Finite Group Databasebeta

← #1394605

Subgroups — #1394605

Order 512.

2374 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic, complemented
2C2×#2360C211yesmaximal, characteristic
3#17025C212yesmaximal, complemented
4#16991C212yesmaximal, complemented
5#16343C214yesmaximal
6#16311C214yesmaximal, complemented
7#15202C211yesmaximal, characteristic, complemented
8#15195C211yesmaximal, characteristic
9#15186C212yesmaximal, complemented
10#15182C212yesmaximal
11#15145C211yesmaximal, characteristic, complemented
12#14969C212yesmaximal
13#14319C212yesmaximal
14#14130C212yesmaximal
15#14029C212yesmaximal, complemented
16#13931C211yesmaximal, characteristic, complemented
17C4×#393C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2582—28no—
23C2×#398C2214yes—
24C2×#398C2214yes—
25C2×#393C2211yescharacteristic
26#2360—28no—
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#387C2212yes—
37C2×#387C2214yes—
38C2×#385C2212yes—
39C2×#385C2214yes—
40C2×#384C2212yes—
41C2×#384C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47#1446C2211yescharacteristic
48#1446C2212yes—
49#1445C2214yes—
50#1445C2214yes—
51#1445C2214yes—
52#1444C2212yes—
53#1444C2212yes—
54#1444C2214yes—
55#1444C2212yes—
56#1444C2214yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2211yescharacteristic
60#1442C2212yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2212yes—
64#1442C2212yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1439C2211yescharacteristic
69#1439C2212yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2212yes—
73#1439C2212yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80#1434C2212yes—
81#1434C2212yes—
82C22⋊#95C2214yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—44no—
94C2×C4×Q8—44no—
95C2×C4×Q8D418yes—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127#398—216no—
128#398—48no—
129#393Q818yescomplemented
130#391—28no—
131#391—24no—
132#391—28no—
133#391—44no—
134#389—28no—
135#389—28no—
136#389—28no—
137#389—24no—
138#389—44no—
139#387—28no—
140#387—48no—
141#385—28no—
142#385—48no—
143#384—28no—
144#384—24no—
145#380—28no—
146C4×C4⋊C4—24no—
147C4×C22⋊C4—24no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—44no—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2311yescharacteristic
181C2×#95C2312yes—
182C25C2411yescharacteristic
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8C2×D414yes—
192C22×Q8C2×Q814yes—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—44no—
200C23×C4—42no—
201C23×C4—44no—
202C23×C4#4114yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C22⋊Q8—232no—
220C4×Q8—416no—
221C4×Q8—416no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—416no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D418yes—
265C2×C4⋊C4#4114yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4C2×Q814yes—
283C2×C22⋊C4C2×Q818yes—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42C2×D414yes—
297C2×C42C2×D418yes—
298#95—22no—
299#95—24no—
300#95—22no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—22no—
313#95—24no—
314#95—44no—
315#95—42no—
316#95—42no—
317#95—44no—
318#95—42no—
319#95—44no—
320#95—42no—
321#95—42no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95C2×Q818yes—
329#95C2×Q814yes—
330C24—21no—
331C24—21no—
332C24—22no—
333C24—22no—
334C24—21no—
335C24—22no—
336C24—22no—
337C24—21no—
338C24C22×Q812yes—
339C24C2×#4112yes—
340C242−1+412yes—
341C24C2511yescharacteristic
342C2×Q8—28no—
343C2×Q8—28no—
344C2×Q8—28no—
345C2×Q8—28no—
346C2×Q8—48no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—48no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—28no—
397C22×C4—216no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—84no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×Q812yes—
443C22×C4C22×Q814yes—
444C22×C4C22×Q814yes—
445C22×C4C22×Q812yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4H(5,𝔽2)14yes—
451C22×C4H(5,𝔽2)14yes—
452C22×C42−1+412yes—
453C22×C42−1+414yes—
454C4⋊C4—216no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—48no—
458C4⋊C4—416no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—88no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4—88no—
470C4⋊C4—88no—
471C22⋊C4—216no—
472C22⋊C4—28no—
473C22⋊C4—48no—
474C22⋊C4—44no—
475C22⋊C4—48no—
476C22⋊C4—416no—
477C42—28no—
478C42—48no—
479C42—48no—
480C42—44no—
481C42—48no—
482C42—84no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23#55714yes—
506C23C23×D411yescharacteristic
507C23C23×Q811yescharacteristic
508C23C22×#4111yescharacteristic
509C23C22×#4111yescharacteristic
510C23C2×H(5,𝔽2)12yes—
511C23C2×H(5,𝔽2)12yes—
512C23C2×2−1+412yes—
513C23C2×2−1+411yescharacteristic
514C23C2×2−1+411yescharacteristic
515C23#58511yescharacteristic
516C23#58512yes—
517Q8—416no—
518Q8—416no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—416no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—816no—
607C2×C4#52514yes—
608C2×C4#53414yes—
609C2×C4#53614yes—
610C2×C4#53918yes—
611C2×C4#54218yes—
612C2×C4D4×Q814yes—
613C22—24no—
614C22—28no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—42no—
624C22C2×#52511yescharacteristic
625C22C2×#53411yescharacteristic
626C22C2×#53611yescharacteristic
627C22C2×#53912yes—
628C22C2×#54212yes—
629C22C2×D4×Q811yescharacteristic
630C22C2×#55711yescharacteristic
631C22Q8×#4111yescharacteristic
632C22#348711yescharacteristic
633C22#348911yescharacteristic
634C22#349811yescharacteristic
635C22#350112yes—
636C22#350312yes—
637C22#350612yes—
638C22#350711yescharacteristic
639C22#351312yes—
640C22#352311yescharacteristic
641C22#353612yes—
642C22#354112yes—
643C22#354412yes—
644C22#354612yes—
645C22#356312yes—
646C22#356512yes—
647C4—24no—
648C4—42no—
649C4—44no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—42no—
654C4—44no—
655C4—42no—
656C4—44no—
657C4—84no—
658C4—84no—
659C4—81no—
660C4—82no—
661C4—82no—
662C4#263818yes—
663C2—44no—
664C2C2×#263811yescharacteristic
665C2#3584411yescharacteristic
666C2#3654511yescharacteristic
667C2#3959511yescharacteristic
668C2#3961711yescharacteristic
669C2#3963912yes—
670C2#3965212yes—
671C2#4252112yes—
672C2#4256812yes—
673C2#4288811yescharacteristic
674C2#4298811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2582C2semidirect
3. #17025C2semidirect
4. #16991C2semidirect
6. #16311C2semidirect
7. #15202C2semidirect
9. #15186C2semidirect
11. #15145C2semidirect
15. #14029C2semidirect
16. #13931C2semidirect
129. #393Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#23602
#170252
#169912
#163432
#163112
#152022
#151952
#151862
#151822
#151452
#149692
#143192
#141302
#140292
#139312
C4×#3932