Finite Group Databasebeta

← #6565046

Subgroups — #6565046

Order 512.

1613 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36187C211yesmaximal, characteristic
2#34324C211yesmaximal, characteristic, complemented
3#30108C212yesmaximal, complemented
4#29983C211yesmaximal, characteristic, complemented
5#29337C212yesmaximal, complemented
6#29219C211yesmaximal, characteristic, complemented
7#28201C211yesmaximal, characteristic, complemented
8#27615C212yesmaximal, complemented
9#27350C214yesmaximal, complemented
10#26748C214yesmaximal, complemented
11#24966C212yesmaximal, complemented
12#24119C212yesmaximal, complemented
13#23763C212yesmaximal, complemented
14#23678C211yesmaximal, characteristic, complemented
15#23066C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic, complemented
17#22759C212yesmaximal, complemented
18C2×#556C2211yescharacteristic
19C2×#555—21no—
20C2×#474C2212yes—
21C2×C4×SD16C2212yes—
22#2868C2211yescharacteristic
23#2856C2212yes—
24#2563—21no—
25#2537—21no—
26#2537C2212yes—
27#2500—22no—
28#2491—22no—
29#2491C2214yes—
30#2373—22no—
31#2367—21no—
32#2365C2211yescharacteristic
33#2362C2212yes—
34#2355—21no—
35#2354—21no—
36#2354C2212yes—
37#2354C2211yescharacteristic
38Q8×C4⋊C4C2211yescharacteristic
39#2346C2212yes—
40#2334—22no—
41#2326—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×#127C2211yescharacteristic
44C4×C22⋊Q8C2212yes—
45#2296C2212yes—
46#2295C2211yescharacteristic
47#2285C2212yes—
48#2276—21no—
49C42×D4C2211yescharacteristic
50#2099C2212yes—
51#2099C2212yes—
52#2099C2212yes—
53#2096C2212yes—
54#2084C2211yescharacteristic
55#2083C2212yes—
56#2077C2214yes—
57#2072C2211yescharacteristic
58#2065C2212yes—
59#2058C2212yes—
60#2055C2212yes—
61#2041C2212yes—
62#2041C2211yescharacteristic
63#2036C2214yes—
64#2034C2214yes—
65#2024C2212yes—
66#2014C2211yescharacteristic
67#2000C2214yes—
68#1990C2212yes—
69#1988C2214yes—
70#1954C2212yes—
71#1948C2212yes—
72#1936C2214yes—
73#1936C2214yes—
74#1926C2214yes—
75#1925C2214yes—
76#1924C2214yes—
77#1921C2214yes—
78#1919C2214yes—
79#1880C2214yes—
80#1875C2214yes—
81#1875C2214yes—
82#1854C2214yes—
83#1834C2212yes—
84#1831C2211yescharacteristic
85#1825C2214yes—
86#1817C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#102C2212yes—
94#1749C2212yes—
95C22×SD16C2312yes—
96#556—24no—
97#556—24no—
98#555—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#126—22no—
105C2×#124—21no—
106C2×#124—24no—
107C2×C22⋊Q8C2312yes—
108C2×#121—22no—
109C2×#121—22no—
110C2×#121—22no—
111C2×#121C2314yes—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
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—
125C2×#117C2312yes—
126C2×#117C2312yes—
127C22×C42C2312yes—
128#474—28no—
129C4×SD16—28no—
130C2×#107C2314yes—
131C2×#106C2312yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#102C2312yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144C2×#97C2314yes—
145C2×C4×C8C2312yes—
146#394—24no—
147#394—24no—
148#393—24no—
149#391—21no—
150#391—21no—
151#391C2311yescharacteristic
152#391C2311yescharacteristic
153#390—21no—
154#390—21no—
155#390—21no—
156#390—22no—
157#390—22no—
158#390C2312yes—
159#390C2314yes—
160#389—21no—
161#389—22no—
162#389C2312yes—
163#389C2311yescharacteristic
164#389C2311yescharacteristic
165#389C2312yes—
166#389C2312yes—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—24no—
173#388—24no—
174#387—21no—
175#387—22no—
176#387—21no—
177#387—22no—
178#387—22no—
179#385—22no—
180#385—22no—
181#385—22no—
182#384—21no—
183#384—21no—
184#384—21no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#382—22no—
190#382—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380C2312yes—
195#380C2314yes—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216#376—21no—
217#376—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×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2412yes—
245C2×SD16—24no—
246C2×SD16—42no—
247C2×SD16—42no—
248C22×C8C2412yes—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C4⋊Q8—22no—
252C4⋊Q8—44no—
253C4⋊Q8—44no—
254C4⋊Q8—82no—
