Finite Group Databasebeta

← #6556928

Subgroups — #6556928

Order 512.

1585 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36468C211yesmaximal, characteristic
2#36194C211yesmaximal, characteristic, complemented
3#30357C212yesmaximal, complemented
4#30230C211yesmaximal, characteristic, complemented
5#29313C211yesmaximal, characteristic, complemented
6#29156C212yesmaximal, complemented
7#29043C214yesmaximal, complemented
8#28509C211yesmaximal, characteristic, complemented
9#27857C212yesmaximal, complemented
10#27835C214yesmaximal, complemented
11#25339C212yesmaximal, complemented
12#25339C212yesmaximal, complemented
13#23815C211yesmaximal, characteristic, complemented
14#23755C212yesmaximal, complemented
15#23061C212yesmaximal, complemented
16#22963C212yesmaximal, complemented
17#20555C211yesmaximal, characteristic, complemented
18C2×#553—21no—
19C2×#476C2212yes—
20C2×#476C2212yes—
21C2×C8×D4C2212yes—
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2212yes—
25#2854C2212yes—
26#2602—22no—
27#2571—21no—
28#2563C2212yes—
29#2562—21no—
30#2559—22no—
31#2538—21no—
32#2537C2211yescharacteristic
33#2537C2211yescharacteristic
34#2535—21no—
35#2497C2214yes—
36#2496C2212yes—
37#2492—22no—
38#2439—22no—
39#2428C2212yes—
40#2369C2212yes—
41#2355—21no—
42#2354—21no—
43#2354C2211yescharacteristic
44#2352—21no—
45D4×C4⋊C4C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#121C2212yes—
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51#2100C2214yes—
52#2100C2212yes—
53#2100C2212yes—
54#2099C2212yes—
55#2094C2212yes—
56#2094C2212yes—
57#2084C2212yes—
58#2077C2214yes—
59#2077C2214yes—
60#2071C2211yescharacteristic
61#2071C2211yescharacteristic
62#2058C2212yes—
63#2041C2212yes—
64#2034C2214yes—
65#2034C2214yes—
66#2023C2211yescharacteristic
67#2023C2211yescharacteristic
68#1990C2212yes—
69#1990C2212yes—
70#1957C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1941C2214yes—
74#1941C2214yes—
75#1934C2214yes—
76#1926C2214yes—
77#1920C2214yes—
78#1919C2214yes—
79#1893C2214yes—
80#1893C2214yes—
81#1864C2214yes—
82#1844C2212yes—
83#1807C2214yes—
84#1807C2214yes—
85C4×#106C2214yes—
86C8×C4⋊C4C2212yes—
87C4×C4⋊C8C2211yescharacteristic
88C4×#102C2212yes—
89C4×#102C2212yes—
90C4×#102C2214yes—
91C8×C22⋊C4C2214yes—
92C4×C22⋊C8C2212yes—
93#1744C2211yescharacteristic
94C2×#336C2212yes—
95C23×C8C2314yes—
96#553—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127C2311yescharacteristic
102C2×#126—22no—
103C2×#125—21no—
104C2×#123—22no—
105C2×C22⋊Q8—22no—
106C2×#121—21no—
107C2×#121—21no—
108C2×#121—22no—
109C2×#121C2312yes—
110C2×#121C2312yes—
111C2×C4×Q8—21no—
112C2×C4×D4—22no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2312yes—
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—21no—
123C22×C4⋊C4—21no—
124C22×C42C2312yes—
125#476—28no—
126#476—28no—
127C8×D4—28no—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2312yes—
133C2×#102C2312yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×C22⋊C8C2314yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2314yes—
143#400—22no—
144#400—24no—
145#397—22no—
146#396—24no—
147#394—22no—
148#394—22no—
149#394—22no—
150#393—24no—
151#392—22no—
152#391—21no—
153#391—22no—
154#391—24no—
155#390—22no—
156#390—21no—
157#390—21no—
158#390C2311yescharacteristic
159#390C2311yescharacteristic
160#390C2312yes—
161#389—22no—
162#389—21no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2314yes—
170#389C2312yes—
171#388—24no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#387—22no—
177#387—24no—
178#386—22no—
179#386—22no—
180#386—22no—
181#385—22no—
182#384—21no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#382—22no—
191#382—22no—
192#382—21no—
193#380—22no—
194#380—22no—
195#380—21no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380C2314yes—
200#380C2314yes—
201#380C2312yes—
202#380C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216C43C2311yescharacteristic
217#340C2314yes—
218#340C2312yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2314yes—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4C2×D411yescharacteristic
