Finite Group Databasebeta

← #6572076

Subgroups — #6572076

Order 512.

1475 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36301C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#30093C212yesmaximal, complemented
4#29691C211yesmaximal, characteristic, complemented
5#29368C212yesmaximal, complemented
6#29129C212yesmaximal, complemented
7#27961C211yesmaximal, characteristic, complemented
8#27678C211yesmaximal, characteristic
9#27116C214yesmaximal, complemented
10#26676C214yesmaximal
11#26113C212yesmaximal, complemented
12#24814C212yesmaximal, complemented
13#23780C212yesmaximal, complemented
14#23568C211yesmaximal, characteristic
15#23069C212yesmaximal, complemented
16#22762C212yesmaximal, complemented
17#22643C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#477C2212yes—
20C2×C4×SD16C2212yes—
21#2864C2212yes—
22#2857C2211yescharacteristic
23#2571C2211yescharacteristic
24#2563C2211yescharacteristic
25#2527—22no—
26#2494—21no—
27#2494C2212yes—
28#2485—21no—
29#2477—22no—
30#2420C2212yes—
31#2369C2211yescharacteristic
32#2360—22no—
33#2360C2214yes—
34#2355—21no—
35#2355C2212yes—
36#2352—21no—
37Q8×C4⋊C4—21no—
38Q8×C4⋊C4C2212yes—
39D4×C4⋊C4C2211yescharacteristic
40#2344—21no—
41#2330—22no—
42#2317—21no—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×#121C2211yescharacteristic
46#2293C2212yes—
47C2×#384C2211yescharacteristic
48#2274—21no—
49C4×#117C2211yescharacteristic
50#2101C2214yes—
51#2100C2212yes—
52#2100C2212yes—
53#2100C2212yes—
54#2094C2212yes—
55#2088C2214yes—
56#2085C2214yes—
57#2067C2214yes—
58#2058C2211yescharacteristic
59#2057C2212yes—
60#2043C2211yescharacteristic
61#2043C2212yes—
62#2036C2214yes—
63#2035C2212yes—
64#2023C2211yescharacteristic
65#2013C2211yescharacteristic
66#2000C2212yes—
67#1990C2212yes—
68#1948C2212yes—
69#1939C2212yes—
70#1935C2214yes—
71#1935C2214yes—
72#1927C2214yes—
73#1924C2214yes—
74#1920C2214yes—
75#1918C2214yes—
76#1880C2214yes—
77#1877C2214yes—
78#1874C2212yes—
79#1865C2214yes—
80#1853C2212yes—
81#1839C2212yes—
82#1834C2212yes—
83#1824C2214yes—
84#1805C2214yes—
85#1805C2214yes—
86#1803C2212yes—
87#1802C2212yes—
88#1797C2211yescharacteristic
89#1776C2214yes—
90#1770C2212yes—
91#1767C2212yes—
92#1762C2214yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×SD16C2312yes—
96#533—28no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×#125—21no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8C2312yes—
104C2×#121C2311yescharacteristic
105C2×C4×Q8—21no—
106C2×C4×Q8—22no—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2314yes—
109C2×C4×D4—21no—
110C2×C4×D4C2311yescharacteristic
111C2×C4×D4C2312yes—
112C2×#117—21no—
113C2×#117—22no—
114C2×#117D411yescharacteristic
115C2×#117D411yescharacteristic
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4C2312yes—
120C22×C42C2311yescharacteristic
121#477—28no—
122C4×SD16—28no—
123C2×#107C2314yes—
124C2×#106C2312yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2314yes—
128C2×#103C2314yes—
129C2×#103C2312yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#102C2312yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×#97C2314yes—
138C2×C4×C8C2312yes—
139#401—22no—
140#400—22no—
141#397—22no—
142#397—22no—
143#394—22no—
144#393—22no—
145#391—22no—
146#391—21no—
147#391—22no—
148#391C2314yes—
149#390C2312yes—
150#390C2311yescharacteristic
151#389—21no—
152#389—21no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—21no—
157#389—24no—
158#389—24no—
159#389C2314yes—
160#389C2312yes—
161#389C2311yescharacteristic
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—21no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—24no—
174#387—24no—
175#387—22no—
176#386—21no—
177#386—22no—
178#385—24no—
179#385—22no—
180#384—22no—
181#384—21no—
182#384—21no—
183#384C2312yes—
184#384C2312yes—
185#384C2311yescharacteristic
186#382—21no—
187#382—22no—
188#382C2312yes—
189#382C2312yes—
190#381—22no—
191#380—21no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380C2312yes—
196#380C2311yescharacteristic
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—24no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215#376D412yes—
216C43C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#336C2312yes—
227#336C2311yescharacteristic
228#336C2314yes—
229#336C2314yes—
