Finite Group Databasebeta

← #4930623

Subgroups — #4930623

Order 512.

6448 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
1C22×#518C211yesmaximal, characteristic, complemented
2C2×#2544C212yesmaximal, complemented
3C2×#2435C211yesmaximal, characteristic, complemented
4C2×#2433C211yesmaximal, characteristic
5C2×#2620C211yesmaximal, characteristic, complemented
6#14732C218yesmaximal, complemented
7#14725C218yesmaximal, complemented
8#14138C211yesmaximal, characteristic, complemented
9#14136C211yesmaximal, characteristic, complemented
10#14027C212yesmaximal, complemented
11#14025C212yesmaximal, complemented
12#13906C212yesmaximal, complemented
13#13852C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C22×#127C2212yes—
16C22×#124C2212yes—
17C22×#124C2211yescharacteristic, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×#120C2214yes—
21C2×#518—216no—
22C2×#518—28no—
23C22×C4×D4C2214yes—
24C22×C4×D4C2212yes—
25C22×C4×D4C2212yes—
26C22×#117C2211yescharacteristic
27C22×#117C2211yescharacteristic
28C23×C22⋊C4C2214yes—
29#2620—28no—
30#2544—216no—
31#2435—28no—
32#2433—28no—
33C2×#396C2214yescomplemented
34C2×#394C2214yes—
35C2×#394C2214yes—
36C2×#394C2218yescomplemented
37C2×#392C2214yes—
38C2×#391C2211yescharacteristic, complemented
39C2×#391C2212yescomplemented
40C2×#390C2212yes—
41C2×#390C2212yes—
42C2×#390C2211yescharacteristic, complemented
43C2×#390C2212yescomplemented
44C2×#386C2218yescomplemented
45C2×#386C2218yescomplemented
46C2×#384C2211yescharacteristic, complemented
47C2×#380C2218yescomplemented
48C2×#379C2214yes—
49C2×C4×C4⋊C4C2212yescomplemented
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#1441C2214yescomplemented
53#1440C2214yes—
54#1440C2218yescomplemented
55#1439C2214yes—
56#1439C2218yescomplemented
57#1438C2214yes—
58#1438C2218yescomplemented
59#1437C2214yescomplemented
60#1436C2214yescomplemented
61C4⋊#95C2212yescomplemented
62C4⋊#95C2212yescomplemented
63C22⋊#95C2218yescomplemented
64C4×#95C2212yescomplemented
65C26C2312yes—
66C23×Q8C2311yescharacteristic
67C23×D4—28no—
68C23×D4—28no—
69C23×D4—24no—
70C23×D4—24no—
71C23×D4D418yes—
72C23×D4C2311yescharacteristic
73C23×D4C2312yes—
74C23×D4C2318yes—
75C23×D4C2312yes—
76C23×D4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2314yes—
79C24×C4C2314yes—
80C2×#127—28no—
81C2×#127—28no—
82C2×#127—28no—
83C2×#124—28no—
84C2×#124—28no—
85C2×#124—28no—
86C2×#124—24no—
87C2×#124—28no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×#121—24no—
92C2×#121—24no—
93C2×#121—24no—
94C2×#120—216no—
95C2×#120—416no—
96#518—2128no—
97C2×C4×D4—28no—
98C2×C4×D4—216no—
99C2×C4×D4—28no—
100C2×C4×D4—216no—
101C2×C4×D4—216no—
102C2×C4×D4—28no—
103C2×C4×D4—28no—
104C2×C4×D4—28no—
105C2×C4×D4D4116yescomplemented
106C2×#117—24no—
107C2×#117—28no—
108C2×#117—28no—
109C2×#117D418yescomplemented
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C22⋊C4—216no—
118C22×C22⋊C4—216no—
119C22×C22⋊C4—216no—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2318yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2318yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2318yes—
132C22×C22⋊C4C2318yes—
133C22×C22⋊C4C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#396—216no—
140#394—216no—
141#394—216no—
142#394—232no—
143#392—216no—
144#391—28no—
145#391—24no—
146#390—28no—
147#390—28no—
148#390—28no—
149#390—24no—
150#386—232no—
151#386—416no—
152#384—24no—
153#380—416no—
154#379—216no—
155C4×C4⋊C4—28no—
156C4×C22⋊C4—24no—
157C4×C22⋊C4—24no—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2318yes—
162C2×#95C2314yes—
163C2×#95C2318yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2318yes—
167C25—28no—
168C25—24no—
169C25—28no—
170C25—24no—
171C25—48no—
172C25—48no—
173C25C2×D418yes—
174C25C2412yes—
175C25C2411yescharacteristic
