Finite Group Databasebeta

← #205751

Subgroups — #205751

Order 512.

1352 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3050C211yesmaximal, characteristic
2#21874C212yesmaximal, complemented
3C2×#1807C211yesmaximal, characteristic, complemented
4#20385C211yesmaximal, characteristic, complemented
5#17592C212yesmaximal, complemented
6#9785C212yesmaximal, complemented
7#9784C212yesmaximal, complemented
8#9781C214yesmaximal, complemented
9C22×#135C411yescharacteristic, complemented
10C24×C8C2211yescharacteristic
11C2×#525C2212yes—
12#3050—21no—
13#3050—21no—
14#3050—21no—
15#3050—21no—
16#3050—21no—
17#3050—21no—
18#3050—21no—
19#3050—21no—
20C2×#467C411yescharacteristic, complemented
21C2×#467C2211yescharacteristic
22C2×#466C411yescharacteristic, complemented
23C2×#466C2211yescharacteristic
24C2×#465C412yescomplemented
25C2×#465C2212yes—
26C2×#427C411yescharacteristic, complemented
27C22×#106—21no—
28C2×#416C412yescomplemented
29C2×#407C2211yescharacteristic
30C2×#405—21no—
31#2280C2211yescharacteristic
32C2×#380C2212yes—
33#1996C2214yes—
34#1964C2214yes—
35#1898—24no—
36#1807—21no—
37#1807—21no—
38#1807C2212yes—
39#1807C2212yes—
40#1799—24no—
41#1752—21no—
42#1752C2212yes—
43C8×C22⋊C4—21no—
44C8×C22⋊C4C2212yes—
45C2×#340—21no—
46#1729—21no—
47#1520C2214yes—
48#1519C2214yes—
49#1513—24no—
50#1504—24no—
51C22×#41C2×C412yes—
52C24×C4C2311yescharacteristic
53C2×#135—21no—
54C2×#135—21no—
55C2×#135—21no—
56C2×#135—21no—
57C2×#135—44no—
58C22×Q16C2×C411yescharacteristic, complemented
59C22×SD16C2×C412yescomplemented
60C22×D8C2×C411yescharacteristic, complemented
61C22×M16—21no—
62C23×C8—44no—
63C23×C8C2×C411yescharacteristic
64C23×C8C2×C411yescharacteristic
65C23×C8C2×C411yescharacteristic
66C23×C8C2×C411yescharacteristic
67C23×C8D411yescharacteristic
68C23×C8D411yescharacteristic
69C23×C8D411yescharacteristic
70C23×C8D411yescharacteristic
71C23×C8D411yescharacteristic
72C23×C8D411yescharacteristic
73C23×C8D411yescharacteristic
74C23×C8D411yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77#525—22no—
78#525—22no—
79#525—22no—
80#525—22no—
81#525—22no—
82#525—22no—
83#525—22no—
84#525—22no—
85C2×#123—22no—
86C2×C22⋊Q8C2×C412yes—
87C2×C22⋊Q8C2312yes—
88C2×#121C2×C412yes—
89C2×#121C2312yes—
90C2×#120—22no—
91C2×C4×D4—22no—
92C2×C4×D4—22no—
93C2×#117—21no—
94C2×#117C2×C412yes—
95C22×C4⋊C4—22no—
96C22×C22⋊C4—22no—
97#467—21no—
98#467—21no—
99#467—21no—
100#467—21no—
101#467—21no—
102#467—21no—
103#467—21no—
104#467—21no—
105#466—21no—
106#466—21no—
107#466—21no—
108#466—21no—
109#466—21no—
110#466—21no—
111#466—21no—
112#466—21no—
113#465—22no—
114#465—22no—
115#465—22no—
116#465—22no—
117#465—22no—
118#465—22no—
119#465—22no—
120#465—22no—
121#427—21no—
122#427—21no—
123#427—21no—
124#427—21no—
125C2×#107C2×C411yescharacteristic, complemented
126C2×#107C2×C411yescharacteristic, complemented
127C2×#106—22no—
128C2×#106—22no—
129C2×#106—44no—
130C2×#106C2×C412yescomplemented
131#416—22no—
132#416—22no—
133#416—22no—
134#416—22no—
135C2×#103C2×C412yescomplemented
136C2×#103C2×C412yescomplemented
137C2×#102C2×C412yescomplemented
138C2×#102C2×C412yescomplemented
139C2×#99—28no—
140#407—21no—
141#407—21no—
142#407D412yes—
143#407D412yes—
144C2×C22⋊C8—21no—
145C2×C22⋊C8C2312yes—
146#405—44no—
147C2×#97—22no—
148C2×C4×C8—22no—
149#381—21no—
150#380—21no—
151#380—22no—
152#380—22no—
153#380—22no—
154#380D412yes—
155#380C2314yes—
156#379D412yes—
157C4×C22⋊C4—21no—
158C4×C22⋊C4C2312yes—
159C2×#95—22no—
160#342—28no—
161#340—22no—
162#340—22no—
163#336—22no—
164#336—22no—
165#328—28no—
166#327—28no—
167C25C2×D411yescharacteristic
168C2×#41—22no—
169C2×#41—22no—
170C2×#41—22no—
