Finite Group Databasebeta

← #641060

Subgroups — #641060

Order 512.

4174 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2483C212yesmaximal
2C22×#385C211yesmaximal, characteristic
3C22×#380C211yesmaximal, characteristic
4#16815C214yesmaximal, complemented
5#16399C214yesmaximal, complemented
6#15073C212yesmaximal, complemented
7#15019C212yesmaximal, complemented
8#14935C214yesmaximal, complemented
9#14821C212yesmaximal, complemented
10#14786C212yesmaximal, complemented
11#14680C211yesmaximal, characteristic
12#14339C212yesmaximal, complemented
13#14275C212yesmaximal, complemented
14#13869C212yesmaximal
15C25×C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23#2483—216no—
24C2×#398C2214yescomplemented
25C2×#397C2218yes—
26C2×#385—28no—
27C2×#385C2212yes—
28C2×#385C2212yes—
29C2×#385C2214yes—
30C2×#385C2212yes—
31C2×#385C2214yes—
32C2×#385C2212yes—
33C2×#381C2214yes—
34C2×#381C2214yes—
35C2×#381C2214yes—
36C2×#380—28no—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#379C2212yes—
45C22×#95C2212yes—
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48#1445C2214yescomplemented
49#1444C2214yescomplemented
50#1443C2218yescomplemented
51#1442C2214yescomplemented
52#1442C2214yescomplemented
53#1441C2218yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2218yescomplemented
57#1441C2214yescomplemented
58#1441C2218yescomplemented
59#1440C2218yes—
60C22⋊#95C2214yes—
61C22⋊#95C2214yes—
62C22⋊#95C2214yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C26C2311yescharacteristic
67C24×C4—28no—
68C24×C4—44no—
69C24×C4—44no—
70C24×C4D418yes—
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×C4⋊C4—24no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—28no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2318yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—28no—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—28no—
107C22×C22⋊C4—44no—
108C22×C22⋊C4D418yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118#398—48no—
119#397—232no—
120#385—216no—
121#385—28no—
122#385—28no—
123#385—28no—
124#385—28no—
125#385—216no—
126#381—216no—
127#381—216no—
128#381—48no—
129#380—28no—
130#380—28no—
131#380—28no—
132#380—28no—
133#380—28no—
134#380—44no—
135#380—44no—
136#379—28no—
137C2×#95—28no—
138C2×#95—24no—
139C2×#95—24no—
140C2×#95C2318yes—
141C2×#95C2318yes—
142C2×#95C2318yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2318yes—
152C2×#95C2318yes—
153C2×#95C2314yes—
154C2×#95C2318yes—
155C2×#95C2318yes—
156C25—24no—
157C25—24no—
158C25—24no—
159C25—24no—
160C25—42no—
161C25—42no—
162C25C2×D414yes—
163C25C2×D414yes—
164C25#4114yes—
165C25C2411yescharacteristic
166C25C2411yescharacteristic
167C25C2411yescharacteristic
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—232no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—28no—
192C23×C4—28no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—416no—
198C23×C4—42no—
199C23×C4—416no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4—44no—
214C23×C4—464no—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4#4118yes—
224C23×C4C2418yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—216no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—416no—
254C2×C4⋊C4—416no—
255C2×C4⋊C4—416no—
256C2×C4⋊C4—416no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—48no—
261C2×C22⋊C4—232no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—416no—
284C2×C22⋊C4—432no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4C2×D418yes—
288C2×C22⋊C4C2×D418yes—
289#95—28no—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—216no—
296#95—216no—
297#95—216no—
298#95—28no—
299#95—28no—
300#95—48no—
301#95—44no—
302#95—48no—
303#95—48no—
304#95—48no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—28no—
314C24—28no—
315C24—28no—
316C24—28no—
317C24—216no—
318C24—216no—
319C24—216no—
320C24—216no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—44no—
328C24—44no—
329C24—48no—
