Finite Group Databasebeta

← #6470322

Subgroups — #6470322

Order 512.

2938 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3010C212yesmaximal, complemented
2#37297C211yesmaximal, characteristic, complemented
3#34257C211yesmaximal, characteristic, complemented
4#32912C212yesmaximal, complemented
5#32903C212yesmaximal, complemented
6#29725C212yesmaximal, complemented
7#29716C212yesmaximal, complemented
8#24541C214yesmaximal, complemented
9#24535C214yesmaximal, complemented
10#24370C214yesmaximal, complemented
11#24356C214yesmaximal, complemented
12#21937C211yesmaximal, characteristic, complemented
13#21770C212yesmaximal, complemented
14C22×2−1+4C2211yescharacteristic
15C22×AGL(1,ℤ/8)C2212yescomplemented
16C23×SD16C2212yescomplemented
17C2×#535C2212yes—
18C2×#519C2211yescharacteristic, complemented
19C22×C4×Q8C2211yescharacteristic, complemented
20C2×#460C2214yes—
21C2×#460C2214yes—
22#3010—216no—
23C2×#449C2214yes—
24C2×#449C2214yescomplemented
25C2×#448C2214yes—
26C2×#448C2214yescomplemented
27C2×#419C2212yescomplemented
28C2×#416C2212yescomplemented
29C2×#408C2212yes—
30#2856C2212yes—
31#2733—22no—
32#2679—22no—
33#2669—22no—
34#2669—22no—
35#2619C2212yes—
36#2500C2214yes—
37#2433C2214yes—
38#2341—22no—
39#2341C2214yescomplemented
40Q8×C22⋊C4—22no—
41Q8×C22⋊C4C2214yescomplemented
42#2311—22no—
43#2311C2211yescharacteristic
44#2311C2211yescharacteristic, complemented
45#2297—22no—
46#2297C2211yescharacteristic
47#2297C2211yescharacteristic, complemented
48#2074C2212yescomplemented
49#2073C2212yescomplemented
50#2072C2212yescomplemented
51#2071C2212yescomplemented
52#2055C2214yescomplemented
53#2055C2214yescomplemented
54#2054C2214yescomplemented
55#2054C2214yescomplemented
56#2014C2212yes—
57#2014C2212yes—
58#2011C2212yes—
59#2011C2212yes—
60#1895C2214yes—
61#1893C2214yes—
62#1880C2214yescomplemented
63#1880C2214yescomplemented
64#1880C2214yescomplemented
65#1880C2214yescomplemented
66#1875C2214yescomplemented
67#1874C2214yescomplemented
68#1851C2214yescomplemented
69#1850C2214yescomplemented
70#1825C2214yescomplemented
71#1805C2214yescomplemented
72#1803C2214yescomplemented
73#1798C2212yescomplemented
74#1797C2212yescomplemented
75#1786C2214yescomplemented
76C2×2−1+4—24no—
77C2×2−1+4—44no—
78C2×2−1+4—44no—
79C2×2−1+4D418yescomplemented
80C22×#41—22no—
81C22×#41—21no—
82C22×#41C2312yes—
83C22×#41C2311yescharacteristic
84C22×#41C2311yescharacteristic
85C23×Q8—21no—
86C23×Q8C2311yescharacteristic
87C23×Q8C2312yes—
88C23×D4C2312yes—
89C2×AGL(1,ℤ/8)—28no—
90C2×AGL(1,ℤ/8)—44no—
91C2×AGL(1,ℤ/8)—44no—
92C22×SD16—28no—
93C22×SD16—44no—
94C22×SD16—44no—
95C22×SD16C2314yes—
96C22×SD16C2314yes—
97C22×SD16C2314yes—
98C22×D8C2314yes—
99C22×M16C2312yes—
100C23×C8C2312yes—
101#535—28no—
102#535—28no—
103C2×#127—22no—
104C2×#127C2312yes—
105C2×#127C2312yes—
106C2×#124—22no—
107C2×#121—22no—
108C2×#121—22no—
109C2×#121C2314yes—
110C2×#121C2314yes—
111C2×#120—22no—
112C2×#120—22no—
113C2×#120—24no—
114C2×#120—24no—
115#519—44no—
116C2×C4×Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8—44no—
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2314yes—
121C2×#117—22no—
122C2×#117C2312yes—
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2311yescharacteristic
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129#460—48no—
130#460—48no—
131#449—216no—
132#449—216no—
133#448—216no—
134#448—216no—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2314yes—
137#419—28no—
138#416—28no—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147#408—28no—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2314yes—
150#394—24no—
151#394—24no—
152#394—24no—
153#392—24no—
154#392—24no—
155#392—24no—
156#391—21no—
157#391—24no—
158#391—24no—
159#391D412yes—
160#391C2312yes—
161#391C2312yes—
162#390—21no—
163#390—24no—
164#390—24no—
165#390—22no—
166#390—22no—
167#390D412yes—
168#390C2312yes—
169#390C2312yes—
170#390C2314yes—
171#390C2314yes—
172#388—24no—
173#388—24no—
174#386—22no—
175#386—21no—
176#386—21no—
177#386—22no—
178#386D412yes—
179#386D412yes—
180#384C2312yes—
