Finite Group Databasebeta

← #6986470

Subgroups — #6986470

Order 512.

1485 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#36449C211yesmaximal, characteristic
2#35320C211yesmaximal, characteristic, complemented
3#32842C212yesmaximal, complemented
4#31939C214yesmaximal, complemented
5#31843C214yesmaximal
6#30338C212yesmaximal, complemented
7#30046C212yesmaximal, complemented
8#29664C211yesmaximal, characteristic, complemented
9#29142C212yesmaximal, complemented
10#28403C212yesmaximal, complemented
11#27730C212yesmaximal, complemented
12#25902C211yesmaximal, characteristic, complemented
13#25323C212yesmaximal, complemented
14#24816C212yesmaximal, complemented
15#24298C211yesmaximal, characteristic
16#23568C211yesmaximal, characteristic
17#22639C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#487C2212yes—
21C2×#474C2212yes—
22C2×#462C2212yes—
23C2×C4×SD16C2212yes—
24#2863C2212yes—
25#2862C2212yes—
26#2862C2212yes—
27#2857C2211yescharacteristic
28#2642—22no—
29#2578—22no—
30#2571C2211yescharacteristic
31#2568—21no—
32#2568—21no—
33#2568C2212yes—
34#2568C2212yes—
35#2547C2211yescharacteristic
36#2530—22no—
37#2503—22no—
38#2503C2214yes—
39#2495—21no—
40#2485—21no—
41#2427C2212yes—
42#2421C2212yes—
43#2365C2211yescharacteristic
44#2352—21no—
45D4×C4⋊C4C2211yescharacteristic
46#2345—21no—
47#2310—21no—
48#2310—21no—
49C4×#124C2212yes—
50C4×#121C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2101C2214yes—
53#2097C2214yes—
54#2085C2214yes—
55#2074C2214yes—
56#2074C2214yes—
57#2068C2214yes—
58#2058C2211yescharacteristic
59#2057C2212yes—
60#2056C2214yes—
61#2043C2211yescharacteristic
62#2035C2212yes—
63#2024C2212yes—
64#2024C2212yes—
65#2024C2212yes—
66#2023C2211yescharacteristic
67#2023C2212yes—
68#2016C2212yes—
69#2015C2214yes—
70#2013C2211yescharacteristic
71#2001C2214yes—
72#2000C2212yes—
73#1990C2212yes—
74#1988C2212yes—
75#1969C2214yes—
76#1958C2214yes—
77#1939C2212yes—
78#1933C2214yes—
79#1923C2214yes—
80#1878C2214yes—
81#1877C2214yes—
82#1874C2212yes—
83#1872C2214yes—
84#1852C2212yes—
85#1839C2212yes—
86#1835C2214yes—
87#1805C2214yes—
88#1804C2214yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C22×Q16C2314yes—
97C22×SD16C2312yes—
98#533—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#125—21no—
108C2×#124C2312yes—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×#121C2311yescharacteristic
113C2×C4×Q8—21no—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×D4—21no—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2312yes—
119C2×#117—21no—
120C2×#117D411yescharacteristic
121C2×#117D411yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4C2312yes—
125C22×C42C2311yescharacteristic
126#487—44no—
127#474—44no—
128#462—44no—
129C4×SD16—44no—
130C2×#107C2314yes—
131C2×#106C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2314yes—
147#400—24no—
148#400—24no—
149#400—24no—
150#400—24no—
151#400—22no—
152#400—22no—
153#397—22no—
154#397—22no—
155#397—22no—
156#394—22no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391—22no—
167#391C2312yes—
168#391C2314yes—
169#391C2314yes—
170#391C2312yes—
171#391C2312yes—
172#390C2311yescharacteristic
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389C2311yescharacteristic
178#389C2312yes—
179#389C2312yes—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—21no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#386—21no—
190#386—22no—
191#385—22no—
192#384C2311yescharacteristic
193#383—22no—
194#382—21no—
195#382—22no—
196#382—22no—
197#382C2314yes—
198#382C2314yes—
199#382C2312yes—
200#381—24no—
201#381—24no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—21no—
207#380—22no—
208#380C2311yescharacteristic
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2314yes—
233#336C2311yescharacteristic
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—44no—