171C2×#41—22no—
172C2×#41—44no—
173C2×#41—44no—
174C22×Q8C22×C412yes—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4C22×C412yes—
179C23×C4—22no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4#4111yescharacteristic
193C23×C4#4111yescharacteristic
194C23×C4#4111yescharacteristic
195C23×C4#4111yescharacteristic
196C23×C4C2411yescharacteristic
197#135—44no—
198#135—44no—
199#135—44no—
200#135—44no—
201C2×Q16—21no—
202C2×Q16—21no—
203C2×Q16—42no—
204C2×SD16—22no—
205C2×SD16—22no—
206C2×SD16—44no—
207C2×D8—21no—
208C2×D8—21no—
209C2×D8—42no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—42no—
213C2×M16—42no—
214C22×C8—24no—
215C22×C8—24no—
216C22×C8—24no—
217C22×C8—22no—
218C22×C8—42no—
219C22×C8—42no—
220C22×C8—42no—
221C22×C8—42no—
222C22×C8—42no—
223C22×C8—42no—
224C22×C8—42no—
225C22×C8—42no—
226C22×C8—42no—
227C22×C8—42no—
228C22×C8—42no—
229C22×C8C22⋊C411yescharacteristic
230C22×C8C22⋊C411yescharacteristic
231C22×C8C22⋊C411yescharacteristic
232C22×C8C22⋊C411yescharacteristic
233C22×C8C22⋊C411yescharacteristic
234C22×C8C22⋊C411yescharacteristic
235C22×C8C22⋊C411yescharacteristic
236C22×C8C22⋊C411yescharacteristic
237C22×C8C22⋊C411yescharacteristic
238C22×C8C22⋊C411yescharacteristic
239C22×C8C22⋊C411yescharacteristic
240C22×C8C22⋊C411yescharacteristic
241C22×C8C22⋊C411yescharacteristic
242C22×C8C22⋊C411yescharacteristic
243C22×C8C22⋊C411yescharacteristic
244C22×C8C22⋊C411yescharacteristic
245C22×C8C22×C411yescharacteristic
246C22×C8C22×C411yescharacteristic
247C22×C8C22×C411yescharacteristic
248C22×C8C22×C411yescharacteristic
249C22×C8C2×D411yescharacteristic
250C22×C8C2×D411yescharacteristic
251C22×C8C2×D411yescharacteristic
252C22×C8C2×D411yescharacteristic
253C22×C8C2×D411yescharacteristic
254C22×C8C2×D411yescharacteristic
255C22×C8C2×D411yescharacteristic
256C22×C8C2×D411yescharacteristic
257#123—42no—
258#123—42no—
259#123—42no—
260#123—42no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269#121—22no—
270#121—22no—
271#121—22no—
272#121—22no—
273#121—22no—
274#121—22no—
275#121—22no—
276#121—22no—
277#120—42no—
278#120—42no—
279#120—42no—
280#120—42no—
281C4×D4—42no—
282C4×D4—42no—
283C4×D4—42no—
284C4×D4—42no—
285C4×D4—42no—
286C4×D4—42no—
287C4×D4—42no—
288C4×D4—42no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—42no—
294#117—42no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4C22×C412yes—
303C2×C4⋊C4C22×C412yes—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—84no—
314C2×C22⋊C4C2×D414yes—
315C2×C42—22no—
316C2×C42—22no—
317#107—21no—
318#107—21no—
319#107—21no—
320#107—21no—
321#106—22no—
322#106—22no—
323#106—44no—
324#106—44no—
325#103—22no—
326#103—22no—
327#103—22no—
328#103—22no—
329#102—22no—
330#102—22no—
331#102—22no—
332#102—22no—
333#99—88no—
334C22⋊C8—22no—
335C22⋊C8—22no—
336C22⋊C8—22no—
337C22⋊C8C2×D414yes—
338#97—44no—
339C4×C8—44no—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—42no—
344C24—24no—
345C24—22no—
346C24—42no—
347C24—42no—
348C24C4×D411yescharacteristic
349C24C4×D411yescharacteristic
350C24#12111yescharacteristic
351C24#12111yescharacteristic
352C24#12711yescharacteristic
353C24#12711yescharacteristic
354C24C22×D411yescharacteristic
355#41—44no—
356#41—44no—
357#41—44no—
358#41—44no—
359#41—84no—
360#41—84no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—84no—
378C2×D4—84no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—44no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—84no—
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C4×D411yescharacteristic
416C22×C4C4×D411yescharacteristic
417C22×C4C4×D411yescharacteristic
418C22×C4C4×D411yescharacteristic
419C22×C4C4×D411yescharacteristic
