Finite Group Databasebeta

← #6544575

Subgroups — #6544575

Order 512.

1683 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36480C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic
3#30128C212yesmaximal, complemented
4#29690C211yesmaximal, characteristic
5#29519C212yesmaximal, complemented
6#29276C212yesmaximal, complemented
7#27961C211yesmaximal, characteristic, complemented
8#27681C211yesmaximal, characteristic, complemented
9#27049C214yesmaximal, complemented
10#26739C214yesmaximal, complemented
11#26020C212yesmaximal, complemented
12#24680C212yesmaximal, complemented
13#23822C212yesmaximal, complemented
14#23570C211yesmaximal, characteristic
15#22969C212yesmaximal, complemented
16#22762C212yesmaximal, complemented
17#22636C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#520—21no—
20C2×#476C2212yes—
21C2×C4×SD16C2212yes—
22#2863C2212yes—
23#2857C2211yescharacteristic
24#2605—22no—
25#2567C2211yescharacteristic
26#2551C2211yescharacteristic
27#2509—21no—
28#2497C2212yes—
29#2492C2212yes—
30#2465—22no—
31#2439—21no—
32#2369C2211yescharacteristic
33#2365—21no—
34#2361—22no—
35#2361C2214yes—
36#2355C2211yescharacteristic
37#2354—21no—
38#2354C2212yes—
39D4×C4⋊C4—21no—
40D4×C4⋊C4C2212yes—
41#2347—21no—
42#2326—22no—
43#2311—21no—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×#127C2212yes—
47C4×C22⋊Q8C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2100C2212yes—
51#2100C2212yes—
52#2100C2212yes—
53#2095C2212yes—
54#2071C2214yes—
55#2067C2214yes—
56#2058C2211yescharacteristic
57#2058C2212yes—
58#2055C2214yes—
59#2041C2211yescharacteristic
60#2041C2212yes—
61#2034C2212yes—
62#2034C2214yes—
63#2024C2211yescharacteristic
64#2023C2214yes—
65#2014C2211yescharacteristic
66#2000C2212yes—
67#1990C2212yes—
68#1988C2214yes—
69#1948C2214yes—
70#1947C2212yes—
71#1944C2214yes—
72#1939C2212yes—
73#1935C2214yes—
74#1927C2214yes—
75#1924C2214yes—
76#1920C2214yes—
77#1920C2214yes—
78#1918C2214yes—
79#1877C2214yes—
80#1872C2212yes—
81#1862C2214yes—
82#1852C2212yes—
83#1834C2212yes—
84#1831C2212yes—
85#1824C2214yes—
86#1803C2212yes—
87#1802C2212yes—
88#1797C2211yescharacteristic
89#1770C2212yes—
90#1762C2214yes—
91#1762C2214yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1745C2212yes—
95C22×#41—21no—
96C22×SD16C2312yes—
97#532—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#124—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2311yescharacteristic
106C2×#121—21no—
107C2×#121C2312yes—
108#520—48no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—24no—
113C2×C4×D4—21no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4C2312yes—
117C2×C4×D4C2314yes—
118C2×#117—22no—
119C2×#117D411yescharacteristic
120C2×#117D411yescharacteristic
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4C2312yes—
125C22×C42C2311yescharacteristic
126#476—28no—
127C4×SD16—28no—
128C2×#107C2314yes—
129C2×#106C2312yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#102C2314yes—
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×#97C2314yes—
143C2×C4×C8C2312yes—
144#400—22no—
145#400—22no—
146#396—22no—
147#395—22no—
148#395—22no—
149#393—22no—
150#391—22no—
151#391—22no—
152#391—21no—
153#391—22no—
154#391—22no—
155#391C2312yes—
156#391C2311yescharacteristic
157#391C2312yes—
158#390—22no—
159#390—22no—
160#390C2314yes—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389—21no—
168#389C2311yescharacteristic
169#389C2312yes—
170#389C2314yes—
171#389C2312yes—
172#389C2312yes—
173#388—22no—
174#388—22no—
175#388—24no—
176#388—24no—
177#387—21no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#386—24no—
184#385—21no—
185#385—24no—
186#385—24no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2312yes—
193#383—22no—
194#382—21no—
195#380—21no—
196#380—22no—
197#380—22no—
198#380C2312yes—
199#380C2311yescharacteristic
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
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
232C2×#41—28no—
233C2×#41—44no—
234C2×#41—44no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8C2411yescharacteristic
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2412yes—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—24no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×SD16—24no—
