Finite Group Databasebeta

← #725993

Subgroups — #725993

Order 512.

5550 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#547C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic, complemented
3C2×#2535C211yesmaximal, characteristic, complemented
4C2×#2397C211yesmaximal, characteristic, complemented
5#16600C214yesmaximal, complemented
6#15669C214yesmaximal, complemented
7#15412C214yesmaximal, complemented
8#15123C214yesmaximal, complemented
9#14885C214yesmaximal, complemented
10#14282C211yesmaximal, characteristic, complemented
11#13905C211yesmaximal, characteristic, complemented
12#13872C214yesmaximal, complemented
13#13785C211yesmaximal, characteristic, complemented
14C23×#41C2212yes—
15C2×#547—28no—
16C2×#547—28no—
17C2×#547—28no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×#123C2214yes—
20C22×C22⋊Q8C2212yes—
21C22×#121C2212yes—
22C22×#121C2212yes—
23C22×C4×D4C2211yescharacteristic
24C22×C4×D4C2214yes—
25C22×C4×D4C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C23×C4⋊C4C2212yes—
28#2563—28no—
29#2535—28no—
30#2397—28no—
31C2×#399C2212yes—
32C2×#399C2212yes—
33C2×#397C2212yes—
34C2×#393C2214yes—
35C2×#392C2212yes—
36C2×#392C2212yes—
37C2×#392C2212yes—
38C2×#392C2218yes—
39C2×#390C2212yes—
40C2×#390C2212yes—
41C2×#386C2214yes—
42C2×#386C2218yes—
43C2×#382C2212yes—
44C2×#382C2211yescharacteristic
45C2×#380C2214yes—
46C2×#380C2212yes—
47C2×#380C2214yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×#376C2211yescharacteristic
50#1446C2214yes—
51#1445C2214yes—
52#1445C2212yes—
53#1441C2218yes—
54#1439C2218yes—
55#1439C2218yes—
56#1438C2218yes—
57#1438C2218yes—
58#1438C2212yes—
59#1437C2214yes—
60#1437C2214yes—
61C4⋊#95C2214yes—
62C4⋊#95C2212yes—
63C22⋊#95C2218yes—
64C22⋊#95C2214yes—
65C22×#41—28no—
66C22×#41—28no—
67C22×#41—28no—
68C22×#41—416no—
69C23×Q8C2312yes—
70C23×D4C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C23×D4C2314yes—
73C23×D4C2314yes—
74C23×D4C2311yescharacteristic
75C23×D4C2314yes—
76C24×C4C2312yes—
77C24×C4C2312yes—
78C24×C4C2314yes—
79C24×C4C2312yes—
80#547—2128no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8D418yescomplemented
84C2×#123—216no—
85C2×#123—216no—
86C2×#123—216no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×#121—28no—
91C2×#121—28no—
92C2×#121—28no—
93C2×#121—28no—
94C2×#121—48no—
95C2×C4×D4—216no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4—216no—
104C2×C4×D4—48no—
105C2×C4×D4D418yescomplemented
106C2×C4×D4D418yescomplemented
107C22×C4⋊C4—28no—
108C22×C4⋊C4—28no—
109C22×C4⋊C4—28no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2318yes—
126C22×C22⋊C4C2318yes—
127C22×C22⋊C4C2318yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#399—28no—
134#399—28no—
135#397—28no—
136#393—216no—
137#392—28no—
138#392—28no—
139#392—28no—
140#392—416no—
141#390—28no—
142#390—28no—
143#386—216no—
144#386—416no—
145#382—28no—
146#382—24no—
147#380—216no—
148#380—28no—
149#380—48no—
150C4×C4⋊C4—24no—
151#376—24no—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2318yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C25C2412yes—
164C25C2412yes—
165C25C2412yes—
166C2×#41—264no—
167C2×#41—464no—
168C2×#41—464no—
169C2×#41—464no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—48no—
173C22×Q8C2×D418yes—
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—24no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—216no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—28no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—28no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—28no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—44no—
197C22×D4—42no—
198C22×D4—44no—
199C22×D4—48no—
200C22×D4—48no—
201C22×D4—44no—
202C22×D4—48no—
203C22×D4C2×D414yes—
204C22×D4C2×D414yes—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—216no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—28no—
219C23×C4—28no—
220C23×C4—48no—
221C23×C4—48no—
222C23×C4C2×D418yes—
223C23×C4C2×D418yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C4⋊Q8—232no—
235#123—2128no—
236C22⋊Q8—264no—
237#121—264no—
238#121—432no—
239C4×D4—232no—
240C4×D4—232no—
241C4×D4—464no—
242C4×D4—416no—
243C2×C4⋊C4—264no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D418yes—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—216no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—216no—
288C2×C22⋊C4—216no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4—48no—
299C2×C22⋊C4C2×D418yes—
300C2×C22⋊C4C2×D418yes—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—44no—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308C2×C42#4118yescomplemented
