Finite Group Databasebeta

← #6566088

Subgroups — #6566088

Order 512.

1825 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3340C212yesmaximal
2#36188C211yesmaximal, characteristic, complemented
3#34321C211yesmaximal, characteristic
4#29864C211yesmaximal, characteristic, complemented
5#29759C212yesmaximal, complemented
6#28687C211yesmaximal, characteristic, complemented
7#28571C212yesmaximal, complemented
8#28206C211yesmaximal, characteristic, complemented
9#27650C212yesmaximal, complemented
10#25454C212yesmaximal, complemented
11#25211C214yesmaximal, complemented
12#24816C212yesmaximal, complemented
13#24584C214yesmaximal, complemented
14#23677C211yesmaximal, characteristic, complemented
15#23656C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic, complemented
17#22490C212yesmaximal, complemented
18C2×#556C2212yes—
19C2×#555C2211yescharacteristic
20#3340—216no—
21C2×#498C2212yes—
22C2×#485C2214yes—
23C2×#482C2214yes—
24C2×#477C2212yes—
25C2×#474C2212yes—
26C2×#465C2214yes—
27C2×C4×SD16C2212yes—
28C2×C8×D4C2212yes—
29C2×#427C2214yes—
30#2868C2211yescharacteristic
31#2857C2212yes—
32#2568C2212yes—
33#2547—21no—
34#2547—21no—
35#2499C2214yes—
36#2496—22no—
37#2496—22no—
38#2369C2211yescharacteristic
39#2367—22no—
40#2365—21no—
41#2365—21no—
42#2363C2212yes—
43#2361—21no—
44#2354C2211yescharacteristic
45D4×C4⋊C4C2211yescharacteristic
46#2347—21no—
47#2347C2212yes—
48#2337—22no—
49#2324—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×#121C2212yes—
53#2296C2212yes—
54#2293C2211yescharacteristic
55#2285C2212yes—
56#2274—21no—
57C42×D4C2211yescharacteristic
58#2085C2214yes—
59#2085C2214yes—
60#2084C2212yes—
61#2084C2212yes—
62#2084C2211yescharacteristic
63#2083C2212yes—
64#2083C2212yes—
65#2083C2212yes—
66#2071C2211yescharacteristic
67#2071C2212yes—
68#2071C2212yes—
69#2068C2214yes—
70#2066C2212yes—
71#2066C2214yes—
72#2055C2212yes—
73#2043C2212yes—
74#2043C2211yescharacteristic
75#2013C2211yescharacteristic
76#1990C2212yes—
77#1988C2212yes—
78#1949C2214yes—
79#1948C2214yes—
80#1939C2212yes—
81#1928C2214yes—
82#1927C2214yes—
83#1892C2214yes—
84#1877C2214yes—
85#1873C2214yes—
86#1850C2214yes—
87#1831C2211yescharacteristic
88#1825C2214yes—
89#1804C2214yes—
90#1803C2212yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94#1735C2214yes—
95C2×#336C2212yes—
96C22×SD16C2312yes—
97C23×C8C2314yes—
98#556—28no—
99#556—28no—
100#555—24no—
101#555D418yes—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#126—24no—
108C2×#124—22no—
109C2×#124—22no—
110C2×C22⋊Q8C2314yes—
111C2×#121—21no—
112C2×#121C2312yes—
113C2×C4×Q8—22no—
114C2×C4×Q8—21no—
115C2×C4×Q8C2312yes—
116C2×C4×D4—22no—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2311yescharacteristic
122C2×#117—21no—
123C2×#117—21no—
124C2×#117C2314yes—
125C2×#117C2312yes—
126C2×#117C2312yes—
127C22×C4⋊C4—21no—
128C22×C42C2312yes—
129#498—28no—
130#485—216no—
131#482—216no—
132#477—28no—
133#474—28no—
134#465—216no—
135C4×SD16—28no—
136C8×D4—28no—
137#427—216no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2314yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C4×C8C2312yes—
155#400—24no—
156#397—24no—
157#397—24no—
158#391—22no—
159#391—22no—
160#391—21no—
161#391—22no—
162#391C2312yes—
163#391C2314yes—
164#390—21no—
165#390—22no—
166#390C2311yescharacteristic
167#390C2311yescharacteristic
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389C2312yes—
175#389C2311yescharacteristic
176#389C2312yes—
177#389C2312yes—
178#389C2311yescharacteristic
179#388—22no—
180#388—22no—
181#388—22no—
182#388—21no—
183#388—22no—
184#388—21no—
185#388—22no—
186#387—24no—
187#387—24no—
188#386—22no—
189#386—22no—
190#385—22no—
191#385—22no—
192#385—22no—
193#384—21no—
194#384—21no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2312yes—
199#383—22no—
200#382C2314yes—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380—21no—
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—22no—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4C2312yes—
229C43C2311yescharacteristic
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2312yes—
240#336C2311yescharacteristic
241#336C2314yes—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8C2412yes—
245C22×D4—22no—
246C22×D4C2×D411yescharacteristic
247C22×D4C2411yescharacteristic
