Finite Group Databasebeta

← #4418518

Subgroups — #4418518

Order 512.

2591 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2592C211yesmaximal, characteristic, complemented
2C2×#2637C211yesmaximal, characteristic, complemented
3C2×#2662C211yesmaximal, characteristic
4C2×#2351C211yesmaximal, characteristic, complemented
5#17040C212yesmaximal, complemented
6#16888C212yesmaximal, complemented
7#16565C212yesmaximal, complemented
8#16516C212yesmaximal
9#15677C214yesmaximal, complemented
10#15619C214yesmaximal, complemented
11#15176C212yesmaximal, complemented
12#14883C212yesmaximal, complemented
13#14866C212yesmaximal, complemented
14#14813C212yesmaximal, complemented
15#14662C211yesmaximal, characteristic, complemented
16#14291C211yesmaximal, characteristic, complemented
17#13955C211yesmaximal, characteristic, complemented
18C22×#126C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22#2662—28no—
23#2637—28no—
24#2592—28no—
25C2×#398C2212yes—
26C2×#398C2211yescharacteristic
27C2×#397C2211yescharacteristic
28C2×#397C2212yes—
29C2×#396C2212yes—
30C2×#396C2212yes—
31C2×#395C2211yescharacteristic
32C2×#395C2211yescharacteristic
33C2×#394C2212yes—
34C2×#394C2211yescharacteristic
35C2×#394C2214yes—
36C2×#393C2211yescharacteristic
37C2×#393C2211yescharacteristic
38C2×#393C2212yes—
39C2×#392C2211yescharacteristic
40#2351—28no—
41C2×#391C2212yes—
42C2×#390C2212yes—
43C2×#390C2212yes—
44C2×#389C2214yes—
45C2×#389C2212yes—
46C2×#388C2214yes—
47C2×#388C2212yes—
48C2×#388C2212yes—
49C2×#388C2214yes—
50C2×#387C2212yes—
51C2×#382C2211yescharacteristic
52C2×#382C2211yescharacteristic
53C2×#382C2212yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57#1446C2212yes—
58#1445C2214yes—
59#1443C2214yes—
60#1442C2212yes—
61#1442C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2212yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2212yes—
77#1438C2214yes—
78#1438C2212yes—
79#1437C2212yes—
80#1437C2214yes—
81#1437C2214yes—
82#1436C2212yes—
83C4⋊#95C2214yes—
84#1434C2214yes—
85#1434C2212yes—
86#1434C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×D4C2311yescharacteristic
90C23×D4C2312yes—
91C2×#126—24no—
92C2×#126—24no—
93C2×#126—24no—
94C2×#124—24no—
95C2×#124—24no—
96C2×#124—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—28no—
99C2×C4×Q8—24no—
100C2×C4×Q8—48no—
101C2×C4×Q8D418yescomplemented
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C42C2314yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#398—24no—
138#398—28no—
139#397—28no—
140#397—24no—
141#396—28no—
142#396—28no—
143#395—24no—
144#395—24no—
145#394—28no—
146#394—24no—
147#394—48no—
148#393—28no—
149#393—24no—
150#393—24no—
151#392—24no—
152#391—28no—
153#390—28no—
154#390—28no—
155#389—216no—
156#389—28no—
157#388—216no—
158#388—28no—
159#388—28no—
160#388—48no—
161#387—28no—
162#382—24no—
163#382—24no—
164#382—28no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—28no—
167C4×C4⋊C4D418yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2311yescharacteristic
193C2×#95C2311yescharacteristic
194C25C2411yescharacteristic
195C25C2412yes—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—44no—
204C22×Q8C2×D414yes—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—24no—
208C22×D4—22no—
209C22×D4—44no—
210C22×D4—42no—
211C22×D4—42no—
212C22×D4—44no—
213C22×D4—44no—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229#126—416no—
230#124—416no—
231C4×Q8—232no—
232C4×Q8—432no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
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—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—42no—
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⋊C4C2×D418yes—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D418yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