309#95—24no—
310#95—24no—
311#95—24no—
312#95—216no—
313#95—28no—
314#95—24no—
315#95—24no—
316#95—48no—
317#95—48no—
318#95—48no—
319#95—48no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—28no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24C22×D414yes—
345C24C22×D414yes—
346C24H(5,𝔽2)18yes—
347C242−1+414yes—
348C24C2511yescharacteristic
349#41—4256no—
350C2×Q8—216no—
351C2×Q8—416no—
352C2×Q8—416no—
353C2×Q8—416no—
354C2×D4—216no—
355C2×D4—216no—
356C2×D4—216no—
357C2×D4—232no—
358C2×D4—28no—
359C2×D4—232no—
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—416no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—416no—
368C2×D4—432no—
369C2×D4—44no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—432no—
373C2×D4—416no—
374C2×D4—416no—
375C2×D4—416no—
376C2×D4—416no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—44no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—88no—
383C2×D4—816no—
384C2×D4—816no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—232no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—232no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—416no—
425C22×C4—432no—
426C22×C4—432no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—416no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4C22×D418yes—
440C22×C4C22×D414yes—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C2×#4118yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)14yes—
454C4⋊C4—232no—
455C4⋊C4—28no—
456C4⋊C4—216no—
457C4⋊C4—28no—
458C4⋊C4—216no—
459C4⋊C4—416no—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C22⋊C4—232no—
464C22⋊C4—216no—
465C22⋊C4—216no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—48no—
470C22⋊C4—88no—
471C22⋊C4—816no—
472C22⋊C4—816no—
473C22⋊C4—816no—
474C42—216no—
475C42—48no—
476C42#120116yes—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—48no—
528C23—48no—
529C23—44no—
530C23—44no—
531C23—48no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—48no—
536C23#53518yes—
537C23C23×D411yescharacteristic
538C23C23×D411yescharacteristic
539C23C23×D411yescharacteristic
540C23C22×#4112yes—
541C23C2×H(5,𝔽2)12yes—
542C23C2×H(5,𝔽2)14yes—
543C23C2×2−1+412yes—
544C23#58512yes—
545Q8—432no—
546D4—232no—
547D4—416no—
548D4—464no—
549D4—464no—
550D4—416no—
551D4—832no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—216no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—432no—
569C2×C4—464no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—48no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—432no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—816no—
599C2×C4C2×#12014yes—
600C2×C4C2×#12014yes—
601C2×C4C2×#12014yes—
602C2×C4#53214yes—
603C2×C4#53514yes—
604C2×C4#54618yes—
605C2×C4#55018yes—
606C22—28no—
607C22—216no—
608C22—48no—
609C22—44no—
610C22—48no—
611C22—48no—
612C22—44no—
613C22—48no—
614C22—48no—
615C22—48no—
616C22—44no—
617C22—44no—
618C22—48no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—48no—
623C22—44no—
624C22—48no—
625C22—84no—
626C22—82no—
627C22—84no—
628C22—84no—
629C22—82no—
630C22—82no—
631C22—88no—
632C22—84no—
633C22—82no—
634C22C22×#12011yescharacteristic
635C22C2×#53211yescharacteristic
636C22C2×#53512yes—
637C22C2×#53511yescharacteristic
638C22C2×#54612yes—
639C22C2×#55012yes—
640C22#348412yes—
641C22#348412yes—
642C22#348611yescharacteristic
643C22#348711yescharacteristic
644C22#348912yes—
645C22#348912yes—
646C22#349214yes—
647C22#349414yes—
648C22#353614yes—
649C22#354214yes—
650C4—24no—
651C4—24no—
652C4—44no—
653C4—48no—
654C4—44no—
655C4—48no—
656C4—84no—
657C4—82no—
658C4—84no—
659C4#243118yes—
660C2—48no—
661C2—84no—
662C2—84no—
663C2C2×#243111yescharacteristic
664C2#3522211yescharacteristic
665C2#3565911yescharacteristic
666C2#3592312yes—
667C2#3783914yes—
668C2#4247812yes—
669C2#4493312yes—
670C2#4535612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#547C2semidirect
2. C2×#2563C2semidirect
3. C2×#2535C2semidirect
4. C2×#2397C2semidirect
5. #16600C2semidirect
6. #15669C2semidirect
7. #15412C2semidirect
8. #15123C2semidirect
9. #14885C2semidirect
10. #14282C2semidirect
11. #13905C2semidirect
12. #13872C2semidirect
13. #13785C2semidirect
83. C2×C4⋊Q8D4semidirect
105. C2×C4×D4D4semidirect
106. C2×C4×D4D4semidirect
308. C2×C42#41semidirect

Maximal subgroups

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

subgroupindex
C22×#5472
C2×#25632
C2×#25352
C2×#23972
#166002
#156692
#154122
#151232
#148852
#142822
#139052
#138722
#137852