Finite Group Databasebeta

← #620105

Subgroups — #620105

Order 512.

3594 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic, complemented
2C22×#380C211yesmaximal, characteristic
3#15797C214yesmaximal, complemented
4#15788C214yesmaximal, complemented
5#15770C214yesmaximal, complemented
6#15541C214yesmaximal, complemented
7#15019C211yesmaximal, characteristic, complemented
8#14934C211yesmaximal, characteristic, complemented
9#14820C211yesmaximal, characteristic
10#14717C211yesmaximal, characteristic
11#14681C212yesmaximal
12#14342C212yesmaximal, complemented
13#13972C212yesmaximal, complemented
14#13971C212yesmaximal
15#13776C211yesmaximal, characteristic, complemented
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22#2627—28no—
23C2×#400C2212yescomplemented
24C2×#398C2212yescomplemented
25C2×#398C2212yescomplemented
26C2×#397C2212yes—
27C2×#395C2212yescomplemented
28C2×#393C2212yes—
29C2×#388C2214yes—
30C2×#385C2212yes—
31C2×#381C2212yes—
32C2×#381C2212yes—
33C2×#381C2212yes—
34C2×#381C2212yes—
35C2×#380—28no—
36C2×#380C2214yes—
37C2×#380C2211yescharacteristic
38C2×#380C2214yes—
39C2×#379C2212yes—
40C2×#379C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44#1446C2212yescomplemented
45#1445C2214yescomplemented
46#1445C2214yescomplemented
47#1445C2214yescomplemented
48#1445C2214yescomplemented
49#1445C2212yescomplemented
50#1443C2214yescomplemented
51#1443C2214yescomplemented
52#1442C2214yescomplemented
53#1442C2212yescomplemented
54#1442C2214yescomplemented
55#1442C2212yescomplemented
56#1440C2214yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2212yes—
62#1437C2214yes—
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66#1434C2214yescomplemented
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2211yescharacteristic
77C26C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4—24no—
80C24×C4—24no—
81C24×C4—24no—
82C24×C4—28no—
83C24×C4—44no—
84C24×C4C2311yescharacteristic
85C24×C4C2314yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—44no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4D418yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—44no—
109C22×C22⋊C4—44no—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C42C2314yes—
122#400—28no—
123#398—28no—
124#398—28no—
125#397—44no—
126#395—28no—
127#393—44no—
128#388—48no—
129#385—28no—
130#381—28no—
131#381—28no—
132#381—28no—
133#381—44no—
134#380—24no—
135#380—216no—
136#380—216no—
137#379—28no—
138#379—44no—
139C4×C22⋊C4—44no—
140C2×#95—24no—
141C2×#95—24no—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C25—22no—
172C25—22no—
173C25—24no—
174C25—24no—
175C25—24no—
176C25—24no—
177C25—42no—
178C25—42no—
179C25#4114yes—
180C25C2412yes—
181C25C2411yescharacteristic
182C22×Q8—22no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8—42no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—232no—
190C23×C4—22no—
191C23×C4—28no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—432no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—432no—
217C23×C4—416no—
218C23×C4—416no—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4#4114yes—
223C23×C4#4118yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2412yes—
237C22⋊Q8—416no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—416no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2×D418yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—416no—
290C2×C22⋊C4—416no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—44no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—42no—
315#95—42no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—42no—
326#95—44no—
327#95—42no—
328#95—44no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—28no—
334C24—28no—
335C24—216no—
336C24—28no—
337C24—216no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—216no—
347C24—48no—
348C24—416no—
349C24—416no—
350C24—416no—
351C24—416no—
352C24—48no—
353C24—44no—
354C24—44no—
355C24—44no—
356C24—44no—
357C24—48no—
358C24C22×D412yes—
359C24C2×#4114yes—
360C24C2×#4112yes—
361C24C2×#4112yes—
362C24C2×#4114yes—
363C24H(5,𝔽2)14yes—
364C24H(5,𝔽2)14yes—
365C242−1+414yes—
366C24C2511yescharacteristic
367C2×Q8—44no—
368C2×Q8—82no—
369C2×Q8—82no—
370C2×Q8—88no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—216no—
374C22×C4—216no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—232no—
380C22×C4—28no—
381C22×C4—216no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—4128no—
451C22×C4—416no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—416no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4#124116yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D414yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4114yes—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—416no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4—82no—
482C4⋊C4—84no—
483C22⋊C4—416no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—416no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C42—88no—
495C23—24no—
496C23—24no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—232no—
508C23—24no—
509C23—24no—
510C23—28no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—24no—
515C23—24no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—464no—
521C23—416no—
522C23—416no—
523C23—416no—
524C23—416no—
525C23—464no—
526C23—416no—
527C23—416no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23#52514yes—
539C23C2×#12414yes—
540C23#53414yes—
541C23#53918yes—
542C23#55118yes—
543C23#55118yes—
544C23C23×D411yescharacteristic
545C23C22×#4111yescharacteristic
546C23C22×#4111yescharacteristic
547C23C22×#4112yes—
548C23C2×H(5,𝔽2)12yes—
549C23C2×H(5,𝔽2)12yes—
550C23C2×2−1+412yes—
551C23#58512yes—
552C23#58512yes—
553Q8—88no—
554C2×C4—232no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—464no—
573C2×C4—416no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
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—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
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—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—88no—
626C2×C4—88no—
627C2×C4—88no—
628C2×C4—88no—
629C2×C4C2×#12418yes—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—216no—
636C22—216no—
637C22—24no—
638C22—28no—
639C22—216no—
640C22—464no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—48no—
647C22—48no—
648C22—48no—
649C22—44no—
650C22#256118yes—
651C22C2×#52511yescharacteristic
652C22C22×#12411yescharacteristic
653C22C2×#53411yescharacteristic
654C22C2×#53912yes—
655C22#345512yes—
656C22C2×#55112yes—
657C22C2×#55112yes—
658C22#348411yescharacteristic
659C22#348511yescharacteristic
660C22#348611yescharacteristic
661C22#348711yescharacteristic
662C22#349112yes—
663C22#349212yes—
664C22#349414yes—
665C22#349514yes—
666C22#353414yes—
667C22#353514yes—
668C4—28no—
669C4—44no—
670C4—42no—
671C4—44no—
672C4—42no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—84no—
679C4—84no—
680C4—82no—
681C2—28no—
682C2—48no—
683C2C2×#256111yescharacteristic
684C2#3498911yescharacteristic
685C2#3509411yescharacteristic
686C2#3558411yescharacteristic
687C2#3564411yescharacteristic
688C2#3711711yescharacteristic
689C2#3718611yescharacteristic
690C2#3784914yes—
691C2#3785414yes—
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
1. C2×#2627C2semidirect
3. #15797C2semidirect
4. #15788C2semidirect
5. #15770C2semidirect
6. #15541C2semidirect
7. #15019C2semidirect
8. #14934C2semidirect
12. #14342C2semidirect
13. #13972C2semidirect
15. #13776C2semidirect
23. C2×#400C22semidirect
24. C2×#398C22semidirect
25. C2×#398C22semidirect
27. C2×#395C22semidirect
44. #1446C22semidirect
45. #1445C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1445C22semidirect
49. #1445C22semidirect
50. #1443C22semidirect
51. #1443C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C22×#3802
#157972
#157882
#157702
#155412
#150192
#149342
#148202
#147172
#146812
#143422
#139722
#139712
#137762