Finite Group Databasebeta

← #6993549

Subgroups — #6993549

Order 512.

1527 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36314C211yesmaximal, characteristic, complemented
2#35289C211yesmaximal, characteristic
3#32823C212yesmaximal, complemented
4#32288C214yesmaximal, complemented
5#31943C214yesmaximal, complemented
6#30306C212yesmaximal, complemented
7#29985C212yesmaximal, complemented
8#29687C211yesmaximal, characteristic
9#29555C212yesmaximal, complemented
10#28529C212yesmaximal, complemented
11#27728C212yesmaximal, complemented
12#25811C211yesmaximal, characteristic, complemented
13#25452C212yesmaximal, complemented
14#24814C212yesmaximal, complemented
15#24302C211yesmaximal, characteristic, complemented
16#23570C211yesmaximal, characteristic
17#22632C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#498C2212yes—
21C2×#477C2212yes—
22C2×#474C2212yes—
23C2×C4×SD16C2212yes—
24#2864C2212yes—
25#2862C2212yes—
26#2858C2212yes—
27#2857C2211yescharacteristic
28#2640—22no—
29#2571C2211yescharacteristic
30#2568C2211yescharacteristic
31#2563C2212yes—
32#2563C2212yes—
33#2547—21no—
34#2547—21no—
35#2533—22no—
36#2530—22no—
37#2511—21no—
38#2504C2214yes—
39#2499—21no—
40#2495C2212yes—
41#2436—22no—
42#2421C2212yes—
43#2365C2211yescharacteristic
44#2362—21no—
45#2356—21no—
46Q8×C4⋊C4C2211yescharacteristic
47#2311—21no—
48#2311—21no—
49C4×C22⋊Q8C2211yescharacteristic
50#2295C2212yes—
51C4×#117C2211yescharacteristic
52#2101C2214yes—
53#2097C2214yes—
54#2092C2214yes—
55#2085C2214yes—
56#2085C2214yes—
57#2073C2214yes—
58#2067C2214yes—
59#2065C2214yes—
60#2058C2211yescharacteristic
61#2058C2212yes—
62#2058C2212yes—
63#2057C2212yes—
64#2057C2212yes—
65#2041C2211yescharacteristic
66#2041C2212yes—
67#2034C2212yes—
68#2033C2214yes—
69#2024C2211yescharacteristic
70#2024C2212yes—
71#2014C2211yescharacteristic
72#2000C2212yes—
73#1990C2212yes—
74#1987C2212yes—
75#1968C2214yes—
76#1949C2214yes—
77#1939C2212yes—
78#1934C2214yes—
79#1926C2214yes—
80#1923C2214yes—
81#1880C2214yes—
82#1872C2212yes—
83#1853C2214yes—
84#1852C2212yes—
85#1846C2214yes—
86#1835C2214yes—
87#1831C2212yes—
88#1805C2214yes—
89#1802C2212yes—
90#1797C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yes—
95C22×#41—21no—
96C23×Q8—21no—
97C22×SD16C2312yes—
98#532—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#127C2311yescharacteristic
107C2×#124—21no—
108C2×#124—22no—
109C2×#124—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8C2311yescharacteristic
112C2×#121—21no—
113C2×#121C2312yes—
114C2×C4×Q8—21no—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×D4—21no—
118C2×C4×D4C2312yes—
119C2×#117D411yescharacteristic
120C2×#117D411yescharacteristic
121C22×C4⋊C4—21no—
122C22×C4⋊C4C2312yes—
123C22×C42C2311yescharacteristic
124#498—44no—
125#477—44no—
126#474—44no—
127C4×SD16—44no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×#106C2312yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#102C2312yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C4×C8C2314yes—
145C2×C4×C8C2312yes—
146#400—22no—
147#400—22no—
148#400—22no—
149#400—22no—
150#397—22no—
151#397—22no—
152#397—24no—
153#397—24no—
154#393—24no—
155#393—24no—
156#391—21no—
157#391—21no—
158#391—21no—
159#391—22no—
160#391—22no—
161#391—21no—
162#391—22no—
163#391—22no—
164#391C2312yes—
165#391C2311yescharacteristic
166#390—22no—
167#390—22no—
168#390C2314yes—
169#390C2314yes—
170#389—21no—
171#389—21no—
172#389—22no—
173#389C2311yescharacteristic
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—21no—
187#387—22no—
188#387—22no—
189#386—24no—
190#386—24no—
191#385—21no—
192#384—21no—
193#384C2311yescharacteristic
194#383—22no—
195#382—21no—
196#382C2314yes—
197#382C2314yes—
198#382C2312yes—
199#382C2312yes—
200#380—21no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380C2312yes—
206#380C2311yescharacteristic
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219#376—22no—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2314yes—
234C2×#41—44no—
235C2×#41—44no—
236C2×#41—44no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—44no—
