Finite Group Databasebeta

← #4457148

Subgroups — #4457148

Order 512.

2581 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic, complemented
2C2×#2604C211yesmaximal, characteristic, complemented
3C2×#2603C211yesmaximal, characteristic
4C2×#2320C211yesmaximal, characteristic, complemented
5#17071C212yesmaximal, complemented
6#17039C212yesmaximal, complemented
7#16901C212yesmaximal, complemented
8#16887C212yesmaximal, complemented
9#16677C212yesmaximal, complemented
10#16662C212yesmaximal
11#16519C212yesmaximal, complemented
12#16518C212yesmaximal, complemented
13#14756C214yesmaximal, complemented
14#14752C214yesmaximal, complemented
15#14291C211yesmaximal, characteristic, complemented
16#13963C211yesmaximal, characteristic, complemented
17#13847C211yesmaximal, characteristic, complemented
18C22×#126C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2604—28no—
22#2603—28no—
23#2580—28no—
24C2×#400C2211yescharacteristic
25C2×#399C2211yescharacteristic
26C2×#399C2212yes—
27C2×#398C2211yescharacteristic
28C2×#397C2211yescharacteristic
29C2×#396C2214yes—
30C2×#396C2214yes—
31C2×#395C2211yescharacteristic
32C2×#394C2211yescharacteristic
33C2×#393C2212yes—
34C2×#393C2211yescharacteristic
35C2×#392C2211yescharacteristic
36#2320—28no—
37C2×#391C2212yes—
38C2×#390C2212yes—
39C2×#390C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#388C2212yes—
44C2×#388C2212yes—
45C2×#388C2212yes—
46C2×#387C2214yes—
47C2×#387C2212yes—
48C2×#386C2212yes—
49C2×#385C2212yes—
50C2×#385C2214yes—
51C2×#383C2212yes—
52C2×#382C2212yes—
53C2×#382C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2214yes—
56C2×#376C2211yescharacteristic
57C2×#376C2211yescharacteristic
58#1445C2212yes—
59#1445C2214yes—
60#1444C2214yes—
61#1444C2212yes—
62#1443C2214yes—
63#1442C2212yes—
64#1442C2214yes—
65#1441C2214yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1439C2214yes—
73#1439C2212yes—
74#1439C2214yes—
75#1439C2212yes—
76#1439C2212yes—
77#1439C2214yes—
78#1439C2212yes—
79#1438C2212yes—
80#1438C2214yes—
81#1438C2214yes—
82#1437C2212yes—
83#1436C2214yes—
84#1436C2212yes—
85C4⋊#95C2212yes—
86#1434C2212yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C23×D4C2312yes—
90C24×C4C2312yes—
91C2×#126—24no—
92C2×#126—24no—
93C2×#126—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8D418yescomplemented
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#400—24no—
135#399—28no—
136#399—24no—
137#398—24no—
138#397—24no—
139#396—48no—
140#396—48no—
141#395—24no—
142#394—24no—
143#393—28no—
144#393—24no—
145#392—24no—
146#391—28no—
147#390—28no—
148#390—28no—
149#389—28no—
150#389—28no—
151#389—28no—
152#388—28no—
153#388—28no—
154#388—28no—
155#387—28no—
156#387—216no—
157#386—28no—
158#385—216no—
159#385—28no—
160#383—28no—
161#382—24no—
162#382—28no—
163C4×C4⋊C4—24no—
164C4×C22⋊C4—48no—
165#376—24no—
166#376D418yes—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2311yescharacteristic
192C2×#95C2311yescharacteristic
193C2×#95C2312yes—
194C25C2411yescharacteristic
195C25C2412yes—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—44no—
199C22×Q8C2×D414yes—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—42no—
207C22×D4—42no—
208C22×D4—44no—
209C22×D4—44no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230#126—416no—
231C4×Q8—232no—
232C4×D4—416no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—42no—
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⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2×D414yes—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—44no—
318C2×C42—42no—
319C2×C42—44no—
320C2×C42C2×D418yes—
321C2×C42C2×D418yes—
322C2×C42C2×D414yes—
323C2×C42C2×D414yes—
324C2×C42#4114yes—
325#95—24no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—24no—
335#95—28no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95C2×D414yes—
350#95C2×D414yes—
351#95C2×D418yes—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24—24no—
360C24—24no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—22no—
369C24—22no—
370C242−1+414yes—
371C24C2511yescharacteristic
372C2×Q8—28no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×D4—28no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—44no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—44no—
383C2×D4—84no—
384C2×D4—88no—
385C2×D4—88no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—416no—
452C22×C4—48no—
453C22×C4—88no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4C22×D414yes—
457C22×C4C22×D414yes—
458C22×C4C22×D414yes—
459C22×C4C22×D414yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4H(5,𝔽2)14yes—
470C4⋊C4—28no—
471C4⋊C4—216no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—88no—
481C4⋊C4—88no—
482C4⋊C4—84no—
483C4⋊C4—88no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—416no—
488C22⋊C4—48no—
489C22⋊C4—416no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—88no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—88no—
499C42—216no—
500C42—48no—
501C42—48no—
502C42—84no—
503C42#121116yes—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—28no—
508C23—28no—
509C23—42no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—42no—
515C23—44no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23C23×D411yescharacteristic
544C23C23×D411yescharacteristic
545C23C22×#4111yescharacteristic
546C23C2×H(5,𝔽2)12yes—
547C23C2×2−1+412yes—
548C23#58514yes—
549C23#58512yes—
550C23#58512yes—
551Q8—416no—
552D4—416no—
553D4—816no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—816no—
631C2×C4C2×#12114yes—
632C2×C4C2×#12114yes—
633C2×C4C2×#12114yes—
634C22—48no—
635C22—44no—
636C22—44no—
637C22—48no—
638C22—44no—
639C22—84no—
640C22—82no—
641C22—82no—
642C22—84no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—81no—
648C22—81no—
649C22—82no—
650C22—84no—
651C22—81no—
652C22—82no—
653C22C22×#12111yescharacteristic
654C22#344912yes—
655C22#344912yes—
656C22#345512yes—
657C22#348611yescharacteristic
658C22#348711yescharacteristic
659C22#348812yes—
660C22#348912yes—
661C22#349512yes—
662C22#349811yescharacteristic
663C22#350011yescharacteristic
664C22#350612yes—
665C22#353314yes—
666C22#354114yes—
667C22#356414yes—
668C22#356614yes—
669C4—24no—
670C4—24no—
671C4—24no—
672C4—44no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—84no—
678C4—82no—
679C4—81no—
680C4—81no—
681C4—81no—
682C4—82no—
683C4—84no—
684C2—84no—
685C2—161no—
686C2#3463311yescharacteristic
687C2#3553311yescharacteristic
688C2#3566111yescharacteristic
689C2#3680414yes—
690C2#4195912yes—
691C2#4198712yes—
692C2#4198812yes—
693C2#4239212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2580C2semidirect
2. C2×#2604C2semidirect
4. C2×#2320C2semidirect
5. #17071C2semidirect
6. #17039C2semidirect
7. #16901C2semidirect
8. #16887C2semidirect
9. #16677C2semidirect
11. #16519C2semidirect
12. #16518C2semidirect
13. #14756C2semidirect
14. #14752C2semidirect
15. #14291C2semidirect
16. #13963C2semidirect
17. #13847C2semidirect
96. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
C2×#26042
C2×#26032
C2×#23202
#170712
#170392
#169012
#168872
#166772
#166622
#165192
#165182
#147562
#147522
#142912
#139632
#138472