Finite Group Databasebeta

← #6499935

Subgroups — #6499935

Order 512.

3576 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3537C211yesmaximal, characteristic
2C2×#3200C212yesmaximal, complemented
3#35633C211yesmaximal, characteristic, complemented
4#25548C214yesmaximal, complemented
5#25532C214yesmaximal, complemented
6#24448C214yesmaximal, complemented
7#24432C214yesmaximal, complemented
8#24070C212yesmaximal, complemented
9#24009C212yesmaximal, complemented
10#23940C212yesmaximal, complemented
11#23919C212yesmaximal, complemented
12#21744C211yesmaximal, characteristic, complemented
13#20914C212yesmaximal, complemented
14#3537—216no—
15C2×#560—22no—
16C2×#553C2214yes—
17C2×D42—22no—
18C2×D42C2214yes—
19C2×#540C2212yes—
20C2×#538—22no—
21C2×#536C2211yescharacteristic, complemented
22C2×#535C2212yes—
23C2×#534—22no—
24C2×#525C2211yescharacteristic
25C22×C22⋊Q8C2211yescharacteristic, complemented
26C22×#121C2212yes—
27C2×#520C2211yescharacteristic, complemented
28#3200—216no—
29C2×#485C2214yescomplemented
30C2×#482C2214yescomplemented
31C2×#468C2214yescomplemented
32C2×#465C2214yescomplemented
33C2×#448C2214yescomplemented
34C2×#450C2214yescomplemented
35C2×#450C2214yescomplemented
36C2×#418C2212yescomplemented
37C22×#102C2212yescomplemented
38C2×#408C2212yescomplemented
39#2534—22no—
40#2486—22no—
41#2429—22no—
42#2427—22no—
43#2417C2214yescomplemented
44#2415C2214yescomplemented
45#2316C2211yescharacteristic, complemented
46#2310C2211yescharacteristic, complemented
47#2280C2211yescharacteristic, complemented
48#2059C2214yescomplemented
49#2041C2214yescomplemented
50#2034C2214yescomplemented
51#2033C2214yescomplemented
52#2026C2212yescomplemented
53#2026C2212yescomplemented
54#2014C2212yescomplemented
55#2014C2212yescomplemented
56#2000C2214yescomplemented
57#2000C2214yescomplemented
58#1877C2214yescomplemented
59#1877C2214yescomplemented
60#1875C2214yescomplemented
61#1875C2214yescomplemented
62#1874C2214yescomplemented
63#1874C2214yescomplemented
64#1872C2214yescomplemented
65#1870C2214yescomplemented
66#1869C2214yescomplemented
67#1864C2214yescomplemented
68#1862C2214yescomplemented
69#1853C2212yescomplemented
70#1852C2212yescomplemented
71#1818C2212yescomplemented
72#1786C2212yescomplemented
73#1786C2212yescomplemented
74#1779C2212yescomplemented
75#1736C2212yescomplemented
76#1735C2212yescomplemented
77C22×#41C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4—22no—
80C23×D4—24no—
81C23×D4C2312yes—
82C23×D4C2312yes—
83C24×C4C2311yescharacteristic
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86C22×M16C2312yes—
87C23×C8C2312yes—
88#560—232no—
89#553—216no—
90#553—216no—
91D42—232no—
92D42—216no—
93D42—216no—
94#540—216no—
95#538—232no—
96#536—28no—
97#535—28no—
98#535D4116yes—
99#534—232no—
100C2×C4⋊Q8—21no—
101C2×#127—22no—
102C2×#127C2312yes—
103C2×#126—24no—
104C2×#125C2312yes—
105C2×#124—22no—
106C2×#124—21no—
107#525—28no—
108C2×#123—22no—
109C2×#123—24no—
110C2×#123—24no—
111C2×#123D412yes—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—44no—
114C2×C22⋊Q8D412yes—
115C2×C22⋊Q8C2314yes—
116C2×C22⋊Q8C2314yes—
117C2×#121—24no—
118C2×#121—22no—
119C2×#121—22no—
120C2×#121—216no—
121C2×#121—24no—
122C2×#121—24no—
123C2×#121D412yes—
124C2×#121C2314yes—
125C2×#121C2314yes—
