Finite Group Databasebeta

← #6507909

Subgroups — #6507909

Order 512.

1838 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#35014C211yesmaximal, characteristic, complemented
2C2×#2420C211yesmaximal, characteristic
3#29109C211yesmaximal, characteristic, complemented
4#28988C214yesmaximal, complemented
5#28854C214yesmaximal, complemented
6#28311C211yesmaximal, characteristic, complemented
7#27983C211yesmaximal, characteristic, complemented
8#27956C211yesmaximal, characteristic, complemented
9#26974C214yesmaximal, complemented
10#25040C214yesmaximal, complemented
11#24552C212yesmaximal, complemented
12#24379C212yesmaximal
13#22085C212yesmaximal, complemented
14#21453C212yesmaximal, complemented
15#20526C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#484C2212yes—
19C2×#465C2214yes—
20C2×#450C2212yes—
21C2×#440C2214yescomplemented
22C22×C22⋊C8C2212yes—
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2848C2212yes—
26#2646—21no—
27#2626—21no—
28#2589—22no—
29#2587—22no—
30#2497C2212yescomplemented
31#2492C2212yescomplemented
32#2429C2212yescomplemented
33#2423C2212yescomplemented
34#2420—22no—
35#2420—22no—
36#2420C2214yes—
37#2420C2214yes—
38C2×#397—21no—
39C2×#394—21no—
40#2329—21no—
41#2312—21no—
42C4×C22⋊Q8C2212yes—
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#386—21no—
46C2×#382C2211yescharacteristic
47C2×#381—21no—
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2093C2212yescomplemented
51#2085C2214yescomplemented
52#2076C2214yescomplemented
53#2074C2214yescomplemented
54#2071C2212yescomplemented
55#2036C2214yescomplemented
56#2034C2214yes—
57#2023C2212yescomplemented
58#2013C2212yescomplemented
59#2000C2214yes—
60#1993C2212yescomplemented
61#1973C2214yescomplemented
62#1960C2212yescomplemented
63#1943C2214yescomplemented
64#1936C2214yescomplemented
65#1935C2214yescomplemented
66#1918C2214yescomplemented
67#1893C2214yes—
68#1892C2214yescomplemented
69#1877C2214yescomplemented
70#1872C2212yes—
71#1872C2212yes—
72#1865C2214yescomplemented
73#1862C2214yescomplemented
74#1853C2212yes—
75#1852C2212yes—
76#1847C2212yescomplemented
77#1836C2212yescomplemented
78#1824C2212yes—
79#1808C2214yescomplemented
80#1786C2212yes—
81#1762C2214yescomplemented
82#1752C2214yescomplemented
83#1752C2214yescomplemented
84#1749C2211yescharacteristic
85#1749C2212yes—
86#1749C2211yescharacteristic
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C22×SD16C2314yes—
92C23×C8C2314yes—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×#121—22no—
100C2×#121—44no—
101C2×#121C2314yes—
102C2×C4×Q8C2312yes—
103C2×C4×D4—22no—
104C2×C4×D4—44no—
105C2×C4×D4C2314yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4D411yescharacteristic
116C22×C22⋊C4D411yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#484—28no—
120#465—48no—
121#450—28no—
122#440—48no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×C22⋊C8—28no—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×#97C2312yes—
137C2×#97C2312yes—
138C2×#97C2314yes—
139#400—22no—
140#400—22no—
141#399—22no—
142#398—22no—
143#398—22no—
144#397—24no—
145#397—24no—
146#396—22no—
147#396—22no—
148#394—24no—
149#394—24no—
150#393—22no—
151#391—21no—
152#391—22no—
153#391C2312yes—
154#391C2312yes—
155#390—21no—
156#390—21no—
157#390C2312yes—
158#390C2312yes—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389C2312yes—
165#389C2312yes—
166#389C2314yes—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#386—24no—
175#386—24no—
176#385—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#382—21no—
183#382—21no—
184#382C2312yes—
185#382C2312yes—
186#381—24no—
187#381—24no—
188#380—21no—
189#380—21no—
190#380—21no—
191#380—22no—
192#380C2312yes—
193#380C2314yes—
194#380C2312yes—
195#380C2312yes—
196#379—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—22no—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4C2312yes—
203C4×C22⋊C4C2312yes—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2312yes—
217#336C2312yes—
218#336C2314yes—
219C25—21no—
220C25C2×D411yescharacteristic
221C22×Q8—21no—
222C22×Q8C2412yes—
223C22×D4—24no—
224C22×D4—24no—
225C22×D4—21no—
226C22×D4—24no—
227C22×D4—42no—
228C22×D4—44no—
229C22×D4—44no—
230C22×D4—82no—
231C22×D4C2412yes—
232C23×C4—24no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—24no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—82no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D412yes—
248C23×C4C2412yes—
249C23×C4C2411yescharacteristic
250C2×SD16—28no—
251C2×SD16—28no—
252C2×SD16—44no—
253C22×C8—28no—
254C22×C8—44no—
255C22×C8—44no—
256C22×C8C2414yes—
