Finite Group Databasebeta

← #6512682

Subgroups — #6512682

Order 512.

3562 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3537C211yesmaximal, characteristic
2C2×#3126C212yesmaximal, complemented
3#35661C211yesmaximal, characteristic, complemented
4#24252C214yesmaximal, complemented
5#24240C214yesmaximal, complemented
6#24138C214yesmaximal, complemented
7#24126C214yesmaximal, complemented
8#24015C212yesmaximal, complemented
9#24009C212yesmaximal, complemented
10#23926C212yesmaximal, complemented
11#23919C212yesmaximal, complemented
12#21872C212yesmaximal, complemented
13#21792C211yesmaximal, characteristic, 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×#519C2211yescharacteristic, complemented
28#3126—216no—
29C2×#482C2214yescomplemented
30C2×#479C2214yescomplemented
31C2×#460C2214yescomplemented
32C2×#448C2214yescomplemented
33C2×#450C2214yescomplemented
34C2×#450C2212yescomplemented
35C2×#450C2212yescomplemented
36C2×#440C2214yescomplemented
37C2×C4×SD16C2214yescomplemented
38C2×#433C2212yescomplemented
39#2538C2214yescomplemented
40#2536C2214yescomplemented
41#2429—22no—
42#2426—22no—
43#2413—22no—
44#2411—22no—
45#2317C2211yescharacteristic, complemented
46#2311C2211yescharacteristic, complemented
47#2280C2211yescharacteristic, complemented
48#2034C2214yescomplemented
49#2034C2214yescomplemented
50#2033C2214yescomplemented
51#2033C2214yescomplemented
52#2000C2214yescomplemented
53#2000C2214yescomplemented
54#2000C2214yescomplemented
55#2000C2214yescomplemented
56#1992C2212yescomplemented
57#1991C2212yescomplemented
58#1961C2212yescomplemented
59#1957C2212yescomplemented
60#1955C2214yescomplemented
61#1954C2214yescomplemented
62#1875C2214yescomplemented
63#1874C2214yescomplemented
64#1873C2212yescomplemented
65#1873C2212yescomplemented
66#1872C2214yescomplemented
67#1854C2214yescomplemented
68#1853C2212yescomplemented
69#1852C2212yescomplemented
70#1804C2214yescomplemented
71#1802C2214yescomplemented
72#1786C2212yescomplemented
73#1786C2214yescomplemented
74#1786C2212yescomplemented
75#1780C2212yescomplemented
76#1779C2212yescomplemented
77C22×#41C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4—22no—
80C23×D4—24no—
81C23×D4C2312yes—
82C23×D4C2312yes—
83C24×C4C2311yescharacteristic
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86#560—232no—
87#553—216no—
88#553—216no—
89D42—232no—
90D42—216no—
91D42—216no—
92#540—28no—
93#540D4116yes—
94#538—232no—
95#536—44no—
96#535—216no—
97#534—232no—
98C2×C4⋊Q8—21no—
99C2×#127—22no—
100C2×#127C2312yes—
101C2×#126—24no—
102C2×#125C2312yes—
103C2×#124—22no—
104C2×#124—21no—
105#525—28no—
106C2×#123—22no—
107C2×#123—24no—
108C2×#123—24no—
109C2×#123D412yes—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—44no—
112C2×C22⋊Q8D412yes—
113C2×C22⋊Q8C2314yes—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×#121—24no—
117C2×#121—22no—
118C2×#121—22no—
119C2×#121—216no—
120C2×#121—24no—
121C2×#121—24no—
122C2×#121D412yes—
123C2×#121C2314yes—
124C2×#121C2314yes—
125C2×#121C2314yes—
126C2×#120—22no—
127C2×#120—22no—
128C2×#120—24no—
129C2×#120—24no—
130C2×#120D412yes—
131#519—28no—
132C2×C4×Q8—22no—
133C2×C4×Q8C2314yes—
134C2×C4×D4—24no—
135C2×C4×D4C2314yes—
136C2×C4×D4C2314yes—
137C2×#117—22no—
138C2×#117C2311yescharacteristic
139C2×#117C2312yes—
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C22×C4⋊C4—21no—
143C22×C4⋊C4C2312yes—
144C22×C4⋊C4C2311yescharacteristic
145C22×C22⋊C4—22no—
146C22×C22⋊C4—21no—
147#482—216no—
148#479—216no—
149#460—216no—
150#450—48no—
151#450—44no—
152#450—44no—
153#448—216no—
154#440—216no—
155C4×SD16—216no—
156#433—28no—
157C2×#107C2314yes—
158C2×#106C2314yes—
159C2×C4⋊C8C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×C22⋊C8C2314yes—
169C2×C22⋊C8C2312yes—
170C2×C22⋊C8C2312yes—
171C2×C22⋊C8C2314yes—
172C2×C22⋊C8C2314yes—
173C2×C22⋊C8C2312yes—
174C2×C22⋊C8C2312yes—
175C2×#97C2312yes—
176C2×C4×C8C2312yes—
177#398—24no—
178#396—24no—
179#396—24no—
180#394—24no—
181#391—21no—
