Finite Group Databasebeta

← #6539648

Subgroups — #6539648

Order 512.

2289 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3298C212yesmaximal
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#34308C211yesmaximal, characteristic
4#29798C211yesmaximal, characteristic, complemented
5#29756C212yesmaximal, complemented
6#28748C211yesmaximal, characteristic, complemented
7#28571C212yesmaximal, complemented
8#28205C211yesmaximal, characteristic, complemented
9#25276C212yesmaximal, complemented
10#25078C214yesmaximal, complemented
11#24730C212yesmaximal, complemented
12#24480C214yesmaximal, complemented
13#24209C212yesmaximal, complemented
14#23677C211yesmaximal, characteristic, complemented
15C2×#2043C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic, complemented
17D4×#103C212yesmaximal, complemented
18C2×D4×Q8C2212yes—
19C2×#547C2211yescharacteristic
20C22×C4⋊Q8C2212yes—
21C2×#520—21no—
22#3298—216no—
23C2×#487C2212yes—
24C2×#483C2214yes—
25C2×#475C2212yes—
26C2×#465C2214yes—
27C2×#461C2212yes—
28C2×#461C2214yes—
29C2×#448C2214yes—
30C2×C4×SD16C2212yes—
31C2×C8×D4C2212yes—
32C22×#103C2214yes—
33C22×#103C2214yes—
34#2868C2211yescharacteristic
35#2857C2212yes—
36#2551—21no—
37#2551—21no—
38#2546C2212yes—
39#2440—22no—
40#2440—22no—
41#2434C2214yes—
42#2365C2211yescharacteristic
43#2361C2212yes—
44#2360—21no—
45#2359—22no—
46D4×C4⋊C4—21no—
47D4×C4⋊C4—21no—
48D4×C4⋊C4C2211yescharacteristic
49D4×C4⋊C4C2211yescharacteristic
50#2344C2212yes—
51#2339—21no—
52#2323—21no—
53#2322—22no—
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×#121C2212yes—
57#2293C2211yescharacteristic
58C2×#389C2212yes—
59C42×D4C2211yescharacteristic
60#2084C2212yes—
61#2084C2211yescharacteristic
62#2071C2211yescharacteristic
63#2066C2214yes—
64#2066C2214yes—
65#2066C2212yes—
66#2055C2212yes—
67#2043C2212yes—
68#2043C2212yes—
69#2043C2212yes—
70#2043C2212yes—
71#2043C2212yes—
72#2043C2211yescharacteristic
73#2043C2214yes—
74#2043C2214yes—
75#2013C2212yes—
76#2013C2212yes—
77#2013C2211yescharacteristic
78#1981C2214yes—
79#1981C2212yes—
80#1981C2214yes—
81#1956C2212yes—
82#1939C2212yes—
83#1927C2214yes—
84#1927C2214yes—
85#1892C2214yes—
86#1877C2214yes—
87#1872C2214yes—
88#1850C2214yes—
89#1831C2211yescharacteristic
90#1803C2212yes—
91#1797C2212yes—
92#1787C2214yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C2×#336C2212yes—
96C22×#41—21no—
97C23×Q8C2314yes—
98C22×SD16C2312yes—
99C22×SD16C2314yes—
100C23×C8C2314yes—
101D4×Q8—28no—
102D4×Q8—28no—
103#547—24no—
104#547D418yes—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—22no—
108C2×C4⋊Q8—48no—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2314yes—
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2311yescharacteristic
115C2×#123—22no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8C2314yes—
119C2×#121C2312yes—
120#520—48no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8C2312yes—
124C2×C4×D4—21no—
125C2×C4×D4—22no—
126C2×C4×D4—22no—
127C2×C4×D4—22no—
128C2×C4×D4D411yescharacteristic
129C2×C4×D4D411yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2312yes—
132C2×C4×D4C2311yescharacteristic
133C2×#117—22no—
134C2×#117—21no—
135C22×C4⋊C4—22no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4—22no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2312yes—
141C22×C42C2312yes—
142#487—28no—
143#483—216no—
144#475—28no—
145#465—216no—
146#461—28no—
147#461—48no—
148#448—216no—
149C4×SD16—28no—
150C8×D4—28no—
151C2×#106C2314yes—
152C2×#106C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2314yes—
155C2×C4⋊C8C2312yes—
156C2×#103—216no—
157C2×#103—48no—
158C2×#103C2314yes—
159C2×#103C2312yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2312yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#102C2312yes—
171C2×C22⋊C8C2314yes—
172C2×C4×C8C2312yes—
173#398—24no—
174#395—24no—
175#395—24no—
176#391—21no—
177#391—21no—
178#391C2312yes—
179#391C2314yes—
