Finite Group Databasebeta

← #6566755

Subgroups — #6566755

Order 512.

1603 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#34334C211yesmaximal, characteristic
3#30116C212yesmaximal, complemented
4#29983C211yesmaximal, characteristic, complemented
5#28512C211yesmaximal, characteristic, complemented
6#28313C212yesmaximal, complemented
7#28161C211yesmaximal, characteristic, complemented
8#27583C212yesmaximal, complemented
9#27241C214yesmaximal, complemented
10#26807C214yesmaximal, complemented
11#25014C212yesmaximal
12#24243C212yesmaximal
13#23763C212yesmaximal, complemented
14#23679C211yesmaximal, characteristic, complemented
15#23102C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic
17#22814C212yesmaximal, complemented
18C2×D4×Q8C2211yescharacteristic
19C2×#483C2214yes—
20C2×#475C2214yes—
21C2×#462C2212yes—
22C2×#451C2214yes—
23C2×C4×Q16C2212yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2571—21no—
27#2563—21no—
28#2538C2212yes—
29#2497—22no—
30#2492C2214yes—
31#2439—22no—
32#2369—21no—
33#2369—21no—
34#2368—22no—
35#2365C2211yescharacteristic
36#2360C2212yes—
37#2355C2211yescharacteristic
38#2355C2212yes—
39#2354—21no—
40#2349—21no—
41D4×C4⋊C4C2211yescharacteristic
42Q8×C22⋊C4C2212yes—
43#2331—21no—
44#2329—22no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48#2295C2211yescharacteristic
49C2×#391C2212yes—
50#2285C2212yes—
51#2275—21no—
52C42×D4C2211yescharacteristic
53#2099C2212yes—
54#2096C2212yes—
55#2096C2212yes—
56#2096C2212yes—
57#2083C2211yescharacteristic
58#2077C2214yes—
59#2072C2212yes—
60#2072C2211yescharacteristic
61#2068C2212yes—
62#2058C2212yes—
63#2056C2212yes—
64#2043C2211yescharacteristic
65#2036C2214yes—
66#2035C2214yes—
67#2024C2212yes—
68#2016C2212yes—
69#2016C2211yescharacteristic
70#2001C2214yes—
71#1971C2214yes—
72#1971C2214yes—
73#1956C2212yes—
74#1954C2212yes—
75#1948C2212yes—
76#1927C2214yes—
77#1927C2214yes—
78#1927C2214yes—
79#1924C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1853C2214yes—
83#1844C2212yes—
84#1831C2211yescharacteristic
85#1817C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1787C2214yes—
89#1776C2214yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#103C2212yes—
94#1749C2212yes—
95C23×Q8C2312yes—
96C22×Q16C2312yes—
97C22×Q16C2314yes—
98D4×Q8—24no—
99D4×Q8—24no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2314yes—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×#117—22no—
119C22×C4⋊C4C2312yes—
120C22×C42C2312yes—
121#483—48no—
122#475—48no—
123#462—28no—
124#451—48no—
125C4×Q16—28no—
126C2×#107C2312yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143C2×#97C2314yes—
144C2×C4×C8C2312yes—
145#400—24no—
146#396—24no—
147#393—24no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391—24no—
153#391C2312yes—
154#391C2312yes—
155#391C2311yescharacteristic
156#391C2311yescharacteristic
157#391C2312yes—
158#391C2314yes—
159#391C2312yes—
160#390—22no—
161#389—22no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389C2311yescharacteristic
172#389C2311yescharacteristic
173#388—24no—
174#387—21no—
175#387—22no—
176#387—21no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—24no—
181#385—22no—
182#385—22no—
183#385—22no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#382—22no—
191#382—21no—
192#382—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380C2312yes—
198#380C2314yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4C2312yes—
221#376—21no—
222C2×#95—22no—
223C2×#95—22no—
224C43C2311yescharacteristic
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2314yes—
234#336C2311yescharacteristic
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—44no—
240C22×Q8—44no—
241C22×Q8C2412yes—
242C22×Q8C2412yes—
243C22×Q8C2411yescharacteristic
244C22×D4C2×D411yescharacteristic
245C23×C4—22no—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C2×Q16—24no—
