Finite Group Databasebeta

← #6566086

Subgroups — #6566086

Order 512.

2051 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3335C212yesmaximal
2#36180C211yesmaximal, characteristic, complemented
3#34308C211yesmaximal, characteristic
4#29868C211yesmaximal, characteristic, complemented
5#29756C212yesmaximal, complemented
6#28687C211yesmaximal, characteristic, complemented
7#28571C212yesmaximal, complemented
8#28205C211yesmaximal, characteristic, complemented
9#27645C212yesmaximal, complemented
10#25453C212yesmaximal, complemented
11#25202C214yesmaximal, complemented
12#24815C212yesmaximal, complemented
13#24580C214yesmaximal, complemented
14#23677C211yesmaximal, characteristic, complemented
15C2×#2084C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic, complemented
17#22487C212yesmaximal, complemented
18C2×#554C2212yes—
19C2×#547C2211yescharacteristic
20C2×#520—21no—
21#3335—216no—
22C2×#498C2212yes—
23C2×#481C2214yes—
24C2×#483C2214yes—
25C2×#475C2212yes—
26C2×#476C2212yes—
27C2×#465C2214yes—
28C2×C4×SD16C2212yes—
29C2×C8×D4C2212yes—
30C22×#106C2214yes—
31#2868C2211yescharacteristic
32#2857C2212yes—
33#2567C2212yes—
34#2552—21no—
35#2552—21no—
36#2502C2214yes—
37#2440—22no—
38#2440—22no—
39#2365C2211yescharacteristic
40#2361C2212yes—
41#2360—21no—
42#2359—22no—
43D4×C4⋊C4—21no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2211yescharacteristic
46D4×C4⋊C4C2211yescharacteristic
47#2344C2212yes—
48#2339—21no—
49#2323—21no—
50#2322—22no—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×#121C2212yes—
54#2293C2211yescharacteristic
55C2×#389C2212yes—
56C2×#384C2212yes—
57C42×D4C2211yescharacteristic
58#2084C2212yes—
59#2084C2214yes—
60#2084C2212yes—
61#2084C2212yes—
62#2084C2212yes—
63#2084C2212yes—
64#2084C2211yescharacteristic
65#2084C2214yes—
66#2071C2212yes—
67#2071C2211yescharacteristic
68#2071C2212yes—
69#2066C2212yes—
70#2066C2214yes—
71#2066C2214yes—
72#2055C2212yes—
73#2043C2212yes—
74#2043C2211yescharacteristic
75#2013C2211yescharacteristic
76#1990C2212yes—
77#1987C2212yes—
78#1948C2214yes—
79#1948C2214yes—
80#1939C2212yes—
81#1927C2214yes—
82#1927C2214yes—
83#1892C2214yes—
84#1877C2214yes—
85#1872C2214yes—
86#1850C2214yes—
87#1831C2211yescharacteristic
88#1824C2214yes—
89#1803C2212yes—
90#1802C2214yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94C2×#340C2214yes—
95C2×#336C2212yes—
96C22×#41—21no—
97C22×SD16C2312yes—
98C23×C8C2314yes—
99#554—28no—
100#554—28no—
101#547—24no—
102#547D418yes—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#125—22no—
109C2×#123—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8C2314yes—
113C2×#121C2312yes—
114#520—48no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C2×#117—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2314yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C42C2312yes—
137#498—28no—
138#483—216no—
139#481—216no—
140#476—28no—
141#475—28no—
142#465—216no—
143C4×SD16—28no—
144C8×D4—28no—
145C2×#106—216no—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×#106C2312yes—
150C2×#106C2314yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2314yes—
154C2×#103C2314yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#102C2312yes—
160C2×#102C2314yes—
161C2×C22⋊C8C2314yes—
162C2×C4×C8C2312yes—
163#398—24no—
164#398—24no—
165#395—24no—
166#391—21no—
167#391—21no—
168#391C2312yes—
169#391C2314yes—
170#390C2311yescharacteristic
171#390C2311yescharacteristic
172#389—22no—
173#389—22no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389C2312yes—
183#389C2311yescharacteristic
184#389C2312yes—
185#389C2312yes—
186#389C2311yescharacteristic
187#388—21no—
188#388—22no—
189#388—21no—
190#388—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#386—22no—
195#386—22no—
196#385—22no—
197#385—24no—
198#385—24no—
199#385—22no—
200#385—22no—
201#384—21no—
202#384—21no—
203#384—22no—
204#384—22no—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#384C2314yes—
208#384C2312yes—
209#384C2312yes—
210#381—22no—
211#380—22no—
212#380—22no—
213#380—21no—
214#380C2312yes—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—21no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4C2312yes—
229C2×#95—22no—
230C43C2311yescharacteristic
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#336C2311yescharacteristic
239#336C2312yes—
240#336C2312yes—
241#336C2311yescharacteristic
242#336C2314yes—
243C2×#41—28no—
244C2×#41—44no—
245C2×#41—44no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8C2412yes—
249C22×D4—22no—
250C22×D4C2×D411yescharacteristic
251C22×D4C2411yescharacteristic
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—24no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2412yes—
260C2×SD16—24no—
261C2×SD16—24no—
262C2×SD16—42no—
263C22×C8—28no—
264C22×C8—28no—
265C22×C8—44no—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C4⋊Q8—24no—
270C4⋊Q8—22no—
271C4⋊Q8—24no—
272C4⋊Q8—42no—
273C4⋊Q8—82no—
274C4⋊Q8—82no—
275C4⋊Q8C2×D414yes—
276#125—48no—
277#123—216no—