230#336C2311yescharacteristic
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8C2412yes—
234C22×D4—21no—
235C22×D4C2411yescharacteristic
236C23×C4—22no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2411yescharacteristic
242C2×SD16—24no—
243C2×SD16—24no—
244C2×SD16—42no—
245C22×C8C2412yes—
246C22×C8C2414yes—
247C22×C8C2412yes—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251#125—28no—
252C22⋊Q8—28no—
253C22⋊Q8—24no—
254C22⋊Q8—24no—
255C22⋊Q8—82no—
256#121—22no—
257#121—41no—
258C4×Q8—28no—
259C4×Q8—24no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×Q8—84no—
263C4×Q8#4118yes—
264C4×D4—24no—
265C4×D4—24no—
266C4×D4—42no—
267C4×D4—44no—
268#117—28no—
269#117—24no—
270#117—42no—
271#117—84no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—21no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2412yes—
332C2×C42C2412yes—
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335#107—28no—
336#106—24no—
337#106—44no—
338C4⋊C8—24no—
339C4⋊C8—44no—
340#103—24no—
341#103—28no—
342#103—44no—
343#103—42no—
344#103—42no—
345#102—24no—
346#102—44no—
347C22⋊C8—42no—
348C22⋊C8—42no—
349#97—44no—
350C4×C8—24no—
351#95—24no—
352#95—24no—
353#95—22no—
354#95—21no—
355#95—22no—
356#95—21no—
357#95—22no—
358#95—24no—
359#95—24no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95C2×D412yes—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372C24—21no—
373C24C22×D411yescharacteristic
374C2×Q8—24no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—81no—
384C2×Q8—81no—
385C2×Q8C2×#4114yes—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—21no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—81no—
392C2×D42−1+412yes—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—41no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4#13714yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C2511yescharacteristic
430SD16—48no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—28no—
443C4⋊C4—24no—
444C4⋊C4—28no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C4⋊C4—81no—
474C4⋊C4C2×#4114yes—
475C22⋊C4—28no—
476C22⋊C4—24no—
477C22⋊C4—42no—
478C22⋊C4—44no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C42—22no—
488C42—22no—
489C42—28no—
490C42—24no—
491C42—24no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—24no—
496C42—42no—
497C42—42no—
498C42—42no—
499C42—82no—
500C42—84no—
501C42—84no—
502C42C2×#4114yes—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23C2×#13712yes—
517C23C23×D411yescharacteristic
518Q8—24no—
519Q8—24no—
520Q8—48no—
521Q8—42no—
522Q8—42no—
523Q8—162no—
524D4—24no—
525D4—42no—
526D4—84no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—88no—
577C2×C4—84no—
578C2×C4—164no—
579C2×C4#52512yes—
580C2×C4#52512yes—
581C2×C4C2×#13712yes—
582C2×C4C2×#13712yes—
583C2×C4#57512yes—
584C2×C4#57712yes—
585C2×C4C22×#4112yes—
586C2×C4C2×2−1+411yescharacteristic
587C2×C4#58512yes—
588C2×C4#58511yescharacteristic
589C8—42no—
590C8—42no—
591C8—44no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22C2×#52511yescharacteristic
607C22#348911yescharacteristic
608C22#350011yescharacteristic
609C22C22×#13711yescharacteristic
610C22C2×#57511yescharacteristic
611C22C2×#57711yescharacteristic
612C22#358511yescharacteristic
613C4—22no—
614C4—22no—
615C4—22no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—44no—
620C4—42no—
621C4—41no—
622C4—44no—
623C4—81no—
624C4—84no—
625C4—81no—
626C4—81no—
627C4—82no—
628C4—82no—
629C4#311714yes—
630C4#349512yes—
631C4#351212yes—
632C2—42no—
633C2—82no—
634C2#3608011yescharacteristic
635C2C2×#311711yescharacteristic
636C2#6058811yescharacteristic
637C2#6210811yescharacteristic
638C2#6214411yescharacteristic
639C2#6254011yescharacteristic
640C2#6258011yescharacteristic
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. #30093C2semidirect
4. #29691C2semidirect
5. #29368C2semidirect
6. #29129C2semidirect
7. #27961C2semidirect
9. #27116C2semidirect
11. #26113C2semidirect
12. #24814C2semidirect
13. #23780C2semidirect
15. #23069C2semidirect
16. #22762C2semidirect
17. #22643C2semidirect

Maximal subgroups

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

subgroupindex
#363012
#342142
#300932
#296912
#293682
#291292
#279612
#276782
#271162
#266762
#261132
#248142
#237802
#235682
#230692
#227622
#226432