176C25C2414yes—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—24no—
180C22×Q8—22no—
181C22×Q8—24no—
182C22×D4—232no—
183C22×D4—232no—
184C22×D4—264no—
185C22×D4—28no—
186C22×D4—28no—
187C22×D4—264no—
188C22×D4—24no—
189C22×D4—216no—
190C22×D4—216no—
191C22×D4—216no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—232no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4—24no—
206C22×D4—216no—
207C22×D4—48no—
208C22×D4—48no—
209C22×D4—48no—
210C22×D4—48no—
211C22×D4C2×D418yes—
212C22×D4C2×D418yes—
213C22×D4C2×D418yes—
214C22×D4#4118yes—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—28no—
219C23×C4—28no—
220C23×C4—28no—
221C23×C4—28no—
222C23×C4—48no—
223C23×C4—48no—
224C23×C4—48no—
225C23×C4—48no—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D4116yes—
229C23×C4#4114yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240#127—432no—
241#124—432no—
242#124—416no—
243C22⋊Q8—416no—
244#121—416no—
245#120—464no—
246C4×D4—2128no—
247C4×D4—264no—
248C4×D4—264no—
249#117—232no—
250#117—232no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—48no—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2×D418yes—
271C2×C22⋊C4—2128no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—216no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—216no—
287C2×C22⋊C4—216no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—216no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—216no—
298C2×C22⋊C4—216no—
299C2×C22⋊C4—216no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4—48no—
305C2×C22⋊C4—48no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4C2×D418yes—
309C2×C22⋊C4C2×D4116yes—
310C2×C42—28no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42C2×D414yes—
320C2×C42C2×D418yes—
321C2×C42C2×D414yes—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—216no—
326#95—28no—
327#95—216no—
328#95—28no—
329#95—44no—
330#95—48no—
331C24—216no—
332C24—216no—
333C24—232no—
334C24—22no—
335C24—24no—
336C24—24no—
337C24—22no—
338C24—24no—
339C24—24no—
340C24—28no—
341C24—24no—
342C24—28no—
343C24—24no—
344C24—28no—
345C24—28no—
346C24—232no—
347C24—216no—
348C24—22no—
349C24—24no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—28no—
354C24—28no—
355C24—416no—
356C24—416no—
357C24—816no—
358C24—816no—
359C24—88no—
360C24—88no—
361C24—88no—
362C24—88no—
363C24—816no—
364C24—816no—
365C24C22×D412yes—
366C24C22×D418yes—
367C24C22×D412yes—
368C24C22×D418yes—
369C24C2×#4112yes—
370C24C2×#4118yes—
371C24C2511yescharacteristic
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—48no—
375C2×Q8—44no—
376C2×Q8—48no—
377C2×D4—216no—
378C2×D4—216no—
379C2×D4—232no—
380C2×D4—2256no—
381C2×D4—232no—
382C2×D4—216no—
383C2×D4—216no—
384C2×D4—216no—
385C2×D4—216no—
386C2×D4—216no—
387C2×D4—216no—
388C2×D4—216no—
389C2×D4—432no—
390C2×D4—432no—
391C2×D4—432no—
392C2×D4—416no—
393C2×D4—48no—
394C2×D4—432no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—48no—
398C2×D4—44no—
399C2×D4—48no—
400C2×D4—816no—
401C2×D4—816no—
402C2×D4—816no—
403C2×D4—88no—
404C2×D4—88no—
405C2×D4#121132yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—216no—
412C22×C4—28no—
413C22×C4—216no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—232no—
429C22×C4—232no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—28no—
446C22×C4—24no—
447C22×C4—28no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—432no—
451C22×C4—816no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4—816no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4—816no—
458C22×C4#121116yes—
459C22×C4C22×D418yes—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D414yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4112yes—
472C22×C4H(5,𝔽2)18yes—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—416no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—88no—
480C22⋊C4—216no—
481C22⋊C4—232no—