420C22×C4C4×D411yescharacteristic
421C22×C4#12111yescharacteristic
422C22×C4#12111yescharacteristic
423C22×C4#12111yescharacteristic
424C22×C4#12111yescharacteristic
425C22×C4#12111yescharacteristic
426C22×C4#12111yescharacteristic
427C22×C4#12411yescharacteristic
428C22×C4#12411yescharacteristic
429C22×C4#12411yescharacteristic
430C22×C4#12411yescharacteristic
431C22×C4#12711yescharacteristic
432C22×C4#12711yescharacteristic
433C22×C4C23×C411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439Q16—42no—
440Q16—42no—
441SD16—44no—
442SD16—44no—
443D8—42no—
444D8—42no—
445M16—44no—
446M16—44no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8#9914yescomplemented
478C2×C8C2×C22⋊C411yescharacteristic
479C2×C8C2×C22⋊C411yescharacteristic
480C2×C8C2×C22⋊C411yescharacteristic
481C2×C8C2×C22⋊C411yescharacteristic
482C2×C8C2×C22⋊C411yescharacteristic
483C2×C8C2×C22⋊C411yescharacteristic
484C2×C8C2×C22⋊C411yescharacteristic
485C2×C8C2×C22⋊C411yescharacteristic
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—84no—
498C22⋊C4—28no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—48no—
512C22⋊C4—84no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—42no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—416no—
538C23—42no—
539C23—44no—
540C23—82no—
541C23—82no—
542C23#39011yescharacteristic
543C23#39011yescharacteristic
544C23#39011yescharacteristic
545C23#39011yescharacteristic
546C23C2×C4×D411yescharacteristic
547C23C2×#12111yescharacteristic
548C23C2×#12711yescharacteristic
549C23#57714yes—
550Q8—42no—
551Q8—42no—
552Q8—82no—
553D4—42no—
554D4—42no—
555D4—82no—
556D4—84no—
557D4—84no—
558D4—82no—
559D4—82no—
560D4—84no—
561D4—84no—
562D4—84no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—168no—
611C2×C4#39011yescharacteristic
612C2×C4#39011yescharacteristic
613C2×C4#39011yescharacteristic
614C2×C4#39011yescharacteristic
615C2×C4C2×#9912yes—
616C2×C4#41012yes—
617C2×C4C22×C22⋊C411yescharacteristic
618C2×C4C2×C4×D411yescharacteristic
619C2×C4C2×#12111yescharacteristic
620C2×C4C2×#12411yescharacteristic
621C8—21no—
622C8—21no—
623C8—21no—
624C8—21no—
625C8—21no—
626C8—21no—
627C8—44no—
628C8C2×#9912yes—
629C8C2×#9912yes—
630C22—28no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—164no—
657C22#196412yes—
658C22#197612yes—
659C22C2×#39011yescharacteristic
660C22#295212yes—
661C22#306012yes—
662C22#315412yes—
663C4—21no—
664C4—21no—
665C4—21no—
666C4—21no—
667C4—21no—
668C4—21no—
669C4—84no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—164no—
674C4#196412yes—
675C4C22×#9911yescharacteristic
676C4C2×#41011yescharacteristic
677C2—22no—
678C2—22no—
679C2—22no—
680C2—22no—
681C2—22no—
682C2—41no—
683C2—41no—
684C2—162no—
685C2C2×#196411yescharacteristic
686C2#2122911yescharacteristic
687C2C2×#197611yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#3050C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21874C2semidirect
3. C2×#1807C2semidirect
4. #20385C2semidirect
5. #17592C2semidirect
6. #9785C2semidirect
7. #9784C2semidirect
8. #9781C2semidirect
9. C22×#135C4semidirect
20. C2×#467C4semidirect
22. C2×#466C4semidirect
24. C2×#465C4semidirect
26. C2×#427C4semidirect
28. C2×#416C4semidirect
58. C22×Q16C2×C4semidirect
59. C22×SD16C2×C4semidirect
60. C22×D8C2×C4semidirect
125. C2×#107C2×C4semidirect
126. C2×#107C2×C4semidirect
130. C2×#106C2×C4semidirect
135. C2×#103C2×C4semidirect
136. C2×#103C2×C4semidirect
137. C2×#102C2×C4semidirect
138. C2×#102C2×C4semidirect
477. C2×C8#99semidirect

Maximal subgroups

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

subgroupindex
C2×#30502
#218742
C2×#18072
#203852
#175922
#97852
#97842
#97812