Finite Group Databasebeta

← #6472800

Subgroups — #6472800

Order 512.

2454 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35305C211yesmaximal, characteristic
2#34044C211yesmaximal, characteristic
3#32984C214yesmaximal, complemented
4#29897C214yesmaximal, complemented
5#27607C214yesmaximal, complemented
6#24566C214yesmaximal, complemented
7#24564C214yesmaximal, complemented
8#24014C214yesmaximal, complemented
9#21943C211yesmaximal, characteristic, complemented
10#21939C211yesmaximal, characteristic, complemented
11#21768C211yesmaximal, characteristic, complemented
12#21762C211yesmaximal, characteristic, complemented
13#20581C211yesmaximal, characteristic
14C23×#41C2211yescharacteristic
15C2×#536C2212yes—
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117C2212yes—
18C2×#484C2214yescomplemented
19C2×#483C2214yescomplemented
20C2×#408C2212yes—
21C2×#408C2212yes—
22C2×#407C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24#2848C2212yes—
25#2848C2212yes—
26#2616C2214yes—
27#2435—21no—
28#2434—21no—
29#2432—22no—
30#2406—22no—
31#2405—21no—
32#2405—21no—
33#2311C2214yes—
34#2311C2212yes—
35#2311C2214yes—
36#2310—21no—
37#2309—21no—
38#2308—21no—
39#2308—21no—
40#2297—22no—
41#2297—22no—
42#2280C2211yescharacteristic
43#2280C2211yescharacteristic
44#2280C2211yescharacteristic
45#2280C2212yes—
46#2077C2214yescomplemented
47#2077C2214yescomplemented
48#2076C2214yescomplemented
49#2075C2214yescomplemented
50#2036C2214yescomplemented
51#2036C2214yescomplemented
52#2035C2214yescomplemented
53#2035C2214yescomplemented
54#1894C2214yescomplemented
55#1892C2212yescomplemented
56#1892C2212yescomplemented
57#1873C2214yes—
58#1873C2214yes—
59#1873C2214yescomplemented
60#1873C2214yescomplemented
61#1854C2214yes—
62#1854C2214yes—
63#1854C2214yescomplemented
64#1854C2214yescomplemented
65#1854C2214yescomplemented
66#1854C2214yescomplemented
67#1824C2212yescomplemented
68#1824C2214yescomplemented
69#1823C2212yescomplemented
70#1823C2214yescomplemented
71#1809C2214yescomplemented
72#1808C2212yescomplemented
73#1807C2212yescomplemented
74#1787C2212yescomplemented
75#1787C2212yescomplemented
76#1780C2214yes—
77#1779C2212yes—
78#1779C2212yes—
79C22×#41—24no—
80C22×#41—24no—
81C22×#41—44no—
82C22×#41—44no—
83C22×#41D412yes—
84C22×#41D412yes—
85C22×#41C2312yes—
86C22×#41C2312yes—
87C23×Q8D411yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×D4D411yescharacteristic
90C23×D4D411yescharacteristic
91C23×D4D411yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C22×M16C2312yes—
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98#536—44no—
99#536—44no—
100C2×C4⋊Q8—22no—
101C2×#124—22no—
102C2×#123—22no—
103C2×#123—22no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8C2314yes—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2314yes—
112C2×#117—22no—
113C2×#117—22no—
114C2×#117—44no—
115C2×#117—44no—
116C2×#117C2314yes—
117C2×#117C2312yes—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C22×C4⋊C4—21no—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—22no—
124C22×C22⋊C4—21no—
125C22×C42—22no—
126#484—48no—
127#483—48no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136#408—44no—
137#408—44no—
138#407—24no—
139C2×C22⋊C8—24no—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2314yes—
148#395—24no—
149#394—24no—
150#394—24no—
151#394—24no—
152#391—24no—
153#391—24no—
154#391—22no—
155#390—24no—
156#390—24no—
157#390—22no—
158#386—24no—
159#386—24no—
160#386—24no—
161#386—24no—
162#386—22no—
163#386—22no—
164#385—22no—
165#385—22no—
166#385—24no—
167#385—24no—
168#381—22no—
169#381—21no—
170#381—21no—
