Finite Group Databasebeta

← #6565044

Subgroups — #6565044

Order 512.

1649 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36187C211yesmaximal, characteristic
2#34332C211yesmaximal, characteristic, complemented
3#30115C212yesmaximal, complemented
4#29984C211yesmaximal, characteristic, complemented
5#29334C212yesmaximal, complemented
6#29218C211yesmaximal, characteristic, complemented
7#28202C211yesmaximal, characteristic, complemented
8#27613C212yesmaximal, complemented
9#27343C214yesmaximal, complemented
10#26741C214yesmaximal, complemented
11#24965C212yesmaximal, complemented
12#24118C212yesmaximal, complemented
13#23762C212yesmaximal, complemented
14#23678C211yesmaximal, characteristic, complemented
15#23067C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic, complemented
17#22759C212yesmaximal, complemented
18C2×#555—21no—
19C2×#554C2211yescharacteristic
20C2×#476C2212yes—
21C2×C4×SD16C2212yes—
22#2868C2211yescharacteristic
23#2856C2212yes—
24#2563—21no—
25#2537—21no—
26#2537C2212yes—
27#2500—22no—
28#2491—22no—
29#2491C2214yes—
30#2370—21no—
31#2369C2211yescharacteristic
32#2360—22no—
33#2360C2212yes—
34#2354—21no—
35#2354C2212yes—
36Q8×C4⋊C4C2211yescharacteristic
37D4×C4⋊C4—21no—
38D4×C4⋊C4C2211yescharacteristic
39#2345C2212yes—
40#2330—21no—
41#2329—22no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×#127C2211yescharacteristic
44C4×C22⋊Q8C2212yes—
45#2295C2211yescharacteristic
46C2×#389C2212yes—
47#2285C2212yes—
48#2275—21no—
49C42×D4C2211yescharacteristic
50#2100C2212yes—
51#2100C2212yes—
52#2100C2212yes—
53#2095C2212yes—
54#2084C2212yes—
55#2084C2211yescharacteristic
56#2075C2214yes—
57#2072C2211yescharacteristic
58#2065C2212yes—
59#2057C2212yes—
60#2055C2212yes—
61#2041C2212yes—
62#2041C2211yescharacteristic
63#2035C2214yes—
64#2033C2214yes—
65#2024C2212yes—
66#2014C2211yescharacteristic
67#2000C2214yes—
68#1990C2212yes—
69#1987C2214yes—
70#1954C2212yes—
71#1947C2212yes—
72#1936C2214yes—
73#1936C2214yes—
74#1926C2214yes—
75#1925C2214yes—
76#1924C2214yes—
77#1919C2214yes—
78#1919C2214yes—
79#1880C2214yes—
80#1874C2214yes—
81#1874C2214yes—
82#1853C2214yes—
83#1834C2212yes—
84#1831C2211yescharacteristic
85#1823C2214yes—
86#1817C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#102C2212yes—
94#1749C2212yes—
95C22×SD16C2312yes—
96#555—48no—
97#554—24no—
98#554—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#125—21no—
105C2×#124—24no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2312yes—
108C2×#121—22no—
109C2×#121—22no—
110C2×#121—22no—
111C2×#121C2314yes—
112C2×C4×Q8—21no—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C2×#117—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C42C2312yes—
129#476—28no—
130C4×SD16—28no—
131C2×#107C2314yes—
132C2×#106C2312yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#102C2312yes—
139C2×#102C2314yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×#97C2314yes—
146C2×C4×C8C2312yes—
147#394—24no—
148#394—24no—
149#393—24no—
150#391—21no—
151#391—22no—
152#391C2311yescharacteristic
153#391C2311yescharacteristic
154#390—21no—
155#390—21no—
156#390—22no—
157#390—22no—
158#390C2312yes—
159#390C2314yes—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2311yescharacteristic
169#389C2311yescharacteristic
170#389C2312yes—
171#389C2312yes—
172#388—24no—
173#388—24no—
174#387—21no—
175#387—22no—
176#387—21no—
177#387—22no—
178#387—22no—
179#387—22no—
180#385—22no—
181#385—22no—
182#385—22no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#382—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380C2312yes—
196#380C2314yes—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216#376—21no—
217C2×#95—22no—
218C43C2311yescharacteristic
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2314yes—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8C2411yescharacteristic
236C22×D4—24no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4C2×D411yescharacteristic
240C22×D4C2412yes—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2412yes—
247C2×SD16—24no—
248C2×SD16—42no—
249C2×SD16—42no—