182#390—21no—
183#390—24no—
184#389—22no—
185#389—24no—
186#388—24no—
187#388—24no—
188#386—22no—
189#385—22no—
190#384C2312yes—
191#382C2312yes—
192#381—21no—
193#380C2314yes—
194#380C2312yes—
195#380C2312yes—
196#379—21no—
197C4×C22⋊C4—22no—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202C25—22no—
203C25—22no—
204C25C2×D411yescharacteristic
205C2×#41—24no—
206C2×#41—42no—
207C2×#41—42no—
208C22×Q8—24no—
209C22×Q8—21no—
210C22×Q8—24no—
211C22×Q8—42no—
212C22×Q8C2×D411yescharacteristic
213C22×Q8C2412yes—
214C22×D4—24no—
215C22×D4—216no—
216C22×D4—232no—
217C22×D4—24no—
218C22×D4—24no—
219C22×D4—28no—
220C22×D4—28no—
221C22×D4—22no—
222C22×D4—24no—
223C22×D4—22no—
224C22×D4—42no—
225C22×D4—42no—
226C22×D4—44no—
227C22×D4—44no—
228C22×D4—42no—
229C22×D4—42no—
230C22×D4—88no—
231C22×D4—84no—
232C22×D4—88no—
233C22×D4—84no—
234C22×D4C2×D414yes—
235C22×D4C2×D411yescharacteristic
236C22×D4C2×D412yes—
237C22×D4C2414yes—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4—24no—
241C23×C4—24no—
242C23×C4—21no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2411yescharacteristic
250C2×SD16—28no—
251C2×SD16—28no—
252C2×SD16—28no—
253C2×SD16—28no—
254C2×SD16—44no—
255C2×SD16—44no—
256C22×C8C2414yes—
257C22×C8C2414yes—
258C4⋊Q8—44no—
259#127—216no—
260#127—24no—
261#127C2×D418yes—
262#126—232no—
263#125—24no—
264#125C2×D418yes—
265#124—216no—
266#124—44no—
267#123—232no—
268#123—232no—
269#123—28no—
270#123—48no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8—44no—
276C22⋊Q8—42no—
277C22⋊Q8—42no—
278C22⋊Q8—48no—
279#121—216no—
280#121—216no—
281#121—232no—
282#121—232no—
283#121—28no—
284#121—28no—
285#121—28no—
286#121—28no—
287#121—28no—
288#121—28no—
289#121—232no—
290#121C2×D4116yes—
291#120—216no—
292#120—232no—
293#120—216no—
294#120—232no—
295#120—28no—
296C4×Q8—28no—
297C4×Q8—28no—
298C4×Q8—48no—
299C4×D4—232no—
300C4×D4—28no—
301C4×D4—28no—
302C4×D4—28no—
303C4×D4C2×D4116yes—
304#117—28no—
305#117—24no—
306#117—42no—
307#117—48no—
308#117C2×D418yes—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4C2×D414yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2414yes—
327C2×C22⋊C4—216no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4C2×D414yes—
343C2×C22⋊C4C2×D414yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—22no—
346C2×C42—24no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42C2412yes—
350#107—28no—
351#106—28no—
352C4⋊C8—28no—
353#103—28no—
354#103—28no—
355#102—28no—
356#102—28no—
357#102—44no—
358#102—44no—
359#102—44no—
360#102—44no—
361C22⋊C8—28no—
362C22⋊C8—42no—
363C22⋊C8—42no—
364C22⋊C8—44no—
365C22⋊C8—44no—
366C22⋊C8—42no—
367C22⋊C8—42no—
368#97—24no—
369C4×C8—24no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374C24—28no—
375C24—28no—
376C24—24no—
377C24—22no—
378C24—24no—
379C24—41no—
380C24—42no—
381C24—41no—
382C24—44no—
383C24—44no—
384C24—82no—
385C24—82no—
386C24—82no—
387C24—82no—
388C24—82no—
389C24—82no—
390C24—84no—
391C24—84no—
392C24C22×D412yes—
393C24C22×D411yescharacteristic
394C24C22×D412yes—
395#41—88no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—41no—
402C2×Q8—41no—
403C2×Q8—416no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8H(5,𝔽2)14yes—
407C2×D4—216no—
408C2×D4—28no—
409C2×D4—28no—
410C2×D4—216no—
411C2×D4—28no—
412C2×D4—24no—
413C2×D4—216no—
414C2×D4—28no—
415C2×D4—22no—
416C2×D4—216no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—42no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4—84no—
434C2×D4—84no—
435C2×D4—88no—
436C2×D4—88no—
437C2×D4—88no—
438C2×D4—88no—
439C2×D4—84no—
440C2×D4—84no—
441C2×D4—84no—
442C2×D4—84no—
443C2×D4—864no—
444C2×D4—832no—
445C2×D4C22×D418yes—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—28no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—216no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—41no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—41no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—88no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D414yes—
