Finite Group Databasebeta

← #6503182

Subgroups — #6503182

Order 512.

2086 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35000C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic, complemented
3#27859C214yesmaximal, complemented
4#26208C214yesmaximal, complemented
5#26136C214yesmaximal, complemented
6#24945C214yesmaximal, complemented
7#24873C214yesmaximal, complemented
8#24555C214yesmaximal, complemented
9#21940C211yesmaximal, characteristic, complemented
10#21758C211yesmaximal, characteristic, complemented
11C2×#1893C211yesmaximal, characteristic, complemented
12C2×#1869C211yesmaximal, characteristic, complemented
13C2×#1807C211yesmaximal, characteristic, complemented
14C24×C8C2211yescharacteristic
15C2×#534—21no—
16C22×#121C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#482C2214yescomplemented
19C22×#106C2212yescomplemented
20C22×#102C2212yescomplemented
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2212yescomplemented
23#2854C2212yes—
24#2847C2212yes—
25#2618—21no—
26#2617—21no—
27#2543—22no—
28#2534—22no—
29#2486C2212yes—
30#2416C2214yes—
31#2408C2212yes—
32#2353—22no—
33#2353—22no—
34#2352C2214yes—
35#2352C2214yescomplemented
36#2313—21no—
37#2312—21no—
38#2308—21no—
39#2308—21no—
40#2283C2212yescomplemented
41#2280C2212yescomplemented
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45#2273—21no—
46#2101C2214yescomplemented
47#2092C2214yescomplemented
48#2085C2214yescomplemented
49#2077C2214yescomplemented
50#2076C2214yescomplemented
51#2073C2214yescomplemented
52#2034C2214yescomplemented
53#2033C2214yescomplemented
54#1926C2214yescomplemented
55#1925C2214yescomplemented
56#1921C2214yescomplemented
57#1920C2214yescomplemented
58#1893C2214yes—
59#1893C2214yes—
60#1893C2212yescomplemented
61#1893C2212yescomplemented
62#1875C2214yescomplemented
63#1874C2214yescomplemented
64#1869C2214yes—
65#1869C2212yescomplemented
66#1869C2212yescomplemented
67#1854C2214yescomplemented
68#1852C2214yescomplemented
69#1852C2214yescomplemented
70#1825C2214yescomplemented
71#1823C2212yescomplemented
72#1808C2214yescomplemented
73#1807C2214yescomplemented
74#1807C2214yescomplemented
75#1786C2212yescomplemented
76C2×#340C2212yescomplemented
77C2×#340C2212yescomplemented
78C2×#336C2212yescomplemented
79C23×D4—21no—
80C23×D4D411yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C23×C8—24no—
85C23×C8—44no—
86C23×C8C2311yescharacteristic
87C23×C8C2312yes—
88C23×C8C2312yes—
89C23×C8C2312yes—
90C23×C8C2311yescharacteristic
91#534—48no—
92C2×#123—22no—
93C2×#123—22no—
94C2×#121—24no—
95C2×#121—21no—
96C2×#121—21no—
97C2×#121—24no—
98C2×#121—24no—
99C2×#121C2312yes—
100C2×#121C2312yes—
101C2×#121C2314yes—
102C2×#120—22no—
103C2×#120—22no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4D412yes—
107C2×C4×D4D412yes—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×#117C2314yes—
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4D411yescharacteristic
121C22×C22⋊C4D411yescharacteristic
122C22×C42C2311yescharacteristic
123#482—48no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106—44no—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×#102—44no—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×C22⋊C8—24no—
135C2×C22⋊C8—44no—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2314yes—
140#398—22no—
141#397—22no—
142#396—24no—
143#394—22no—
144#394—24no—
145#392—22no—
146#390C2314yes—
147#389C2314yes—
148#388—24no—
149#388—24no—
150#388—24no—
151#387—24no—
152#387—24no—
153#386—24no—
154#386—24no—
155#386—22no—
156#386—22no—
157#386—24no—
158#386—24no—
159#383—22no—
160#382C2314yes—
161#381—22no—
162#381—22no—
163#381—22no—
164#380—21no—
165#380—21no—
166#380—21no—
167#380—21no—
168#380—21no—
169#380—21no—
170#380C2312yes—
171#380C2312yes—
172#380C2312yes—
173#380C2312yes—
174#380C2312yes—
175#380C2314yes—
176#380C2314yes—
177#380C2312yes—
178#379—22no—
179#379—22no—
180C4×C4⋊C4C2314yes—
181C4×C22⋊C4—22no—
182C4×C22⋊C4—22no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—24no—
185C4×C22⋊C4—24no—
186#376—22no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C25C2×D411yescharacteristic
202C25C2×D411yescharacteristic
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—24no—
209C22×D4—21no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—42no—
213C22×D4—42no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4—44no—
217C22×D4C2×D412yes—
218C22×D4C2×D412yes—
219C22×D4C2412yes—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—21no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D412yes—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4#4114yes—
238C23×C4#4114yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C22×C8—22no—
244C22×C8—24no—
245C22×C8—24no—
246C22×C8—24no—
247C22×C8—22no—
248C22×C8—24no—
249C22×C8—24no—
250C22×C8—24no—
251C22×C8—42no—
252C22×C8—42no—
253C22×C8—42no—
254C22×C8—42no—
255C22×C8—42no—
256C22×C8—416no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260#123—48no—
261#123—48no—
262#121—24no—
