Finite Group Databasebeta

← #6534733

Subgroups — #6534733

Order 512.

1521 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36301C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#30080C212yesmaximal, complemented
4#29691C211yesmaximal, characteristic, complemented
5#29388C212yesmaximal, complemented
6#29317C212yesmaximal, complemented
7#27950C211yesmaximal, characteristic, complemented
8#27645C211yesmaximal, characteristic
9#27052C214yesmaximal, complemented
10#26567C214yesmaximal
11#26093C212yesmaximal, complemented
12#24679C212yesmaximal, complemented
13#23683C212yesmaximal, complemented
14#23568C211yesmaximal, characteristic
15#22971C212yesmaximal, complemented
16#22671C212yesmaximal, complemented
17#22643C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#481C2212yes—
20C2×#477C2212yes—
21C2×#440C2214yes—
22C2×C4×SD16C2212yes—
23#2864C2212yes—
24#2857C2211yescharacteristic
25#2571C2211yescharacteristic
26#2563C2211yescharacteristic
27#2527—22no—
28#2494—21no—
29#2494C2212yes—
30#2485—21no—
31#2477—22no—
32#2420C2212yes—
33#2369C2211yescharacteristic
34#2360—22no—
35#2360C2214yes—
36#2355—21no—
37#2355C2212yes—
38#2352—21no—
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4C2212yes—
41D4×C4⋊C4C2211yescharacteristic
42#2344—21no—
43#2330—22no—
44#2317—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×#121C2211yescharacteristic
48#2293C2212yes—
49C2×#384C2211yescharacteristic
50#2274—21no—
51C4×#117C2211yescharacteristic
52#2097C2214yes—
53#2084C2212yes—
54#2083C2212yes—
55#2083C2212yes—
56#2074C2214yes—
57#2071C2212yes—
58#2068C2214yes—
59#2058C2211yescharacteristic
60#2057C2212yes—
61#2055C2214yes—
62#2043C2211yescharacteristic
63#2043C2212yes—
64#2036C2214yes—
65#2035C2212yes—
66#2023C2211yescharacteristic
67#2013C2211yescharacteristic
68#2000C2212yes—
69#1990C2212yes—
70#1949C2214yes—
71#1949C2214yes—
72#1948C2212yes—
73#1944C2214yes—
74#1943C2214yes—
75#1939C2212yes—
76#1925C2214yes—
77#1924C2214yes—
78#1918C2214yes—
79#1918C2214yes—
80#1865C2214yes—
81#1863C2214yes—
82#1862C2214yes—
83#1852C2214yes—
84#1845C2212yes—
85#1839C2212yes—
86#1824C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1797C2211yescharacteristic
90#1770C2212yes—
91#1767C2212yes—
92#1763C2214yes—
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#481—44no—
122#477—28no—
123#440—48no—
124C4×SD16—28no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#106C2312yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#102C2314yes—
138C2×#102C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×#97C2314yes—
142C2×C4×C8C2312yes—
143#401—22no—
144#400—22no—
145#397—22no—
146#397—22no—
147#394—22no—
148#393—22no—
149#391—22no—
150#391—21no—
151#391—22no—
152#391C2314yes—
153#390C2312yes—
154#390C2311yescharacteristic
155#389—21no—
156#389—21no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—21no—
161#389—24no—
162#389—24no—
163#389C2314yes—
164#389C2312yes—
165#389C2311yescharacteristic
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—21no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—24no—
178#387—24no—
179#387—22no—
180#386—21no—
181#386—22no—
182#385—24no—
183#385—22no—
184#384—22no—
185#384—21no—
186#384—21no—
187#384C2312yes—
188#384C2312yes—
189#384C2311yescharacteristic
190#382—21no—
191#382—22no—
192#382C2312yes—
193#382C2312yes—
194#381—22no—
195#380—21no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380C2311yescharacteristic
200#380C2312yes—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—24no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219#376D412yes—
220C43C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2314yes—
231#336C2311yescharacteristic
232#336C2314yes—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8C2412yes—
236C22×D4—21no—
237C22×D4C2411yescharacteristic
