Finite Group Databasebeta

← #6565772

Subgroups — #6565772

Order 512.

1889 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3375C212yesmaximal, complemented
2#36595C211yesmaximal, characteristic
3#34335C211yesmaximal, characteristic, complemented
4#29801C212yesmaximal, complemented
5#29796C211yesmaximal, characteristic
6#28753C212yesmaximal, complemented
7#28749C211yesmaximal, characteristic
8#28206C211yesmaximal, characteristic, complemented
9#28201C212yesmaximal, complemented
10#25285C214yesmaximal, complemented
11#25277C212yesmaximal, complemented
12#24639C214yesmaximal, complemented
13#24633C212yesmaximal, complemented
14#23684C212yesmaximal, complemented
15#23679C211yesmaximal, characteristic, complemented
16#22736C211yesmaximal, characteristic, complemented
17#22546C212yesmaximal, complemented
18C2×#559C2212yes—
19C2×Q82C2211yescharacteristic
20#3375—216no—
21C2×#489C2214yes—
22C2×#486C2212yes—
23C2×#477C2212yes—
24C2×#474C2214yes—
25C2×#464C2214yes—
26C2×#460C2212yes—
27C2×#446C2212yes—
28C2×#440C2214yes—
29C2×C4×Q16C2212yes—
30#2867C2211yescharacteristic
31#2858C2212yes—
32#2548—21no—
33#2548—21no—
34#2548C2212yes—
35#2544C2214yes—
36#2445—22no—
37#2445—22no—
38#2369C2211yescharacteristic
39#2369C2212yes—
40#2366—21no—
41#2366—21no—
42#2365—21no—
43#2365—22no—
44#2355C2211yescharacteristic
45#2355C2212yes—
46#2351—21no—
47Q8×C4⋊C4C2211yescharacteristic
48#2331—21no—
49#2331—22no—
50C4×C4⋊Q8C2212yes—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×#127C2212yes—
54#2295C2211yescharacteristic
55#2295C2212yes—
56#2276—21no—
57C42×Q8C2211yescharacteristic
58#2083C2211yescharacteristic
59#2083C2212yes—
60#2072C2211yescharacteristic
61#2068C2212yes—
62#2067C2214yes—
63#2056C2212yes—
64#2055C2214yes—
65#2043C2212yes—
66#2043C2211yescharacteristic
67#2042C2214yes—
68#2041C2212yes—
69#2041C2212yes—
70#2041C2212yes—
71#2041C2212yes—
72#2016C2211yescharacteristic
73#2015C2214yes—
74#2014C2212yes—
75#2014C2212yes—
76#1983C2214yes—
77#1983C2214yes—
78#1983C2214yes—
79#1982C2212yes—
80#1982C2214yes—
81#1959C2214yes—
82#1957C2212yes—
83#1940C2214yes—
84#1939C2212yes—
85#1928C2214yes—
86#1925C2214yes—
87#1925C2214yes—
88#1831C2212yes—
89#1831C2211yescharacteristic
90#1803C2212yes—
91#1802C2214yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94#1762C2214yes—
95C4×#103C2212yes—
96C22×Q16C2312yes—
97C22×SD16C2314yes—
98#559—28no—
99#559—28no—
100Q82—24no—
101Q82D418yescomplemented
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×#127—21no—
108C2×#127—21no—
109C2×#127—21no—
110C2×#127—21no—
111C2×#127C2312yes—
112C2×#127C2312yes—
113C2×#127C2312yes—
114C2×#124—22no—
115C2×#124C2314yes—
116C2×#124C2314yes—
117C2×#123—24no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—22no—
121C2×C4×Q8—21no—
122C2×C4×Q8D411yescharacteristic
123C2×C4×Q8D411yescharacteristic
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2311yescharacteristic
128C2×C4×D4—22no—
129C2×C4×D4—22no—
130C2×C4×D4—22no—
131C2×C4×D4C2314yes—
132#489—216no—
133#486—28no—
134#477—28no—
135#474—216no—
136#464—216no—
137#460—28no—
138#446—28no—
139#440—216no—
140C4×Q16—28no—
141C2×#107C2314yes—
142C2×#107C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2314yes—
146C2×C4⋊C8C2312yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#97C2314yes—
161C2×C4×C8C2312yes—
162#394—24no—
163#394—24no—
164#394—24no—
165#391—22no—
166#391—21no—
167#391—22no—
168#391—22no—
169#391—22no—
170#391C2312yes—
171#391C2312yes—
172#391C2311yescharacteristic
173#391C2311yescharacteristic
174#390—22no—
175#390—22no—
176#390—22no—
177#390—22no—
178#390—22no—
179#390—22no—
180#390—22no—
181#390—22no—
182#390C2314yes—
183#389—21no—
184#389—22no—
185#389—21no—
186#389—22no—
187#389—21no—
188#389—22no—
189#389—21no—
190#389—21no—
191#389C2311yescharacteristic
192#389C2312yes—
193#389C2312yes—
194#389C2311yescharacteristic
195#387—22no—
196#387—21no—
197#387—22no—
198#387—21no—
199#387—22no—
200#387—22no—
201#387—22no—
202#386—24no—
203#386—24no—
204#384C2311yescharacteristic
205#384C2312yes—
206#384C2311yescharacteristic
207#382—21no—
208#382—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D412yes—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4D412yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2312yes—
230#376—22no—
231#376—21no—
232C43C2312yes—
233C43C2311yescharacteristic
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2311yescharacteristic
240#336C2314yes—
241#336C2312yes—
242C22×Q8—22no—
243C22×Q8—21no—
244C22×Q8C2×D411yescharacteristic