126C2×#121C2314yes—
127C2×#120—22no—
128C2×#120—22no—
129C2×#120—24no—
130C2×#120—24no—
131C2×#120D412yes—
132#520—44no—
133C2×C4×Q8—21no—
134C2×C4×D4—21no—
135C2×C4×D4—22no—
136C2×C4×D4—24no—
137C2×C4×D4C2314yes—
138C2×C4×D4C2314yes—
139C2×#117C2311yescharacteristic
140C2×#117C2312yes—
141C2×#117C2312yes—
142C2×#117C2311yescharacteristic
143C2×#117C2311yescharacteristic
144C22×C4⋊C4—21no—
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2312yes—
148C22×C22⋊C4—22no—
149C22×C22⋊C4—21no—
150#485—216no—
151#482—216no—
152#468—216no—
153#465—216no—
154#450—216no—
155#450—48no—
156#448—216no—
157C2×#107C2314yes—
158C2×#106C2314yes—
159#418—44no—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#102—44no—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×#102C2314yes—
169#408—28no—
170C2×C22⋊C8C2314yes—
171C2×C22⋊C8C2314yes—
172C2×C22⋊C8C2312yes—
173C2×C22⋊C8C2312yes—
174C2×C22⋊C8C2314yes—
175#398—24no—
176#397—24no—
177#396—24no—
178#394—24no—
179#391—22no—
180#391C2312yes—
181#391C2312yes—
182#389—22no—
183#389—24no—
184#389C2314yes—
185#388—24no—
186#386—24no—
187#381—21no—
188#381—22no—
189#380—22no—
190#380—24no—
191#380C2312yes—
192#380C2312yes—
193#379—21no—
194C4×C22⋊C4—22no—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2312yes—
201#336C2312yes—
202C25—22no—
203C25—22no—
204C25C2×D411yescharacteristic
205C2×#41—42no—
206C2×#41—41no—
207C2×#41—41no—
208C2×#41C2×D418yes—
209C22×Q8—21no—
210C22×Q8—24no—
211C22×Q8—42no—
212C22×Q8—41no—
213C22×Q8—41no—
214C22×Q8C2×D411yescharacteristic
215C22×Q8C2412yes—
216C22×D4—24no—
217C22×D4—216no—
218C22×D4—232no—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—28no—
222C22×D4—22no—
223C22×D4—24no—
224C22×D4—22no—
225C22×D4—24no—
226C22×D4—42no—
227C22×D4—42no—
228C22×D4—42no—
229C22×D4—42no—
230C22×D4—42no—
231C22×D4—42no—
232C22×D4—44no—
233C22×D4—88no—
234C22×D4—84no—
235C22×D4—88no—
236C22×D4—84no—
237C22×D4C2×D414yes—
238C22×D4C2×D418yes—
239C22×D4C2×D411yescharacteristic
240C22×D4C2×D412yes—
241C22×D4C2414yes—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4—24no—
245C23×C4—24no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—42no—
249C23×C4—41no—
250C23×C4—41no—
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255C23×C4C2412yes—
256C2×SD16—28no—
257C2×SD16—28no—
258C2×SD16—44no—
259C2×SD16—44no—
260C2×SD16—44no—
261C2×SD16—44no—
262C2×M16—24no—
263C2×M16—42no—
264C2×M16—42no—
265C22×C8—24no—
266C22×C8—42no—
267C22×C8—42no—
268C22×C8C2414yes—
269C22×C8C2414yes—
270C4⋊Q8—44no—
271#127—216no—
272#127—28no—
273#126—232no—
274#125—28no—
275#124—216no—
276#124—44no—
277#123—232no—
278#123—232no—
279#123—28no—
280#123—48no—
281C22⋊Q8—28no—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C22⋊Q8—44no—
286C22⋊Q8—48no—
287#121—216no—
288#121—216no—
289#121—232no—
290#121—232no—
291#121—232no—
292#121—28no—
293#121—28no—
294#121—28no—
295#121—28no—
296#121—28no—
297#121C2×D4116yes—
298#121C2×D4116yes—
299#120—232no—
300#120—232no—
301#120—216no—
302#120—216no—
303#120—28no—
304C4×Q8—44no—
305C4×D4—232no—
306C4×D4—28no—
307C4×D4—28no—
308C4×D4—28no—
309C4×D4—28no—
310C4×D4—44no—
311C4×D4—48no—
312#117—24no—
313#117—24no—
314#117—24no—
315#117—24no—
316#117—42no—
317#117—42no—
