Finite Group Databasebeta

← #6572058

Subgroups — #6572058

Order 512.

1583 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36272C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#30094C212yesmaximal, complemented
4#29688C211yesmaximal, characteristic, complemented
5#29368C212yesmaximal, complemented
6#29123C212yesmaximal, complemented
7#27961C211yesmaximal, characteristic, complemented
8#27678C211yesmaximal, characteristic
9#27098C214yesmaximal, complemented
10#26660C214yesmaximal
11#26108C212yesmaximal, complemented
12#24813C212yesmaximal, complemented
13#23775C212yesmaximal, complemented
14#23568C211yesmaximal, characteristic
15#23062C212yesmaximal, complemented
16#22770C212yesmaximal, complemented
17#22639C211yesmaximal, characteristic, complemented
18C2×#552—21no—
19C2×#533C2211yescharacteristic
20C2×#475C2212yes—
21C2×C4×SD16C2212yes—
22#2859C2212yes—
23#2857C2211yescharacteristic
24#2611—22no—
25#2553—22no—
26#2552C2211yescharacteristic
27#2542C2211yescharacteristic
28#2496C2212yes—
29#2489—21no—
30#2422C2212yes—
31#2415—21no—
32#2365C2211yescharacteristic
33#2360—21no—
34#2360—22no—
35#2360C2214yes—
36#2355—21no—
37#2355C2212yes—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4C2212yes—
40D4×C4⋊C4C2211yescharacteristic
41#2330—22no—
42#2317—21no—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×#121C2211yescharacteristic
46#2293C2212yes—
47C2×#384C2211yescharacteristic
48#2274—21no—
49C4×#117C2211yescharacteristic
50#2100C2212yes—
51#2100C2212yes—
52#2100C2212yes—
53#2099C2214yes—
54#2094C2212yes—
55#2088C2214yes—
56#2084C2214yes—
57#2065C2214yes—
58#2058C2211yescharacteristic
59#2058C2212yes—
60#2043C2211yescharacteristic
61#2043C2212yes—
62#2035C2212yes—
63#2035C2214yes—
64#2023C2211yescharacteristic
65#2013C2211yescharacteristic
66#2000C2212yes—
67#1990C2212yes—
68#1948C2212yes—
69#1939C2212yes—
70#1933C2214yes—
71#1933C2214yes—
72#1927C2214yes—
73#1923C2214yes—
74#1919C2214yes—
75#1919C2214yes—
76#1880C2214yes—
77#1877C2214yes—
78#1874C2212yes—
79#1865C2214yes—
80#1853C2212yes—
81#1839C2212yes—
82#1834C2212yes—
83#1824C2214yes—
84#1803C2214yes—
85#1803C2214yes—
86#1803C2212yes—
87#1802C2212yes—
88#1797C2211yescharacteristic
89C4×#106C2214yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2212yes—
92C4×#103C2212yes—
93C4×#102C2214yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×SD16C2312yes—
96#552—48no—
97#533—28no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#126—22no—
102C2×#125—21no—
103C2×#125—21no—
104C2×#125—22no—
105C2×#123—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8C2312yes—
111C2×#121C2311yescharacteristic
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2314yes—
117C2×C4×D4—21no—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2312yes—
120C2×#117—21no—
121C2×#117—22no—
122C2×#117D411yescharacteristic
123C2×#117D411yescharacteristic
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2312yes—
128C22×C42C2311yescharacteristic
129#475—28no—
130C4×SD16—28no—
131C2×#106C2314yes—
132C2×#106C2312yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2314yes—
147#400—22no—
148#399—22no—
149#398—22no—
150#397—22no—
151#397—22no—
152#394—22no—
153#391—21no—
154#391—22no—
155#391—21no—
156#391C2314yes—
157#390C2312yes—
158#390C2311yescharacteristic
159#389—21no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389—24no—
167#389—24no—
168#389C2314yes—
169#389C2312yes—
170#389C2311yescharacteristic
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#388—21no—
175#388—22no—
176#388—22no—
177#387—24no—
178#387—24no—
179#387—22no—
180#387—22no—
181#386—21no—
182#385—24no—
183#385—22no—
184#385—22no—
185#384—22no—
186#384—21no—
187#384—21no—
188#384C2312yes—
189#384C2312yes—
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2312yes—
193#382—22no—
194#382—22no—
195#380—21no—
196#380—21no—
197#380C2312yes—
198#380C2311yescharacteristic
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—24no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216#376D412yes—
217C43C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2314yes—
230#336C2314yes—
231#336C2311yescharacteristic