242C22×Q8—82no—
243C22×Q8—82no—
244C22×Q8C2411yescharacteristic
245C22×D4—22no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C2412yes—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—42no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—44no—
264C4⋊Q8—42no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267#127—42no—
268#124—44no—
269#124—84no—
270#124—84no—
271C22⋊Q8—22no—
272C22⋊Q8—41no—
273C22⋊Q8—44no—
274#121—42no—
275#121—42no—
276#121—44no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—42no—
280C4×Q8—44no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—44no—
284#117—24no—
285#117—42no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
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—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42C2×D412yes—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340#107—44no—
341#107—44no—
342#106—42no—
343#106—44no—
344#106—44no—
345C4⋊C8—24no—
346C4⋊C8—44no—
347#103—42no—
348#103—44no—
349#102—24no—
350#102—44no—
351#102—42no—
352#102—42no—
353#102—44no—
354C22⋊C8—24no—
355C22⋊C8—42no—
356C4×C8—28no—
357C4×C8—24no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—21no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—22no—
384C24C22×D411yescharacteristic
385#41—816no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8—816no—
402C2×Q82−1+412yes—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—41no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4#13514yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2511yescharacteristic
452SD16—48no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C82−1+414yes—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C22⋊C4—24no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4—81no—
503C42—22no—
504C42—22no—
505C42—24no—
506C42—24no—
507C42—22no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—42no—
512C42—42no—
513C42—44no—
514C42—44no—
515C42—42no—
516C42—42no—
517C42H(5,𝔽2)14yes—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23C2×#13512yes—
532C23C23×D411yescharacteristic
533Q8—24no—
534Q8—42no—
535Q8—88no—
536Q8—84no—
537Q8—88no—
538Q8—84no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—84no—
544D4—84no—
545D4—88no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
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—22no—
568C2×C4—22no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—88no—
596C2×C4—88no—
597C2×C4#53512yes—
598C2×C4#53512yes—
599C2×C4C2×#13512yes—
600C2×C4C2×#13512yes—
601C2×C4#57512yes—
602C2×C4#57712yes—
603C2×C4C2×H(5,𝔽2)12yes—
604C2×C4C2×2−1+411yescharacteristic
605C2×C4C2×2−1+412yes—
606C2×C4#58511yescharacteristic
607C8—24no—
608C8—42no—
609C8—44no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22C2×#53511yescharacteristic
625C22#348911yescharacteristic
626C22#349911yescharacteristic
627C22C22×#13511yescharacteristic
628C22C2×#57511yescharacteristic
629C22C2×#57711yescharacteristic
630C22#358511yescharacteristic
631C4—22no—
632C4—22no—
633C4—22no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—41no—
639C4—42no—
640C4—81no—
641C4—82no—
642C4—82no—
643C4—81no—
644C4—84no—
645C4—82no—
646C4—81no—
647C4#321114yes—
648C4#353812yes—
649C4#354712yes—
650C2—42no—
651C2—162no—
652C2#4299411yescharacteristic
653C2C2×#321111yescharacteristic
654C2#6081211yescharacteristic
655C2#6210211yescharacteristic
656C2#6214511yescharacteristic
657C2#6247011yescharacteristic
658C2#6250811yescharacteristic
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. #36314C2semidirect
3. #32823C2semidirect
4. #32288C2semidirect
5. #31943C2semidirect
6. #30306C2semidirect
7. #29985C2semidirect
9. #29555C2semidirect
10. #28529C2semidirect
11. #27728C2semidirect
12. #25811C2semidirect
13. #25452C2semidirect
14. #24814C2semidirect
15. #24302C2semidirect
17. #22632C2semidirect

Maximal subgroups

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

subgroupindex
#363142
#352892
#328232
#322882
#319432
#303062
#299852
#296872
#295552
#285292
#277282
#258112
#254522
#248142
#243022
#235702
#226322