318#117C2×D418yes—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—28no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D414yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2412yes—
338C2×C22⋊C4—216no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D414yes—
354C2×C22⋊C4C2×D414yes—
355C2×C42—22no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42C2412yes—
360#107—28no—
361#106—28no—
362#103—28no—
363#103—28no—
364#102—28no—
365#102—28no—
366#102—28no—
367#102—28no—
368#102—44no—
369#102—44no—
370C22⋊C8—28no—
371C22⋊C8—28no—
372C22⋊C8—42no—
373C22⋊C8—42no—
374C22⋊C8—44no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380C24—28no—
381C24—24no—
382C24—22no—
383C24—24no—
384C24—28no—
385C24—41no—
386C24—42no—
387C24—41no—
388C24—44no—
389C24—44no—
390C24—82no—
391C24—82no—
392C24—82no—
393C24—82no—
394C24—82no—
395C24—84no—
396C24—82no—
397C24—84no—
398C24C22×D412yes—
399C24C22×D411yescharacteristic
400C24C22×D412yes—
401#41—48no—
402#41—48no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—41no—
408C2×Q8—41no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—84no—
413C2×Q8—84no—
414C2×Q8#12014yes—
415C2×D4—216no—
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—216no—
420C2×D4—216no—
421C2×D4—216no—
422C2×D4—28no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—41no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—41no—
434C2×D4—42no—
435C2×D4—48no—
436C2×D4—48no—
437C2×D4—84no—
438C2×D4—84no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—84no—
444C2×D4—84no—
445C2×D4—88no—
446C2×D4—88no—
447C2×D4—84no—
448C2×D4—84no—
449C2×D4—84no—
450C2×D4—84no—
451C2×D4—864no—
452C2×D4—832no—
453C2×D4#12014yes—
454C2×D4#12018yes—
455C2×D4C22×D418yes—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—28no—
466C22×C4—24no—
467C22×C4—216no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—41no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—41no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—84no—
487C22×C4—84no—
488C22×C4—84no—
489C22×C4—84no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4—84no—
494C22×C4C22×D414yes—
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D412yes—
498C22×C4C22×D414yes—
499C22×C4C22×D414yes—
500C22×C4C22×D414yes—
501C22×C4C2511yescharacteristic
502SD16—88no—
503SD16—88no—
504M16—84no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—24no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—84no—
512C4⋊C4—24no—
513C4⋊C4—28no—
514C4⋊C4—216no—
515C4⋊C4—24no—
516C4⋊C4—28no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—44no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—84no—
526C4⋊C4C22×D418yes—
527C4⋊C4H(5,𝔽2)14yes—
528C22⋊C4—216no—
529C22⋊C4—216no—
530C22⋊C4—28no—
531C22⋊C4—24no—
532C22⋊C4—28no—
533C22⋊C4—216no—
534C22⋊C4—216no—
535C22⋊C4—28no—
536C22⋊C4—48no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—48no—
541C22⋊C4—44no—
542C22⋊C4—82no—
543C42—28no—
544C42—24no—
545C42—44no—
546C42—42no—
547C42—42no—
548C23—22no—
549C23—24no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—21no—
555C23—22no—
556C23—21no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—24no—
561C23—22no—
562C23—28no—
563C23—48no—
564C23—44no—
565C23—416no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—82no—
573C23—84no—
574C23—88no—
575C23—816no—