239C22×D4C2411yescharacteristic
240C22×D4C2411yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2412yes—
247C22×C8—28no—
248C22×C8—44no—
249C22×C8—44no—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—42no—
256C4⋊Q8—82no—
257C4⋊Q8—82no—
258#127—22no—
259#127—41no—
260#126—48no—
261#125—44no—
262#123—48no—
263C22⋊Q8—84no—
264#121—24no—
265#121—42no—
266#121—44no—
267#121—44no—
268#121—48no—
269#121—44no—
270C4×Q8—44no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—44no—
276C4×D4—48no—
277C4×D4—44no—
278C4×D4—44no—
279C4×D4SD1618yes—
280#117—44no—
281#117—84no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—44no—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#106—28no—
345#106—28no—
346#106—44no—
347#106—44no—
348C4⋊C8#4118yes—
349#102—24no—
350#102—28no—
351#102—28no—
352#102—42no—
353#102—42no—
354#102—42no—
355C22⋊C8—28no—
356C4×C8—24no—
357C4×C8—42no—
358C4×C8—44no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95—22no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C24—21no—
375C24—21no—
376C24C22×D412yes—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—44no—
380C2×Q8—42no—
381C2×Q8—82no—
382C2×Q8—82no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—81no—
400C2×D4—81no—
401C2×D4—82no—
402C2×D4—82no—
403C2×D4—82no—
404C2×D4C2×SD1614yes—
405C2×D42−1+412yes—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4C2×SD1618yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2511yescharacteristic
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—88no—
451C2×C8C2×#4118yes—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—84no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4C2×SD1614yes—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—44no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4C2×SD1618yes—
503C42—21no—
504C42—21no—
505C42—24no—
506C42—22no—
507C42—42no—
508C42—42no—
509C42—41no—
510C42—41no—
511C42—44no—
512C42—44no—
513C42—84no—
514C42C2×SD1614yes—
515C42#13514yes—
516C42C2×#4114yes—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C22×SD1614yes—
538C23C23×D411yescharacteristic
539Q8—84no—
540Q8—164no—
541D4—28no—
542D4—24no—
543D4—42no—
544D4—42no—
545D4—42no—
546D4—84no—
547D4—84no—
548D4—88no—
549D4—164no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—164no—
596C2×C4#53212yes—
597C2×C4#53212yes—
598C2×C4C22×SD1614yes—
599C2×C4C22×SD1612yes—
600C2×C4C22×SD1612yes—
601C2×C4C22×SD1612yes—
602C2×C4C2×#13512yes—
603C2×C4C2×#13512yes—
604C2×C4C2×#13512yes—
605C2×C4C22×#4112yes—
606C2×C4C2×2−1+411yescharacteristic
607C2×C4#58512yes—
608C2×C4#58511yescharacteristic
609C8—28no—
610C8—42no—
611C8—42no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C2×#53211yescharacteristic
634C22#348511yescharacteristic
635C22#348911yescharacteristic
636C22C23×SD1611yescharacteristic
637C22C22×#13511yescharacteristic
638C22#358511yescharacteristic
639C22#358511yescharacteristic
640C4—22no—
641C4—24no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—42no—
647C4—42no—
648C4—82no—
649C4—82no—
650C4—84no—
651C4—82no—
652C4—82no—
653C4—81no—
654C4—81no—
655C4#316714yes—
656C4#351612yes—
657C4#352112yes—
658C2—44no—
659C2—82no—
660C2—161no—
661C2#3585611yescharacteristic
662C2C2×#316711yescharacteristic
663C2#6072511yescharacteristic
664C2#6169711yescharacteristic
665C2#6170011yescharacteristic
666C2#6209611yescharacteristic
667C2#6209811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36468C2
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. #36194C2semidirect
3. #30357C2semidirect
4. #30230C2semidirect
5. #29313C2semidirect
6. #29156C2semidirect
7. #29043C2semidirect
8. #28509C2semidirect
9. #27857C2semidirect
10. #27835C2semidirect
11. #25339C2semidirect
12. #25339C2semidirect
13. #23815C2semidirect
14. #23755C2semidirect
15. #23061C2semidirect
16. #22963C2semidirect
17. #20555C2semidirect

Maximal subgroups

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

subgroupindex
#364682
#361942
#303572
#302302
#293132
#291562
#290432
#285092
#278572
#278352
#253392
#238152
#237552
#230612
#229632
#205552