Finite Group Databasebeta

← #4397204

Subgroups — #4397204

Order 512.

2838 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2816C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic, complemented
3C2×#2501C211yesmaximal, characteristic
4C2×#2333C211yesmaximal, characteristic
5#17062C212yesmaximal, complemented
6#16803C212yesmaximal, complemented
7#15629C214yesmaximal, complemented
8#15479C214yesmaximal, complemented
9#15271C214yesmaximal, complemented
10#15269C214yesmaximal, complemented
11#15161C212yesmaximal, complemented
12#15135C212yesmaximal, complemented
13#14342C211yesmaximal, characteristic, complemented
14#14292C211yesmaximal, characteristic, complemented
15#13953C211yesmaximal, characteristic, complemented
16C22×#123C2214yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2816—28no—
21#2502—28no—
22#2501—28no—
23C2×#399C2212yes—
24C2×#399C2212yes—
25C2×#399C2212yes—
26C2×#398C2212yes—
27C2×#398C2214yes—
28C2×#398C2214yes—
29C2×#397C2214yes—
30C2×#394C2212yes—
31C2×#394C2214yes—
32C2×#393C2212yes—
33#2333—28no—
34C2×#391C2211yescharacteristic
35C2×#391C2211yescharacteristic
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#389C2214yes—
39C2×#388C2212yes—
40C2×#387C2212yes—
41C2×#386C2212yes—
42C2×#386C2212yes—
43C2×#385C2214yes—
44C2×#384C2212yes—
45C2×#384C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#381C2211yescharacteristic
48C2×#381C2211yescharacteristic
49C2×#380C2214yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53#1446C2212yes—
54#1445C2214yes—
55#1445C2212yes—
56#1445C2212yes—
57#1445C2212yes—
58#1444C2214yes—
59#1444C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1440C2212yes—
66#1440C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2212yes—
75#1434C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2214yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C2×#123—216no—
86C2×#123—48no—
87C2×#123—48no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×#121—24no—
92C2×#121—24no—
93C2×#121D418yes—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#399—28no—
129#399—28no—
130#399—28no—
131#398—28no—
132#398—216no—
133#398—48no—
134#397—48no—
135#394—28no—
136#394—216no—
137#393—28no—
138#391—24no—
139#391—24no—
140#390—28no—
141#390—28no—
142#389—48no—
143#388—28no—
144#387—28no—
145#386—28no—
146#386—28no—
147#385—216no—
148#384—28no—
149#384—24no—
150#384—24no—
151#381—24no—
152#381—24no—
153#380—48no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—28no—
156C4×C22⋊C4D418yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C25C2412yes—
180C25C2411yescharacteristic
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—42no—
185C22×Q8—42no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—42no—
193C22×D4—42no—
194C22×D4C2×D418yes—
195C22×D4C2×D414yes—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4#4114yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224#123—464no—
225C22⋊Q8—416no—
226#121—232no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—416no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C2×D414yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4C2×D418yes—
302C2×C22⋊C4C2×D418yes—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—44no—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95C2×D418yes—
334C24—24no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24C22×D414yes—
350C24C22×D412yes—
351C24C22×D412yes—
352C24C2×#4112yes—
353C24C2511yescharacteristic
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—84no—
358C2×Q8—84no—
359C2×D4—216no—
360C2×D4—28no—
361C2×D4—416no—
362C2×D4—48no—
363C2×D4—82no—
364C2×D4—82no—
365C2×D4—84no—
366C2×D4—84no—
367C2×D4—84no—
368C2×D4—88no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—48no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—416no—
431C22×C4—416no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4—88no—
437C22×C4—88no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4#121116yes—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4H(5,𝔽2)18yes—
451C22×C42−1+414yes—
452C22×C42−1+414yes—
453C4⋊C4—28no—
454C4⋊C4—416no—
455C4⋊C4—416no—
456C4⋊C4—416no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—88no—
463C4⋊C4—88no—
464C4⋊C4—84no—
465C4⋊C4—88no—
466C22⋊C4—216no—
467C22⋊C4—216no—
468C22⋊C4—216no—
469C22⋊C4—416no—
470C22⋊C4—416no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—84no—
474C22⋊C4—84no—
475C22⋊C4—88no—
476C22⋊C4—82no—
477C22⋊C4—84no—
478C22⋊C4—88no—
479C22⋊C4—82no—
480C42—44no—
481C42—44no—
482C42—48no—
483C42—84no—
484C23—24no—
485C23—24no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—24no—
499C23—24no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23C2×#12114yes—
523C23C23×D411yescharacteristic
524C23C23×D411yescharacteristic
525C23C22×#4111yescharacteristic
526C23C2×H(5,𝔽2)14yes—
527C23C2×2−1+412yes—
528C23C2×2−1+412yes—
529C23#58512yes—
530C23#58512yes—
531Q8—88no—
532D4—88no—
533D4—816no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—416no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—816no—
602C2×C4—816no—
603C2×C4C2×#12114yes—
604C2×C4C2×#12114yes—
605C2×C4#53618yes—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—48no—
614C22—44no—
615C22—48no—
616C22—48no—
617C22—44no—
618C22—44no—
619C22—84no—
620C22—84no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22C22×#12111yescharacteristic
626C22C2×#53612yes—
627C22#348412yes—
628C22#348511yescharacteristic
629C22#348511yescharacteristic
630C22#348512yes—
631C22#349112yes—
632C22#349812yes—
633C22#349912yes—
634C22#350812yes—
635C22#350911yescharacteristic
636C22#350911yescharacteristic
637C22#353514yes—
638C22#354114yes—
639C22#355414yes—
640C22#355914yes—
641C4—24no—
642C4—24no—
643C4—28no—
644C4—44no—
645C4—48no—
646C4—44no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4—84no—
651C4—82no—
652C4—82no—
653C4—82no—
654C2—44no—
655C2—84no—
656C2#3531611yescharacteristic
657C2#3544711yescharacteristic
658C2#3552811yescharacteristic
659C2#4137012yes—
660C2#4265112yes—
661C2#4685412yes—
662C2#4722412yes—
663C2#4882214yes—
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
2. C2×#2502C2semidirect
5. #17062C2semidirect
6. #16803C2semidirect
7. #15629C2semidirect
8. #15479C2semidirect
9. #15271C2semidirect
10. #15269C2semidirect
11. #15161C2semidirect
12. #15135C2semidirect
13. #14342C2semidirect
14. #14292C2semidirect
15. #13953C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28162
C2×#25022
C2×#25012
C2×#23332
#170622
#168032
#156292
#154792
#152712
#152692
#151612
#151352
#143422
#142922
#139532