Finite Group Databasebeta

← #6459554

Subgroups — #6459554

Order 512.

2638 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3056C212yesmaximal
2C2×#2857C211yesmaximal, characteristic, complemented
3C4×#525C211yesmaximal, characteristic
4#28581C214yesmaximal, complemented
5#28577C214yesmaximal, complemented
6#28575C214yesmaximal, complemented
7#28571C214yesmaximal, complemented
8#23658C211yesmaximal, characteristic, complemented
9#23630C211yesmaximal, characteristic, complemented
10#23507C211yesmaximal, characteristic, complemented
11#23432C211yesmaximal, characteristic, complemented
12#21061C212yesmaximal, complemented
13#20972C212yesmaximal, complemented
14#20927C212yesmaximal, complemented
15C2×#1728C211yesmaximal, characteristic, complemented
16C22×#137C2212yes—
17C23×M16C2212yes—
18C22×C4⋊Q8C2211yescharacteristic
19C2×#525C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C2×C4×#41C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#3056—216no—
24C2×#470C2214yes—
25C2×#468C2214yes—
26C2×#467C2214yes—
27C2×#465C2214yes—
28C2×#428C2212yes—
29C2×#417C2212yes—
30C2×#419C2212yes—
31C2×#419C2212yes—
32C2×#419C2212yes—
33#2857—21no—
34#2857—21no—
35#2857—21no—
36#2857—21no—
37#2857C2214yes—
38#2857C2214yes—
39C4×C22⋊Q8C2212yes—
40C4×C22⋊Q8C2212yes—
41C4×#123—22no—
42C4×#121C2212yes—
43C4×#121C2212yes—
44C4×#120—22no—
45#2297C2211yescharacteristic
46#2296C2211yescharacteristic
47C2×#391C2212yes—
48C2×#389—22no—
49#2290—21no—
50#2290—21no—
51C2×#384C2211yescharacteristic
52#2280C2211yescharacteristic
53C2×#380—22no—
54C42×D4—21no—
55C42×D4—21no—
56C4×#117C2211yescharacteristic
57#2084C2214yescomplemented
58#2083C2214yescomplemented
59#2072C2214yescomplemented
60#2071C2214yescomplemented
61#2043C2214yescomplemented
62#2043C2214yescomplemented
63#2016C2214yescomplemented
64#2013C2214yescomplemented
65#1928C2218yescomplemented
66#1927C2218yescomplemented
67#1894C2218yes—
68#1892C2218yes—
69#1878C2218yescomplemented
70#1877C2218yescomplemented
71#1736C2212yes—
72#1736C2212yes—
73#1728C2212yes—
74#1728C2211yescharacteristic
75#1728C2211yescharacteristic
76#1728C2214yes—
77C2×#336C2214yes—
78C22×#41C2311yescharacteristic
79C23×Q8C2312yes—
80C24×C4D412yes—
81C24×C4C2311yescharacteristic
82C2×#137—28no—
83C2×#137—28no—
84C2×#137—44no—
85C22×Q16C2314yes—
86C22×SD16C2314yes—
87C22×M16—28no—
88C22×M16—28no—
89C22×M16—44no—
90C22×M16C2312yes—
91C22×M16C2314yes—
92C22×M16C2312yes—
93C23×C8C2314yes—
94C2×C4⋊Q8—21no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—44no—
97C2×C4⋊Q8C2314yes—
98#525—24no—
99#525D418yes—
100C2×#123—21no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—28no—
104C2×C22⋊Q8—28no—
105C2×C22⋊Q8C2314yes—
106C2×C22⋊Q8C2314yes—
107C2×C22⋊Q8C2311yescharacteristic
108C2×C22⋊Q8C2311yescharacteristic
109C2×#121C2311yescharacteristic
110C2×#121C2311yescharacteristic
111C2×#120—21no—
112C4×#41—44no—
113C2×C4×Q8C2312yes—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4C2312yes—
120C2×#117—21no—
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4—22no—
129C22×C42—21no—
130C22×C42—44no—
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42D412yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139C22×C42C2312yes—
140#470—216no—
141#468—216no—
142#467—216no—
143#465—216no—
144#428—28no—
145C2×#107C2314yes—
146C2×#106C2314yes—
147#419—28no—
148#419—28no—
149#419—44no—
150#417—28no—
151C2×#103C2314yes—
152C2×#103C2318yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2318yes—