255C4⋊Q8#4114yes—
256#127—42no—
257#126—216no—
258#124—28no—
259#124—416no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262#121—48no—
263#121—48no—
264#121—44no—
265#121—44no—
266#121—84no—
267C4×Q8—28no—
268C4×Q8—22no—
269C4×Q8—44no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8#4114yes—
273C4×D4—24no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—42no—
279C4×D4—44no—
280C4×D4—44no—
281#117—24no—
282#117—24no—
283#117—48no—
284#117—48no—
285#117—44no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
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—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—44no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—28no—
348#106—24no—
349#106—44no—
350C4⋊C8—24no—
351C4⋊C8—42no—
352#103—24no—
353#103—44no—
354#102—24no—
355#102—28no—
356#102—42no—
357#102—42no—
358#102—44no—
359C22⋊C8—44no—
360C22⋊C8—44no—
361#97—44no—
362C4×C8—24no—
363#95—21no—
364#95—22no—
365#95—21no—
366#95—21no—
367#95—22no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—21no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381C24—22no—
382C24C22×D412yes—
383C2×Q8—24no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×Q8—82no—
391C2×Q8—82no—
392C2×Q8C2×#4112yes—
393C2×D4—22no—
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—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
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
442SD16—48no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C4⋊C4—28no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—81no—
477C4⋊C4—81no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4C2×#4112yes—
483C4⋊C4C2×#4114yes—
484C22⋊C4—28no—
485C22⋊C4—28no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—42no—
489C22⋊C4—48no—
490C22⋊C4—44no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C42—28no—
499C42—24no—
500C42—22no—
501C42—22no—
502C42—24no—
503C42—22no—
504C42—22no—
505C42—42no—
506C42—42no—
507C42—41no—
508C42—48no—
509C42—44no—
510C42—44no—
511C42—42no—
512C42—82no—
513C42—82no—
514C42AGL(1,ℤ/8)14yes—
515C42C2×#4112yes—
516C42C2×#4112yes—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23C23×D411yescharacteristic
532Q8—24no—
533Q8—48no—
534Q8—42no—
535Q8—164no—
536D4—44no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—84no—
542D4—84no—
543D4—168no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—48no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—84no—
592C2×C4—88no—
593C2×C4#52512yes—
594C2×C4#52512yes—
595C2×C4C2×AGL(1,ℤ/8)12yes—
596C2×C4C2×AGL(1,ℤ/8)12yes—
597C2×C4C2×AGL(1,ℤ/8)12yes—
598C2×C4#57512yes—
599C2×C4#57712yes—
600C2×C4C22×#4111yescharacteristic
601C2×C4C22×#4111yescharacteristic
602C2×C4#58512yes—
603C2×C4#58512yes—
604C8—42no—
605C8—42no—
606C8—44no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22C2×#52511yescharacteristic
622C22#348511yescharacteristic
623C22#350011yescharacteristic
624C22C22×AGL(1,ℤ/8)11yescharacteristic
625C22C2×#57511yescharacteristic
626C22C2×#57711yescharacteristic
627C22#358611yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—41no—
635C4—44no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—82no—
640C4—81no—
641C4—84no—
642C4—81no—
643C4—82no—
644C4#311514yes—
645C4#349312yes—
646C4#352412yes—
647C2—82no—
648C2—162no—
649C2#3601111yescharacteristic
650C2C2×#311511yescharacteristic
651C2#6059911yescharacteristic
652C2#6171711yescharacteristic
653C2#6173111yescharacteristic
654C2#6253911yescharacteristic
655C2#6258111yescharacteristic
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. #34324C2semidirect
3. #30108C2semidirect
4. #29983C2semidirect
5. #29337C2semidirect
6. #29219C2semidirect
7. #28201C2semidirect
8. #27615C2semidirect
9. #27350C2semidirect
10. #26748C2semidirect
11. #24966C2semidirect
12. #24119C2semidirect
13. #23763C2semidirect
14. #23678C2semidirect
15. #23066C2semidirect
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
#343242
#301082
#299832
#293372
#292192
#282012
#276152
#273502
#267482
#249662
#241192
#237632
#236782
#230662
#228392
#227592