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244C2×SD16—24no—
245C2×SD16—24no—
246C2×SD16—24no—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253#125—28no—
254C22⋊Q8—28no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—82no—
258#121—22no—
259#121—41no—
260C4×Q8—28no—
261C4×Q8—24no—
262C4×Q8—44no—
263C4×Q8—44no—
264C4×Q8—84no—
265C4×Q8#4118yes—
266C4×D4—24no—
267C4×D4—24no—
268C4×D4—42no—
269C4×D4—44no—
270#117—28no—
271#117—24no—
272#117—42no—
273#117—84no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336C2×C42C2412yes—
337C2×C42C2411yescharacteristic
338#107—28no—
339#107—44no—
340#107—44no—
341#106—24no—
342#106—44no—
343C4⋊C8—28no—
344C4⋊C8—24no—
345#103—24no—
346#103—28no—
347#103—24no—
348#103—42no—
349#103—44no—
350#102—28no—
351#102—24no—
352C22⋊C8—24no—
353C22⋊C8—42no—
354#97—44no—
355C4×C8—24no—
356#95—24no—
357#95—24no—
358#95—22no—
359#95—21no—
360#95—22no—
361#95—21no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24—21no—
378C24C22×D411yescharacteristic
379C2×Q8—24no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—81no—
389C2×Q8—81no—
390C2×Q8C2×#4114yes—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—81no—
397C2×D4H(5,𝔽2)12yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—41no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4#13714yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C2511yescharacteristic
435SD16—216no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C82−1+414yes—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—28no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—81no—
479C4⋊C4C2×#4114yes—
480C22⋊C4—28no—
481C22⋊C4—24no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C42—22no—
493C42—22no—
494C42—28no—
495C42—24no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—24no—
500C42—22no—
501C42—42no—
502C42—42no—
503C42—42no—
504C42—82no—
505C42—84no—
506C42—84no—
507C42C2×#4114yes—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×#13712yes—
522C23C23×D411yescharacteristic
523Q8—24no—
524Q8—24no—
525Q8—48no—
526Q8—42no—
527Q8—42no—
528Q8—162no—
529D4—24no—
530D4—42no—
531D4—84no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—28no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—88no—
582C2×C4—84no—
583C2×C4—164no—
584C2×C4#52512yes—
585C2×C4#52512yes—
586C2×C4C2×#13712yes—
587C2×C4C2×#13712yes—
588C2×C4#57512yes—
589C2×C4#57612yes—
590C2×C4C22×#4112yes—
591C2×C4C2×H(5,𝔽2)11yescharacteristic
592C2×C4C2×2−1+412yes—
593C2×C4#58511yescharacteristic
594C8—24no—
595C8—44no—
596C8—42no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22C2×#52511yescharacteristic
612C22#348811yescharacteristic
613C22#349911yescharacteristic
614C22C22×#13711yescharacteristic
615C22C2×#57511yescharacteristic
616C22C2×#57611yescharacteristic
617C22#358611yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—41no—
626C4—44no—
627C4—44no—
628C4—81no—
629C4—81no—
630C4—82no—
631C4—84no—
632C4—81no—
633C4—82no—
634C4#311714yes—
635C4#349212yes—
636C4#350712yes—
637C2—42no—
638C2—82no—
639C2#3568611yescharacteristic
640C2C2×#311711yescharacteristic
641C2#6058411yescharacteristic
642C2#6196011yescharacteristic
643C2#6199511yescharacteristic
644C2#6248011yescharacteristic
645C2#6250711yescharacteristic
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. #30080C2semidirect
4. #29691C2semidirect
5. #29388C2semidirect
6. #29317C2semidirect
7. #27950C2semidirect
9. #27052C2semidirect
11. #26093C2semidirect
12. #24679C2semidirect
13. #23683C2semidirect
15. #22971C2semidirect
16. #22671C2semidirect
17. #22643C2semidirect

Maximal subgroups

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

subgroupindex
#363012
#342142
#300802
#296912
#293882
#293172
#279502
#276452
#270522
#265672
#260932
#246792
#236832
#235682
#229712
#226712
#226432