250C2×SD16—42no—
251C2×SD16—42no—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—42no—
259C4⋊Q8—44no—
260#127—24no—
261#124—28no—
262C22⋊Q8—22no—
263C22⋊Q8—41no—
264C22⋊Q8—82no—
265#121—28no—
266#121—24no—
267#121—24no—
268C4×Q8—24no—
269C4×Q8—48no—
270C4×Q8—84no—
271C4×D4—28no—
272C4×D4—24no—
273C4×D4—48no—
274C4×D4—44no—
275C4×D4—44no—
276C4×D4—416no—
277C4×D4—84no—
278C4×D4#4118yes—
279#117—24no—
280#117—24no—
281#117—48no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—48no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342#107—44no—
343#106—24no—
344#106—28no—
345C4⋊C8—24no—
346C4⋊C8—44no—
347#103—24no—
348#103—44no—
349#102—24no—
350#102—28no—
351#102—24no—
352#102—44no—
353#102—42no—
354C22⋊C8—42no—
355C22⋊C8—42no—
356#97—44no—
357C4×C8—24no—
358#95—21no—
359#95—21no—
360#95—22no—
361#95—24no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—21no—
366#95—24no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—22no—
380C24C22×D411yescharacteristic
381#41—432no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—21no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—82no—
393C2×Q8—81no—
394C2×Q8—81no—
395C2×D4—24no—
396C2×D4—28no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4C2×#4114yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—28no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—41no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4#13714yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2511yescharacteristic
447SD16—48no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—81no—
488C4⋊C4AGL(1,ℤ/8)14yes—
489C4⋊C4C2×#4114yes—
490C4⋊C42−1+412yes—
491C22⋊C4—28no—
492C22⋊C4—24no—
493C22⋊C4—28no—
494C22⋊C4—24no—
495C22⋊C4—42no—
496C22⋊C4—48no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C22⋊C4—81no—
500C22⋊C4—82no—
501C42—22no—
502C42—22no—
503C42—24no—
504C42—24no—
505C42—24no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—22no—
510C42—42no—
511C42—42no—
512C42—48no—
513C42—42no—
514C42—82no—
515C42—82no—
516C42C2×#4114yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—24no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23C2×#13712yes—
531C23C23×D411yescharacteristic
532Q8—48no—
533Q8—42no—
534Q8—42no—
535Q8—88no—
536Q8—162no—
537D4—24no—
538D4—24no—
539D4—48no—
540D4—48no—
541D4—416no—
542D4—42no—
543D4—42no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—416no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—168no—
594C2×C4—164no—
595C2×C4#52512yes—
596C2×C4#52512yes—
597C2×C4C2×AGL(1,ℤ/8)14yes—
598C2×C4C2×AGL(1,ℤ/8)12yes—
599C2×C4C2×#13712yes—
600C2×C4C2×#13712yes—
601C2×C4#57712yes—
602C2×C4C22×#4112yes—
603C2×C4C2×2−1+411yescharacteristic
604C2×C4#58511yescharacteristic
605C2×C4#58512yes—
606C8—44no—
607C8—42no—
608C8—42no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22C2×#52511yescharacteristic
624C22#348911yescharacteristic
625C22#350011yescharacteristic
626C22C22×AGL(1,ℤ/8)11yescharacteristic
627C22C22×#13711yescharacteristic
628C22C2×#57711yescharacteristic
629C22#358511yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—24no—
634C4—42no—
635C4—42no—
636C4—44no—
637C4—42no—
638C4—41no—
639C4—42no—
640C4—81no—
641C4—84no—
642C4—81no—
643C4—82no—
644C4—81no—
645C4—82no—
646C4#311014yes—
647C4#349312yes—
648C4#350712yes—
649C2—42no—
650C2—84no—
651C2#3575911yescharacteristic
652C2C2×#311011yescharacteristic
653C2#6058811yescharacteristic
654C2#6211411yescharacteristic
655C2#6212811yescharacteristic
656C2#6254211yescharacteristic
657C2#6258811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
1. #36480C2semidirect
3. #30128C2semidirect
5. #29519C2semidirect
6. #29276C2semidirect
7. #27961C2semidirect
8. #27681C2semidirect
9. #27049C2semidirect
10. #26739C2semidirect
11. #26020C2semidirect
12. #24680C2semidirect
13. #23822C2semidirect
15. #22969C2semidirect
16. #22762C2semidirect
17. #22636C2semidirect

Maximal subgroups

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

subgroupindex
#364802
#342072
#301282
#296902
#295192
#292762
#279612
#276812
#270492
#267392
#260202
#246802
#238222
#235702
#229692
#227622
#226362