Finite Group Databasebeta

← #374062

Subgroups — #374062

Order 512.

3570 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊C8C211yesmaximal, characteristic
2C2×#2848C211yesmaximal, characteristic
3C22×#380C211yesmaximal, characteristic
4C2×#1892C212yesmaximal, complemented
5C2×#1872C212yesmaximal, complemented
6#11200C212yesmaximal, complemented
7#10970C212yesmaximal, complemented
8#10730C212yesmaximal, complemented
9#10721C212yesmaximal, complemented
10C25×C4C2211yescharacteristic
11C24×C8C2212yes—
12C22×C22⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C23×C4⋊C4C2212yes—
16C23×C22⋊C4—21no—
17C22×#103C2214yescomplemented
18C22×C22⋊C8—28no—
19C22×C22⋊C8—24no—
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22#2848—24no—
23#2848C418yescomplemented
24C2×#385—21no—
25C2×#385—21no—
26C2×#385—21no—
27C2×#380—28no—
28C2×#380—21no—
29C2×#380—21no—
30C2×#380C2212yes—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×#379—21no—
36C22×#95—21no—
37#1892—28no—
38#1872—28no—
39C2×#340C2214yescomplemented
40C2×#340C2214yescomplemented
41C2×#340C2214yescomplemented
42C26D411yescharacteristic
43C23×Q8C2311yescharacteristic
44C24×C4—24no—
45C24×C4—24no—
46C24×C4—24no—
47C24×C4—24no—
48C24×C4—22no—
49C24×C4—44no—
50C24×C4C2×C414yes—
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D414yes—
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C23×C8—24no—
59C23×C8—28no—
60C23×C8—24no—
61C23×C8—24no—
62C23×C8—44no—
63C23×C8C2314yes—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—44no—
72C2×C22⋊Q8C2×C414yes—
73C2×C22⋊Q8C2×C414yescomplemented
74C2×C22⋊Q8C2×C414yescomplemented
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—28no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—44no—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2314yes—
87C22×C22⋊C4—28no—
88C22×C22⋊C4—22no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—44no—
97C22×C22⋊C4—44no—
98C2×#103—28no—
99C2×#103—28no—
100C2×#103—44no—
101C2×C22⋊C8—232no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8—28no—
106C2×C22⋊C8D418yes—
107#385—28no—
108#385—28no—
109#385—28no—
110#380—28no—
111#380—28no—
112#380—22no—
113#380—22no—
114#380—22no—
115#380—24no—
116#380—28no—
117#380C2×C414yes—
118#380C2×C414yescomplemented
119#380C2×C414yescomplemented
120#380D418yes—
121#379—28no—
122C2×#95—28no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128#340—28no—
129#340—28no—
130#340—44no—
131C25—22no—
132C25—24no—
133C25—22no—
134C25—24no—
135C25—22no—
136C25—22no—
137C25—42no—
138C25—42no—
139C25C22⋊C414yes—
140C25SD1614yes—
141C25Q1614yes—
142C25C2×D411yescharacteristic
143C25C2×D411yescharacteristic
144C25C2×D411yescharacteristic
145C22×Q8—21no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—24no—
151C22×Q8C22×C412yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—216no—
162C23×C4—216no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—28no—
166C23×C4—24no—
167C23×C4—216no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—28no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—28no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—24no—
183C23×C4—48no—
184C23×C4—48no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—416no—
189C23×C4—416no—
190C23×C4—416no—
191C23×C4—416no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4C22⋊C414yes—
199C23×C4C22⋊C414yes—
200C23×C4C22⋊C414yes—
201C23×C4C22×C412yes—
202C23×C4C22×C412yes—
203C23×C4C22×C412yes—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D412yes—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D412yes—
209C23×C4C2×D412yes—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4#4114yes—
213C23×C4#4114yes—
214C23×C4C2411yescharacteristic
215C22×C8—216no—
216C22×C8—24no—
217C22×C8—24no—
218C22×C8—24no—
219C22×C8—24no—
220C22×C8—24no—
221C22×C8—24no—
222C22×C8—416no—
223C22×C8—416no—
224C22×C8—48no—
225C22×C8—48no—
226C22×C8C2×D418yes—
227C22⋊Q8—216no—
228C22⋊Q8—216no—
229C22⋊Q8—48no—
230C22⋊Q8—48no—
231C22⋊Q8—44no—
232C22⋊Q8—48no—
233C22⋊Q8—44no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—416no—
252C2×C4⋊C4—416no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4C22×C412yes—
260C2×C4⋊C4C22×C412yes—
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C2×D418yes—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—432no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—42no—
284#103—416no—
285C22⋊C8—216no—
286C22⋊C8—216no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—28no—
292C24—21no—
293C24—22no—
294C24—21no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—21no—
299C24—21no—
300C24—22no—
301C24—22no—
302C24—21no—
303C24—22no—
304C24—21no—
305C24—22no—
306C24—22no—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—28no—
311C24—28no—
312C24—28no—
313C24—28no—
314C24—28no—
315C24—28no—
316C24—216no—
317C24—216no—
318C24—28no—
319C24—28no—
320C24—44no—
321C24—44no—
322C24—48no—
323C24—48no—
324C24—48no—
325C24—48no—
326C24—48no—
327C24—48no—
328C24—48no—
329C24—48no—
330C24—48no—
331C24—48no—
332C24#103116yescomplemented
333C24C2×C22⋊C412yes—
334C24C2×C22⋊C412yes—
335C24C2×C22⋊C412yes—
336C24#12012yes—
337C24#12114yes—
