Finite Group Databasebeta

← #6993037

Subgroups — #6993037

Order 512.

1660 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36473C211yesmaximal, characteristic
2#35281C211yesmaximal, characteristic, complemented
3#32800C212yesmaximal, complemented
4#31288C214yesmaximal, complemented
5#30522C214yesmaximal
6#30339C212yesmaximal, complemented
7#30060C212yesmaximal, complemented
8#29688C211yesmaximal, characteristic, complemented
9#28748C212yesmaximal, complemented
10#28400C212yesmaximal, complemented
11#27769C212yesmaximal, complemented
12#25856C211yesmaximal, characteristic, complemented
13#25276C212yesmaximal, complemented
14#24630C212yesmaximal, complemented
15#24209C211yesmaximal, characteristic
16#23568C211yesmaximal, characteristic
17#22639C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#533C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C2×#487C2212yes—
22C2×#487C2214yes—
23C2×#475C2212yes—
24C2×#461C2212yes—
25C2×#448C2212yes—
26C2×C4×SD16C2212yes—
27#2868C2212yes—
28#2862C2212yes—
29#2860C2212yes—
30#2857C2211yescharacteristic
31#2660—22no—
32#2604—22no—
33#2567—21no—
34#2567—21no—
35#2551C2211yescharacteristic
36#2546C2212yes—
37#2546C2212yes—
38#2542C2211yescharacteristic
39#2521—22no—
40#2502—22no—
41#2496C2212yes—
42#2492—21no—
43#2491C2212yes—
44#2434C2214yes—
45#2415—21no—
46#2365C2211yescharacteristic
47#2362—21no—
48#2354C2211yescharacteristic
49#2317—21no—
50#2317—21no—
51C4×C4⋊Q8C2212yes—
52C4×#121C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2096C2214yes—
55#2066C2214yes—
56#2058C2211yescharacteristic
57#2058C2212yes—
58#2055C2214yes—
59#2043C2212yes—
60#2043C2212yes—
61#2043C2211yescharacteristic
62#2043C2214yes—
63#2043C2214yes—
64#2043C2212yes—
65#2043C2212yes—
66#2035C2212yes—
67#2035C2214yes—
68#2024C2214yes—
69#2023C2211yescharacteristic
70#2013C2211yescharacteristic
71#2013C2214yes—
72#2013C2214yes—
73#2013C2212yes—
74#2000C2212yes—
75#1981C2214yes—
76#1981C2214yes—
77#1981C2212yes—
78#1968C2214yes—
79#1956C2212yes—
80#1956C2214yes—
81#1939C2212yes—
82#1933C2214yes—
83#1927C2214yes—
84#1927C2214yes—
85#1923C2214yes—
86#1839C2212yes—
87#1803C2212yes—
88#1797C2211yescharacteristic
89#1787C2214yes—
90#1787C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C22×SD16C2314yes—
97C22×SD16C2312yes—
98#556—48no—
99#533—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2314yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2311yescharacteristic
112C2×#126—22no—
113C2×#126—22no—
114C2×#125—21no—
115C2×#124—22no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8C2312yes—
121C2×#121C2311yescharacteristic
122C2×C4×Q8—21no—
123C2×C4×Q8—22no—
124C2×C4×Q8—21no—
125C2×C4×Q8C2312yes—
126C2×C4×D4—21no—
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117—21no—
130C2×#117D411yescharacteristic
131C2×#117D411yescharacteristic
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C42C2311yescharacteristic
135#487—44no—
136#487—48no—
137#475—44no—
138#461—44no—
139#448—44no—
140C4×SD16—44no—
141C2×#106C2314yes—
142C2×#106C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#102C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C4×C8C2312yes—
161C2×C4×C8C2314yes—
162#400—22no—
163#399—22no—
164#398—24no—
165#398—24no—
166#397—22no—
167#395—24no—
168#395—24no—
169#395—22no—
170#394—22no—
171#393—22no—
172#391—21no—
173#391—22no—
174#391—22no—
175#391—22no—
176#391—21no—
177#391C2314yes—
178#391C2314yes—
179#390C2312yes—
180#390C2311yescharacteristic
181#390C2312yes—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—21no—
187#389—22no—
188#389—21no—
189#389C2311yescharacteristic
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#388—21no—
194#388—22no—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#387—22no—
200#387—22no—
201#386—21no—
202#385—24no—
203#385—24no—
204#384—22no—
205#384—22no—
206#384—22no—
207#384—22no—
208#384C2312yes—
209#384C2311yescharacteristic
210#382—22no—
211#382—22no—
212#380—21no—
213#380—21no—
214#380C2312yes—
215#380C2311yescharacteristic
216C4×C4⋊C4—22no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—21no—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4C2312yes—
227#376D412yes—
228C43C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2314yes—
235#336C2314yes—
236#336C2311yescharacteristic
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8—42no—
244C22×Q8—42no—
245C22×Q8C2412yes—
246C22×Q8C2414yes—
247C22×D4—21no—
