Finite Group Databasebeta

← #759990

Subgroups — #759990

Order 512.

5682 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C2×#2583C212yesmaximal, complemented
3C22×#385C211yesmaximal, characteristic
4#16606C214yesmaximal, complemented
5#14914C212yesmaximal, complemented
6#14874C212yesmaximal, complemented
7#14786C212yesmaximal, complemented
8#14763C214yesmaximal, complemented
9#14720C212yesmaximal, complemented
10#14159C212yesmaximal, complemented
11#14127C212yesmaximal
12#14127C212yesmaximal
13#14043C214yesmaximal, complemented
14C4×#379C211yesmaximal, characteristic
15C24×Q8C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8—28no—
18C22×C22⋊Q8—216no—
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C22⋊Q8C2212yes—
23C23×C4⋊C4C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2212yes—
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4C2212yes—
28C23×C22⋊C4C2211yescharacteristic
29C23×C22⋊C4C2211yescharacteristic
30#2583—216no—
31C2×#398C2214yescomplemented
32C2×#395C2214yescomplemented
33C2×#395C2214yescomplemented
34C2×#389C2218yescomplemented
35C2×#385—28no—
36C2×#385C2214yes—
37C2×#385C2214yes—
38C2×#385C2212yes—
39C2×#385C2214yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2214yes—
44C2×#379C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50#1446C2214yescomplemented
51#1446C2214yescomplemented
52#1445C2214yescomplemented
53#1444C2214yescomplemented
54#1444C2218yescomplemented
55#1441C2218yescomplemented
56#1441C2214yescomplemented
57#1439C2218yes—
58#1439C2218yes—
59#1436C2218yescomplemented
60C22⋊#95C2214yes—
61C22⋊#95C2212yes—
62C22⋊#95C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C26C2311yescharacteristic
68C23×Q8—28no—
69C23×Q8—24no—
70C23×Q8—44no—
71C23×Q8—44no—
72C23×Q8C2312yes—
73C23×Q8C2312yes—
74C24×C4—28no—
75C24×C4—44no—
76C24×C4—44no—
77C24×C4D418yes—
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—2128no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—216no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—216no—
100C2×C22⋊Q8D4116yes—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—216no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—44no—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2318yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2318yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—28no—
120C22×C22⋊C4—24no—
121C22×C22⋊C4—216no—
122C22×C22⋊C4—28no—
123C22×C22⋊C4—24no—
124C22×C22⋊C4—44no—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C42C2314yes—
135#398—216no—
136#395—216no—
137#395—48no—
138#389—232no—
139#385—28no—
140#385—28no—
141#385—28no—
142#385—28no—
143#385—216no—
144#385—216no—
145#385—216no—
146#385—48no—
147#379Q8116yescomplemented
148C4×C22⋊C4—28no—
149C4×C22⋊C4—44no—
150C4×C22⋊C4—44no—
151C2×#95—24no—
152C2×#95—24no—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2318yes—
156C2×#95C2318yes—
157C2×#95C2318yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2318yes—
163C2×#95C2318yes—
164C2×#95C2314yes—
165C2×#95C2318yes—
166C25—24no—
167C25—24no—
168C25—24no—
169C25—24no—
170C25—42no—
171C25—42no—
172C25C2×D414yes—
173C25C2×Q818yes—
174C25C2411yescharacteristic
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C22×Q8—232no—
178C22×Q8—28no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—28no—
183C22×Q8—44no—
184C22×Q8—44no—
185C22×Q8—44no—
186C22×Q8—464no—
187C22×Q8—416no—
188C22×Q8—416no—
189C22×Q8C2×D418yes—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—232no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—416no—
218C23×C4—42no—
219C23×C4—416no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—44no—
228C23×C4—42no—
229C23×C4—44no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—464no—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4C2×D414yes—
241C23×C4C2×D418yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2414yes—
245C23×C4C2414yes—
246C23×C4C2411yescharacteristic
247C23×C4C2418yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C22⋊Q8—264no—
254C22⋊Q8—264no—
255C22⋊Q8—264no—
256C22⋊Q8—264no—
257C2×C4⋊C4—232no—
258C2×C4⋊C4—264no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—216no—
275C2×C4⋊C4—216no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—216no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—48no—
280C2×C4⋊C4—416no—
281C2×C4⋊C4—416no—
282C2×C4⋊C4—48no—
283C2×C4⋊C4—48no—
284C2×C4⋊C4C2×D4116yes—
285C2×C4⋊C4C2×D418yes—
286C2×C22⋊C4—232no—
287C2×C22⋊C4—264no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—416no—
309C2×C22⋊C4C2×D4116yes—
310C2×C22⋊C4C2×Q8116yes—
311C2×C22⋊C4C2×Q8116yescomplemented
312C2×C22⋊C4C2×Q8116yescomplemented