180#390C2311yescharacteristic
181#390C2311yescharacteristic
182#389—21no—
183#389—22no—
184#389—22no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—22no—
189#389—22no—
190#389—22no—
191#389—22no—
192#389C2312yes—
193#389C2311yescharacteristic
194#389C2312yes—
195#389C2312yes—
196#389C2311yescharacteristic
197#388—21no—
198#388—22no—
199#388—21no—
200#388—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#386—22no—
205#386—22no—
206#385—22no—
207#385—24no—
208#385—24no—
209#385—22no—
210#385—22no—
211#384—21no—
212#384—21no—
213#384—21no—
214#384—21no—
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#381—22no—
218#380—22no—
219#380—22no—
220#380—21no—
221#380C2312yes—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4D411yescharacteristic
226C4×C4⋊C4D411yescharacteristic
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2311yescharacteristic
230C4×C4⋊C4C2311yescharacteristic
231C4×C22⋊C4—21no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4D412yes—
234C4×C22⋊C4D412yes—
235C4×C22⋊C4C2312yes—
236C2×#95—22no—
237C43C2311yescharacteristic
238#340C2312yes—
239#340C2314yes—
240#340C2312yes—
241#336C2311yescharacteristic
242#336C2312yes—
243#336C2311yescharacteristic
244#336C2312yes—
245#336C2314yes—
246C2×#41—28no—
247C2×#41—44no—
248C2×#41—44no—
249C22×Q8—21no—
250C22×Q8—24no—
251C22×Q8—28no—
252C22×Q8—28no—
253C22×Q8—28no—
254C22×Q8—44no—
255C22×Q8—44no—
256C22×Q8—44no—
257C22×Q8C2414yes—
258C22×Q8C2412yes—
259C22×D4—22no—
260C22×D4C2×D411yescharacteristic
261C22×D4C2411yescharacteristic
262C23×C4—22no—
263C23×C4—21no—
264C23×C4—24no—
265C23×C4—22no—
266C23×C4—22no—
267C23×C4C2×D412yes—
268C23×C4C2×D412yes—
269C23×C4C2412yes—
270C2×SD16—24no—
271C2×SD16—24no—
272C2×SD16—28no—
273C2×SD16—42no—
274C2×SD16—44no—
275C2×SD16—44no—
276C22×C8—28no—
277C22×C8—44no—
278C22×C8—44no—
279C22×C8C2412yes—
280C22×C8C2414yes—
281C22×C8C2412yes—
282C4⋊Q8—24no—
283C4⋊Q8—28no—
284C4⋊Q8—24no—
285C4⋊Q8—22no—
286C4⋊Q8—24no—
287C4⋊Q8—44no—
288C4⋊Q8—44no—
289C4⋊Q8—44no—
290C4⋊Q8—48no—
291C4⋊Q8—48no—
292C4⋊Q8—48no—
293C4⋊Q8C2×D414yes—
294#123—216no—
295C22⋊Q8—28no—
296C22⋊Q8—28no—
297C22⋊Q8—28no—
298C22⋊Q8—416no—
299#121—24no—
300#121C2×D418yes—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—44no—
304C4×Q8—44no—
305C4×D4—24no—
306C4×D4—22no—
307C4×D4—44no—
308C4×D4—48no—
309C4×D4—44no—
310C4×D4—48no—
311C4×D4—48no—
312C4×D4SD1618yescomplemented
313C4×D4C2×D414yes—
314C4×D4C2×D414yes—
315C4×D4#4114yes—
316#117—48no—
317#117—44no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—44no—
344C2×C4⋊C4—48no—
345C2×C4⋊C4—48no—
346C2×C4⋊C4—48no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D418yes—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2412yes—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4C2×D412yes—
368C2×C22⋊C4C2×D412yes—
369C2×C22⋊C4C2×D412yes—
370C2×C42—22no—
371C2×C42—21no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—44no—
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D412yes—
378C2×C42C2×D412yes—
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385#106—24no—
386#106—28no—
387C4⋊C8—24no—
388C4⋊C8—42no—
389C4⋊C8—44no—
390#103—24no—
391#103—28no—
392#103—24no—
393#103—28no—
394#103—28no—
395#103—28no—
396#103—28no—
397#103—44no—
398#103—44no—
399#103—42no—
400#103—44no—
401#103—44no—
402#102—24no—
403C22⋊C8—28no—
404C4×C8—24no—
405#95—21no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—21no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—21no—
416#95—24no—
417#95—24no—
418#95C2×D412yes—
419#95C2×D412yes—
420#95C2×D412yes—
421C24—21no—
422C24C22×D412yes—
423#41—432no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—24no—
435C2×Q8—24no—
436C2×Q8—48no—
437C2×Q8—416no—
438C2×Q8—416no—
439C2×Q8—48no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—84no—
443C2×Q8—88no—
444C2×Q8—88no—
445C2×Q8H(5,𝔽2)14yes—
446C2×D4—28no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4#12114yes—