181#382C2312yes—
182#380—22no—
183#380—24no—
184#380—24no—
185#380C2314yes—
186C4×C22⋊C4—24no—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—24no—
189C4×C22⋊C4C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#336C2312yes—
195#336C2312yes—
196#336C2312yes—
197#336C2312yes—
198C25—22no—
1992−1+4—432no—
2002−1+4—432no—
201C2×#41—216no—
202C2×#41—28no—
203C2×#41—24no—
204C2×#41—22no—
205C2×#41—24no—
206C2×#41—24no—
207C2×#41—42no—
208C2×#41—42no—
209C2×#41—42no—
210C2×#41—42no—
211C2×#41—42no—
212C2×#41—42no—
213C2×#41—48no—
214C2×#41—48no—
215C2×#41—44no—
216C2×#41—44no—
217C2×#41C2×D418yes—
218C2×#41C2×D414yes—
219C2×#41C2×D414yes—
220C2×#41#4114yes—
221C22×Q8—24no—
222C22×Q8—24no—
223C22×Q8—28no—
224C22×Q8—21no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—24no—
228C22×Q8—24no—
229C22×Q8—24no—
230C22×Q8—42no—
231C22×Q8—42no—
232C22×Q8—44no—
233C22×Q8—44no—
234C22×Q8C2×D414yes—
235C22×Q8C2×D418yes—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2411yescharacteristic
238C22×Q8C2412yes—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—24no—
242C22×D4—24no—
243C22×D4—24no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—24no—
248C22×D4—24no—
249C22×D4—42no—
250C22×D4—42no—
251C22×D4—42no—
252C22×D4—42no—
253C22×D4C2×D411yescharacteristic
254C22×D4C2×D412yes—
255C22×D4C2414yes—
256C22×D4C2411yescharacteristic
257C23×C4—21no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2411yescharacteristic
265C23×C4C2412yes—
266AGL(1,ℤ/8)—432no—
267C2×SD16—28no—
268C2×SD16—28no—
269C2×SD16—28no—
270C2×SD16—28no—
271C2×SD16—28no—
272C2×SD16—28no—
273C2×SD16—28no—
274C2×SD16—44no—
275C2×SD16—44no—
276C2×SD16—432no—
277C2×D8—28no—
278C2×D8—44no—
279C2×D8—44no—
280C2×M16—24no—
281C2×M16—42no—
282C2×M16—42no—
283C22×C8—24no—
284C22×C8—42no—
285C22×C8—42no—
286C22×C8C2414yes—
287C22×C8C2414yes—
288#127—44no—
289#127—44no—
290#127—48no—
291#124—48no—
292#121—28no—
293#121—28no—
294#121—28no—
295#121—28no—
296#121—84no—
297#121—84no—
298#120—48no—
299#120—48no—
300#120—88no—
301#120—88no—
302C4×Q8—28no—
303C4×Q8—28no—
304C4×Q8—44no—
305C4×Q8—44no—
306C4×Q8—48no—
307C4×Q8—48no—
308#117—24no—
309#117—24no—
310#117—42no—
311#117—42no—
312#117—48no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2414yes—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4C2×D414yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D414yes—
339C2×C42—24no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—42no—
344C2×C42—44no—
345C2×C42C2×D412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348C4⋊C8—44no—
349C4⋊C8—44no—
350#103—28no—
351#103—28no—
352#103—28no—
353#102—28no—
354#102—44no—
355#102—44no—
356#102—44no—
357#102—44no—
358C22⋊C8—28no—
359C22⋊C8—28no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95C2×D414yes—
366#95C2×D414yes—
367C24—21no—
368C24—24no—
369C24—44no—
370C24—82no—
371C24—82no—
372C24—84no—
373C24—84no—
374C24C22×D412yes—
375C24C22×D411yescharacteristic
376#41—216no—
377#41—48no—
378#41—48no—
379#41—48no—
380#41—48no—
381#41—48no—
382#41—48no—
383#41—464no—
384#41—48no—
385#41—432no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—216no—
389C2×Q8—216no—
390C2×Q8—24no—
391C2×Q8—28no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—216no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—41no—
400C2×Q8—41no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—48no—
404C2×Q8—48no—
405C2×Q8—48no—
406C2×Q8—432no—
407C2×Q8#12114yes—
408C2×Q8C22×D412yes—
409C2×Q8C22×D414yes—
410C2×Q8H(5,𝔽2)14yes—
411C2×Q8H(5,𝔽2)14yes—
412C2×D4—28no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—48no—
435C2×D4—48no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—84no—
439C2×D4—84no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—84no—
443C2×D4—84no—
444C2×D4—82no—
445C2×D4—82no—
446C2×D4—82no—
447C2×D4—84no—
448C2×D4—82no—
449C2×D4—84no—
450C2×D4—84no—
451C2×D4—84no—
452C2×D4—84no—
453C2×D4#12114yes—
454C2×D4#12118yes—
455C2×D4C22×D412yes—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—28no—
467C22×C4—21no—