257C22×C8C2414yes—
258C4⋊Q8—42no—
259C4⋊Q8—42no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—44no—
263C22⋊Q8—44no—
264C22⋊Q8—44no—
265#121—28no—
266#121—44no—
267#121—48no—
268C4×Q8—44no—
269C4×D4—28no—
270C4×D4—44no—
271C4×D4—48no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—82no—
317C2×C22⋊C4—82no—
318C2×C22⋊C4—82no—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42C2×D412yes—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2412yes—
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333#107—28no—
334#107—44no—
335#106—44no—
336#103—28no—
337#103—44no—
338#103—44no—
339#102—28no—
340#102—28no—
341#102—44no—
342C22⋊C8—24no—
343C22⋊C8—28no—
344C22⋊C8#4118yes—
345#97—42no—
346#97—42no—
347#97—44no—
348#95—24no—
349#95—22no—
350#95—22no—
351#95—24no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95C2×D412yes—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368C24—22no—
369C24—22no—
370C24—24no—
371C24—22no—
372C24—41no—
373C24—41no—
374C24—41no—
375C24—41no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24#13714yes—
383C24C22×D412yes—
384C24C22×D411yescharacteristic
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—42no—
390C2×Q8—81no—
391C2×Q8—81no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—24no—
396C2×D4—416no—
397C2×D4—42no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—42no—
401C2×D4—84no—
402C2×D4—816no—
403C2×D4H(5,𝔽2)14yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—41no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—88no—
447C22×C4—84no—
448C22×C4—816no—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C2×#4114yes—
458C22×C4C2511yescharacteristic
459SD16—416no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—88no—
466C2×C8C2×#4118yes—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—82no—
484C4⋊C4—84no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—84no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—41no—
512C42—41no—
513C42—42no—
514C42—82no—
515C42—82no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—28no—
527C23—24no—
528C23—44no—
529C23—44no—
530C23—48no—
531C23—48no—
532C23—48no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23#53212yes—
548C23C2×#13714yes—
549C23C2×#13712yes—
550C23#57512yes—
551C23C23×D411yescharacteristic
552Q8—44no—
553Q8—44no—
554Q8—162no—
555D4—28no—
556D4—48no—
557D4—44no—
558D4—816no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—84no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—816no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4#53212yes—
617C2×C4C22×#4112yes—
618C2×C4C2×H(5,𝔽2)12yes—
619C2×C4#58512yes—
620C8—28no—
621C8—44no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—88no—
639C22—816no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22#318114yes—
645C22C2×#53211yescharacteristic
646C22#348411yescharacteristic
647C22#348511yescharacteristic
648C22C22×#13711yescharacteristic
649C22C2×#57511yescharacteristic
650C22#358511yescharacteristic
651C22#358511yescharacteristic
652C4—22no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4#351314yes—
668C2—24no—
669C2—44no—
670C2—84no—
671C2#3539411yescharacteristic
672C2C2×#318111yescharacteristic
673C2#6070411yescharacteristic
674C2#6157011yescharacteristic
675C2#6157611yescharacteristic
676C2#6170811yescharacteristic
677C2#6171011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35014C2semidirect
3. #29109C2semidirect
4. #28988C2semidirect
5. #28854C2semidirect
6. #28311C2semidirect
7. #27983C2semidirect
8. #27956C2semidirect
9. #26974C2semidirect
10. #25040C2semidirect
11. #24552C2semidirect
13. #22085C2semidirect
14. #21453C2semidirect
15. #20526C2semidirect
21. C2×#440C22semidirect
30. #2497C22semidirect
31. #2492C22semidirect
32. #2429C22semidirect
33. #2423C22semidirect
50. #2093C22semidirect
51. #2085C22semidirect
52. #2076C22semidirect
53. #2074C22semidirect
54. #2071C22semidirect
55. #2036C22semidirect
57. #2023C22semidirect
58. #2013C22semidirect
60. #1993C22semidirect
61. #1973C22semidirect
62. #1960C22semidirect
63. #1943C22semidirect
64. #1936C22semidirect
65. #1935C22semidirect
66. #1918C22semidirect
68. #1892C22semidirect
69. #1877C22semidirect
72. #1865C22semidirect
73. #1862C22semidirect
76. #1847C22semidirect
77. #1836C22semidirect
79. #1808C22semidirect
81. #1762C22semidirect
82. #1752C22semidirect
83. #1752C22semidirect

Maximal subgroups

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

subgroupindex
#350142
C2×#24202
#291092
#289882
#288542
#283112
#279832
#279562
#269742
#250402
#245522
#243792
#220852
#214532
#205262