576C23—82no—
577C23—88no—
578C23—88no—
579C23—164no—
580C23—164no—
581C23—164no—
582C23—164no—
583C23—168no—
584C23—164no—
585C23—164no—
586C23C2×#12014yes—
587C23C2×#12012yes—
588C23#57714yes—
589C23C23×D411yescharacteristic
590Q8—44no—
591Q8—44no—
592Q8—44no—
593Q8—84no—
594D4—28no—
595D4—28no—
596D4—44no—
597D4—48no—
598D4—416no—
599D4—432no—
600D4—44no—
601D4—44no—
602D4—88no—
603D4—816no—
604D4—816no—
605D4—88no—
606D4—816no—
607C2×C4—21no—
608C2×C4—22no—
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—24no—
619C2×C4—28no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—84no—
641C2×C4—82no—
642C2×C4—84no—
643C2×C4—816no—
644C2×C4—88no—
645C2×C4—84no—
646C2×C4—164no—
647C2×C4C2×#12014yes—
648C2×C4C2×#12012yes—
649C2×C4C23×D412yes—
650C2×C4C2×H(5,𝔽2)12yes—
651C2×C4#58511yescharacteristic
652C2×C4#58511yescharacteristic
653C8—44no—
654C8—44no—
655C22—22no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—28no—
661C22—24no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—44no—
666C22—41no—
667C22—42no—
668C22—41no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—48no—
673C22—48no—
674C22—48no—
675C22—88no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—1616no—
681C22—1616no—
682C22C22×#12011yescharacteristic
683C22#348411yescharacteristic
684C22#348511yescharacteristic
685C22C2×#57712yes—
686C22#358512yes—
687C4—22no—
688C4—24no—
689C4—42no—
690C4—44no—
691C4—44no—
692C4—48no—
693C4—82no—
694C4—81no—
695C4—81no—
696C4—82no—
697C4#349612yes—
698C4#349612yes—
699C2—42no—
700C2—44no—
701C2—44no—
702C2—88no—
703C2#3534711yescharacteristic
704C2#6037312yes—
705C2#6163311yescharacteristic
706C2#6164811yescharacteristic
707C2#6175612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3200C2semidirect
3. #35633C2semidirect
4. #25548C2semidirect
5. #25532C2semidirect
6. #24448C2semidirect
7. #24432C2semidirect
8. #24070C2semidirect
9. #24009C2semidirect
10. #23940C2semidirect
11. #23919C2semidirect
12. #21744C2semidirect
13. #20914C2semidirect
21. C2×#536C22semidirect
25. C22×C22⋊Q8C22semidirect
27. C2×#520C22semidirect
29. C2×#485C22semidirect
30. C2×#482C22semidirect
31. C2×#468C22semidirect
32. C2×#465C22semidirect
33. C2×#448C22semidirect
34. C2×#450C22semidirect
35. C2×#450C22semidirect
36. C2×#418C22semidirect
37. C22×#102C22semidirect
38. C2×#408C22semidirect
43. #2417C22semidirect
44. #2415C22semidirect
45. #2316C22semidirect
46. #2310C22semidirect
47. #2280C22semidirect
48. #2059C22semidirect
49. #2041C22semidirect
50. #2034C22semidirect
51. #2033C22semidirect
52. #2026C22semidirect
53. #2026C22semidirect
54. #2014C22semidirect
55. #2014C22semidirect
56. #2000C22semidirect
57. #2000C22semidirect
58. #1877C22semidirect
59. #1877C22semidirect
60. #1875C22semidirect
61. #1875C22semidirect
62. #1874C22semidirect
63. #1874C22semidirect
64. #1872C22semidirect
65. #1870C22semidirect
66. #1869C22semidirect
67. #1864C22semidirect
68. #1862C22semidirect
69. #1853C22semidirect
70. #1852C22semidirect
71. #1818C22semidirect
72. #1786C22semidirect
73. #1786C22semidirect
74. #1779C22semidirect
75. #1736C22semidirect
76. #1735C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35372
C2×#32002
#356332
#255482
#255322
#244482
#244322
#240702
#240092
#239402
#239192
#217442
#209142