Finite Group Databasebeta

← #4419100

Subgroups — #4419100

Order 512.

2481 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2498C211yesmaximal, characteristic
2C2×#2625C211yesmaximal, characteristic, complemented
3C2×#2663C211yesmaximal, characteristic
4C2×#2346C211yesmaximal, characteristic, complemented
5#17074C212yesmaximal, complemented
6#16899C212yesmaximal, complemented
7#16567C212yesmaximal, complemented
8#16566C212yesmaximal, complemented
9#15404C214yesmaximal
10#15290C214yesmaximal, complemented
11#15149C212yesmaximal, complemented
12#15081C212yesmaximal, complemented
13#14910C212yesmaximal, complemented
14#14901C212yesmaximal, complemented
15#14658C211yesmaximal, characteristic
16#14281C211yesmaximal, characteristic, complemented
17#13965C211yesmaximal, characteristic, complemented
18C22×#123C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#117C2211yescharacteristic
21#2663—28no—
22#2625—28no—
23#2498—28no—
24C2×#400C2212yes—
25C2×#400C2211yescharacteristic
26C2×#400C2211yescharacteristic
27C2×#399C2211yescharacteristic
28C2×#399C2211yescharacteristic
29C2×#398C2211yescharacteristic
30C2×#398C2212yes—
31C2×#397C2212yes—
32C2×#397C2214yes—
33C2×#397C2211yescharacteristic
34C2×#395C2211yescharacteristic
35C2×#394C2212yes—
36C2×#393C2211yescharacteristic
37#2346—28no—
38C2×#391C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2214yes—
43C2×#388C2212yes—
44C2×#388C2214yes—
45C2×#388C2212yes—
46C2×#388C2214yes—
47C2×#387C2212yes—
48C2×#386C2212yes—
49C2×#385C2214yes—
50C2×#382C2212yes—
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×#380C2212yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×#376C2212yes—
57C22×#95C2211yescharacteristic
58#1446C2212yes—
59#1445C2214yes—
60#1445C2212yes—
61#1444C2212yes—
62#1444C2212yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2212yes—
71#1441C2212yes—
72#1441C2214yes—
73#1441C2212yes—
74#1441C2214yes—
75#1440C2212yes—
76#1439C2214yes—
77#1439C2214yes—
78#1439C2212yes—
79#1439C2214yes—
80#1439C2212yes—
81#1438C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2214yes—
84C4⋊#95C2212yes—
85#1434C2214yes—
86C22⋊#95C2214yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C2×#123—24no—
92C2×#123—24no—
93C2×#123—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117D418yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C42C2314yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132#400—28no—
133#400—24no—
134#400—24no—
135#399—24no—
136#399—24no—
137#398—28no—
138#398—24no—
139#397—24no—
140#397—216no—
141#397—28no—
142#395—24no—
143#394—28no—
144#393—24no—
145#391—28no—
146#389—28no—
147#389—28no—
148#389—28no—
149#389—48no—
150#388—216no—
151#388—28no—
152#388—28no—
153#388—48no—
154#387—28no—
155#386—28no—
156#385—48no—
157#382—28no—
158#380—24no—
159#380—24no—
160#380—28no—
161C4×C4⋊C4—28no—
162C4×C22⋊C4D418yes—
163#376—28no—
164C2×#95—24no—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2311yescharacteristic
193C2×#95C2311yescharacteristic
194C25C2411yescharacteristic
195C25C2411yescharacteristic
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—44no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—44no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—42no—
209C23×C4—44no—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4#4114yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228#123—416no—
229C22⋊Q8—416no—
230#117—232no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
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⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D414yes—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—42no—
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⋊C4C2×D418yes—
309C2×C22⋊C4C2×D418yes—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—28no—
313C2×C42—28no—
314C2×C42—28no—
315C2×C42—44no—
316C2×C42C2×D418yes—
317C2×C42C2×D418yes—
318#95—24no—
319#95—22no—
320#95—22no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95C2×D414yes—
346#95C2×D414yes—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—21no—
353C24—21no—
354C24—21no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24C22×D412yes—
361C24C22×D412yes—
362C24C2×#4112yes—
363C24C2511yescharacteristic
364C2×Q8—48no—
365C2×Q8—44no—
366C2×Q8—84no—
367C2×Q8—84no—
368C2×D4—48no—
369C2×D4—44no—
370C2×D4—84no—
371C2×D4—84no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
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—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—88no—
445C22×C4—84no—
446C22×C4—88no—
447C22×C4#121116yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D414yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C42−1+414yes—
460C22×C42−1+414yes—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—416no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C4⋊C4—88no—
475C4⋊C4—88no—
476C4⋊C4—88no—
477C22⋊C4—216no—
478C22⋊C4—216no—
479C22⋊C4—48no—
480C22⋊C4—416no—
481C22⋊C4—48no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—48no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C22⋊C4—84no—
490C42—216no—
491C42—48no—
492C42—416no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23C2×#12114yes—
525C23C23×D411yescharacteristic
526C23C23×D411yescharacteristic
527C23C22×#4111yescharacteristic
528C23C2×H(5,𝔽2)12yes—
529C23C2×2−1+412yes—
530C23C2×2−1+412yes—
531C23#58514yes—
532C23#58512yes—
533Q8—88no—
534D4—88no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—416no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—816no—
614C2×C4C2×#12114yes—
615C2×C4C2×#12114yes—
616C2×C4#53718yes—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—48no—
625C22—81no—
626C22—81no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—81no—
631C22—82no—
632C22—84no—
633C22C22×#12111yescharacteristic
634C22C2×#53712yes—
635C22#344912yes—
636C22#348512yes—
637C22#348912yes—
638C22#349112yes—
639C22#349811yescharacteristic
640C22#349911yescharacteristic
641C22#349911yescharacteristic
642C22#350011yescharacteristic
643C22#350712yes—
644C22#351112yes—
645C22#353514yes—
646C22#354014yes—
647C22#356314yes—
648C22#356514yes—
649C4—24no—
650C4—24no—
651C4—28no—
652C4—44no—
653C4—44no—
654C4—44no—
655C4—44no—
656C4—81no—
657C4—82no—
658C4—81no—
659C4—84no—
660C4—81no—
661C4—82no—
662C4—84no—
663C4—82no—
664C2—44no—
665C2—161no—
666C2#3508911yescharacteristic
667C2#3527311yescharacteristic
668C2#3562111yescharacteristic
669C2#4137312yes—
670C2#4196012yes—
671C2#4231612yes—
672C2#4265012yes—
673C2#4887314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2625C2semidirect
4. C2×#2346C2semidirect
5. #17074C2semidirect
6. #16899C2semidirect
7. #16567C2semidirect
8. #16566C2semidirect
10. #15290C2semidirect
11. #15149C2semidirect
12. #15081C2semidirect
13. #14910C2semidirect
14. #14901C2semidirect
16. #14281C2semidirect
17. #13965C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24982
C2×#26252
C2×#26632
C2×#23462
#170742
#168992
#165672
#165662
#154042
#152902
#151492
#150812
#149102
#149012
#146582
#142812
#139652