Finite Group Databasebeta

← #6508771

Subgroups — #6508771

Order 512.

2066 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3187C211yesmaximal, characteristic
2C2×#3094C211yesmaximal, characteristic
3C2×#2499C211yesmaximal, characteristic, complemented
4#34246C211yesmaximal, characteristic
5#28118C212yesmaximal, complemented
6#28115C212yesmaximal, complemented
7#27651C212yesmaximal, complemented
8#27645C212yesmaximal, complemented
9#25211C214yesmaximal, complemented
10#25205C214yesmaximal, complemented
11#24590C214yesmaximal, complemented
12#24538C214yesmaximal, complemented
13#21932C211yesmaximal, characteristic, complemented
14#21768C211yesmaximal, characteristic, complemented
15#20699C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#3187—28no—
20#3094—28no—
21C2×#481C2212yes—
22C2×#481C2212yes—
23C2×#482C2214yes—
24C2×#483C2212yes—
25C2×#480C2212yes—
26C2×#465C2214yes—
27C2×#448C2212yes—
28C2×#450C2212yes—
29C2×#428C2212yes—
30C2×#418C2212yes—
31C2×#407C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33#2499—21no—
34#2499—21no—
35#2499—21no—
36#2499—21no—
37#2499C2214yes—
38#2499C2214yescomplemented
39C2×#397—22no—
40#2364—22no—
41#2363C2212yes—
42#2363C2212yes—
43#2345C2212yes—
44D4×C22⋊C4—22no—
45D4×C22⋊C4C2212yes—
46#2333—21no—
47#2333—21no—
48#2312—21no—
49C4×#121C2212yes—
50C4×#121C2212yes—
51C2×#391C2211yescharacteristic, complemented
52C2×#391C2211yescharacteristic, complemented
53C2×#387—22no—
54C2×#384C2211yescharacteristic, complemented
55C2×#382C2211yescharacteristic, complemented
56#2280C2211yescharacteristic
57#2273—21no—
58C2×C4×C22⋊C4C2211yescharacteristic
59#2085C2214yescomplemented
60#2084C2214yescomplemented
61#2073C2214yescomplemented
62#2071C2214yescomplemented
63#2068C2214yescomplemented
64#2066C2214yescomplemented
65#2055C2214yescomplemented
66#2055C2214yescomplemented
67#1949C2214yescomplemented
68#1948C2214yescomplemented
69#1928C2214yescomplemented
70#1927C2214yescomplemented
71#1894C2214yes—
72#1877C2214yescomplemented
73#1877C2214yescomplemented
74#1873C2212yescomplemented
75#1873C2212yescomplemented
76#1851C2214yes—
77#1850C2214yescomplemented
78#1825C2214yes—
79#1824C2212yescomplemented
80#1824C2212yescomplemented
81#1823C2212yescomplemented
82#1804C2214yescomplemented
83#1802C2214yescomplemented
84#1797C2212yescomplemented
85#1797C2212yescomplemented
86#1787C2212yescomplemented
87#1735C2212yescomplemented
88#1735C2212yescomplemented
89C23×Q8C2311yescharacteristic
90C23×D4—21no—
91C23×D4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C22×SD16C2314yes—
94C22×M16C2312yes—
95C23×C8C2312yes—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121—24no—
105C2×#121D418yes—
106C2×#121C2312yes—
107C2×#121C2312yes—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×#117—21no—
113C2×#117—21no—
114C2×#117—24no—
115C2×#117—24no—
116C2×#117C2312yes—
117C2×#117C2312yes—
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C22×C4⋊C4—22no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—22no—
130C22×C22⋊C4D412yes—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#483—28no—
135#482—216no—
136#481—28no—
137#481—28no—
138#480—28no—
139#465—216no—
140#450—28no—
141#448—28no—
142#428—28no—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148#418—28no—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155#407—24no—
156C2×C22⋊C8—24no—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161#397—24no—
162#397—24no—
163#397—24no—
164#397—24no—
165#391—21no—
166#391—21no—
167#391—21no—
168#391—21no—
169#391C2314yes—
170#391C2314yes—
171#390—22no—
172#390—22no—
173#390C2312yes—
174#390C2312yes—
175#389C2312yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#387—22no—
182#387—22no—
183#387—24no—
184#387—24no—
185#387—24no—
186#387—24no—
187#386—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#384—21no—
194#384—21no—
195#384C2314yes—
196#382—21no—
197#382—21no—
198#382C2314yes—
199#381—21no—
200#380—22no—
201#380C2312yes—
202#380C2312yes—
203#379—21no—
204#379—22no—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C22⋊C4—21no—
208C4×C22⋊C4—21no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4D411yescharacteristic
215C4×C22⋊C4D411yescharacteristic
216C4×C22⋊C4D411yescharacteristic
217C4×C22⋊C4D411yescharacteristic
218C4×C22⋊C4C2312yes—
219C4×C22⋊C4C2312yes—
220C2×#95—22no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232C25—21no—
233C25C2×D411yescharacteristic
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—42no—
238C22×Q8—42no—
239C22×Q8C2412yes—
240C22×D4—22no—
241C22×D4—28no—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—42no—
246C22×D4—42no—
247C22×D4—84no—
248C22×D4C2×D414yes—
249C22×D4C2412yes—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—42no—
258C23×C4—42no—
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D414yes—
265C23×C4#4114yes—
266C23×C4C2411yescharacteristic
267C23×C4C2411yescharacteristic
268C23×C4C2411yescharacteristic
269C2×SD16—28no—
270C2×SD16—44no—
271C2×SD16—44no—
272C2×M16—24no—
273C2×M16—42no—
