Finite Group Databasebeta

← #726430

Subgroups — #726430

Order 512.

4498 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3C2×#2535C211yesmaximal, characteristic
4C2×#2326C211yesmaximal, characteristic, complemented
5#16682C214yesmaximal, complemented
6#15951C218yesmaximal, complemented
7#15215C212yesmaximal, complemented
8#15182C212yesmaximal, complemented
9#14796C214yesmaximal, complemented
10#14282C211yesmaximal, characteristic, complemented
11#14137C212yesmaximal, complemented
12#13942C211yesmaximal, characteristic, complemented
13#13880C212yesmaximal, complemented
14#13847C211yesmaximal, characteristic, complemented
15C2×#554—28no—
16C2×#554—28no—
17C2×#554—28no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2212yes—
20C22×#124C2211yescharacteristic
21C22×C22⋊Q8C2212yes—
22C22×C22⋊Q8C2214yes—
23C22×C22⋊Q8C2214yes—
24C22×#121C2212yes—
25C22×C4×Q8C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C22×C4×D4C2212yes—
28C22×C4×D4C2211yescharacteristic
29C23×C4⋊C4C2212yes—
30#2542—28no—
31#2535—28no—
32C2×#399C2214yes—
33C2×#397C2212yes—
34C2×#392C2212yes—
35#2326—28no—
36C2×#391C2211yescharacteristic
37C2×#391C2212yes—
38C2×#390C2211yescharacteristic
39C2×#390C2212yes—
40C2×#388C2214yes—
41C2×#386C2214yes—
42C2×#386C2218yes—
43C2×#385C2214yes—
44C2×#385C2218yes—
45C2×#384C2211yescharacteristic
46C2×#384C2212yes—
47C2×#382C2211yescharacteristic
48C2×#380C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×#376C2211yescharacteristic
53#1446C2214yes—
54#1445C2212yes—
55#1445C2214yes—
56#1444C2218yes—
57#1441C2218yes—
58#1441C2214yes—
59#1439C2214yes—
60#1439C2218yes—
61#1439C2218yes—
62#1439C2218yes—
63#1438C2212yes—
64#1438C2218yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68#1434C2214yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2312yes—
71C23×D4C2311yescharacteristic
72C23×D4C2311yescharacteristic
73C23×D4C2314yes—
74C24×C4C2312yes—
75C24×C4C2314yes—
76C24×C4C2312yes—
77#554—2128no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8D418yescomplemented
81C2×#125—28no—
82C2×#125—28no—
83C2×#125—28no—
84C2×#124—24no—
85C2×#124—24no—
86C2×#124—24no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—216no—
91C2×C22⋊Q8—216no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—216no—
94C2×C22⋊Q8—416no—
95C2×#121—28no—
96C2×#121—28no—
97C2×#121—44no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8D418yescomplemented
101C2×C4×D4—28no—
102C2×C4×D4—28no—
103C2×C4×D4—28no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4—24no—
109C2×C4×D4D418yes—
110C22×C4⋊C4—28no—
111C22×C4⋊C4—28no—
112C22×C4⋊C4—28no—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2318yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2318yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#399—216no—
141#397—28no—
142#392—28no—
143#391—24no—
144#391—44no—
145#390—24no—
146#390—28no—
147#388—216no—
148#386—216no—
149#386—416no—
150#385—216no—
151#385—416no—
152#384—24no—
153#384—28no—
154#382—24no—
155#380—28no—
156C4×C4⋊C4—24no—
157C4×C22⋊C4—28no—
158C4×C22⋊C4—44no—
159#376—24no—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2318yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2318yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2314yes—
173C25C2412yes—
174C25C2412yes—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—48no—
184C22×Q8C2×D414yes—
185C22×Q8C2×D418yes—
186C22×D4—28no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—216no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—44no—
199C22×D4—48no—
200C22×D4C2×D414yes—
201C22×D4#4114yes—
202C23×C4—28no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—48no—
215C23×C4—48no—
216C23×C4—48no—
217C23×C4C2×D418yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C4⋊Q8—232no—
231#125—264no—
232#124—232no—
233C22⋊Q8—264no—
234C22⋊Q8—2128no—
235C22⋊Q8—464no—
236#121—432no—
237C4×Q8—232no—
238C4×D4—264no—
239C4×D4—232no—
240C4×D4—416no—
241C2×C4⋊C4—264no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—216no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—216no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—48no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D418yes—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
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—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—216no—