485C22×C4C2511yescharacteristic
486SD16—416no—
487SD16—416no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C4⋊C4—28no—
495C4⋊C4—216no—
496C4⋊C4—28no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—48no—
507C4⋊C4—84no—
508C4⋊C4#12018yes—
509C4⋊C4C22×D418yes—
510C22⋊C4—216no—
511C22⋊C4—28no—
512C22⋊C4—216no—
513C22⋊C4—216no—
514C22⋊C4—216no—
515C22⋊C4—28no—
516C22⋊C4—28no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—82no—
524C22⋊C4—84no—
525C22⋊C4#12018yes—
526C42—28no—
527C42—42no—
528C42—42no—
529C42—48no—
530C42C22×D414yes—
531C42C22×D414yes—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—22no—
544C23—28no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—48no—
549C23—44no—
550C23—416no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—82no—
558C23—84no—
559C23—82no—
560C23—88no—
561C23—88no—
562C23—88no—
563C23—816no—
564C23—164no—
565C23—164no—
566C23—164no—
567C23—164no—
568C23—168no—
569C23—164no—
570C23—164no—
571C23C2×#12012yes—
572C23#57714yes—
573C23C23×D411yescharacteristic
574Q8—28no—
575Q8—48no—
576Q8—44no—
577Q8—82no—
578D4—28no—
579D4—28no—
580D4—416no—
581D4—432no—
582D4—44no—
583D4—44no—
584D4—82no—
585D4—816no—
586D4—84no—
587D4—816no—
588D4—88no—
589D4—88no—
590D4—816no—
591C2×C4—24no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—22no—
597C2×C4—28no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—84no—
623C2×C4—816no—
624C2×C4—82no—
625C2×C4—88no—
626C2×C4—84no—
627C2×C4C2×#12018yes—
628C2×C4C2×#12012yes—
629C2×C4C23×D412yes—
630C2×C4C2×H(5,𝔽2)12yes—
631C2×C4#58512yes—
632C8—44no—
633C8—44no—
634C22—24no—
635C22—22no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—41no—
643C22—41no—
644C22—42no—
645C22—42no—
646C22—44no—
647C22—42no—
648C22—48no—
649C22—42no—
650C22—42no—
651C22—48no—
652C22—44no—
653C22—48no—
654C22—88no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—1616no—
660C22—1616no—
661C22C22×#12011yescharacteristic
662C22#348411yescharacteristic
663C22#348511yescharacteristic
664C22C2×#57712yes—
665C22#358512yes—
666C4—22no—
667C4—28no—
668C4—42no—
669C4—42no—
670C4—44no—
671C4—48no—
672C4—81no—
673C4—81no—
674C4—82no—
675C4—82no—
676C4#348814yes—
677C2—42no—
678C2—44no—
679C2—82no—
680C2—88no—
681C2#3551211yescharacteristic
682C2#6037312yes—
683C2#6163311yescharacteristic
684C2#6165111yescharacteristic
685C2#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×#3126C2semidirect
3. #35661C2semidirect
4. #24252C2semidirect
5. #24240C2semidirect
6. #24138C2semidirect
7. #24126C2semidirect
8. #24015C2semidirect
9. #24009C2semidirect
10. #23926C2semidirect
11. #23919C2semidirect
12. #21872C2semidirect
13. #21792C2semidirect
21. C2×#536C22semidirect
25. C22×C22⋊Q8C22semidirect
27. C2×#519C22semidirect
29. C2×#482C22semidirect
30. C2×#479C22semidirect
31. C2×#460C22semidirect
32. C2×#448C22semidirect
33. C2×#450C22semidirect
34. C2×#450C22semidirect
35. C2×#450C22semidirect
36. C2×#440C22semidirect
37. C2×C4×SD16C22semidirect
38. C2×#433C22semidirect
39. #2538C22semidirect
40. #2536C22semidirect
45. #2317C22semidirect
46. #2311C22semidirect
47. #2280C22semidirect
48. #2034C22semidirect
49. #2034C22semidirect
50. #2033C22semidirect
51. #2033C22semidirect
52. #2000C22semidirect
53. #2000C22semidirect
54. #2000C22semidirect
55. #2000C22semidirect
56. #1992C22semidirect
57. #1991C22semidirect
58. #1961C22semidirect
59. #1957C22semidirect
60. #1955C22semidirect
61. #1954C22semidirect
62. #1875C22semidirect
63. #1874C22semidirect
64. #1873C22semidirect
65. #1873C22semidirect
66. #1872C22semidirect
67. #1854C22semidirect
68. #1853C22semidirect
69. #1852C22semidirect
70. #1804C22semidirect
71. #1802C22semidirect
72. #1786C22semidirect
73. #1786C22semidirect
74. #1786C22semidirect
75. #1780C22semidirect
76. #1779C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35372
C2×#31262
#356612
#242522
#242402
#241382
#241262
#240152
#240092
#239262
#239192
#218722
#217922