Finite Group Databasebeta

← #626410

Subgroups — #626410

Order 512.

3048 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic, complemented
2C2×#2501C211yesmaximal, characteristic
3C2×#2639C211yesmaximal, characteristic
4C22×#387C211yesmaximal, characteristic, complemented
5#15886C214yesmaximal, complemented
6#15855C214yesmaximal, complemented
7#15815C214yesmaximal, complemented
8#15789C214yesmaximal, complemented
9#14927C212yesmaximal, complemented
10#14927C212yesmaximal, complemented
11#14706C212yesmaximal, complemented
12#14256C212yesmaximal, complemented
13#14159C211yesmaximal, characteristic, complemented
14#14152C211yesmaximal, characteristic
15#13778C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#123C2212yes—
18C22×C22⋊Q8C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2639—28no—
24#2503—28no—
25#2501—28no—
26C2×#400C2212yes—
27C2×#400C2214yes—
28C2×#399C2214yes—
29C2×#397C2212yes—
30C2×#394C2212yes—
31C2×#393C2212yes—
32C2×#391C2212yes—
33C2×#390C2212yes—
34C2×#387—28no—
35C2×#387C2214yes—
36C2×#387C2214yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#384C2212yes—
42C2×#382C2212yes—
43C2×#381C2212yes—
44C2×#381C2212yes—
45C2×#380C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51C22×#95C2211yescharacteristic
52#1446C2212yes—
53#1445C2212yes—
54#1445C2212yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1438C2212yes—
68#1437C2214yes—
69#1437C2214yes—
70#1437C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×#123—28no—
90C2×#123—28no—
91C2×#123—28no—
92C2×#123—28no—
93C2×#123—44no—
94C2×#123—44no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×#121—24no—
99C2×#121—24no—
100C2×#121—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—44no—
105C22×C4⋊C4D418yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42—24no—
128C22×C42—44no—
129C22×C42C2314yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132#400—216no—
133#400—28no—
134#399—216no—
135#397—28no—
136#394—28no—
137#393—28no—
138#391—28no—
139#390—28no—
140#387—216no—
141#387—216no—
142#386—28no—
143#386—28no—
144#386—44no—
145#386—44no—
146#384—28no—
147#382—28no—
148#381—28no—
149#381—28no—
150#380—28no—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—24no—
153C2×#95—24no—
154C2×#95—24no—
155C2×#95—24no—
156C2×#95D418yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C25C2412yes—
183C25C2411yescharacteristic
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—42no—
200C22×D4—42no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4#4114yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245#123—432no—
246#123—432no—
247C22⋊Q8—416no—
248#121—416no—
249C2×C4⋊C4—232no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—48no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—416no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C2×D418yes—
276C2×C4⋊C4C2×D418yes—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4#4118yes—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—28no—
312C2×C42—48no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—416no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95C2×D418yes—
342#95C2×D418yes—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24C22×D412yes—
359C24C22×D412yes—
360C24C2×#4112yes—
361C24C2×#4114yes—
362C24H(5,𝔽2)14yes—
363C24C2511yescharacteristic
364C2×Q8—48no—
365C2×Q8—44no—
366C2×Q8—82no—
367C2×Q8—82no—
368C2×Q8—84no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—44no—
378C2×D4—82no—
379C2×D4—82no—
380C2×D4—84no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—216no—
390C22×C4—216no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—416no—
455C22×C4—44no—
456C22×C4—416no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4#121116yes—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C2×#4118yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C42−1+414yes—
482C4⋊C4—216no—
483C4⋊C4—216no—
484C4⋊C4—416no—
485C4⋊C4—416no—
486C4⋊C4—416no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—88no—
491C22⋊C4—216no—
492C22⋊C4—416no—
493C22⋊C4—416no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C42—48no—
504C42—48no—
505C42—416no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23C2×#12114yes—
542C23#53214yes—
543C23#53514yes—
544C23#54618yes—
545C23#55918yes—
546C23C23×D411yescharacteristic
547C23C23×D411yescharacteristic
548C23C22×#4111yescharacteristic
549C23C22×#4112yes—
550C23C2×H(5,𝔽2)12yes—
551C23C2×2−1+412yes—
552C23#58512yes—
553C23#58512yes—
554C23#58512yes—
555Q8—88no—
556D4—88no—
557D4—816no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—416no—
582C2×C4—416no—
583C2×C4—416no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—416no—
601C2×C4—416no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—88no—
632C2×C4—88no—
633C2×C4C2×#12114yes—
634C2×C4C2×#12114yes—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—216no—
639C22—24no—
640C22—24no—
641C22—216no—
642C22—24no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—84no—
648C22—84no—
649C22—82no—
650C22—82no—
651C22—84no—
652C22—84no—
653C22—82no—
654C22#254418yes—
655C22C22×#12111yescharacteristic
656C22C2×#53211yescharacteristic
657C22C2×#53511yescharacteristic
658C22#344912yes—
659C22#346112yes—
660C22C2×#54612yes—
661C22C2×#55912yes—
662C22#348511yescharacteristic
663C22#348511yescharacteristic
664C22#348511yescharacteristic
665C22#348611yescharacteristic
666C22#349112yes—
667C22#349712yes—
668C22#351514yes—
669C22#352114yes—
670C22#353314yes—
671C22#353514yes—
672C4—24no—
673C4—24no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—44no—
678C4—48no—
679C4—44no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—84no—
685C2—28no—
686C2—162no—
687C2C2×#254411yescharacteristic
688C2#3508911yescharacteristic
689C2#3515311yescharacteristic
690C2#3546711yescharacteristic
691C2#3546811yescharacteristic
692C2#3695611yescharacteristic
693C2#3711811yescharacteristic
694C2#3808714yes—
695C2#3926614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2503C2semidirect
4. C22×#387C2semidirect
5. #15886C2semidirect
6. #15855C2semidirect
7. #15815C2semidirect
8. #15789C2semidirect
9. #14927C2semidirect
10. #14927C2semidirect
11. #14706C2semidirect
12. #14256C2semidirect
13. #14159C2semidirect
15. #13778C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#25012
C2×#26392
C22×#3872
#158862
#158552
#158152
#157892
#149272
#147062
#142562
#141592
#141522
#137782