Finite Group Databasebeta

← #6489398

Subgroups — #6489398

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic, complemented
2#34275C211yesmaximal, characteristic
3#32963C212yesmaximal, complemented
4#32962C212yesmaximal, complemented
5#30083C212yesmaximal, complemented
6#30075C212yesmaximal, complemented
7#28722C211yesmaximal, characteristic, complemented
8#28720C211yesmaximal, characteristic, complemented
9#28519C211yesmaximal, characteristic, complemented
10#28508C211yesmaximal, characteristic, complemented
11#28211C212yesmaximal, complemented
12#28210C212yesmaximal, complemented
13#27958C212yesmaximal, complemented
14#27587C212yesmaximal, complemented
15#27582C212yesmaximal, complemented
16#25867C212yesmaximal
17#22232C212yesmaximal, complemented
18#22228C212yesmaximal, complemented
19#20781C211yesmaximal, characteristic, complemented
20C2×#535C2211yescharacteristic
21C2×#520C2211yescharacteristic
22C2×#518C2211yescharacteristic
23C2×#490C2212yes—
24C2×#488C2212yes—
25C2×#477C2212yescomplemented
26C2×#476C2212yes—
27C2×#474C2212yes—
28C2×#474C2212yescomplemented
29#2868C2211yescharacteristic, complemented
30#2859C2212yescomplemented
31#2859C2211yescharacteristic, complemented
32#2629C2212yes—
33#2628C2212yes—
34#2570—22no—
35#2550—22no—
36#2441—22no—
37#2439—22no—
38#2363—22no—
39#2361—22no—
40#2358—22no—
41#2358—22no—
42#2331C2212yescomplemented
43#2330C2212yescomplemented
44#2328C2211yescharacteristic
45#2328C2211yescharacteristic
46#2326C2211yescharacteristic
47#2326C2211yescharacteristic
48#2317C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C4×#127C2211yescharacteristic
51C4×#124C2211yescharacteristic
52C4×#124C2211yescharacteristic
53#2297C2211yescharacteristic
54C2×#390C2211yescharacteristic
55#2285C2212yes—
56C4×#117C2211yescharacteristic
57#2096C2212yescomplemented
58#2093C2212yes—
59#2093C2212yes—
60#2093C2212yescomplemented
61#2085C2214yescomplemented
62#2083C2214yescomplemented
63#2072C2212yescomplemented
64#2071C2212yes—
65#2071C2212yes—
66#2071C2212yescomplemented
67#2042C2214yes—
68#2041C2214yes—
69#1992C2212yescomplemented
70#1991C2212yescomplemented
71#1959C2212yes—
72#1957C2212yes—
73#1928C2214yescomplemented
74#1926C2214yescomplemented
75#1925C2214yes—
76#1925C2214yes—
77#1864C2214yes—
78#1863C2214yescomplemented
79#1862C2214yes—
80#1862C2214yescomplemented
81#1854C2214yes—
82#1852C2214yes—
83#1847C2212yescomplemented
84#1845C2212yescomplemented
85#1839C2212yescomplemented
86#1832C2212yescomplemented
87#1817C2212yes—
88#1805C2212yescomplemented
89#1804C2212yes—
90#1804C2212yes—
91#1804C2212yescomplemented
92#1798C2212yes—
93#1788C2214yes—
94#1786C2214yes—
95#1767C2211yescharacteristic, complemented
96C4×C4⋊C8C2211yescharacteristic, complemented
97#1750C2212yes—
98#1746C2212yescomplemented
99C22×#41C2311yescharacteristic
100C23×D4C2311yescharacteristic
101#535—24no—
102#535—24no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#125—22no—
108C2×#125C2312yes—
109C2×#124C2311yescharacteristic
110C2×#124C2311yescharacteristic
111C2×#121—22no—
112C2×#120—22no—
113#520—44no—
114#518—44no—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×#117—22no—
130C2×#117—21no—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2311yescharacteristic
136C22×C22⋊C4—22no—
137C22×C22⋊C4—21no—
138C22×C22⋊C4—21no—
139C22×C42C2311yescharacteristic
140#490—28no—
141#488—28no—
142#477—44no—
143#476—44no—
144#474—44no—
145#474—44no—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×C22⋊C8C2314yes—
161C2×#97C2312yes—
162C2×C4×C8C2312yes—
163#400—22no—
164#399—22no—
165#398—24no—
166#396—24no—
167#395—22no—
168#393—22no—
169#391—24no—
170#391C2311yescharacteristic
171#391C2311yescharacteristic
172#390—22no—
173#390—22no—
174#390—22no—
175#390C2311yescharacteristic
176#390C2311yescharacteristic
177#390C2311yescharacteristic
178#390C2311yescharacteristic
179#390C2311yescharacteristic
180#390C2311yescharacteristic
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—24no—
187#389C2314yes—
188#389C2314yes—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#387—24no—
194#387—24no—
195#386—22no—
196#386—22no—
197#386—22no—
198#386—22no—
199#386—22no—
200#385—22no—
201#385—22no—
202#385—22no—
203#384—22no—
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2312yes—
207#384C2312yes—
208#384C2312yes—
209#382—22no—
210#382—22no—
211#382C2312yes—
212#382C2311yescharacteristic
213#382C2311yescharacteristic
214#380—22no—
215#380C2314yes—
216C4×C4⋊C4—22no—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4D412yes—
219C4×C4⋊C4C2314yes—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225#376C2311yescharacteristic
226#376C2311yescharacteristic
227C43C2311yescharacteristic
228C43C2311yescharacteristic
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238C25—21no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—24no—
242C2×#41—42no—
243C22×Q8—22no—
244C22×Q8C2411yescharacteristic
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—24no—
252C22×D4—42no—
253C22×D4C2411yescharacteristic
254C22×D4C2411yescharacteristic
255C22×D4C2411yescharacteristic
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4C2×D412yes—
265C23×C4C2411yescharacteristic
