C411yescharacteristic
549C23C2×C4×D412yes—
550C23C23×C811yescharacteristic
551C23C22×M1611yescharacteristic
552C23C2×#13112yes—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—216no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4#38612yes—
616C2×C4#38612yes—
617C2×C4C2×#9914yes—
618C2×C4C2×#12011yescharacteristic
619C2×C4C2×#12112yes—
620C2×C4C2×#12311yescharacteristic
621C8—48no—
622C8—84no—
623C22—232no—
624C22—216no—
625C22—216no—
626C22—24no—
627C22—21no—
628C22—21no—
629C22—22no—
630C22—21no—
631C22—21no—
632C22—22no—
633C22—24no—
634C22—22no—
635C22—22no—
636C22—24no—
637C22—22no—
638C22—22no—
639C22—24no—
640C22—24no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—432no—
650C22—416no—
651C22—416no—
652C22—416no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22C2×#32414yes—
658C22C2×#32418yescomplemented
659C22#145914yes—
660C22#146814yes—
661C22#147214yes—
662C22#186914yes—
663C22#186914yes—
664C22#189814yes—
665C22#190114yes—
666C22C2×#38611yescharacteristic
667C22C22×C22⋊C811yescharacteristic
668C22C2×#40811yescharacteristic
669C22C22×#9911yescharacteristic
670C22C22×#9911yescharacteristic
671C22C22×#10111yescharacteristic
672C22C2×#41311yescharacteristic
673C22C2×C8×D412yes—
674C22C2×#43512yes—
675C4—21no—
676C4—22no—
677C4—21no—
678C4—22no—
679C4—24no—
680C4—22no—
681C4—44no—
682C4—44no—
683C4—42no—
684C4—82no—
685C4—82no—
686C2—24no—
687C2—22no—
688C2—22no—
689C2—28no—
690C2—44no—
691C2#908118yesdirect product factor
692C2C22×#32411yescharacteristic
693C2C2×#145911yescharacteristic
694C2C2×#146811yescharacteristic
695C2C2×#147211yescharacteristic
696C2C2×#186911yescharacteristic
697C2C2×#189811yescharacteristic
698C2C2×#190111yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
LCSG ▷ C23 ▷ C2 ▷ 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
4. C22×#324C2semidirect
5. #9081C2direct (N × H)
6. C2×#1280C2semidirect
34. C2×#324C22semidirect
35. #1280C22semidirect
82. C22×C22⋊C4C8semidirect
195. C23×C4C2×C8semidirect
196. C23×C4C2×C8semidirect
257. C2×C22⋊C4C2×C8semidirect
447. C22×C4C22×C8semidirect
448. C22×C4C22×C8semidirect
531. C23#324semidirect
658. C22C2×#324semidirect
691. C2#9081direct (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
C22×#3852
C2×#17792
C22×#3242
#90812
C2×#12802