313C2×C22⋊C4#4118yes—
314C2×C42—28no—
315C2×C42—44no—
316C2×C42—44no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—216no—
323#95—216no—
324#95—216no—
325#95—28no—
326#95—48no—
327#95—44no—
328#95—48no—
329#95—48no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—28no—
339C24—28no—
340C24—28no—
341C24—28no—
342C24—232no—
343C24—232no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—44no—
351C24—44no—
352C24—416no—
353C24—416no—
354C24—416no—
355C24—416no—
356C24—416no—
357C24C4⋊Q8116yes—
358C24C22×D412yes—
359C24C22×D412yes—
360C24C22×D414yes—
361C24C22×D412yes—
362C24C22×Q814yes—
363C24C22×Q814yes—
364C24C22×Q814yes—
365C24C2×#4114yes—
366C24H(5,𝔽2)14yes—
367C242−1+414yes—
368C24C2511yescharacteristic
369C2×Q8—216no—
370C2×Q8—216no—
371C2×Q8—416no—
372C2×Q8—432no—
373C2×Q8—4128no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×Q8—416no—
378C2×Q8—48no—
379C22×C4—232no—
380C22×C4—28no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—216no—
390C22×C4—216no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—432no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—432no—
448C22×C4—4128no—
449C22×C4—416no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4#120116yes—
467C22×C4C22×D418yes—
468C22×C4C22×D418yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×Q8116yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4114yes—
478C22×C4H(5,𝔽2)14yes—
479C4⋊C4—216no—
480C4⋊C4—232no—
481C4⋊C4—216no—
482C4⋊C4—232no—
483C4⋊C4—432no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—816no—
489C22⋊C4—216no—
490C22⋊C4—232no—
491C22⋊C4—232no—
492C22⋊C4—216no—
493C22⋊C4—232no—
494C22⋊C4—232no—
495C22⋊C4—416no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C42—88no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—232no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—232no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—416no—
531C23—416no—
532C23—416no—
533C23—4128no—
534C23—416no—
535C23—416no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23C2×#12014yes—
541C23#52514yes—
542C23C2×C4⋊Q814yes—
543C23C2×C4⋊Q814yes—
544C23C2×C4⋊Q814yes—
545C23#53414yes—
546C23#53614yes—
547C23D4×Q8116yes—
548C23#554116yes—
549C23C23×D411yescharacteristic
550C23C23×D412yes—
551C23C23×Q812yes—
552C23C22×#4112yes—
553C23C2×H(5,𝔽2)12yes—
554C23C2×H(5,𝔽2)12yes—
555C23C2×2−1+412yes—
556C23#58512yes—
557Q8—432no—
558Q8—432no—
559C2×C4—232no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—216no—
577C2×C4—28no—
578C2×C4—216no—
579C2×C4—216no—
580C2×C4—216no—
581C2×C4—216no—
582C2×C4—28no—
583C2×C4—416no—
584C2×C4—416no—
585C2×C4—416no—
586C2×C4—416no—
587C2×C4—416no—
588C2×C4—416no—
589C2×C4—432no—
590C2×C4—416no—
591C2×C4—432no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—88no—
618C2×C4—88no—
619C2×C4C2×#12014yes—
620C2×C4C2×#12014yes—
621C2×C4#53414yes—
622C22—24no—
623C22—24no—
624C22—232no—
625C22—232no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—464no—
640C22—42no—
641C22—416no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22#243418yes—
646C22#258318yescomplemented
647C22C22×#12011yescharacteristic
648C22C2×#52511yescharacteristic
649C22C22×C4⋊Q811yescharacteristic
650C22C2×#53411yescharacteristic
651C22C2×#53411yescharacteristic
652C22C2×#53611yescharacteristic
653C22#344911yescharacteristic
654C22C2×D4×Q814yes—
655C22D4×#4114yes—
656C22C2×#55414yes—
657C22#348414yes—
658C22#350114yes—
659C22#352014yes—
660C22#353514yes—
661C4—24no—
662C4—24no—
663C4—24no—
664C4—44no—
665C4—416no—
666C4—42no—
667C4—48no—
668C4—48no—
669C4—44no—
670C4—82no—
671C4—82no—
672C2—28no—
673C2—28no—
674C2—44no—
675C2C2×#243411yescharacteristic
676C2#3473611yescharacteristic
677C2C2×#258311yescharacteristic
678C2#3498011yescharacteristic
679C2#3501111yescharacteristic
680C2#3508411yescharacteristic
681C2#3744111yescharacteristic
682C2#5915018yes—
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
2. C2×#2583C2semidirect
4. #16606C2semidirect
5. #14914C2semidirect
6. #14874C2semidirect
7. #14786C2semidirect
8. #14763C2semidirect
9. #14720C2semidirect
10. #14159C2semidirect
13. #14043C2semidirect
31. C2×#398C22semidirect
32. C2×#395C22semidirect
33. C2×#395C22semidirect
34. C2×#389C22semidirect
50. #1446C22semidirect
51. #1446C22semidirect
52. #1445C22semidirect
53. #1444C22semidirect
54. #1444C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
59. #1436C22semidirect
147. #379Q8semidirect
311. C2×C22⋊C4C2×Q8semidirect
312. C2×C22⋊C4C2×Q8semidirect
646. C22#2583semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#25832
C22×#3852
#166062
#149142
#148742
#147862
#147632
#147202
#141592
#141272
#140432
C4×#3792