Finite Group Databasebeta

← #6492378

Subgroups — #6492378

Order 512.

1919 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3248C211yesmaximal, characteristic
2C2×#3209C211yesmaximal, characteristic, complemented
3#37617C211yesmaximal, characteristic
4#34235C211yesmaximal, characteristic, complemented
5#33169C212yesmaximal, complemented
6#33041C212yesmaximal, complemented
7#29837C212yesmaximal, complemented
8#29805C212yesmaximal, complemented
9#25370C212yesmaximal, complemented
10#25365C212yesmaximal, complemented
11#25276C212yesmaximal, complemented
12#25275C212yesmaximal, complemented
13#25086C214yesmaximal
14#25082C214yesmaximal, complemented
15#21973C211yesmaximal, characteristic, complemented
16#21388C211yesmaximal, characteristic
17C2×#1981C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#529C2211yescharacteristic
21#3248—28no—
22#3209—28no—
23C2×#494C2212yes—
24C2×#487C2212yes—
25C2×#479C2212yes—
26C2×#475C2211yescharacteristic
27C2×#474C2211yescharacteristic
28C2×#467C2212yes—
29C2×#465C2212yes—
30C2×#464C2212yes—
31C2×#464C2214yes—
32C2×#462C2212yes—
33C2×#462C2211yescharacteristic
34C2×#462C2212yes—
35C2×#462C2214yes—
36C2×#461C2211yescharacteristic
37C2×#451C2214yes—
38C2×#431C2211yescharacteristic
39C22×C4⋊C8C2211yescharacteristic
40C22×#103C2214yes—
41#2821—22no—
42#2658C2212yes—
43#2642C2212yes—
44#2628—21no—
45#2628—21no—
46#2623—22no—
47#2547C2212yes—
48#2546C2212yes—
49#2487—22no—
50#2436C2212yes—
51#2434C2212yes—
52#2365C2212yes—
53#2365C2212yes—
54#2360C2214yes—
55#2356—21no—
56#2351—21no—
57#2343—22no—
58#2336—22no—
59#2320C2212yes—
60C4×C22⋊Q8—22no—
61#2283C2211yescharacteristic
62#2276C2212yes—
63C2×C4×C4⋊C4C2211yescharacteristic
64#2068C2214yes—
65#2066C2214yes—
66#2056C2214yes—
67#2056C2214yes—
68#2043C2214yes—
69#2042C2214yes—
70#2016C2214yes—
71#2015C2214yes—
72#1989C2211yescharacteristic
73#1989C2211yescharacteristic
74#1983C2211yescharacteristic
75#1983C2211yescharacteristic
76#1983C2214yes—
77#1981C2212yes—
78#1981C2212yes—
79#1981C2212yes—
80#1981C2212yes—
81#1981C2214yes—
82#1939C2212yes—
83#1939C2212yes—
84#1928C2214yes—
85#1927C2214yes—
86#1878C2214yes—
87#1878C2214yes—
88#1874C2212yes—
89#1872C2212yes—
90#1850C2214yes—
91#1843C2212yes—
92#1832C2212yes—
93C4×#103C2212yes—
94C4×#102C2212yes—
95C23×Q8C2312yes—
96C22×Q16C2314yes—
97C22×Q16C2314yes—
98C22×Q16C2312yes—
99C22×SD16C2312yes—
100C23×C8C2312yes—
101#533—24no—
102#533D418yes—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—24no—
106C2×C4⋊Q8—24no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2314yes—
111C2×C4⋊Q8C2311yescharacteristic
112#529—24no—
113#529—24no—
114C2×#126—21no—
115C2×#125—21no—
116C2×#125C2311yescharacteristic
117C2×#124C2311yescharacteristic
118C2×#124C2314yes—
119C2×#123—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8C2311yescharacteristic
122C2×C22⋊Q8C2312yes—
123C2×#121C2311yescharacteristic
124C2×C4×Q8—22no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2311yescharacteristic
127C2×C4×Q8C2314yes—
128C2×C4×Q8C2312yes—
129C2×C4×D4—21no—
130C2×C4×D4C2311yescharacteristic
131C2×#117—21no—
132C2×#117—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2312yes—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138#494—28no—
139#487—28no—
140#479—28no—
141#475—24no—
142#474—24no—
143#467—28no—
144#465—28no—
145#464—28no—
