Finite Group Databasebeta

← #6562253

Subgroups — #6562253

Order 512.

1557 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36301C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#30080C212yesmaximal, complemented
4#29939C211yesmaximal, characteristic, complemented
5#29368C212yesmaximal, complemented
6#29129C212yesmaximal, complemented
7#27540C214yesmaximal, complemented
8#26531C212yesmaximal, complemented
9#25856C211yesmaximal, characteristic, complemented
10#25319C212yesmaximal, complemented
11#24914C214yesmaximal
12#24209C211yesmaximal, characteristic
13#23755C211yesmaximal, characteristic
14#23683C212yesmaximal, complemented
15#23221C212yesmaximal, complemented
16#23002C212yesmaximal, complemented
17#22643C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#487C2212yes—
20C2×#461C2212yes—
21C2×#448C2212yes—
22C2×#440C2214yes—
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#2369—21no—
34#2369—21no—
35#2369C2211yescharacteristic
36#2365C2212yes—
37#2365C2212yes—
38#2352—21no—
39D4×C4⋊C4C2211yescharacteristic
40#2345—22no—
41#2344—21no—
42Q8×C22⋊C4C2214yes—
43#2331—22no—
44#2316—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×#121C2211yescharacteristic
48#2293C2212yes—
49#2285C2211yescharacteristic
50#2274—21no—
51C4×#117C2211yescharacteristic
52#2100C2212yes—
53#2100C2211yescharacteristic
54#2097C2214yes—
55#2094C2211yescharacteristic
56#2084C2211yescharacteristic
57#2083C2212yes—
58#2077C2212yes—
59#2076C2214yes—
60#2074C2214yes—
61#2071C2211yescharacteristic
62#2068C2214yes—
63#2055C2214yes—
64#2043C2212yes—
65#2043C2212yes—
66#2043C2212yes—
67#2034C2212yes—
68#2013C2212yes—
69#1987C2212yes—
70#1981C2212yes—
71#1956C2212yes—
72#1947C2212yes—
73#1939C2212yes—
74#1935C2214yes—
75#1935C2214yes—
76#1927C2214yes—
77#1924C2214yes—
78#1880C2214yes—
79#1877C2214yes—
80#1865C2214yes—
81#1865C2214yes—
82#1863C2214yes—
83#1839C2212yes—
84#1805C2214yes—
85#1805C2214yes—
86#1797C2211yescharacteristic
87#1787C2214yes—
88#1787C2212yes—
89C4×#106C2212yes—
90#1770C2212yes—
91#1767C2212yes—
92#1763C2214yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C22×SD16C2314yes—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C22⋊Q8—21no—
104C2×#121C2311yescharacteristic
105C2×C4×Q8C2314yes—
106C2×C4×Q8C2314yes—
107C2×C4×D4—21no—
108C2×C4×D4C2311yescharacteristic
109C2×C4×D4C2312yes—
110C2×#117—21no—
111C2×#117—22no—
112C2×#117D411yescharacteristic
113C2×#117D411yescharacteristic
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C42C2311yescharacteristic
118#487—28no—
119#461—28no—
120#448—44no—
121#440—48no—
122C2×#107C2314yes—
123C2×#106C2312yes—
124C2×#106C2312yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102C2312yes—
135C2×#102C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×#97C2314yes—
139C2×C4×C8C2312yes—
140#401—22no—
141#400—22no—
142#397—22no—
143#397—22no—
144#394—22no—
145#393—22no—
146#391—21no—
147#391—24no—
148#391—24no—
149#391—22no—
150#391C2314yes—
151#391C2314yes—
152#390C2312yes—
153#390C2311yescharacteristic
154#389—21no—
155#389—22no—
156#389—22no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—21no—
163#389C2312yes—
164#389C2312yes—
165#389C2311yescharacteristic
166#389C2312yes—
167#389C2312yes—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—21no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—24no—
177#387—24no—
178#387—22no—
179#386—21no—
180#386—22no—
181#385—22no—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#382—21no—
186#382—21no—
187#382C2312yes—
188#382C2312yes—
189#381—22no—
190#380—21no—
191#380—21no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380—24no—
196#380C2312yes—
197#380C2312yes—
198#380C2311yescharacteristic
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
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#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2314yes—
231#336C2311yescharacteristic
232#336C2314yes—
233C22×Q8—28no—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8—44no—