156C2×#102C2314yes—
157#391—22no—
158#391—22no—
159#391C2312yes—
160#391C2318yes—
161#390C2312yes—
162#389—24no—
163#389—24no—
164#389—24no—
165#389—24no—
166#389C2312yes—
167#389C2312yes—
168#388—22no—
169#388—22no—
170#387—22no—
171#387—22no—
172#386—22no—
173#386—22no—
174#386—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#384—21no—
179#384—21no—
180#384C2314yes—
181#381—21no—
182#380—24no—
183#380—24no—
184#380—24no—
185#380—24no—
186#380C2312yes—
187#380C2312yes—
188#379—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—21no—
197C4×C22⋊C4—21no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—21no—
201C4×C22⋊C4—21no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204#376—21no—
205C2×#95—24no—
206C43—21no—
207#340C2318yes—
208#340C2318yes—
209#336C2314yes—
210#336C2314yes—
211#336C2318yes—
212C25C2×D411yescharacteristic
213C2×#41—22no—
214C2×#41—24no—
215C2×#41—24no—
216C2×#41C2×D414yes—
217C22×Q8—22no—
218C22×Q8—24no—
219C22×Q8—24no—
220C22×Q8—24no—
221C22×Q8—24no—
222C22×Q8—24no—
223C22×Q8—24no—
224C22×Q8C2414yes—
225C22×Q8C2411yescharacteristic
226C22×D4—21no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C2411yescharacteristic
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—28no—
233C23×C4—21no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D414yes—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4#4114yes—
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253#137—432no—
254C2×Q16—28no—
255C2×Q16—28no—
256C2×Q16—44no—
257C2×SD16—28no—
258C2×SD16—28no—
259C2×SD16—44no—
260C2×M16—24no—
261C2×M16—24no—
262C2×M16—24no—
263C2×M16—28no—
264C2×M16—24no—
265C2×M16—24no—
266C2×M16—28no—
267C2×M16—44no—
268C2×M16—432no—
269C2×M16#4118yes—
270C22×C8—28no—
271C22×C8—28no—
272C22×C8—44no—
273C22×C8C2414yes—
274C22×C8C2414yes—
275C4⋊Q8—48no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278#123—28no—
279C22⋊Q8—22no—
280C22⋊Q8—22no—
281C22⋊Q8—28no—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C22⋊Q8—48no—
286C22⋊Q8—48no—
287C22⋊Q8C2×D414yes—
288C22⋊Q8C2×D414yes—
289#121—22no—
290#121—22no—
291#121C2×D414yes—
292#121C2×D414yes—
293#120—28no—
294C4×Q8—44no—
295C4×D4—28no—
296C4×D4—28no—
297C4×D4—44no—
298C4×D4—48no—
299C4×D4—48no—
300C4×D4—48no—
301#117—22no—
302#117—24no—
303#117—24no—
304#117—44no—
305#117C2×D414yes—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4—48no—
322C2×C4⋊C4—48no—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—48no—
343C2×C42—21no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—42no—
352C2×C42—42no—
353C2×C42—42no—
354C2×C42—42no—
355C2×C42—44no—
356C2×C42—44no—
357C2×C42—44no—
358C2×C42—44no—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2412yes—
368C2×C42C2412yes—
369#107—28no—
370#106—28no—
371#103—28no—
372#103—216no—
373#103—28no—
374#103—28no—
375#103—48no—
376#102—28no—
377#95—28no—
378#95—28no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387C24—22no—
388C24—22no—
389C24—24no—
390C24—21no—
391C24—42no—
392C24#12114yes—
393C24C22×D411yescharacteristic
394C24C22×D411yescharacteristic
395C24C22×D411yescharacteristic
396#41—216no—
397#41—48no—
398C2×Q8—21no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—48no—
413C2×Q8—48no—
414C2×Q8C22×D412yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—44no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4C22×D412yes—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—216no—