274C2×M16—42no—
275C22×C8—24no—
276C22×C8—42no—
277C22×C8—42no—
278C22×C8C2414yes—
279C22×C8C2414yes—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8—24no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286#121—28no—
287#121—28no—
288#121—24no—
289#121—24no—
290#121C2×D418yes—
291#121C2×D418yes—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—84no—
295C4×D4—84no—
296#117—24no—
297#117—24no—
298#117—24no—
299#117—24no—
300#117—44no—
301#117—44no—
302#117—42no—
303#117—42no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—48no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—48no—
349C2×C22⋊C4—44no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—42no—
362C2×C42—42no—
363C2×C42—44no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2412yes—
369C2×C42C2412yes—
370#107—28no—
371#106—28no—
372#106—28no—
373C4⋊C8—44no—
374C4⋊C8—44no—
375#103—28no—
376#103—44no—
377#103—44no—
378#102—28no—
379#102—28no—
380#102—28no—
381C22⋊C8—24no—
382C22⋊C8—24no—
383C22⋊C8—24no—
384C22⋊C8—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401C24—24no—
402C24—22no—
403C24—24no—
404C24—42no—
405C24—41no—
406C24—41no—
407C24—44no—
408C24—81no—
409C24—81no—
410C24—82no—
411C24#12114yes—
412C24C22×D411yescharacteristic
413C24C22×D411yescharacteristic
414C24C22×D411yescharacteristic
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—84no—
423C2×Q8—84no—
424C2×D4—28no—
425C2×D4—28no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—48no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—84no—
434C2×D4—81no—
435C2×D4—81no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4—816no—
439C2×D4C22×D414yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—42no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4#12114yes—
485C22×C4#12114yes—
486C22×C4#12114yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4C2511yescharacteristic
501SD16—88no—
502M16—48no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—24no—
508C2×C8—24no—
509C2×C8—84no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—48no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—48no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4C22×D414yes—
530C22⋊C4—28no—
531C22⋊C4—28no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C22⋊C4—81no—
549C22⋊C4—81no—
550C42—24no—
551C42—24no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—44no—
557C42—44no—
558C42—81no—
559C42—81no—
560C42—82no—
561C42—82no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—28no—
574C23—24no—
575C23—44no—
576C23—44no—
577C23—48no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23—164no—
589C23—164no—
590C23C2×#12112yes—
591C23C2×#12112yes—
592C23C2×#12112yes—
593C23#57714yes—
594C23C23×D411yescharacteristic
595Q8—44no—
596Q8—44no—
597Q8—84no—
598D4—28no—
599D4—416no—
600D4—44no—
601D4—84no—
602D4—88no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—48no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—164no—
655C2×C4—164no—
656C2×C4C2×#12112yes—
657C2×C4C2×#12112yes—
658C2×C4C2×#12112yes—
659C2×C4#57514yes—
660C2×C4C23×D411yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C2×C4#58512yes—
663C2×C4#58512yes—
664C8—44no—
665C8—44no—
666C22—24no—
667C22—24no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—88no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—168no—
689C22#304214yes—
690C22C22×#12111yescharacteristic
691C22#348511yescharacteristic
692C22#348511yescharacteristic
693C22C2×#57512yes—
694C22C2×#57712yes—
695C4—24no—
696C4—22no—
697C4—22no—
698C4—22no—
699C4—44no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4#348512yes—
710C4#349112yes—
711C2—24no—
712C2—42no—
713C2—42no—
714C2—84no—
715C2#3544711yescharacteristic
716C2#6041511yescharacteristic
717C2C2×#304211yescharacteristic
718C2#6172912yes—
719C2#6173912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2499C2semidirect
5. #28118C2semidirect
6. #28115C2semidirect
7. #27651C2semidirect
8. #27645C2semidirect
9. #25211C2semidirect
10. #25205C2semidirect
11. #24590C2semidirect
12. #24538C2semidirect
13. #21932C2semidirect
14. #21768C2semidirect
15. #20699C2semidirect
38. #2499C22semidirect
51. C2×#391C22semidirect
52. C2×#391C22semidirect
54. C2×#384C22semidirect
55. C2×#382C22semidirect
59. #2085C22semidirect
60. #2084C22semidirect
61. #2073C22semidirect
62. #2071C22semidirect
63. #2068C22semidirect
64. #2066C22semidirect
65. #2055C22semidirect
66. #2055C22semidirect
67. #1949C22semidirect
68. #1948C22semidirect
69. #1928C22semidirect
70. #1927C22semidirect
72. #1877C22semidirect
73. #1877C22semidirect
74. #1873C22semidirect
75. #1873C22semidirect
77. #1850C22semidirect
79. #1824C22semidirect
80. #1824C22semidirect
81. #1823C22semidirect
82. #1804C22semidirect
83. #1802C22semidirect
84. #1797C22semidirect
85. #1797C22semidirect
86. #1787C22semidirect
87. #1735C22semidirect
88. #1735C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31872
C2×#30942
C2×#24992
#342462
#281182
#281152
#276512
#276452
#252112
#252052
#245902
#245382
#219322
#217682
#206992