294C2×C22⋊C4—216no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—48no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4C2×D418yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42C2×D418yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42#4114yes—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—216no—
328#95—44no—
329#95—48no—
330#95—48no—
331#95—48no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—28no—
346C24—22no—
347C24—22no—
348C24—28no—
349C24C22×D414yes—
350C24C2×#4114yes—
351C242−1+418yes—
352C24C2511yescharacteristic
353C2×Q8—28no—
354C2×Q8—216no—
355C2×Q8—28no—
356C2×Q8—28no—
357C2×Q8—416no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—88no—
363C2×Q8—88no—
364C2×D4—216no—
365C2×D4—216no—
366C2×D4—216no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—28no—
371C2×D4—416no—
372C2×D4—432no—
373C2×D4—416no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—88no—
379C2×D4—816no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—232no—
412C22×C4—216no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—416no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4—816no—
438C22×C4—816no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D418yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4H(5,𝔽2)18yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C42−1+412yes—
460C4⋊C4—232no—
461C4⋊C4—232no—
462C4⋊C4—216no—
463C4⋊C4—28no—
464C4⋊C4—216no—
465C4⋊C4—28no—
466C4⋊C4—216no—
467C4⋊C4—216no—
468C4⋊C4—432no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—88no—
472C22⋊C4—232no—
473C22⋊C4—232no—
474C22⋊C4—232no—
475C22⋊C4—216no—
476C22⋊C4—432no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—48no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C42—216no—
483C42—28no—
484C42—44no—
485C42—48no—
486C42#120116yes—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—216no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—42no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—48no—
522C23—44no—
523C23—44no—
524C23—48no—
525C23—48no—
526C23#53618yes—
527C23C23×D411yescharacteristic
528C23C23×D411yescharacteristic
529C23C23×D411yescharacteristic
530C23C22×#4111yescharacteristic
531C23C22×#4111yescharacteristic
532C23C2×H(5,𝔽2)11yescharacteristic
533C23C2×H(5,𝔽2)14yes—
534C23C2×2−1+414yes—
535C23C2×2−1+411yescharacteristic
536Q8—416no—
537Q8—816no—
538D4—232no—
539D4—416no—
540D4—832no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—216no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—432no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—816no—
598C2×C4—832no—
599C2×C4C2×#12014yes—
600C2×C4C2×#12014yes—
601C2×C4C2×#12014yes—
602C2×C4#52514yes—
603C2×C4#53714yes—
604C2×C4#54614yes—
605C2×C4#54714yes—
606C2×C4#54814yes—
607C2×C4#55014yes—
608C22—28no—
609C22—28no—
610C22—28no—
611C22—48no—
612C22—48no—
613C22—48no—
614C22—48no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—48no—
620C22—48no—
621C22—44no—
622C22—48no—
623C22—84no—
624C22—84no—
625C22—82no—
626C22—82no—
627C22—84no—
628C22C22×#12011yescharacteristic
629C22C2×#52511yescharacteristic
630C22C2×#53612yes—
631C22C2×#53711yescharacteristic
632C22C2×#54611yescharacteristic
633C22C2×#54711yescharacteristic
634C22C2×#54811yescharacteristic
635C22C2×#55011yescharacteristic
636C22#348412yes—
637C22#348612yes—
638C22#348612yes—
639C22#349214yes—
640C22#349812yes—
641C22#349812yes—
642C22#352014yes—
643C22#353614yes—
644C22#355214yes—
645C4—24no—
646C4—24no—
647C4—48no—
648C4—44no—
649C4—44no—
650C4—44no—
651C4—44no—
652C4—48no—
653C4—82no—
654C4—84no—
655C4—82no—
656C4#240718yes—
657C2—48no—
658C2—84no—
659C2C2×#240711yescharacteristic
660C2#3524211yescharacteristic
661C2#3552711yescharacteristic
662C2#3570112yes—
663C2#3786212yes—
664C2#3807812yes—
665C2#4279412yes—
666C2#4533712yes—
667C2#4535612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2535C2
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×#554C2semidirect
2. C2×#2542C2semidirect
4. C2×#2326C2semidirect
5. #16682C2semidirect
6. #15951C2semidirect
7. #15215C2semidirect
8. #15182C2semidirect
9. #14796C2semidirect
10. #14282C2semidirect
11. #14137C2semidirect
12. #13942C2semidirect
13. #13880C2semidirect
14. #13847C2semidirect
80. C2×C4⋊Q8D4semidirect
100. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#25422
C2×#25352
C2×#23262
#166822
#159512
#152152
#151822
#147962
#142822
#141372
#139422
#138802
#138472