248C23×C4—21no—
249C23×C4—22no—
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2412yes—
253C2×SD16—24no—
254C2×SD16—24no—
255C2×SD16—42no—
256C22×C8—28no—
257C22×C8—28no—
258C22×C8—44no—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C4⋊Q8—24no—
263C4⋊Q8—22no—
264C4⋊Q8—24no—
265C4⋊Q8—42no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8C2×D414yes—
269#126—416no—
270#124—216no—
271#124—48no—
272C22⋊Q8—28no—
273C22⋊Q8—28no—
274#121—28no—
275#121—24no—
276#121C2×D418yes—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—48no—
280C4×Q8—82no—
281C4×D4—24no—
282C4×D4—22no—
283C4×D4—42no—
284C4×D4—44no—
285C4×D4—84no—
286C4×D4C2×D414yes—
287C4×D4C2×D414yes—
288C4×D4#4114yes—
289#117—24no—
290#117—28no—
291#117—28no—
292#117—28no—
293#117—44no—
294#117—44no—
295#117C2×D418yes—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—44no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—28no—
360#107—28no—
361#106—24no—
362#106—28no—
363#106—28no—
364C4⋊C8—24no—
365C4⋊C8—24no—
366C4⋊C8—44no—
367#103—24no—
368#103—24no—
369#103—28no—
370#103—42no—
371#103—44no—
372#102—24no—
373#102—28no—
374C22⋊C8—28no—
375C4×C8—24no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—24no—
390#95—24no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—21no—
395C24C22×D412yes—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—81no—
407C2×Q8—81no—
408C2×D4—28no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—82no—
415C2×D4—82no—
416C2×D4#12114yes—
417C2×D4C22×D412yes—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D414yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C2×#4114yes—
451C22×C4C2511yescharacteristic
452SD16—48no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—416no—
462C2×C82−1+414yes—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—82no—
492C4⋊C4—81no—
493C4⋊C4#12114yes—
494C4⋊C4C22×D414yes—
495C4⋊C4C22×D412yes—
496C22⋊C4—28no—
497C22⋊C4—28no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—81no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C22⋊C4#12118yes—
511C42—28no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—24no—
516C42—22no—
517C42—42no—
518C42—42no—
519C42—41no—
520C42—48no—
521C42—44no—
522C42—42no—
523C42—82no—
524C42—82no—
525C42—82no—
526C42#13514yes—
527C42C22×D412yes—
528C42C2×#4112yes—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23C2×#12114yes—
543C23C23×D411yescharacteristic
544Q8—42no—
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—84no—
549Q8—88no—
550D4—24no—
551D4—28no—
552D4—42no—
553D4—88no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—48no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—164no—
603C2×C4C2×#12114yes—
604C2×C4C2×#12112yes—
605C2×C4C2×#12112yes—
606C2×C4C2×#13512yes—
607C2×C4C2×#13512yes—
608C2×C4C2×#13512yes—
609C2×C4#57512yes—
610C2×C4#57712yes—
611C2×C4C23×D411yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C2×2−1+412yes—
614C2×C4#58512yes—
615C8—24no—
616C8—42no—
617C8—44no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22C22×#12111yescharacteristic
633C22#348511yescharacteristic
634C22#349911yescharacteristic
635C22C22×#13511yescharacteristic
636C22C2×#57511yescharacteristic
637C22C2×#57711yescharacteristic
638C22#358611yescharacteristic
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—24no—
644C4—42no—
645C4—44no—
646C4—42no—
647C4—41no—
648C4—42no—
649C4—81no—
650C4—81no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—84no—
655C4#303514yes—
656C4#348912yes—
657C4#351512yes—
658C2—44no—
659C2—82no—
660C2#3602011yescharacteristic
661C2C2×#303511yescharacteristic
662C2#6042911yescharacteristic
663C2#6170111yescharacteristic
664C2#6173911yescharacteristic
665C2#6247011yescharacteristic
666C2#6250711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36188C2semidirect
4. #29864C2semidirect
5. #29759C2semidirect
6. #28687C2semidirect
7. #28571C2semidirect
8. #28206C2semidirect
9. #27650C2semidirect
10. #25454C2semidirect
11. #25211C2semidirect
12. #24816C2semidirect
13. #24584C2semidirect
14. #23677C2semidirect
15. #23656C2semidirect
16. #22696C2semidirect
17. #22490C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33402
#361882
#343212
#298642
#297592
#286872
#285712
#282062
#276502
#254542
#252112
#248162
#245842
#236772
#236562
#226962
#224902