245C22×Q8C2412yes—
246C22×Q8C2411yescharacteristic
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4C2414yes—
254C23×C4—24no—
255C2×Q16—24no—
256C2×Q16—42no—
257C2×Q16—42no—
258C2×SD16—28no—
259C2×SD16—44no—
260C2×SD16—44no—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C4⋊Q8—28no—
265C4⋊Q8—216no—
266C4⋊Q8—24no—
267C4⋊Q8—22no—
268C4⋊Q8—42no—
269C4⋊Q8C2×D418yes—
270C4⋊Q8C2×D414yes—
271#127—24no—
272#127—24no—
273#127—44no—
274#127—44no—
275#127—44no—
276#127—44no—
277#127—82no—
278#127—82no—
279#124—28no—
280#124—28no—
281#124—44no—
282#124—44no—
283#124—48no—
284#123—416no—
285C4×Q8—28no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—22no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—42no—
292C4×Q8—44no—
293C4×Q8—84no—
294C4×Q8—82no—
295C4×Q8C2×D414yes—
296C4×Q8C2×D418yes—
297C4×Q8C2×D414yes—
298C4×Q8#4114yes—
299C4×D4—28no—
300C4×D4—28no—
301C4×D4—48no—
302C4×D4—84no—
303C4×D4—84no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361#107—24no—
362#107—28no—
363C4⋊C8—24no—
364C4⋊C8—28no—
365C4⋊C8—44no—
366C4⋊C8—42no—
367#103—24no—
368#103—24no—
369#103—28no—
370#103—42no—
371#103—44no—
372#103—42no—
373#102—28no—
374#102—28no—
375#102—28no—
376#102—44no—
377#102—44no—
378#102—44no—
379#102—44no—
380#97—28no—
381C4×C8—24no—
382#95—21no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—21no—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402C24—24no—
403C2×Q8—28no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—82no—
417C2×Q8—81no—
418C2×Q8—82no—
419C2×Q8—81no—
420C2×Q8#12114yes—
421C2×Q8C22×D412yes—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4—82no—
445C2×D4—82no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—48no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2511yescharacteristic
468Q16—48no—
469SD16—416no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C4⋊C4—28no—
480C4⋊C4—24no—
481C4⋊C4—28no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—82no—
504C4⋊C4—81no—
505C4⋊C4—82no—
506C4⋊C4—81no—
507C4⋊C4#12114yes—
508C4⋊C4#12118yes—
509C4⋊C4C22×D414yes—
510C4⋊C4C22×D412yes—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C42—28no—
519C42—24no—
520C42—24no—
521C42—22no—
522C42—24no—
523C42—22no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—42no—
528C42—42no—
529C42—41no—
530C42—44no—
531C42—44no—
532C42—42no—
533C42—82no—
534C42—82no—
535C42—82no—
536C42#13714yes—
537C42C22×D412yes—
538C42C22×D414yes—
539C42C2×#4114yes—
540C42C2×#4112yes—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23C23×D411yescharacteristic
549Q8—28no—
550Q8—24no—
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—88no—
557Q8—84no—
558D4—44no—
559D4—44no—
560D4—44no—
561D4—44no—
562D4—88no—
563D4—88no—
564D4—88no—
565D4—164no—
566D4—164no—
567D4—164no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—168no—
614C2×C4C2×#12114yes—
615C2×C4C2×#12114yes—
616C2×C4C2×#12112yes—
617C2×C4C2×#12112yes—
618C2×C4C2×#13712yes—
619C2×C4C2×#13712yes—
620C2×C4C2×#13712yes—
621C2×C4#57512yes—
622C2×C4#57612yes—
623C2×C4C23×D411yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4#58512yes—
626C2×C4#58512yes—
627C8—42no—
628C8—44no—
629C8—42no—
630C22—84no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22C22×#12111yescharacteristic
638C22#348711yescharacteristic
639C22#350011yescharacteristic
640C22C22×#13711yescharacteristic
641C22C2×#57511yescharacteristic
642C22C2×#57611yescharacteristic
643C22#358511yescharacteristic
644C4—24no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—24no—
650C4—42no—
651C4—44no—
652C4—42no—
653C4—42no—
654C4—41no—
655C4—42no—
656C4—82no—
657C4—81no—
658C4—82no—
659C4—82no—
660C4—81no—
661C4#303714yes—
662C4#348712yes—
663C4#352412yes—
664C2—164no—
665C2#3601811yescharacteristic
666C2C2×#303711yescharacteristic
667C2#6042811yescharacteristic
668C2#6189211yescharacteristic
669C2#6189811yescharacteristic
670C2#6253711yescharacteristic
671C2#6258011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2867C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3375C2semidirect
3. #34335C2semidirect
4. #29801C2semidirect
6. #28753C2semidirect
8. #28206C2semidirect
9. #28201C2semidirect
10. #25285C2semidirect
11. #25277C2semidirect
12. #24639C2semidirect
13. #24633C2semidirect
14. #23684C2semidirect
15. #23679C2semidirect
16. #22736C2semidirect
17. #22546C2semidirect
101. Q82D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33752
#365952
#343352
#298012
#297962
#287532
#287492
#282062
#282012
#252852
#252772
#246392
#246332
#236842
#236792
#227362
#225462