238C22×Q8—44no—
239C22×Q8—44no—
240C22×Q8C2412yes—
241C22×Q8C2412yes—
242C22×D4—21no—
243C22×D4C2411yescharacteristic
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×Q16—44no—
251C2×Q16—44no—
252C2×Q16—44no—
253C2×SD16—24no—
254C2×SD16—42no—
255C2×SD16—42no—
256C22×C8C2412yes—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—44no—
263C4⋊Q8—42no—
264C4⋊Q8—42no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267#125—44no—
268#124—24no—
269#124—42no—
270C22⋊Q8—42no—
271C22⋊Q8—42no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274#121—22no—
275#121—41no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×D4—24no—
281C4×D4—44no—
282C4×D4—44no—
283#117—24no—
284#117—42no—
285#117—44no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—44no—
343#106—42no—
344C4⋊C8—24no—
345C4⋊C8—44no—
346#103—42no—
347#103—44no—
348#103—42no—
349#103—42no—
350#103—44no—
351#103—44no—
352#103—44no—
353#102—42no—
354#102—44no—
355C22⋊C8—42no—
356C22⋊C8—42no—
357C4×C8—24no—
358C4×C8—44no—
359#95—22no—
360#95—22no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—21no—
385C24C22×D411yescharacteristic
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—816no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—81no—
412C2×D42−1+412yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—41no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4#13514yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C2511yescharacteristic
450Q16—88no—
451SD16—48no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—81no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C22⋊C4—24no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C42—22no—
504C42—22no—
505C42—24no—
506C42—24no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—24no—
511C42—42no—
512C42—42no—
513C42—44no—
514C42—42no—
515C42—82no—
516C42H(5,𝔽2)14yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—22no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23C2×#13512yes—
531C23C23×D411yescharacteristic
532Q8—42no—
533Q8—44no—
534Q8—42no—
535Q8—42no—
536Q8—44no—
537Q8—42no—
538Q8—84no—
539Q8—88no—
540Q8—88no—
541D4—24no—
542D4—42no—
543D4—84no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—88no—
594C2×C4—84no—
595C2×C4—164no—
596C2×C4#53512yes—
597C2×C4#53512yes—
598C2×C4C2×#13512yes—
599C2×C4C2×#13512yes—
600C2×C4#57512yes—
601C2×C4#57812yes—
602C2×C4C2×H(5,𝔽2)12yes—
603C2×C4C2×2−1+411yescharacteristic
604C2×C4#58512yes—
605C2×C4#58511yescharacteristic
606C8—42no—
607C8—42no—
608C8—44no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22C2×#53511yescharacteristic
624C22#348911yescharacteristic
625C22#350011yescharacteristic
626C22C22×#13511yescharacteristic
627C22C2×#57511yescharacteristic
628C22C2×#57811yescharacteristic
629C22#358611yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—41no—
637C4—42no—
638C4—81no—
639C4—81no—
640C4—84no—
641C4—82no—
642C4—82no—
643C4—81no—
644C4—82no—
645C4—82no—
646C4#321114yes—
647C4#354112yes—
648C4#355112yes—
649C2—42no—
650C2—82no—
651C2#4291011yescharacteristic
652C2C2×#321111yescharacteristic
653C2#6081411yescharacteristic
654C2#6209711yescharacteristic
655C2#6213311yescharacteristic
656C2#6252811yescharacteristic
657C2#6259311yescharacteristic
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. #35320C2semidirect
3. #32842C2semidirect
4. #31939C2semidirect
6. #30338C2semidirect
7. #30046C2semidirect
8. #29664C2semidirect
9. #29142C2semidirect
10. #28403C2semidirect
11. #27730C2semidirect
12. #25902C2semidirect
13. #25323C2semidirect
14. #24816C2semidirect
17. #22639C2semidirect

Maximal subgroups

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

subgroupindex
#364492
#353202
#328422
#319392
#318432
#303382
#300462
#296642
#291422
#284032
#277302
#259022
#253232
#248162
#242982
#235682
#226392