251C2×Q16—28no—
252C2×Q16—24no—
253C2×Q16—42no—
254C2×Q16—44no—
255C2×Q16—44no—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—28no—
260C4⋊Q8—22no—
261C4⋊Q8—44no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264C4⋊Q8—82no—
265C4⋊Q8#4114yes—
266C22⋊Q8—216no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271C4×Q8—22no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8#4114yes—
277C4×D4—28no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—44no—
283#117—84no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—44no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—24no—
342#106—28no—
343#106—44no—
344#106—44no—
345C4⋊C8—24no—
346C4⋊C8—24no—
347#103—24no—
348#103—28no—
349#103—28no—
350#103—24no—
351#103—24no—
352#103—42no—
353#103—44no—
354#103—44no—
355#103—44no—
356C22⋊C8—28no—
357C22⋊C8—44no—
358#97—44no—
359C4×C8—24no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—24no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—22no—
379#95—21no—
380#95—24no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24C22×D412yes—
385C2×Q8—28no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—44no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8—82no—
402C2×Q8—82no—
403C2×Q8—88no—
404C2×Q8—88no—
405C2×Q8C2×#4112yes—
406C2×Q8C2×#4114yes—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4#13714yes—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2511yescharacteristic
445Q16—48no—
446Q16—88no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8H(5,𝔽2)14yes—
456C4⋊C4—28no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—84no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4C2×#4112yes—
490C22⋊C4—28no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—44no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—24no—
502C42—22no—
503C42—42no—
504C42—42no—
505C42—41no—
506C42—44no—
507C42—42no—
508C42—82no—
509C42—84no—
510C42—82no—
511C42—82no—
512C42#13714yes—
513C42C2×#4112yes—
514C42C2×#4112yes—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23C2×#13714yes—
522C23C23×D411yescharacteristic
523Q8—24no—
524Q8—28no—
525Q8—44no—
526Q8—42no—
527Q8—42no—
528Q8—42no—
529Q8—42no—
530Q8—42no—
531Q8—88no—
532Q8—164no—
533D4—28no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—48no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—88no—
586C2×C4—164no—
587C2×C4#52512yes—
588C2×C4#52512yes—
589C2×C4C2×#13712yes—
590C2×C4C2×#13712yes—
591C2×C4C2×#13712yes—
592C2×C4C2×#13712yes—
593C2×C4#57712yes—
594C2×C4C22×#4111yescharacteristic
595C2×C4C22×#4111yescharacteristic
596C2×C4C2×H(5,𝔽2)12yes—
597C2×C4#58512yes—
598C8—24no—
599C8—42no—
600C8—44no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22C2×#52511yescharacteristic
609C22#348511yescharacteristic
610C22#349811yescharacteristic
611C22C22×#13711yescharacteristic
612C22C22×#13711yescharacteristic
613C22C2×#57711yescharacteristic
614C22#358611yescharacteristic
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—42no—
623C4—44no—
624C4—42no—
625C4—41no—
626C4—81no—
627C4—82no—
628C4—82no—
629C4—84no—
630C4—82no—
631C4—81no—
632C4#311314yes—
633C4#349412yes—
634C4#351412yes—
635C2—44no—
636C2#3602211yescharacteristic
637C2C2×#311311yescharacteristic
638C2#6059911yescharacteristic
639C2#6172311yescharacteristic
640C2#6173511yescharacteristic
641C2#6242211yescharacteristic
642C2#6244811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #30116C2semidirect
4. #29983C2semidirect
5. #28512C2semidirect
6. #28313C2semidirect
7. #28161C2semidirect
8. #27583C2semidirect
9. #27241C2semidirect
10. #26807C2semidirect
13. #23763C2semidirect
14. #23679C2semidirect
15. #23102C2semidirect
17. #22814C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#343342
#301162
#299832
#285122
#283132
#281612
#275832
#272412
#268072
#250142
#242432
#237632
#236792
#231022
#228392
#228142