468C22×C4—28no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—41no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—41no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—82no—
485C22×C4C22×D414yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D412yes—
490C22×C4C22×D414yes—
491C22×C4C22×D412yes—
492C22×C4C2×#4114yes—
493C22×C4C2×#4112yes—
494C22×C4C2511yescharacteristic
495SD16—416no—
496SD16—416no—
497SD16—416no—
498D8—416no—
499M16—48no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—48no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—42no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—42no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C22⋊C4—28no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—44no—
526C22⋊C4—48no—
527C22⋊C4—84no—
528C22⋊C4—82no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C42—24no—
533C42—24no—
534C42—42no—
535C42—48no—
536C42—48no—
537C42—42no—
538C42—44no—
539C23—24no—
540C23—22no—
541C23—21no—
542C23—21no—
543C23—22no—
544C23—22no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—42no—
549C23—41no—
550C23—41no—
551C23—42no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—88no—
560C23—168no—
561C23—164no—
562C23—164no—
563C23—168no—
564C23C2×#12114yes—
565C23C2×#12112yes—
566C23C23×D411yescharacteristic
567Q8—216no—
568Q8—24no—
569Q8—24no—
570Q8—24no—
571Q8—24no—
572Q8—44no—
573Q8—416no—
574Q8—42no—
575Q8#53418yes—
576D4—24no—
577D4—48no—
578D4—44no—
579D4—416no—
580D4—48no—
581D4—44no—
582D4—44no—
583D4—42no—
584D4—44no—
585D4—44no—
586D4—88no—
587D4—164no—
588D4—164no—
589D4—168no—
590D4—168no—
591D4—168no—
592C2×C4—216no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—44no—
616C2×C4—416no—
617C2×C4—416no—
618C2×C4—48no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—82no—
630C2×C4—84no—
631C2×C4C2×#12114yes—
632C2×C4C2×#12112yes—
633C2×C4#57714yes—
634C2×C4C23×D411yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C2×H(5,𝔽2)12yes—
637C2×C4C2×H(5,𝔽2)12yes—
638C8—44no—
639C8—44no—
640C22—24no—
641C22—28no—
642C22—22no—
643C22—24no—
644C22—41no—
645C22—42no—
646C22—41no—
647C22—42no—
648C22—42no—
649C22—44no—
650C22—44no—
651C22—42no—
652C22—81no—
653C22—81no—
654C22—82no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—1616no—
663C22C22×#12111yescharacteristic
664C22#348411yescharacteristic
665C22#348411yescharacteristic
666C22C2×#57712yes—
667C22#358512yes—
668C4—28no—
669C4—24no—
670C4—22no—
671C4—24no—
672C4—42no—
673C4—42no—
674C4—44no—
675C4—84no—
676C4—82no—
677C4—82no—
678C4C2×#53412yes—
679C4#349012yes—
680C2—42no—
681C2—44no—
682C2—82no—
683C2—164no—
684C2#3498111yescharacteristic
685C2#6043112yes—
686C2#6163311yescharacteristic
687C2#6165111yescharacteristic
688C2#6168212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3010C2semidirect
2. #37297C2semidirect
3. #34257C2semidirect
4. #32912C2semidirect
5. #32903C2semidirect
6. #29725C2semidirect
7. #29716C2semidirect
8. #24541C2semidirect
9. #24535C2semidirect
10. #24370C2semidirect
11. #24356C2semidirect
12. #21937C2semidirect
13. #21770C2semidirect
15. C22×AGL(1,ℤ/8)C22semidirect
16. C23×SD16C22semidirect
18. C2×#519C22semidirect
19. C22×C4×Q8C22semidirect
24. C2×#449C22semidirect
26. C2×#448C22semidirect
27. C2×#419C22semidirect
28. C2×#416C22semidirect
39. #2341C22semidirect
41. Q8×C22⋊C4C22semidirect
44. #2311C22semidirect
47. #2297C22semidirect
48. #2074C22semidirect
49. #2073C22semidirect
50. #2072C22semidirect
51. #2071C22semidirect
52. #2055C22semidirect
53. #2055C22semidirect
54. #2054C22semidirect
55. #2054C22semidirect
62. #1880C22semidirect
63. #1880C22semidirect
64. #1880C22semidirect
65. #1880C22semidirect
66. #1875C22semidirect
67. #1874C22semidirect
68. #1851C22semidirect
69. #1850C22semidirect
70. #1825C22semidirect
71. #1805C22semidirect
72. #1803C22semidirect
73. #1798C22semidirect
74. #1797C22semidirect
75. #1786C22semidirect
79. C2×2−1+4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30102
#372972
#342572
#329122
#329032
#297252
#297162
#245412
#245352
#243702
#243562
#219372
#217702