266C22×C8C2414yes—
267C22×C8C2414yes—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—84no—
271#127—22no—
272#127—22no—
273#127—22no—
274#127#4114yes—
275#125—24no—
276#125—24no—
277#125—84no—
278#124—22no—
279#124—22no—
280#124—22no—
281#124#4114yes—
282#121—216no—
283#120—216no—
284C4×Q8—24no—
285C4×Q8—42no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—42no—
290C4×D4—42no—
291C4×D4—42no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—84no—
297C4×D4—84no—
298#117—22no—
299#117—22no—
300#117—24no—
301#117—42no—
302#117—44no—
303#117—84no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2414yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—84no—
348C2×C22⋊C4C2×D411yescharacteristic
349C2×C22⋊C4C2×D411yescharacteristic
350C2×C22⋊C4C2×D412yes—
351C2×C42—24no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—24no—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—44no—
368#107—44no—
369#107—44no—
370#106—44no—
371C4⋊C8—24no—
372C4⋊C8—24no—
373#103—28no—
374#103—44no—
375#102—28no—
376#102—28no—
377#102—28no—
378#102—44no—
379#102—44no—
380#102—44no—
381C22⋊C8—28no—
382#97—24no—
383C4×C8—24no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—22no—
393#95—22no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—24no—
399C24—21no—
400C24—21no—
401C24—21no—
402C24—42no—
403C24—42no—
404C24—82no—
405C24—82no—
406C24C22×D411yescharacteristic
407#41—48no—
408#41—48no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—44no—
413C2×Q8—84no—
414C2×Q8C2×#4112yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—82no—
435C2×D4—82no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4C2×#4112yes—
439C2×D4C2×#4112yes—
440C2×D4C2×#4112yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C42−1+414yes—
481C22×C4C2511yescharacteristic
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—48no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C22⋊C4—28no—
514C22⋊C4—28no—
515C22⋊C4—24no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—84no—
532C42—24no—
533C42—24no—
534C42—24no—
535C42—21no—
536C42—21no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—42no—
541C42—42no—
542C42—42no—
543C42—42no—
544C42—42no—
545C42—84no—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C42H(5,𝔽2)14yes—
549C23—22no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—48no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—88no—
570C23—164no—
571C23—162no—
572C23—162no—
573C23#53612yes—
574C23C23×D411yescharacteristic
575Q8—24no—
576Q8—42no—
577Q8—164no—
578D4—24no—
579D4—24no—
580D4—24no—
581D4—48no—
582D4—48no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—88no—
587D4—88no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—48no—
610C2×C4—48no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—162no—
628C2×C4—162no—
629C2×C4—162no—
630C2×C4—162no—
631C2×C4—162no—
632C2×C4—162no—
633C2×C4—162no—
634C2×C4#52514yes—
635C2×C4#53512yes—
636C2×C4#53512yes—
637C2×C4#53612yes—
638C2×C4#57614yes—
639C2×C4C22×#4111yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C2×H(5,𝔽2)12yes—
642C2×C4C2×2−1+412yes—
643C8—44no—
644C8—44no—
645C22—22no—
646C22—24no—
647C22—41no—
648C22—41no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—168no—
666C22C2×#52511yescharacteristic
667C22C2×#53511yescharacteristic
668C22C2×#53611yescharacteristic
669C22C2×#57612yes—
670C22#358612yes—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—24no—
675C4—24no—
676C4—41no—
677C4—41no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—81no—
682C4—81no—
683C4—81no—
684C4—84no—
685C4—81no—
686C4—84no—
687C4#352012yes—
688C4#352012yes—
689C2—42no—
690C2—82no—
691C2—82no—
692C2—82no—
693C2#3524011yescharacteristic
694C2#6059612yes—
695C2#6081711yescharacteristic
696C2#6082611yescharacteristic
697C2#6087112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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. #35294C2semidirect
3. #32963C2semidirect
4. #32962C2semidirect
5. #30083C2semidirect
6. #30075C2semidirect
7. #28722C2semidirect
8. #28720C2semidirect
9. #28519C2semidirect
10. #28508C2semidirect
11. #28211C2semidirect
12. #28210C2semidirect
13. #27958C2semidirect
14. #27587C2semidirect
15. #27582C2semidirect
17. #22232C2semidirect
18. #22228C2semidirect
19. #20781C2semidirect
25. C2×#477C22semidirect
28. C2×#474C22semidirect
29. #2868C22semidirect
30. #2859C22semidirect
31. #2859C22semidirect
42. #2331C22semidirect
43. #2330C22semidirect
57. #2096C22semidirect
60. #2093C22semidirect
61. #2085C22semidirect
62. #2083C22semidirect
63. #2072C22semidirect
66. #2071C22semidirect
69. #1992C22semidirect
70. #1991C22semidirect
73. #1928C22semidirect
74. #1926C22semidirect
78. #1863C22semidirect
80. #1862C22semidirect
83. #1847C22semidirect
84. #1845C22semidirect
85. #1839C22semidirect
86. #1832C22semidirect
88. #1805C22semidirect
91. #1804C22semidirect
95. #1767C22semidirect
96. C4×C4⋊C8C22semidirect
98. #1746C22semidirect

Maximal subgroups

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

subgroupindex
#352942
#342752
#329632
#329622
#300832
#300752
#287222
#287202
#285192
#285082
#282112
#282102
#279582
#275872
#275822
#258672
#222322
#222282
#207812