453C2×D4C2×SD1614yes—
454C2×D4#13714yes—
455C2×D4C22×D412yes—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—28no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—48no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—48no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4—84no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4C2×SD1618yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D412yes—
491C22×C4C22×D414yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D412yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C2×#4114yes—
498C22×C4C2511yescharacteristic
499SD16—48no—
500SD16—88no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—24no—
509C2×C8—22no—
510C2×C8—88no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—28no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—48no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—81no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4#12114yes—
544C4⋊C4C2×SD1614yes—
545C4⋊C4C22×D412yes—
546C4⋊C4C22×D414yes—
547C22⋊C4—24no—
548C22⋊C4—24no—
549C22⋊C4—28no—
550C22⋊C4—48no—
551C22⋊C4—44no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—81no—
556C22⋊C4—82no—
557C22⋊C4#12118yes—
558C22⋊C4C2×SD1618yes—
559C42—22no—
560C42—22no—
561C42—24no—
562C42—22no—
563C42—22no—
564C42—42no—
565C42—42no—
566C42—41no—
567C42—44no—
568C42—44no—
569C42—44no—
570C42—42no—
571C42C2×SD1614yes—
572C42C22×D412yes—
573C42C2×#4112yes—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23C2×#12114yes—
586C23C22×SD1614yes—
587C23C2×#13714yes—
588C23C23×D411yescharacteristic
589Q8—24no—
590Q8—24no—
591Q8—42no—
592Q8—42no—
593Q8—48no—
594Q8—44no—
595Q8—44no—
596Q8—44no—
597Q8—44no—
598Q8—816no—
599D4—24no—
600D4—416no—
601D4—42no—
602D4#461116yescomplemented
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—48no—
627C2×C4—48no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—416no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—48no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—44no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—816no—
652C2×C4—164no—
653C2×C4—164no—
654C2×C4C2×#12112yes—
655C2×C4C2×#12114yes—
656C2×C4C2×#12112yes—
657C2×C4C22×SD1612yes—
658C2×C4C22×SD1614yes—
659C2×C4C22×SD1612yes—
660C2×C4C22×SD1612yes—
661C2×C4C2×#13712yes—
662C2×C4#57812yes—
663C2×C4C23×D411yescharacteristic
664C2×C4C22×#4111yescharacteristic
665C2×C4C2×H(5,𝔽2)12yes—
666C2×C4#58512yes—
667C8—42no—
668C8—44no—
669C8—42no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22C2×#46118yes—
684C22C22×#12111yescharacteristic
685C22#348411yescharacteristic
686C22#350011yescharacteristic
687C22C23×SD1611yescharacteristic
688C22C22×#13711yescharacteristic
689C22C2×#57811yescharacteristic
690C22#358611yescharacteristic
691C4—22no—
692C4—22no—
693C4—22no—
694C4—24no—
695C4—24no—
696C4—22no—
697C4—41no—
698C4—44no—
699C4—42no—
700C4—42no—
701C4—44no—
702C4—82no—
703C4—81no—
704C4—81no—
705C4—82no—
706C4—84no—
707C4C2×#46114yes—
708C4#348812yes—
709C4#351312yes—
710C2—28no—
711C2—82no—
712C2#3572811yescharacteristic
713C2C22×#46111yescharacteristic
714C2#6043211yescharacteristic
715C2#6156211yescharacteristic
716C2#6160211yescharacteristic
717C2#6252611yescharacteristic
718C2#6257711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. D4×C4⋊Q8C2semidirect
4. #29798C2semidirect
5. #29756C2semidirect
6. #28748C2semidirect
7. #28571C2semidirect
8. #28205C2semidirect
9. #25276C2semidirect
10. #25078C2semidirect
11. #24730C2semidirect
12. #24480C2semidirect
13. #24209C2semidirect
14. #23677C2semidirect
15. C2×#2043C2semidirect
16. #22696C2semidirect
17. D4×#103C2semidirect
312. C4×D4SD16semidirect
602. D4#461semidirect

Maximal subgroups

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

subgroupindex
C2×#32982
D4×C4⋊Q82
#343082
#297982
#297562
#287482
#285712
#282052
#252762
#250782
#247302
#244802
#242092
#236772
C2×#20432
#226962
D4×#1032