434C22×C4—216no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—432no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4#12114yes—
475C22×C4#12114yes—
476C22×C4#12114yes—
477C22×C4#13718yescomplemented
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4114yes—
492C22×C4H(5,𝔽2)14yes—
493C22×C4C2511yescharacteristic
494Q16—416no—
495SD16—416no—
496M16—216no—
497M16—416no—
498M16—48no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—416no—
505C2×C8C2×#4118yes—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C4⋊C4—44no—
524C4⋊C4C22×D412yes—
525C4⋊C4C22×D412yes—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—24no—
546C42—24no—
547C42—42no—
548C42—42no—
549C42—42no—
550C42—42no—
551C42—44no—
552C42—44no—
553C42—42no—
554C42—42no—
555C42—44no—
556C23—21no—
557C23—21no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—21no—
562C23—21no—
563C23—28no—
564C23—28no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—44no—
569C23—48no—
570C23—48no—
571C23—41no—
572C23—41no—
573C23—42no—
574C23—42no—
575C23C2×#12112yes—
576C23C2×#12112yes—
577C23C2×#12112yes—
578C23#53212yes—
579C23#53412yes—
580C23C2×#13714yes—
581C23C23×D411yescharacteristic
582Q8—24no—
583Q8—42no—
584Q8—44no—
585Q8—44no—
586Q8—44no—
587Q8—44no—
588Q8—88no—
589D4—24no—
590D4—48no—
591D4—48no—
592D4—48no—
593D4—42no—
594C2×C4—216no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—48no—
622C2×C4—48no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—48no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—48no—
639C2×C4—88no—
640C2×C4—88no—
641C2×C4—88no—
642C2×C4C2×#12112yes—
643C2×C4C2×#12112yes—
644C2×C4C2×#12112yes—
645C2×C4#53212yes—
646C2×C4#53412yes—
647C2×C4#54618yes—
648C2×C4#55318yes—
649C2×C4C2×#13714yes—
650C2×C4C2×#13714yes—
651C2×C4#57514yes—
652C2×C4C23×D411yescharacteristic
653C2×C4C22×#4112yes—
654C2×C4C22×#4111yescharacteristic
655C2×C4C2×H(5,𝔽2)12yes—
656C8—28no—
657C8—44no—
658C22—22no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—416no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—82no—
676C22—82no—
677C22#254314yes—
678C22#303914yes—
679C22C22×#12111yescharacteristic
680C22C2×#53211yescharacteristic
681C22C2×#53411yescharacteristic
682C22C22×#13712yes—
683C22C2×#57512yes—
684C4—22no—
685C4—22no—
686C4—24no—
687C4—24no—
688C4—24no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—42no—
693C4—84no—
694C4—82no—
695C4—82no—
696C4—84no—
697C4#254314yes—
698C4C2×#54612yes—
699C4C2×#55312yes—
700C2—24no—
701C2—24no—
702C2—42no—
703C2—82no—
704C2C2×#254311yescharacteristic
705C2C2×#303911yescharacteristic
706C2#6041511yescharacteristic
707C2#6070112yes—
708C2#6075112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2857C2semidirect
4. #28581C2semidirect
5. #28577C2semidirect
6. #28575C2semidirect
7. #28571C2semidirect
8. #23658C2semidirect
9. #23630C2semidirect
10. #23507C2semidirect
11. #23432C2semidirect
12. #21061C2semidirect
13. #20972C2semidirect
14. #20927C2semidirect
15. C2×#1728C2semidirect
57. #2084C22semidirect
58. #2083C22semidirect
59. #2072C22semidirect
60. #2071C22semidirect
61. #2043C22semidirect
62. #2043C22semidirect
63. #2016C22semidirect
64. #2013C22semidirect
65. #1928C22semidirect
66. #1927C22semidirect
69. #1878C22semidirect
70. #1877C22semidirect
477. C22×C4#137semidirect

Maximal subgroups

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

subgroupindex
C2×#30562
C2×#28572
C4×#5252
#285812
#285772
#285752
#285712
#236582
#236302
#235072
#234322
#210612
#209722
#209272
C2×#17282