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×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—28no—
317C2×C42—28no—
318C2×C42—24no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42C2×D418yes—
323C2×C42C2×D418yes—
324C2×C42C2×D414yes—
325C2×C42C2×D414yes—
326C2×C42#4114yes—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—24no—
332#95—22no—
333#95—22no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—28no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95C2×D414yes—
352#95C2×D414yes—
353C24—21no—
354C24—22no—
355C24—21no—
356C24—22no—
357C24—21no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24C2511yescharacteristic
373C2×Q8—28no—
374C2×Q8—28no—
375C2×Q8—216no—
376C2×Q8—28no—
377C2×Q8—216no—
378C2×Q8—48no—
379C2×Q8—44no—
380C2×Q8—84no—
381C2×Q8—84no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—44no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4—84no—
388C2×D4—84no—
389C2×D4—88no—
390C2×D4—88no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
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—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D414yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4H(5,𝔽2)14yes—
469C22×C42−1+414yes—
470C4⋊C4—216no—
471C4⋊C4—28no—
472C4⋊C4—48no—
473C4⋊C4—416no—
474C4⋊C4—48no—
475C4⋊C4—416no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C22⋊C4—48no—
485C22⋊C4—416no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—416no—
492C22⋊C4—48no—
493C22⋊C4—84no—
494C22⋊C4—88no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C42—216no—
501C42—48no—
502C42—416no—
503C42—84no—
504C42#121116yes—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—42no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23C23×D411yescharacteristic
545C23C23×D411yescharacteristic
546C23C22×#4111yescharacteristic
547C23C2×H(5,𝔽2)12yes—
548C23C2×2−1+412yes—
549C23#58512yes—
550C23#58512yes—
551C23#58514yes—
552Q8—232no—
553Q8—88no—
554D4—88no—
555D4—816no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
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—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
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—44no—
631C2×C4C2×#12114yes—
632C2×C4C2×#12114yes—
633C2×C4C2×#12114yes—
634C2×C4#53718yes—
635C22—81no—
636C22—82no—
637C22—84no—
638C22—84no—
639C22—84no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—81no—
645C22—84no—
646C22—82no—
647C22—84no—
648C22—82no—
649C22—81no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22C22×#12111yescharacteristic
655C22C2×#53712yes—
656C22#344912yes—
657C22#348411yescharacteristic
658C22#348611yescharacteristic
659C22#348712yes—
660C22#349212yes—
661C22#350011yescharacteristic
662C22#350011yescharacteristic
663C22#350012yes—
664C22#351012yes—
665C22#351212yes—
666C22#354014yes—
667C22#355714yes—
668C22#356214yes—
669C22#356514yes—
670C4—24no—
671C4—24no—
672C4—24no—
673C4—28no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—81no—
678C4—84no—
679C4—81no—
680C4—82no—
681C4—84no—
682C4—82no—
683C4—82no—
684C4—81no—
685C2—161no—
686C2—162no—
687C2#3522311yescharacteristic
688C2#3532811yescharacteristic
689C2#3567711yescharacteristic
690C2#4050712yes—
691C2#4215412yes—
692C2#4287712yes—
693C2#4288212yes—
694C2#5124214yes—
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×#2592C2semidirect
2. C2×#2637C2semidirect
4. C2×#2351C2semidirect
5. #17040C2semidirect
6. #16888C2semidirect
7. #16565C2semidirect
9. #15677C2semidirect
10. #15619C2semidirect
11. #15176C2semidirect
12. #14883C2semidirect
13. #14866C2semidirect
14. #14813C2semidirect
15. #14662C2semidirect
16. #14291C2semidirect
17. #13955C2semidirect
101. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25922
C2×#26372
C2×#26622
C2×#23512
#170402
#168882
#165652
#165162
#156772
#156192
#151762
#148832
#148662
#148132
#146622
#142912
#139552