250C22×C8C2412yes—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C4⋊Q8—22no—
254C4⋊Q8—44no—
255C4⋊Q8—44no—
256C4⋊Q8—82no—
257C4⋊Q8#4114yes—
258#127—42no—
259#125—28no—
260#124—416no—
261C22⋊Q8—216no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264#121—48no—
265#121—48no—
266#121—44no—
267#121—44no—
268#121—84no—
269C4×Q8—22no—
270C4×Q8—44no—
271C4×Q8—82no—
272C4×Q8#4114yes—
273C4×D4—28no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—44no—
283#117—48no—
284#117—48no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—84no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—21no—
332C2×C42—24no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—44no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—44no—
349#106—24no—
350#106—28no—
351C4⋊C8—24no—
352C4⋊C8—42no—
353#103—24no—
354#103—44no—
355#102—24no—
356#102—28no—
357#102—42no—
358#102—42no—
359#102—44no—
360C22⋊C8—44no—
361C22⋊C8—44no—
362#97—44no—
363C4×C8—24no—
364#95—21no—
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—22no—
374#95—21no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382C24—22no—
383C24C22×D412yes—
384C2×Q8—24no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—44no—
389C2×Q8—81no—
390C2×Q8—81no—
391C2×Q8—82no—
392C2×Q8—82no—
393C2×Q8C2×#4112yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—48no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—84no—
410C2×D4—84no—
411C2×D4AGL(1,ℤ/8)14yes—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2511yescharacteristic
449SD16—48no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—81no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4C2×#4112yes—
492C4⋊C4C2×#4114yes—
493C22⋊C4—28no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—48no—
498C22⋊C4—44no—
499C22⋊C4—48no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C42—24no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—42no—
512C42—42no—
513C42—41no—
514C42—48no—
515C42—42no—
516C42—82no—
517C42—82no—
518C42—82no—
519C42#13714yes—
520C42C2×#4112yes—
521C42C2×#4112yes—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23C2×AGL(1,ℤ/8)14yes—
536C23C23×D411yescharacteristic
537Q8—24no—
538Q8—42no—
539Q8—84no—
540Q8—164no—
541D4—28no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—42no—
546D4—84no—
547D4—84no—
548D4—168no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—84no—
597C2×C4—88no—
598C2×C4—88no—
599C2×C4—164no—
600C2×C4#52512yes—
601C2×C4#52512yes—
602C2×C4C2×AGL(1,ℤ/8)12yes—
603C2×C4C2×#13712yes—
604C2×C4C2×#13712yes—
605C2×C4C2×#13712yes—
606C2×C4#57712yes—
607C2×C4C22×#4111yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4#58512yes—
610C2×C4#58512yes—
611C8—42no—
612C8—42no—
613C8—44no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C2×#52511yescharacteristic
629C22#348511yescharacteristic
630C22#350011yescharacteristic
631C22C22×AGL(1,ℤ/8)11yescharacteristic
632C22C22×#13711yescharacteristic
633C22C2×#57711yescharacteristic
634C22#358511yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—42no—
640C4—42no—
641C4—41no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—44no—
646C4—82no—
647C4—81no—
648C4—81no—
649C4—84no—
650C4—82no—
651C4#311214yes—
652C4#349312yes—
653C4#352412yes—
654C2—44no—
655C2—162no—
656C2#3601111yescharacteristic
657C2C2×#311211yescharacteristic
658C2#6059711yescharacteristic
659C2#6172111yescharacteristic
660C2#6172811yescharacteristic
661C2#6254211yescharacteristic
662C2#6257611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
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. #34332C2semidirect
3. #30115C2semidirect
4. #29984C2semidirect
5. #29334C2semidirect
6. #29218C2semidirect
7. #28202C2semidirect
8. #27613C2semidirect
9. #27343C2semidirect
10. #26741C2semidirect
11. #24965C2semidirect
12. #24118C2semidirect
13. #23762C2semidirect
14. #23678C2semidirect
15. #23067C2semidirect
16. #22839C2semidirect
17. #22759C2semidirect

Maximal subgroups

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

subgroupindex
#361872
#343322
#301152
#299842
#293342
#292182
#282022
#276132
#273432
#267412
#249652
#241182
#237622
#236782
#230672
#228392
#227592