171#381—21no—
172#380C2312yes—
173#380C2312yes—
174#380C2314yes—
175#380C2314yes—
176#380C2312yes—
177#380C2312yes—
178#380C2314yes—
179#380C2314yes—
180#380C2312yes—
181#380C2312yes—
182#379—22no—
183#379—21no—
184#379—21no—
185#379—22no—
186#379—22no—
187#379—21no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—21no—
193C4×C22⋊C4—21no—
194C4×C22⋊C4—21no—
195C4×C22⋊C4—21no—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204C25C2×D411yescharacteristic
205C25C2×D411yescharacteristic
206C25C2×D411yescharacteristic
207C2×#41—28no—
208C2×#41—28no—
209C2×#41—24no—
210C2×#41—24no—
211C2×#41—24no—
212C2×#41—24no—
213C2×#41—24no—
214C2×#41—24no—
215C2×#41—24no—
216C2×#41—24no—
217C2×#41—42no—
218C2×#41—42no—
219C2×#41—42no—
220C2×#41—42no—
221C2×#41—44no—
222C2×#41—44no—
223C2×#41—44no—
224C2×#41—44no—
225C2×#41—416no—
226C2×#41—416no—
227C2×#41—432no—
228C22×Q8—24no—
229C22×Q8—21no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—44no—
233C22×Q8—42no—
234C22×Q8—42no—
235C22×Q8C2×D412yes—
236C22×Q8C2×D412yes—
237C22×Q8C2412yes—
238C22×D4—24no—
239C22×D4—24no—
240C22×D4—24no—
241C22×D4—44no—
242C22×D4—44no—
243C22×D4—44no—
244C22×D4C2×D412yes—
245C22×D4C2×D412yes—
246C22×D4C2×D412yes—
247C22×D4C2×D412yes—
248C22×D4C2×D412yes—
249C22×D4C2×D412yes—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—24no—
253C23×C4—24no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—24no—
259C23×C4—42no—
260C23×C4—42no—
261C23×C4—42no—
262C23×C4—42no—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D412yes—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4#4114yes—
271C23×C4#4114yes—
272C23×C4C2412yes—
273C23×C4C2411yescharacteristic
274C23×C4C2411yescharacteristic
275C23×C4C2411yescharacteristic
276C23×C4C2412yes—
277C2×M16—24no—
278C2×M16—42no—
279C2×M16—42no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8—42no—
283C22×C8—42no—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2414yes—
287C4⋊Q8—48no—
288#124—48no—
289#123—48no—
290#123—48no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—24no—
294C22⋊Q8—24no—
295C22⋊Q8—28no—
296C22⋊Q8—28no—
297C22⋊Q8—44no—
298C22⋊Q8—44no—
299C22⋊Q8—44no—
300C22⋊Q8—44no—
301#117—42no—
302#117—42no—
303#117—44no—
304#117—42no—
305#117—42no—
306#117—44no—
307#117—42no—
308#117—42no—
309#117—48no—
310#117—48no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2414yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—48no—
345C2×C22⋊C4—44no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—48no—
351#107—44no—
352#107—44no—
353#106—44no—
354#106—44no—
355#103—44no—
356#103—44no—
357#103—44no—
358#103—44no—
359C22⋊C8—24no—
360C22⋊C8—24no—
361C22⋊C8—24no—
362C22⋊C8—24no—
363C22⋊C8—28no—
364C22⋊C8—28no—
365C22⋊C8—44no—
366C22⋊C8—44no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—24no—
377C24—22no—
378C24—24no—
379C24—22no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24#12314yes—
389C24C22×D412yes—
390C24C22×D411yescharacteristic
391C24C22×D411yescharacteristic
392C24C22×D411yescharacteristic
393C24C22×D412yes—
394#41—48no—
395#41—48no—
396#41—48no—
397#41—48no—
398#41—48no—
399#41—48no—
400#41—48no—
401#41—48no—
402#41—432no—
403#41—432no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—416no—
415C2×Q8—44no—
416C2×Q8—42no—
417C2×Q8—82no—
418C2×Q8—82no—
419C2×Q8—84no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—416no—
439C2×D4—416no—
440C2×D4—416no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—216no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—48no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—48no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—48no—