237C22×Q8—42no—
238C22×Q8—42no—
239C22×Q8C2414yes—
240C22×D4—21no—
241C22×D4C2411yescharacteristic
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C2×SD16—28no—
249C2×SD16—28no—
250C2×SD16—44no—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C4⋊Q8—28no—
255C4⋊Q8—28no—
256C4⋊Q8—28no—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8#4118yes—
260C22⋊Q8—82no—
261#121—22no—
262#121—41no—
263C4×Q8—44no—
264C4×Q8—44no—
265C4×Q8—44no—
266C4×Q8—44no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—42no—
270C4×D4—44no—
271#117—24no—
272#117—42no—
273#117—44no—
274#117—84no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
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—24no—
320C2×C42—22no—
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—44no—
336#106—42no—
337#106—42no—
338C4⋊C8—24no—
339C4⋊C8—44no—
340#103—28no—
341#103—28no—
342#103—28no—
343#103—44no—
344#103—44no—
345#103—44no—
346#103—44no—
347#102—24no—
348#102—42no—
349C22⋊C8—42no—
350C22⋊C8—42no—
351#97—44no—
352C4×C8—24no—
353#95—24no—
354#95—24no—
355#95—22no—
356#95—21no—
357#95—22no—
358#95—21no—
359#95—24no—
360#95—24no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376C24—21no—
377C24C22×D411yescharacteristic
378C2×Q8—28no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—42no—
385C2×Q8—82no—
386C2×Q8—82no—
387C2×Q8—81no—
388C2×Q8—81no—
389C2×Q8—88no—
390C2×Q8—88no—
391C2×Q8C2×#4118yes—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—81no—
398C2×D4H(5,𝔽2)12yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4#13714yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C2511yescharacteristic
436SD16—416no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—81no—
479C22⋊C4—24no—
480C22⋊C4—42no—
481C22⋊C4—44no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C42—22no—
492C42—22no—
493C42—24no—
494C42—24no—
495C42—22no—
496C42—22no—
497C42—24no—
498C42—22no—
499C42—42no—
500C42—42no—
501C42—42no—
502C42—44no—
503C42—82no—
504C42—84no—
505C42—84no—
506C42C2×#4114yes—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23C2×#13712yes—
521C23C23×D411yescharacteristic
522Q8—28no—
523Q8—28no—
524Q8—44no—
525Q8—44no—
526Q8—88no—
527Q8—162no—
528D4—24no—
529D4—42no—
530D4—84no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—88no—
581C2×C4—88no—
582C2×C4—84no—
583C2×C4#52512yes—
584C2×C4#52512yes—
585C2×C4C2×#13712yes—
586C2×C4C2×#13712yes—
587C2×C4#57512yes—
588C2×C4#57712yes—
589C2×C4C22×#4112yes—
590C2×C4C2×H(5,𝔽2)11yescharacteristic
591C2×C4#58512yes—
592C2×C4#58511yescharacteristic
593C8—42no—
594C8—44no—
595C8—42no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22C2×#52511yescharacteristic
611C22#348811yescharacteristic
612C22#350011yescharacteristic
613C22C22×#13711yescharacteristic
614C22C2×#57511yescharacteristic
615C22C2×#57711yescharacteristic
616C22#358611yescharacteristic
617C4—22no—
618C4—22no—
619C4—22no—
620C4—42no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—41no—
625C4—48no—
626C4—81no—
627C4—82no—
628C4—82no—
629C4—81no—
630C4—82no—
631C4—81no—
632C4—82no—
633C4#311714yes—
634C4#351312yes—
635C4#352112yes—
636C2—42no—
637C2—82no—
638C2#3598211yescharacteristic
639C2C2×#311711yescharacteristic
640C2#6058211yescharacteristic
641C2#6196011yescharacteristic
642C2#6199611yescharacteristic
643C2#6254011yescharacteristic
644C2#6258111yescharacteristic
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. #34216C2semidirect
3. #30080C2semidirect
4. #29939C2semidirect
5. #29368C2semidirect
6. #29129C2semidirect
7. #27540C2semidirect
8. #26531C2semidirect
9. #25856C2semidirect
10. #25319C2semidirect
14. #23683C2semidirect
15. #23221C2semidirect
16. #23002C2semidirect
17. #22643C2semidirect

Maximal subgroups

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

subgroupindex
#363012
#342162
#300802
#299392
#293682
#291292
#275402
#265312
#258562
#253192
#249142
#242092
#237552
#236832
#232212
#230022
#226432