278C22⋊Q8—28no—
279C22⋊Q8—28no—
280C22⋊Q8—28no—
281C22⋊Q8—416no—
282#121—24no—
283#121C2×D418yes—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×D4—24no—
289C4×D4—22no—
290C4×D4—44no—
291C4×D4—48no—
292C4×D4—44no—
293C4×D4—48no—
294C4×D4—48no—
295C4×D4SD1618yescomplemented
296C4×D4C2×D414yes—
297C4×D4C2×D414yes—
298C4×D4#4114yes—
299#117—48no—
300#117—44no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—48no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—84no—
332C2×C4⋊C4—84no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D418yes—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2414yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2412yes—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—22no—
357C2×C42—21no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—44no—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D411yescharacteristic
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#106—24no—
372#106—28no—
373#106—28no—
374#106—28no—
375#106—28no—
376C4⋊C8—24no—
377C4⋊C8—24no—
378C4⋊C8—44no—
379#103—24no—
380#103—24no—
381#103—28no—
382#103—42no—
383#103—44no—
384#102—24no—
385#102—28no—
386C22⋊C8—28no—
387C4×C8—24no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—21no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—21no—
399#95—24no—
400#95—24no—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404C24—21no—
405C24C22×D412yes—
406#41—432no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—44no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—82no—
418C2×Q8—82no—
419C2×D4—28no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4#12114yes—
426C2×D4C2×SD1614yes—
427C2×D4#13714yes—
428C2×D4C22×D412yes—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—28no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—84no—
457C22×C4—84no—
458C22×C4—82no—
459C22×C4C2×SD1618yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D414yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4114yes—
471C22×C4C2511yescharacteristic
472SD16—48no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—416no—
482C2×C82−1+414yes—
483C4⋊C4—28no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—81no—
512C4⋊C4—82no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4#12114yes—
518C4⋊C4C2×SD1614yes—
519C4⋊C4C22×D412yes—
520C4⋊C4C22×D414yes—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—28no—
524C22⋊C4—48no—
525C22⋊C4—44no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—81no—
530C22⋊C4—82no—
531C22⋊C4#12118yes—
532C22⋊C4C2×SD1618yes—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—24no—
537C42—22no—
538C42—42no—
539C42—42no—
540C42—41no—
541C42—44no—
542C42—44no—
543C42—44no—
544C42—42no—
545C42C2×SD1614yes—
546C42C22×D412yes—
547C42C2×#4112yes—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×#12114yes—
560C23C22×SD1614yes—
561C23C2×#13714yes—
562C23C23×D411yescharacteristic
563Q8—48no—
564Q8—42no—
565Q8—42no—
566Q8—42no—
567Q8—42no—
568Q8—164no—
569D4—24no—
570D4—416no—
571D4—42no—
572D4#461116yes—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—416no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—48no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—164no—
622C2×C4—164no—
623C2×C4—168no—
624C2×C4C2×#12112yes—
625C2×C4C2×#12114yes—
626C2×C4C2×#12112yes—
627C2×C4C22×SD1612yes—
628C2×C4C22×SD1614yes—
629C2×C4C22×SD1612yes—
630C2×C4C22×SD1612yes—
631C2×C4C2×#13712yes—
632C2×C4#57712yes—
633C2×C4C23×D411yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×2−1+412yes—
636C2×C4#58512yes—
637C8—24no—
638C8—42no—
639C8—44no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22C2×#46118yes—
654C22C22×#12111yescharacteristic
655C22#348511yescharacteristic
656C22#349911yescharacteristic
657C22C23×SD1611yescharacteristic
658C22C22×#13711yescharacteristic
659C22C2×#57711yescharacteristic
660C22#358511yescharacteristic
661C4—22no—
662C4—24no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—24no—
667C4—42no—
668C4—41no—
669C4—42no—
670C4—44no—
671C4—81no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4—84no—
676C4—82no—
677C4C2×#46114yes—
678C4#348912yes—
679C4#351512yes—
680C2—28no—
681C2—82no—
682C2#3602011yescharacteristic
683C2C22×#46111yescharacteristic
684C2#6043111yescharacteristic
685C2#6169711yescharacteristic
686C2#6173511yescharacteristic
687C2#6246911yescharacteristic
688C2#6250411yescharacteristic
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. #36180C2semidirect
4. #29868C2semidirect
5. #29756C2semidirect
6. #28687C2semidirect
7. #28571C2semidirect
8. #28205C2semidirect
9. #27645C2semidirect
10. #25453C2semidirect
11. #25202C2semidirect
12. #24815C2semidirect
13. #24580C2semidirect
14. #23677C2semidirect
15. C2×#2084C2semidirect
16. #22696C2semidirect
17. #22487C2semidirect
295. C4×D4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#33352
#361802
#343082
#298682
#297562
#286872
#285712
#282052
#276452
#254532
#252022
#248152
#245802
#236772
C2×#20842
#226962
#224872