338C24#12312yes—
339C24C2×SD1612yes—
340C24C2×SD1612yes—
341C24C2×SD1612yes—
342C24C2×Q1612yes—
343C24C2×Q1612yes—
344C24C2×Q1612yes—
345C24#13514yes—
346C24AGL(1,ℤ/8)12yes—
347C24#13712yes—
348C24C22×D411yescharacteristic
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—41no—
353C2×Q8—42no—
354C2×Q8—41no—
355C2×Q8—42no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—44no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—464no—
410C22×C4—432no—
411C22×C4—432no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—416no—
421C22×C4—416no—
422C22×C4—416no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4C2×C22⋊C412yes—
450C22×C4C2×C22⋊C412yes—
451C22×C4C2×C22⋊C412yes—
452C22×C4C4×D418yes—
453C22×C4C4×D418yes—
454C22×C4#12014yes—
455C22×C4#12012yes—
456C22×C4#12118yes—
457C22×C4#12114yes—
458C22×C4#12314yes—
459C22×C4#12312yes—
460C22×C4C23×C411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4H(5,𝔽2)14yes—
467C2×C8—28no—
468C2×C8—28no—
469C2×C8—28no—
470C2×C8—432no—
471C2×C8—44no—
472C2×C8—44no—
473C2×C8—44no—
474C2×C8—44no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4—41no—
483C4⋊C4—42no—
484C4⋊C4—41no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—416no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—416no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—232no—
538C23—22no—
539C23—232no—
540C23—24no—
541C23—24no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—416no—
549C23—416no—
550C23—416no—
551C23—432no—
552C23—432no—
553C23—416no—
554C23—432no—
555C23—432no—
556C23—416no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23#38614yes—
566C23#38614yes—
567C23C2×#10314yes—
568C23C2×#10314yes—
569C23C2×#10314yes—
570C23#41614yes—
571C23#41714yes—
572C23#45014yes—
573C23#45114yes—
574C23#46518yes—
575C23#46718yes—
576C23#48314yes—
577C23#48414yes—
578C23C22×C22⋊C411yescharacteristic
579C23C2×#12011yescharacteristic
580C23C2×#12112yes—
581C23C2×#12311yescharacteristic
582C23#53412yes—
583C23C22×SD1611yescharacteristic
584C23C22×Q1611yescharacteristic
585C23C2×#13512yes—
586C23C2×AGL(1,ℤ/8)11yescharacteristic
587C23C2×#13711yescharacteristic
588C23#57512yes—
589Q8—48no—
590Q8—82no—
591Q8—82no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—216no—
601C2×C4—216no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—416no—
610C2×C4—416no—
611C2×C4—416no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—48no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—48no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—832no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—84no—
637C2×C4—82no—
638C2×C4—84no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—84no—
642C2×C4—82no—
643C2×C4—84no—
644C2×C4—82no—
645C2×C4#38614yes—
646C2×C4#38614yes—
647C2×C4C2×C4×D412yes—
648C2×C4C2×C4×D412yes—
649C2×C4#51812yes—
650C2×C4C2×#12011yescharacteristic
651C2×C4C2×#12112yes—
652C2×C4C2×#12311yescharacteristic
653C2×C4#53412yes—
654C2×C4D4214yes—
655C2×C4#54614yes—
656C2×C4#54614yes—
657C2×C4#55114yes—
658C8—48no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—28no—
664C22—28no—
665C22—28no—
666C22—28no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—28no—
672C22—28no—
673C22—28no—
674C22—28no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—24no—
681C22—24no—
682C22—464no—
683C22—42no—
684C22—42no—
685C22—44no—
686C22—44no—
687C22—42no—
688C22—44no—
689C22—44no—
690C22—42no—
691C22#187218yescomplemented
692C22#189218yescomplemented
693C22C2×#38611yescharacteristic
694C22C2×#38611yescharacteristic
695C22#230811yescharacteristic
696C22#241012yes—
697C22#241212yes—
698C22C22×#10311yescharacteristic
699C22C2×#41611yescharacteristic
700C22C2×#41711yescharacteristic
701C22#289611yescharacteristic
702C22C2×#45011yescharacteristic
703C22C2×#45111yescharacteristic
704C22#300212yes—
705C22C2×#46512yes—
706C22C2×#46712yes—
707C22#305512yes—
708C22#305612yes—
709C22C2×#48311yescharacteristic
710C22C2×#48411yescharacteristic
711C22#309112yes—
712C22#318712yes—
713C22#318812yes—
714C22#318912yes—
715C22#319012yes—
716C4—24no—
717C4—24no—
718C4—42no—
719C4—82no—
720C4—82no—
721C4—88no—
722C4—82no—
723C4—82no—
724C4#235814yes—
725C4#241012yes—
726C4#241212yes—
727C2—28no—
728C2—28no—
729C2—44no—
730C2#1413411yescharacteristic
731C2C2×#187211yescharacteristic
732C2#2090811yescharacteristic
733C2C2×#189211yescharacteristic
734C2#2097111yescharacteristic
735C2#2175911yescharacteristic
736C2#2176311yescharacteristic
737C2#2437912yes—
738C2#2438012yes—
739C2#2455112yes—
740C2#2455212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1892C2semidirect
5. C2×#1872C2semidirect
6. #11200C2semidirect
7. #10970C2semidirect
8. #10730C2semidirect
9. #10721C2semidirect
17. C22×#103C22semidirect
23. #2848C4semidirect
39. C2×#340C22semidirect
40. C2×#340C22semidirect
41. C2×#340C22semidirect
73. C2×C22⋊Q8C2×C4semidirect
74. C2×C22⋊Q8C2×C4semidirect
118. #380C2×C4semidirect
119. #380C2×C4semidirect
332. C24#103semidirect
691. C22#1872semidirect
692. C22#1892semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊C82
C2×#28482
C22×#3802
C2×#18922
C2×#18722
#112002
#109702
#107302
#107212