Finite Group Databasebeta

← #6489204

Subgroups — #6489204

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35319C211yesmaximal, characteristic, complemented
2C4×#536C211yesmaximal, characteristic
3#32907C212yesmaximal
4#32906C212yesmaximal, complemented
5#30054C212yesmaximal, complemented
6#30052C212yesmaximal, complemented
7#29150C211yesmaximal, characteristic, complemented
8#29140C211yesmaximal, characteristic, complemented
9#28449C211yesmaximal, characteristic, complemented
10#28440C211yesmaximal, characteristic, complemented
11#28161C212yesmaximal, complemented
12#28159C212yesmaximal, complemented
13#27985C212yesmaximal, complemented
14#25735C212yesmaximal
15#23944C212yesmaximal
16#23943C212yesmaximal, complemented
17#22278C212yesmaximal, complemented
18#22277C212yesmaximal, complemented
19#20786C211yesmaximal, characteristic, complemented
20C2×#541C2212yes—
21C2×#536C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C2×#497C2212yes—
25C2×#494C2212yes—
26C2×#462C2212yes—
27C2×#462C2212yes—
28C2×#462C2212yes—
29C2×#461C2212yes—
30C2×#452C2214yes—
31C2×#451C2214yes—
32#2868C2211yescharacteristic
33#2859C2211yescharacteristic
34#2857C2212yes—
35#2642C2212yes—
36#2621C2212yes—
37#2506—22no—
38#2506—22no—
39#2496—22no—
40#2496—22no—
41#2327C2211yescharacteristic
42#2327C2212yes—
43#2327C2211yescharacteristic
44#2326C2211yescharacteristic
45#2326C2211yescharacteristic
46#2323—22no—
47#2322—22no—
48#2317C2211yescharacteristic
49#2311C2211yescharacteristic
50#2310C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×#124C2211yescharacteristic
55C4×#124C2211yescharacteristic
56C4×C22⋊Q8—22no—
57C4×#123—22no—
58C4×#117C2211yescharacteristic
59#2074C2214yes—
60#2072C2214yes—
61#2026C2212yes—
62#2026C2212yes—
63#2026C2212yes—
64#2025C2212yes—
65#2016C2214yes—
66#2016C2212yes—
67#2016C2212yes—
68#2016C2212yes—
69#2015C2214yes—
70#2013C2212yes—
71#1960C2212yes—
72#1960C2212yes—
73#1958C2212yes—
74#1956C2212yes—
75#1880C2214yes—
76#1878C2214yes—
77#1878C2214yes—
78#1878C2214yes—
79#1878C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1877C2214yes—
83#1873C2214yes—
84#1872C2214yes—
85#1840C2212yes—
86#1836C2212yes—
87#1835C2212yes—
88#1831C2212yes—
89#1818C2212yes—
90#1803C2212yes—
91#1803C2212yes—
92#1803C2212yes—
93#1802C2212yes—
94#1797C2212yes—
95#1767C2211yescharacteristic
96C4×C4⋊C8C2211yescharacteristic
97#1750C2212yes—
98#1746C2212yes—
99C22×#41C2311yescharacteristic
100C23×Q8C2311yescharacteristic
101C22×Q16C2314yes—
102C22×Q16C2314yes—
103C22×Q16C2314yes—
104C22×SD16C2314yes—
105#541—48no—
106#536—24no—
107#536—24no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2312yes—
113C2×#126—24no—
114C2×#125—24no—
115C2×#125C2312yes—
116C2×#124C2311yescharacteristic
117C2×#124C2311yescharacteristic
118C2×#123—22no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8C2314yes—
121C4×#41—44no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—44no—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2311yescharacteristic
127C2×C4×Q8C2311yescharacteristic
128C2×C4×Q8C2314yes—
129C2×C4×Q8C2311yescharacteristic
130C2×C4×Q8C2311yescharacteristic
131C2×C4×Q8C2311yescharacteristic
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4C2311yescharacteristic
135C2×C4×D4C2311yescharacteristic
136C2×#117—21no—
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C22×C4⋊C4—21no—
141C22×C4⋊C4C2311yescharacteristic
142C22×C42—21no—
143C22×C42—21no—
144C22×C42C2311yescharacteristic
145#497—28no—
146#494—28no—
147#462—44no—
148#462—44no—
149#462—44no—
150#461—44no—
151#452—48no—
152#451—48no—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#102C2314yes—
163C2×C22⋊C8C2314yes—
164C2×#97C2312yes—
165C2×C4×C8C2312yes—
166#400—24no—
167#400—24no—
168#398—24no—
169#397—24no—
170#391—22no—
171#391C2311yescharacteristic
172#391C2311yescharacteristic
173#391C2311yescharacteristic
174#391C2311yescharacteristic
175#391C2314yes—
176#391C2311yescharacteristic
177#391C2311yescharacteristic
178#391C2314yes—
179#390C2311yescharacteristic
180#390C2311yescharacteristic
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—24no—
199#387—24no—
200#386—22no—
201#385—22no—
202#385—22no—
203#385—22no—
204#384—22no—
205#384—22no—
206#384C2312yes—
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#382C2311yescharacteristic
210#382C2311yescharacteristic
211#382C2312yes—
212#381—22no—
213#380—22no—
214#380—22no—
215#380—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4D412yes—
221C4×C4⋊C4D412yes—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226#376C2311yescharacteristic
227#376C2311yescharacteristic
228C43C2311yescharacteristic
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238#336C2312yes—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—24no—