146#464—48no—
147#462—24no—
148#462—28no—
149#462—28no—
150#462—48no—
151#461—24no—
152#451—216no—
153#431—24no—
154C2×#107C2312yes—
155C2×#106C2312yes—
156C2×C4⋊C8—24no—
157C2×C4⋊C8C2311yescharacteristic
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2311yescharacteristic
160C2×C4⋊C8C2311yescharacteristic
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2314yes—
163C2×C4⋊C8C2314yes—
164C2×#103—48no—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2312yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#103C2314yes—
171C2×#103C2314yes—
172C2×#103C2312yes—
173C2×#103C2314yes—
174C2×#103C2314yes—
175C2×#103C2312yes—
176C2×#102C2312yes—
177C2×#102C2312yes—
178C2×#102C2312yes—
179C2×C22⋊C8C2312yes—
180C2×C4×C8C2312yes—
181#400—24no—
182#400—22no—
183#400—22no—
184#399—24no—
185#399—22no—
186#399—22no—
187#398—24no—
188#397—24no—
189#395—22no—
190#395—22no—
191#393—22no—
192#393—22no—
193#391—22no—
194#391C2314yes—
195#391C2314yes—
196#391C2314yes—
197#389—22no—
198#389—22no—
199#389—22no—
200#389—24no—
201#389C2314yes—
202#387—24no—
203#387—24no—
204#387—24no—
205#387—24no—
206#386—24no—
207#385—24no—
208#385—24no—
209#383—22no—
210#383D412yes—
211#382—22no—
212#382D411yescharacteristic
213#382D411yescharacteristic
214#381D412yes—
215#380—22no—
216#380—22no—
217#380D412yes—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—24no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2314yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2312yes—
228C4×C22⋊C4—24no—
229C4×C22⋊C4C2312yes—
230#376—22no—
231#376C2312yes—
232C2×#95—21no—
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236#340C2312yes—
237#336C2314yes—
238#336C2314yes—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—24no—
242C22×Q8—24no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8C2411yescharacteristic
246C22×Q8C2414yes—
247C22×Q8C2412yes—
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×Q16—28no—
257C2×Q16—28no—
258C2×Q16—24no—
259C2×Q16—44no—
260C2×Q16—44no—
261C2×Q16—44no—
262C2×Q16—44no—
263C2×Q16—44no—
264C2×SD16—24no—
265C2×SD16—44no—
266C22×C8—24no—
267C22×C8—44no—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276C4⋊Q8—24no—
277C4⋊Q8—24no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282C4⋊Q8C2×D414yes—
283C4⋊Q8C2×D414yes—
284#126—44no—
285#125—28no—
286#125—22no—
287#125—22no—
288#124—22no—
289#124—22no—
290#124—44no—
291#124—44no—
292#123—44no—
293C22⋊Q8—28no—
294C22⋊Q8—24no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297C22⋊Q8C2×D418yes—
298#121—22no—
299#121—22no—
300C4×Q8—28no—
301C4×Q8—24no—
302C4×Q8—28no—
303C4×Q8—22no—
304C4×Q8—22no—
305C4×Q8—44no—
306C4×Q8—84no—
307C4×Q8C2×D418yes—
308C4×D4—22no—
309C4×D4—22no—
310C4×D4—44no—
311#117—28no—
312#117—44no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4C2×D414yes—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—82no—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D414yes—
362C2×C42#4114yes—
363C2×C42C2414yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—24no—
367#106—24no—
368C4⋊C8—22no—
369C4⋊C8—22no—
370C4⋊C8—24no—
371C4⋊C8—22no—
372C4⋊C8—22no—
373C4⋊C8—44no—
374C4⋊C8—44no—
375#103—24no—
376#103—24no—
377#103—28no—
378#103—28no—
379#103—24no—
380#103—28no—
381#103—28no—
382#103—24no—
383#103—44no—
384#103—44no—
385#103—42no—
386#102—24no—
387#102—24no—
388#102—42no—
