Finite Group Databasebeta

← #664659

Subgroups — #664659

Order 512.

4746 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic, complemented
2C2×#2435C211yesmaximal, characteristic
3C2×D4×C22⋊C4C211yesmaximal, characteristic
4C22×#390C211yesmaximal, characteristic
5#15479C218yesmaximal, complemented
6#15217C212yesmaximal, complemented
7#15136C212yesmaximal, complemented
8#14727C214yesmaximal, complemented
9#14702C211yesmaximal, characteristic, complemented
10#14027C212yesmaximal, complemented
11#13924C211yesmaximal, characteristic, complemented
12#13906C212yesmaximal, complemented
13#13877C214yesmaximal, complemented
14C4×#385C211yesmaximal, characteristic, complemented
15C24×D4C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2214yes—
23C23×C42C2211yescharacteristic
24#2502—28no—
25#2435—28no—
26C2×#398C2214yescomplemented
27C2×#397C2218yescomplemented
28C2×#394C2214yes—
29D4×C22⋊C4—28no—
30C2×#391C2212yescomplemented
31C2×#391C2212yescomplemented
32C2×#390—28no—
33C2×#390C2211yescharacteristic
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2211yescharacteristic
37C2×#390C2212yes—
38C2×#390C2214yes—
39C2×#386C2214yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#384C2212yescomplemented
44C2×#380C2211yescharacteristic
45C2×#380C2218yescomplemented
46C2×#379C2212yes—
47C2×#379C2214yes—
48C2×C4×C4⋊C4C2212yescomplemented
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51#1445C2214yescomplemented
52#1440C2218yescomplemented
53#1440C2218yescomplemented
54#1439C2218yescomplemented
55#1439C2218yescomplemented
56#1439C2218yescomplemented
57#1436C2214yescomplemented
58#1436C2214yescomplemented
59C4⋊#95C2212yescomplemented
60C4⋊#95C2212yescomplemented
61C4⋊#95C2212yescomplemented
62C22⋊#95C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2212yes—
65C22⋊#95C2218yescomplemented
66C4×#95C2212yescomplemented
67C26C2312yes—
68C23×Q8C2311yescharacteristic
69C23×D4—28no—
70C23×D4—24no—
71C23×D4—24no—
72C23×D4—44no—
73C23×D4D418yes—
74C23×D4C2311yescharacteristic
75C23×D4C2312yes—
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2314yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×#124—24no—
82C2×#124—24no—
83C2×#124—28no—
84C2×#124—44no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C4×D4—28no—
89C2×C4×D4—28no—
90C2×C4×D4—44no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4—216no—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—216no—
105C22×C22⋊C4—416no—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C42—24no—
121C22×C42—44no—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127#398—216no—
128#397—416no—
129#394—216no—
130#391—28no—
131#391—44no—
132#390—24no—
133#390—24no—
134#390—28no—
135#390—216no—
136#390—28no—
137#390—28no—
138#386—216no—
139#385—28no—
140#385—28no—
141#385—28no—
142#384—28no—
143#380—24no—
144#380—416no—
145#379—216no—
146#379—28no—
147C4×C4⋊C4—44no—
148C4×C22⋊C4—24no—
149C4×C22⋊C4D418yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2318yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2318yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C25—28no—
163C25—24no—
164C25—28no—
165C25—24no—
166C25—28no—
167C25—48no—
168C25—44no—
169C25C2×D418yes—
170C25C2412yes—
171C25C2412yes—
172C25C2411yescharacteristic
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×D4—28no—
179C22×D4—232no—
180C22×D4—232no—
181C22×D4—22no—
182C22×D4—24no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—24no—
186C22×D4—22no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—416no—
191C22×D4—432no—
192C22×D4—432no—
193C22×D4—44no—
194C22×D4C2×D414yes—
195C22×D4C2×D418yes—
196C22×D4C2×D414yes—
197C22×D4#4118yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—28no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—216no—
210C23×C4—44no—
211C23×C4—48no—
212C23×C4—44no—
213C23×C4—48no—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4#4114yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228#124—416no—
229#124—416no—
230C22⋊Q8—416no—
231C4×D4—432no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—416no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4—48no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—216no—
258C2×C22⋊C4—216no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—216no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—216no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—464no—
283C2×C22⋊C4—432no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4C2×D414yes—
288C2×C22⋊C4C2×D414yes—
289C2×C22⋊C4#4118yes—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—44no—
301C2×C42—416no—
302C2×C42C2×D418yes—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—216no—
309#95—28no—
310#95—44no—
311#95—48no—
312#95—48no—
313#95—48no—
314#95C2×D418yes—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—22no—
319C24—24no—
320C24—22no—
321C24—28no—
322C24—24no—
323C24—24no—
324C24—216no—
325C24—22no—
326C24—24no—