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8C2412yes—
235C22×D4—21no—
236C22×D4C2411yescharacteristic
237C23×C4—22no—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×SD16—24no—
244C2×SD16—24no—
245C2×SD16—42no—
246C22×C8C2412yes—
247C22×C8C2414yes—
248C22×C8C2412yes—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252#126—48no—
253#125—28no—
254#125—44no—
255#125—48no—
256#123—48no—
257C22⋊Q8—28no—
258C22⋊Q8—24no—
259C22⋊Q8—24no—
260C22⋊Q8—44no—
261C22⋊Q8—44no—
262C22⋊Q8—44no—
263#121—22no—
264#121—41no—
265C4×Q8—28no—
266C4×Q8—24no—
267C4×Q8—44no—
268C4×Q8—44no—
269C4×Q8—44no—
270C4×Q8—84no—
271C4×Q8#4118yes—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—42no—
275C4×D4—44no—
276#117—28no—
277#117—42no—
278#117—48no—
279#117SD1618yescomplemented
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#106—24no—
348#106—28no—
349#106—44no—
350C4⋊C8—24no—
351C4⋊C8—44no—
352#103—24no—
353#103—28no—
354#103—44no—
355#103—42no—
356#103—42no—
357#102—24no—
358#102—44no—
359C22⋊C8—42no—
360C22⋊C8—42no—
361C4×C8—24no—
362C4×C8—44no—
363#95—24no—
364#95—24no—
365#95—22no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—21no—
381C24C22×D411yescharacteristic
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8C2×#4114yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—81no—
400C2×D42−1+412yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—41no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4C2×SD1614yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2511yescharacteristic
438SD16—48no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—28no—
451C4⋊C4—24no—
452C4⋊C4—28no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—81no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4C2×SD1614yes—
485C4⋊C4C2×SD1614yes—
486C4⋊C4C2×#4114yes—
487C22⋊C4—28no—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—42no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4C2×SD1618yes—
499C42—22no—
500C42—22no—
501C42—28no—
502C42—24no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—24no—
507C42—42no—
508C42—42no—
509C42—44no—
510C42—42no—
511C42—84no—
512C42—84no—
513C42C2×SD1618yes—
514C42C2×#4114yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23C22×SD1612yes—
529C23C23×D411yescharacteristic
530Q8—24no—
531Q8—24no—
532Q8—48no—
533Q8—42no—
534Q8—42no—
535Q8—84no—
536D4—24no—
537D4—42no—
538D4—84no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—88no—
587C2×C4—84no—
588C2×C4—164no—
589C2×C4#52512yes—
590C2×C4#52512yes—
591C2×C4C22×SD1614yes—
592C2×C4C22×SD1612yes—
593C2×C4C22×SD1614yes—
594C2×C4C22×SD1612yes—
595C2×C4#57512yes—
596C2×C4#57812yes—
597C2×C4C22×#4112yes—
598C2×C4C2×2−1+411yescharacteristic
599C2×C4#58512yes—
600C2×C4#58511yescharacteristic
601C8—42no—
602C8—42no—
603C8—44no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22C2×#52511yescharacteristic
619C22#348911yescharacteristic
620C22#350011yescharacteristic
621C22C23×SD1611yescharacteristic
622C22C2×#57511yescharacteristic
623C22C2×#57811yescharacteristic
624C22#358511yescharacteristic
625C4—24no—
626C4—22no—
627C4—24no—
628C4—22no—
629C4—22no—
630C4—42no—
631C4—44no—
632C4—42no—
633C4—41no—
634C4—44no—
635C4—81no—
636C4—81no—
637C4—82no—
638C4—84no—
639C4—82no—
640C4—81no—
641C4#310314yes—
642C4#349512yes—
643C4#351212yes—
644C2—42no—
645C2—82no—
646C2#3608011yescharacteristic
647C2C2×#310311yescharacteristic
648C2#6058911yescharacteristic
649C2#6209611yescharacteristic
650C2#6213311yescharacteristic
651C2#6252611yescharacteristic
652C2#6259011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34214C2semidirect
3. #30094C2semidirect
4. #29688C2semidirect
5. #29368C2semidirect
6. #29123C2semidirect
7. #27961C2semidirect
9. #27098C2semidirect
11. #26108C2semidirect
12. #24813C2semidirect
13. #23775C2semidirect
15. #23062C2semidirect
16. #22770C2semidirect
17. #22639C2semidirect
279. #117SD16semidirect

Maximal subgroups

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

subgroupindex
#362722
#342142
#300942
#296882
#293682
#291232
#279612
#276782
#270982
#266602
#261082
#248132
#237752
#235682
#230622
#227702
#226392