263#121—24no—
264#121—24no—
265#121—24no—
266#121—416no—
267#121—44no—
268#121—44no—
269#121—44no—
270#121—44no—
271#120—48no—
272#120—48no—
273C4×D4—28no—
274C4×D4—28no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—24no—
279#117—44no—
280#117—44no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2414yes—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42C2×D412yes—
330C2×C42C2412yes—
331#107—44no—
332#107—44no—
333#106—28no—
334#106—44no—
335#102—28no—
336#102—44no—
337#102—44no—
338#102—44no—
339C22⋊C8—24no—
340C22⋊C8—24no—
341C22⋊C8—28no—
342C22⋊C8—44no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—82no—
367C24—82no—
368C24#12314yes—
369C24#13514yes—
370C24AGL(1,ℤ/8)14yes—
371C24C22×D412yes—
372C24C22×D411yescharacteristic
373C24C22×D411yescharacteristic
374C24C22×D411yescharacteristic
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—48no—
386C2×D4—416no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—48no—
394C2×D4—42no—
395C2×D4—82no—
396C2×D4—82no—
397C2×D4—82no—
398C2×D4—84no—
399C2×D4—84no—
400C2×D4—816no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
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—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—84no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4#12314yes—
446C22×C4#12314yes—
447C22×C4#12314yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2511yescharacteristic
461C2×C8—28no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—48no—
471C2×C8—48no—
472C2×C8—44no—
473C2×C8—84no—
474C2×C8H(5,𝔽2)14yes—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—48no—
486C4⋊C4—42no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—84no—
512C42—84no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—44no—
525C23—44no—
526C23—48no—
527C23—48no—
528C23—48no—
529C23—44no—
530C23—44no—
531C23—42no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—82no—
536C23—84no—
537C23—82no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23—164no—
542C23—164no—
543C23#482116yescomplemented
544C23C2×#12312yes—
545C23C2×#12312yes—
546C23C2×#12312yes—
547C23C2×#13512yes—
548C23C2×#13512yes—
549C23C2×#13512yes—
550C23C2×AGL(1,ℤ/8)12yes—
551C23C2×AGL(1,ℤ/8)12yes—
552C23C2×AGL(1,ℤ/8)12yes—
553C23C23×D411yescharacteristic
554D4—48no—
555D4—48no—
556D4—42no—
557D4—42no—
558D4—44no—
559D4—84no—
560D4—88no—
561D4—88no—
562D4—168no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—84no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—84no—
605C2×C4—164no—
606C2×C4—164no—
607C2×C4—164no—
608C2×C4C2×#12312yes—
609C2×C4C2×#12312yes—
610C2×C4C2×#12312yes—
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C2×H(5,𝔽2)12yes—
614C2×C4#58512yes—
615C8—24no—
616C8—42no—
617C8—44no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—416no—
631C22—42no—
632C22—44no—
633C22—44no—
634C22—42no—
635C22—42no—
636C22—88no—
637C22—82no—
638C22—84no—
639C22—84no—
640C22—82no—
641C22—82no—
642C22—168no—
643C22C2×#48214yes—
644C22C2×#48214yes—
645C22C2×#48214yes—
646C22C22×#12311yescharacteristic
647C22#349612yes—
648C22C22×#13511yescharacteristic
649C22C22×AGL(1,ℤ/8)11yescharacteristic
650C22#358412yes—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—84no—
658C4—84no—
659C4—84no—
660C4—82no—
661C4#349011yescharacteristic
662C4#349111yescharacteristic
663C4#350512yes—
664C2—24no—
665C2—24no—
666C2—24no—
667C2—82no—
668C2—162no—
669C2#3536011yescharacteristic
670C2C22×#48211yescharacteristic
671C2#6051611yescharacteristic
672C2#6224412yes—
673C2#6225312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35000C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34185C2semidirect
3. #27859C2semidirect
4. #26208C2semidirect
5. #26136C2semidirect
6. #24945C2semidirect
7. #24873C2semidirect
8. #24555C2semidirect
9. #21940C2semidirect
10. #21758C2semidirect
11. C2×#1893C2semidirect
12. C2×#1869C2semidirect
13. C2×#1807C2semidirect
18. C2×#482C22semidirect
19. C22×#106C22semidirect
20. C22×#102C22semidirect
22. C22×C22⋊C8C22semidirect
35. #2352C22semidirect
40. #2283C22semidirect
41. #2280C22semidirect
46. #2101C22semidirect
47. #2092C22semidirect
48. #2085C22semidirect
49. #2077C22semidirect
50. #2076C22semidirect
51. #2073C22semidirect
52. #2034C22semidirect
53. #2033C22semidirect
54. #1926C22semidirect
55. #1925C22semidirect
56. #1921C22semidirect
57. #1920C22semidirect
60. #1893C22semidirect
61. #1893C22semidirect
62. #1875C22semidirect
63. #1874C22semidirect
65. #1869C22semidirect
66. #1869C22semidirect
67. #1854C22semidirect
68. #1852C22semidirect
69. #1852C22semidirect
70. #1825C22semidirect
71. #1823C22semidirect
72. #1808C22semidirect
73. #1807C22semidirect
74. #1807C22semidirect
75. #1786C22semidirect
76. C2×#340C22semidirect
77. C2×#340C22semidirect
78. C2×#336C22semidirect
543. C23#482semidirect

Maximal subgroups

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

subgroupindex
#350002
#341852
#278592
#262082
#261362
#249452
#248732
#245552
#219402
#217582
C2×#18932
C2×#18692
C2×#18072