327C24—22no—
328C24—24no—
329C24—28no—
330C24—24no—
331C24—24no—
332C24—216no—
333C24—232no—
334C24—416no—
335C24—48no—
336C24—416no—
337C24—416no—
338C24—432no—
339C24—432no—
340C24—48no—
341C24—48no—
342C24—416no—
343C24—88no—
344C24—88no—
345C24—816no—
346C24—816no—
347C24C22×D414yes—
348C24C22×D414yes—
349C24C22×D412yes—
350C24C22×D412yes—
351C24C2×#4118yes—
352C24C2×#4112yes—
353C24C2×#4114yes—
354C242−1+414yes—
355C24C2511yescharacteristic
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×D4—216no—
362C2×D4—28no—
363C2×D4—216no—
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—216no—
369C2×D4—216no—
370C2×D4—4128no—
371C2×D4—48no—
372C2×D4—416no—
373C2×D4—416no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4#123116yes—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—216no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—432no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—416no—
433C22×C4—48no—
434C22×C4—416no—
435C22×C4—88no—
436C22×C4—816no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4#121116yes—
440C22×C4C22×D414yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4118yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4114yes—
450C22×C4H(5,𝔽2)18yes—
451C22×C4H(5,𝔽2)14yes—
452C4⋊C4—416no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—88no—
459C4⋊C4—88no—
460C22⋊C4—28no—
461C22⋊C4—432no—
462C22⋊C4—432no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—48no—
466C22⋊C4—416no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—416no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4#124116yescomplemented
474C42—216no—
475C42—48no—
476C42—44no—
477C42—48no—
478C42—44no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—24no—
483C23—28no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—216no—
496C23—28no—
497C23—216no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—28no—
502C23—28no—
503C23—432no—
504C23—464no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—48no—
511C23—48no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—48no—
518C23—44no—
519C23—44no—
520C23—48no—
521C23—48no—
522C23—48no—
523C23—432no—
524C23—832no—
525C23—816no—
526C23—864no—
527C23—832no—
528C23—832no—
529C23C2×#12114yes—
530C23C2×#12318yes—
531C23C2×#12414yes—
532C23#53614yes—
533C23#54718yes—
534C23#55618yes—
535C23C23×D411yescharacteristic
536C23C23×D411yescharacteristic
537C23C22×#4111yescharacteristic
538C23C22×#4112yes—
539C23C22×#4112yes—
540C23C2×H(5,𝔽2)12yes—
541C23C2×H(5,𝔽2)14yes—
542C23C2×2−1+412yes—
543Q8—88no—
544D4—232no—
545D4—432no—
546D4—416no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—232no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—432no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—832no—
599C2×C4—816no—
600C2×C4C2×#12114yes—
601C2×C4C2×#12114yes—
602C2×C4C2×#12314yes—
603C2×C4C2×#12418yes—
604C2×C4#53414yes—
605C2×C4#54618yes—
606C2×C4#55118yes—
607C22—28no—
608C22—216no—
609C22—24no—
610C22—28no—
611C22—28no—
612C22—28no—
613C22—24no—
614C22—28no—
615C22—216no—
616C22—48no—
617C22—416no—
618C22—44no—
619C22—48no—
620C22—48no—
621C22—44no—
622C22—48no—
623C22—44no—
624C22—44no—
625C22—48no—
626C22—44no—
627C22—864no—
628C22—88no—
629C22—84no—
630C22—84no—
631C22#254718yes—
632C22C22×#12111yescharacteristic
633C22C22×#12311yescharacteristic
634C22C22×#12411yescharacteristic
635C22C2×#53411yescharacteristic
636C22C2×#53611yescharacteristic
637C22C2×#54612yes—
638C22C2×#54712yes—
639C22C2×#55112yes—
640C22C2×#55612yes—
641C22#348114yes—
642C22#348412yes—
643C22#349812yes—
644C22#350812yes—
645C22#351914yes—
646C22#352014yes—
647C22#353614yes—
648C4—24no—
649C4—28no—
650C4—48no—
651C4—44no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—84no—
656C4—82no—
657C4—84no—
658C4#241218yes—
659C2—28no—
660C2—48no—
661C2—84no—
662C2C2×#241211yescharacteristic
663C2C2×#254711yescharacteristic
664C2#3552811yescharacteristic
665C2#3623712yes—
666C2#3628912yes—
667C2#3807812yes—
668C2#3891612yes—
669C2#5942414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2502C2semidirect
5. #15479C2semidirect
6. #15217C2semidirect
7. #15136C2semidirect
8. #14727C2semidirect
9. #14702C2semidirect
10. #14027C2semidirect
11. #13924C2semidirect
12. #13906C2semidirect
13. #13877C2semidirect
14. C4×#385C2semidirect
26. C2×#398C22semidirect
27. C2×#397C22semidirect
30. C2×#391C22semidirect
31. C2×#391C22semidirect
43. C2×#384C22semidirect
45. C2×#380C22semidirect
48. C2×C4×C4⋊C4C22semidirect
51. #1445C22semidirect
52. #1440C22semidirect
53. #1440C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1436C22semidirect
58. #1436C22semidirect
59. C4⋊#95C22semidirect
60. C4⋊#95C22semidirect
61. C4⋊#95C22semidirect
65. C22⋊#95C22semidirect
66. C4×#95C22semidirect
473. C22⋊C4#124semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
C2×#24352
C2×D4×C22⋊C42
C22×#3902
#154792
#152172
#151362
#147272
#147022
#140272
#139242
#139062
#138772
C4×#3852