242C2×#41—42no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—21no—
246C22×Q8—24no—
247C22×Q8—42no—
248C22×Q8C2414yes—
249C22×Q8C2411yescharacteristic
250C22×Q8C2411yescharacteristic
251C22×Q8C2411yescharacteristic
252C22×D4—21no—
253C22×D4C2411yescharacteristic
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262C2×Q16—28no—
263C2×Q16—28no—
264C2×Q16—28no—
265C2×Q16—44no—
266C2×Q16—44no—
267C2×Q16—44no—
268C2×Q16—44no—
269C2×Q16—44no—
270C2×Q16—44no—
271C2×SD16—28no—
272C2×SD16—44no—
273C2×SD16—44no—
274C22×C8C2414yes—
275C22×C8C2414yes—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—24no—
283C4⋊Q8—44no—
284C4⋊Q8#4114yes—
285#126—416no—
286#125—44no—
287#125—416no—
288#124—22no—
289#124—22no—
290#124—22no—
291#124#4114yes—
292#123—216no—
293C22⋊Q8—216no—
294C22⋊Q8—28no—
295C22⋊Q8—28no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—42no—
300C4×Q8—42no—
301C4×Q8—42no—
302C4×Q8—44no—
303C4×Q8—44no—
304C4×Q8—44no—
305C4×Q8—44no—
306C4×D4—24no—
307C4×D4—42no—
308C4×D4—44no—
309C4×D4—44no—
310#117—24no—
311#117—22no—
312#117—22no—
313#117—44no—
314#117—42no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—42no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2414yes—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—24no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—24no—
363C2×C42—44no—
364C2×C42—44no—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D412yes—
368C2×C42C2×D412yes—
369C2×C42C2×D412yes—
370C2×C42C2×D412yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C4⋊C8—24no—
378C4⋊C8—24no—
379#103—44no—
380#103—44no—
381#103—44no—
382#103—44no—
383#103—44no—
384#103—44no—
385#103—44no—
386#102—44no—
387C22⋊C8—28no—
388#97—24no—
389C4×C8—24no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407C24—21no—
408C24C22×D411yescharacteristic
409#41—48no—
410#41—48no—
411C2×Q8—24no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—24no—
427C2×Q8—48no—
428C2×Q8—48no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8C2×#4112yes—
432C2×Q8C2×#4112yes—
433C2×Q8C2×#4112yes—
434C2×D4—24no—
435C2×D4—21no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4C2×#4112yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4H(5,𝔽2)14yes—
479C22×C4C2511yescharacteristic
480Q16—88no—
481Q16—88no—
482Q16—88no—
483SD16—88no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—48no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C22⋊C4—28no—
519C22⋊C4—24no—
520C22⋊C4—28no—
521C22⋊C4—24no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—48no—
525C22⋊C4—42no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C42—22no—
529C42—22no—
530C42—24no—
531C42—24no—
532C42—21no—
533C42—21no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—42no—
538C42—42no—
539C42—42no—
540C42—42no—
541C42—42no—
542C42—42no—
543C42—48no—
544C42—42no—
545C42—42no—
546C42—42no—
547C42C2×#4112yes—
548C42C2×#4112yes—
549C42H(5,𝔽2)14yes—
550C23—22no—
551C23—21no—
552C23—21no—
553C23—22no—
554C23—22no—
555C23—42no—
556C23—41no—
557C23—41no—
558C23—42no—
559C23#53512yes—
560C23C23×D411yescharacteristic
561Q8—24no—
562Q8—24no—
563Q8—24no—
564Q8—48no—
565Q8—42no—
566Q8—42no—
567Q8—42no—
568Q8—44no—
569Q8—44no—
570Q8—44no—
571Q8—44no—
572D4—24no—
573D4—48no—
574D4—42no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—82no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4#52514yes—
626C2×C4#53512yes—
627C2×C4#53512yes—
628C2×C4#53512yes—
629C2×C4#57814yes—
630C2×C4C22×#4111yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×H(5,𝔽2)12yes—
633C2×C4C2×H(5,𝔽2)12yes—
634C8—44no—
635C8—44no—
636C22—22no—
637C22—24no—
638C22—41no—
639C22—41no—
640C22—42no—
641C22—42no—
642C22—44no—
643C22—81no—
644C22—81no—
645C22—82no—
646C22C2×#52511yescharacteristic
647C22C2×#53511yescharacteristic
648C22C2×#53511yescharacteristic
649C22C2×#57812yes—
650C22#358612yes—
651C4—22no—
652C4—22no—
653C4—24no—
654C4—41no—
655C4—41no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—84no—
667C4—84no—
668C4#322314yes—
669C4#351912yes—
670C4#351912yes—
671C2—42no—
672C2—82no—
673C2#3523711yescharacteristic
674C2#6059412yes—
675C2C2×#322311yescharacteristic
676C2#6080911yescharacteristic
677C2#6083712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35319C2semidirect
4. #32906C2semidirect
5. #30054C2semidirect
6. #30052C2semidirect
7. #29150C2semidirect
8. #29140C2semidirect
9. #28449C2semidirect
10. #28440C2semidirect
11. #28161C2semidirect
12. #28159C2semidirect
13. #27985C2semidirect
16. #23943C2semidirect
17. #22278C2semidirect
18. #22277C2semidirect
19. #20786C2semidirect

Maximal subgroups

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

subgroupindex
#353192
C4×#5362
#329072
#329062
#300542
#300522
#291502
#291402
#284492
#284402
#281612
#281592
#279852
#257352
#239442
#239432
#222782
#222772
#207862