389C22⋊C8—24no—
390C4×C8—24no—
391#95—21no—
392#95—21no—
393#95—22no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95C2×D412yes—
403#95C2×D414yes—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D414yes—
408C24—21no—
409C24C22×D411yescharacteristic
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—24no—
414C2×Q8—21no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—416no—
425C2×Q8—44no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—84no—
429C2×Q8—84no—
430C2×Q8—84no—
431C2×Q8C22×D414yes—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—42no—
437C2×D4—44no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—82no—
458C22×C4—84no—
459C22×C4#12114yes—
460C22×C4#12114yes—
461C22×C4C22×D414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C2×#4112yes—
470C22×C4C2511yescharacteristic
471Q16—84no—
472Q16—88no—
473Q16—88no—
474SD16—84no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—84no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4C22×D414yes—
515C4⋊C4H(5,𝔽2)14yes—
516C22⋊C4—28no—
517C22⋊C4—28no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C42—28no—
526C42—24no—
527C42—24no—
528C42—48no—
529C42—42no—
530C42—44no—
531C42—81no—
532C42—81no—
533C42—84no—
534C42#12114yes—
535C42#12114yes—
536C42C22×D412yes—
537C42C22×D412yes—
538C42C2×#4112yes—
539C42C2×#4112yes—
540C23—22no—
541C23—21no—
542C23—21no—
543C23—41no—
544C23—42no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23C2×#12112yes—
549C23#57814yes—
550C23C23×D411yescharacteristic
551Q8—24no—
552Q8—24no—
553Q8—44no—
554Q8—48no—
555Q8—42no—
556Q8—44no—
557Q8—44no—
558Q8—41no—
559Q8—44no—
560Q8—44no—
561Q8—41no—
562Q8—88no—
563D4—42no—
564D4—41no—
565D4—41no—
566D4—84no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—88no—
605C2×C4—88no—
606C2×C4—84no—
607C2×C4C2×#12112yes—
608C2×C4C2×#12112yes—
609C2×C4C2×#12112yes—
610C2×C4C2×#12112yes—
611C2×C4C2×#12112yes—
612C2×C4#57712yes—
613C2×C4#57812yes—
614C2×C4C23×D411yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4C2×H(5,𝔽2)12yes—
617C2×C4#58512yes—
618C8—42no—
619C8—42no—
620C8—42no—
621C8—42no—
622C22—22no—
623C22—24no—
624C22—24no—
625C22—22no—
626C22—81no—
627C22—81no—
628C22—82no—
629C22—82no—
630C22—81no—
631C22#304414yes—
632C22C22×#12111yescharacteristic
633C22#348812yes—
634C22C2×#57711yescharacteristic
635C22C2×#57812yes—
636C22C2×#57811yescharacteristic
637C4—22no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—44no—
642C4—44no—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—81no—
647C4—81no—
648C4—82no—
649C4—84no—
650C4—81no—
651C4#304614yes—
652C4#346012yes—
653C4#349811yescharacteristic
654C4#350011yescharacteristic
655C2—24no—
656C2—161no—
657C2#3525611yescharacteristic
658C2C2×#304411yescharacteristic
659C2C2×#304611yescharacteristic
660C2#6205412yes—
661C2#6205612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3209C2semidirect
4. #34235C2semidirect
5. #33169C2semidirect
6. #33041C2semidirect
7. #29837C2semidirect
8. #29805C2semidirect
9. #25370C2semidirect
10. #25365C2semidirect
11. #25276C2semidirect
12. #25275C2semidirect
14. #25082C2semidirect
15. #21973C2semidirect
17. C2×#1981C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32482
C2×#32092
#376172
#342352
#331692
#330412
#298372
#298052
#253702
#253652
#252762
#252752
#250862
#250822
#219732
#213882
C2×#19812