488C22×C4—82no—
489C22×C4—84no—
490C22×C4—84no—
491C22×C4#12314yes—
492C22×C4#12314yes—
493C22×C4#12314yes—
494C22×C4C22×D412yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D412yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4112yes—
505C22×C4H(5,𝔽2)14yes—
506C22×C4H(5,𝔽2)14yes—
507C22×C4C2511yescharacteristic
508M16—84no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—82no—
517C2×C8—82no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—48no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—48no—
536C22⋊C4—44no—
537C22⋊C4—48no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—48no—
541C22⋊C4—82no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C22⋊C4—84no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—84no—
550C42—44no—
551C42—44no—
552C42—48no—
553C42—44no—
554C23—28no—
555C23—22no—
556C23—24no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—24no—
569C23—24no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—88no—
578C23—82no—
579C23—82no—
580C23—82no—
581C23—82no—
582C23—88no—
583C23—88no—
584C23C2×#12312yes—
585C23C2×#12312yes—
586C23C2×#12312yes—
587C23#53414yes—
588C23C23×D411yescharacteristic
589Q8—48no—
590Q8—48no—
591Q8—42no—
592Q8—42no—
593Q8—44no—
594Q8—84no—
595D4—48no—
596D4—48no—
597D4—48no—
598D4—48no—
599D4—48no—
600D4—48no—
601C2×C4—28no—
602C2×C4—216no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—48no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—84no—
647C2×C4—816no—
648C2×C4—84no—
649C2×C4C2×#12312yes—
650C2×C4C2×#12312yes—
651C2×C4C2×#12312yes—
652C2×C4#53414yes—
653C2×C4C22×#4111yescharacteristic
654C2×C4C22×#4111yescharacteristic
655C2×C4C2×H(5,𝔽2)12yes—
656C2×C4C2×H(5,𝔽2)12yes—
657C8—42no—
658C8—42no—
659C8—44no—
660C22—28no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—24no—
669C22—22no—
670C22—22no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—42no—
676C22—42no—
677C22—44no—
678C22—44no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—88no—
683C22—88no—
684C22—88no—
685C22C22×#12311yescharacteristic
686C22C2×#53412yes—
687C22#358412yes—
688C22#358612yes—
689C4—22no—
690C4—24no—
691C4—22no—
692C4—22no—
693C4—42no—
694C4—84no—
695C4—84no—
696C4—84no—
697C4—82no—
698C4—82no—
699C4#349011yescharacteristic
700C4#349011yescharacteristic
701C4#349012yes—
702C2—44no—
703C2—42no—
704C2—42no—
705C2—42no—
706C2—82no—
707C2#3499211yescharacteristic
708C2#6051611yescharacteristic
709C2#6051911yescharacteristic
710C2#6076114yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32984C2semidirect
4. #29897C2semidirect
5. #27607C2semidirect
6. #24566C2semidirect
7. #24564C2semidirect
8. #24014C2semidirect
9. #21943C2semidirect
10. #21939C2semidirect
11. #21768C2semidirect
12. #21762C2semidirect
18. C2×#484C22semidirect
19. C2×#483C22semidirect
46. #2077C22semidirect
47. #2077C22semidirect
48. #2076C22semidirect
49. #2075C22semidirect
50. #2036C22semidirect
51. #2036C22semidirect
52. #2035C22semidirect
53. #2035C22semidirect
54. #1894C22semidirect
55. #1892C22semidirect
56. #1892C22semidirect
59. #1873C22semidirect
60. #1873C22semidirect
63. #1854C22semidirect
64. #1854C22semidirect
65. #1854C22semidirect
66. #1854C22semidirect
67. #1824C22semidirect
68. #1824C22semidirect
69. #1823C22semidirect
70. #1823C22semidirect
71. #1809C22semidirect
72. #1808C22semidirect
73. #1807C22semidirect
74. #1787C22semidirect
75. #1787C22semidirect

Maximal subgroups

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

subgroupindex
#353052
#340442
#329842
#298972
#276072
#245662
#245642
#240142
#219432
#219392
#217682
#217622
#205812