330C24—48no—
331C24—416no—
332C24—416no—
333C24—416no—
334C24—416no—
335C24#121116yes—
336C24C22×D412yes—
337C24C22×D412yes—
338C24C22×D412yes—
339C24C22×D412yes—
340C24C22×D412yes—
341C24C22×D412yes—
342C24C2×#4112yes—
343C24C2×#4112yes—
344C24C2×#4112yes—
345C24C2×#4114yes—
346C24H(5,𝔽2)14yes—
347C242−1+414yes—
348C242−1+414yes—
349C24C2511yescharacteristic
350C22×C4—232no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—216no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—28no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—28no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—416no—
414C22×C4—48no—
415C22×C4—416no—
416C22×C4—416no—
417C22×C4—416no—
418C22×C4—4128no—
419C22×C4—416no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—48no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4—88no—
444C22×C4#123116yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)14yes—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—88no—
463C4⋊C4—816no—
464C22⋊C4—216no—
465C22⋊C4—216no—
466C22⋊C4—48no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—48no—
470C22⋊C4—416no—
471C22⋊C4—416no—
472C22⋊C4—88no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—24no—
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—232no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—232no—
511C23—416no—
512C23—416no—
513C23—416no—
514C23—416no—
515C23—464no—
516C23—416no—
517C23—464no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23C2×#12114yes—
523C23C2×#12114yes—
524C23C2×#12114yes—
525C23C2×#12314yes—
526C23#53414yes—
527C23#53614yes—
528C23#53614yes—
529C23#53718yes—
530C23#53918yes—
531C23#54718yes—
532C23#55218yes—
533C23C23×D411yescharacteristic
534C23C23×D411yescharacteristic
535C23C22×#4111yescharacteristic
536C23C22×#4112yes—
537C23C2×H(5,𝔽2)12yes—
538C23C2×H(5,𝔽2)12yes—
539C23C2×2−1+412yes—
540C23C2×2−1+412yes—
541C23#58512yes—
542C2×C4—232no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—416no—
560C2×C4—416no—
561C2×C4—416no—
562C2×C4—416no—
563C2×C4—416no—
564C2×C4—416no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—416no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—88no—
599C2×C4—88no—
600C2×C4—816no—
601C2×C4—816no—
602C2×C4C2×#12314yes—
603C2×C4C2×#12314yes—
604C2×C4#53414yes—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—216no—
615C22—216no—
616C22—24no—
617C22—24no—
618C22—216no—
619C22—216no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—464no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—48no—
629C22—42no—
630C22—48no—
631C22#248618yes—
632C22#261818yes—
633C22C22×#12111yescharacteristic
634C22C22×#12311yescharacteristic
635C22C2×#53411yescharacteristic
636C22C2×#53411yescharacteristic
637C22C2×#53611yescharacteristic
638C22C2×#53611yescharacteristic
639C22C2×#53712yes—
640C22#344911yescharacteristic
641C22C2×#53912yes—
642C22C2×#54712yes—
643C22C2×#55212yes—
644C22#349014yes—
645C22#349514yes—
646C22#349612yes—
647C22#350512yes—
648C22#353514yes—
649C22#353814yes—
650C4—24no—
651C4—24no—
652C4—24no—
653C4—42no—
654C4—44no—
655C4—44no—
656C4—82no—
657C4—82no—
658C4—84no—
659C4—88no—
660C4—84no—
661C2—28no—
662C2—28no—
663C2—44no—
664C2C2×#261811yescharacteristic
665C2C2×#248611yescharacteristic
666C2#3499211yescharacteristic
667C2#3500011yescharacteristic
668C2#3531111yescharacteristic
669C2#3744111yescharacteristic
670C2#3745311yescharacteristic
671C2#3786114yes—
672C2#3788014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #16815C2semidirect
5. #16399C2semidirect
6. #15073C2semidirect
7. #15019C2semidirect
8. #14935C2semidirect
9. #14821C2semidirect
10. #14786C2semidirect
12. #14339C2semidirect
13. #14275C2semidirect
24. C2×#398C22semidirect
48. #1445C22semidirect
49. #1444C22semidirect
50. #1443C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24832
C22×#3852
C22×#3802
#168152
#163992
#150732
#150192
#149352
#148212
#147862
#146802
#143392
#142752
#138692