248C22×D4C2411yescharacteristic
249C23×C4—22no—
250C23×C4—21no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×SD16—42no—
255C2×SD16—42no—
256C2×SD16—42no—
257C2×SD16—44no—
258C2×SD16—44no—
259C2×SD16—44no—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—42no—
268C4⋊Q8—42no—
269C4⋊Q8—44no—
270C4⋊Q8—42no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278#126—48no—
279#126—48no—
280#125—44no—
281#124—48no—
282C22⋊Q8—42no—
283C22⋊Q8—42no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288#121—22no—
289#121—41no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×Q8—48no—
295C4×D4—24no—
296C4×D4—44no—
297#117—24no—
298#117—44no—
299#117—44no—
300#117SD1618yescomplemented
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364#106—42no—
365#106—44no—
366C4⋊C8—24no—
367C4⋊C8—44no—
368#103—28no—
369#103—24no—
370#103—42no—
371#103—44no—
372#103—44no—
373#103—42no—
374#103—44no—
375#103—44no—
376#103—44no—
377#103—44no—
378#103—44no—
379#103—44no—
380#102—42no—
381C22⋊C8—42no—
382C22⋊C8—42no—
383C4×C8—24no—
384C4×C8—44no—
385#95—21no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—22no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402C24—21no—
403C24C22×D411yescharacteristic
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—22no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—24no—
416C2×Q8—44no—
417C2×Q8—42no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—44no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—82no—
426C2×Q8—82no—
427C2×Q8—82no—
428C2×Q8—88no—
429C2×Q8—88no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—81no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—41no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4C2×SD1614yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2511yescharacteristic
471SD16—48no—
472SD16—88no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—81no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4C2×SD1614yes—
515C4⋊C4C2×SD1614yes—
516C4⋊C4#13714yes—
517C4⋊C4H(5,𝔽2)12yes—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—82no—
530C22⋊C4C2×SD1618yes—
531C42—22no—
532C42—22no—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—24no—
538C42—22no—
539C42—42no—
540C42—44no—
541C42—44no—
542C42—42no—
543C42—44no—
544C42C2×SD1618yes—
545C42H(5,𝔽2)14yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C22×SD1612yes—
560C23C23×D411yescharacteristic
561Q8—42no—
562Q8—42no—
563Q8—44no—
564Q8—44no—
565Q8—44no—
566Q8—42no—
567Q8—42no—
568Q8—44no—
569Q8—88no—
570Q8—84no—
571Q8—88no—
572Q8—84no—
573D4—42no—
574D4—42no—
575D4—84no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—28no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—88no—
623C2×C4—84no—
624C2×C4#53512yes—
625C2×C4#53512yes—
626C2×C4C22×SD1614yes—
627C2×C4C22×SD1612yes—
628C2×C4C22×SD1614yes—
629C2×C4C22×SD1612yes—
630C2×C4C2×#13714yes—
631C2×C4C2×#13712yes—
632C2×C4#57812yes—
633C2×C4C2×H(5,𝔽2)12yes—
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4#58511yescharacteristic
636C2×C4#58512yes—
637C8—42no—
638C8—42no—
639C8—44no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22C2×#53511yescharacteristic
655C22#348811yescharacteristic
656C22#350011yescharacteristic
657C22C23×SD1611yescharacteristic
658C22C22×#13711yescharacteristic
659C22C2×#57811yescharacteristic
660C22#358611yescharacteristic
661C4—24no—
662C4—24no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—24no—
667C4—42no—
668C4—41no—
669C4—42no—
670C4—81no—
671C4—81no—
672C4—82no—
673C4—82no—
674C4—84no—
675C4—82no—
676C4—82no—
677C4#320614yes—
678C4#354012yes—
679C4#354612yes—
680C2—42no—
681C2—161no—
682C2#4298711yescharacteristic
683C2C2×#320611yescharacteristic
684C2#6081411yescharacteristic
685C2#6194811yescharacteristic
686C2#6198411yescharacteristic
687C2#6252611yescharacteristic
688C2#6258911yescharacteristic
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. #35281C2semidirect
3. #32800C2semidirect
4. #31288C2semidirect
6. #30339C2semidirect
7. #30060C2semidirect
8. #29688C2semidirect
9. #28748C2semidirect
10. #28400C2semidirect
11. #27769C2semidirect
12. #25856C2semidirect
13. #25276C2semidirect
14. #24630C2semidirect
17. #22639C2semidirect
300. #117SD16semidirect

Maximal subgroups

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

subgroupindex
#364732
#352812
#328002
#312882
#305222
#303392
#300602
#296882
#287482
#284002
#277692
#258562
#252762
#246302
#242092
#235682
#226392