482C22⋊C4—216no—
483C22⋊C4—216no—
484C22⋊C4—232no—
485C22⋊C4—432no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—432no—
489C22⋊C4—416no—
490C22⋊C4—48no—
491C22⋊C4—816no—
492C22⋊C4—88no—
493C22⋊C4—816no—
494C42—216no—
495C42—28no—
496C42—48no—
497C42—48no—
498C42#120116yescomplemented
499C23—264no—
500C23—22no—
501C23—28no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—22no—
506C23—24no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—24no—
511C23—216no—
512C23—24no—
513C23—216no—
514C23—216no—
515C23—216no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—216no—
521C23—216no—
522C23—48no—
523C23—44no—
524C23—48no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—48no—
531C23—48no—
532C23—48no—
533C23—44no—
534C23—48no—
535C23—44no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—48no—
544C23—48no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—48no—
550C23—48no—
551C23—816no—
552C23—864no—
553C23—816no—
554C23—832no—
555C23—832no—
556C23—864no—
557C23C2×#12114yes—
558C23C2×#121116yes—
559C23C23×D412yes—
560C23C23×D411yescharacteristic
561C23C23×D411yescharacteristic
562C23C22×#4111yescharacteristic
563C23C22×#4112yes—
564C23C2×H(5,𝔽2)14yes—
565C23#58514yes—
566Q8—88no—
567D4—264no—
568D4—264no—
569D4—864no—
570D4—816no—
571D4—88no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—232no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—216no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—832no—
619C2×C4—832no—
620C2×C4C2×#12018yes—
621C2×C4C2×#12014yes—
622C2×C4C2×#12114yes—
623C2×C4C2×#12118yes—
624C2×C4C2×#12114yes—
625C2×C4#52514yes—
626C2×C4#52518yes—
627C22—24no—
628C22—216no—
629C22—24no—
630C22—232no—
631C22—24no—
632C22—48no—
633C22—416no—
634C22—48no—
635C22—48no—
636C22—48no—
637C22—44no—
638C22—48no—
639C22—48no—
640C22—48no—
641C22—44no—
642C22—416no—
643C22—416no—
644C22—48no—
645C22—864no—
646C22—82no—
647C22—82no—
648C22—88no—
649C22—84no—
650C22—82no—
651C22—84no—
652C22—88no—
653C22C22×#12011yescharacteristic
654C22C22×#12111yescharacteristic
655C22C22×#12112yes—
656C22C2×#52511yescharacteristic
657C22C2×#52512yes—
658C22#348412yes—
659C22#348512yes—
660C22#348814yes—
661C22#349614yes—
662C22#349812yes—
663C22#350012yes—
664C22#350812yes—
665C22#351012yes—
666C22#351314yes—
667C22#351414yes—
668C4—24no—
669C4—28no—
670C4—48no—
671C4—48no—
672C4—44no—
673C4—82no—
674C4—84no—
675C4—82no—
676C4—82no—
677C4#284718yes—
678C2—416no—
679C2—44no—
680C2—162no—
681C2#3551312yes—
682C2#3552811yescharacteristic
683C2#3553612yes—
684C2#3561711yescharacteristic
685C2#3576114yes—
686C2#3580814yes—
687C2C2×#284711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2433C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#518C2semidirect
2. C2×#2544C2semidirect
3. C2×#2435C2semidirect
5. C2×#2620C2semidirect
6. #14732C2semidirect
7. #14725C2semidirect
8. #14138C2semidirect
9. #14136C2semidirect
10. #14027C2semidirect
11. #14025C2semidirect
12. #13906C2semidirect
13. #13852C2semidirect
17. C22×#124C22semidirect
33. C2×#396C22semidirect
36. C2×#394C22semidirect
38. C2×#391C22semidirect
39. C2×#391C22semidirect
42. C2×#390C22semidirect
43. C2×#390C22semidirect
44. C2×#386C22semidirect
45. C2×#386C22semidirect
46. C2×#384C22semidirect
47. C2×#380C22semidirect
49. C2×C4×C4⋊C4C22semidirect
52. #1441C22semidirect
54. #1440C22semidirect
56. #1439C22semidirect
58. #1438C22semidirect
59. #1437C22semidirect
60. #1436C22semidirect
61. C4⋊#95C22semidirect
62. C4⋊#95C22semidirect
63. C22⋊#95C22semidirect
64. C4×#95C22semidirect
105. C2×C4×D4D4semidirect
109. C2×#117D4semidirect
498. C42#120semidirect

Maximal subgroups

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

subgroupindex
C22×#5182
C2×#25442
C2×#24352
C2×#24332
C2×#26202
#147322
#147252
#141382
#141362
#140272
#140252
#139062
#138522