Finite Group Databasebeta

← #6477615

Subgroups — #6477615

Order 512.

2050 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3092C212yesmaximal, complemented
2#35032C211yesmaximal, characteristic
3C2×#2345C211yesmaximal, characteristic, complemented
4#33110C212yesmaximal, complemented
5#33109C211yesmaximal, characteristic, complemented
6#33107C211yesmaximal, characteristic, complemented
7#29761C212yesmaximal, complemented
8#29757C211yesmaximal, characteristic, complemented
9#29755C211yesmaximal, characteristic, complemented
10#24591C214yesmaximal, complemented
11#24590C214yesmaximal, complemented
12#24491C214yesmaximal, complemented
13#24490C214yesmaximal, complemented
14#21934C211yesmaximal, characteristic, complemented
15#21769C212yesmaximal, complemented
16C2×#525C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18#3092—216no—
19C2×#484C2212yes—
20C2×#484C2212yes—
21C2×#481C2212yes—
22C2×#481C2212yes—
23C2×#483C2214yes—
24C2×#480C2214yes—
25C2×#428C2214yes—
26C2×#417C2214yes—
27C2×#407C2212yes—
28#2857C2211yescharacteristic
29#2856C2211yescharacteristic
30#2850C2212yes—
31#2617—21no—
32#2617—21no—
33#2602—22no—
34#2602—22no—
35#2500C2212yes—
36#2499C2212yes—
37#2432C2212yes—
38#2431C2212yes—
39#2345—22no—
40#2345—22no—
41#2345C2214yes—
42#2345C2214yes—
43#2325—21no—
44#2325—21no—
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#387—21no—
48C2×#387—21no—
49C2×#380—21no—
50C2×#380C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4—21no—
54C2×C4×C22⋊C4C2211yescharacteristic
55#2085C2214yescomplemented
56#2084C2212yescomplemented
57#2083C2212yescomplemented
58#2068C2214yescomplemented
59#2067C2214yescomplemented
60#2066C2214yescomplemented
61#2065C2214yescomplemented
62#2043C2212yescomplemented
63#2042C2214yescomplemented
64#2041C2212yescomplemented
65#1983C2214yescomplemented
66#1982C2212yescomplemented
67#1981C2212yescomplemented
68#1949C2214yescomplemented
69#1948C2212yescomplemented
70#1947C2212yescomplemented
71#1928C2214yescomplemented
72#1927C2214yescomplemented
73#1926C2214yescomplemented
74#1925C2214yescomplemented
75#1895C2214yes—
76#1894C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1872C2212yes—
80#1872C2212yes—
81#1851C2214yes—
82#1851C2214yes—
83#1825C2214yes—
84#1797C2211yescharacteristic
85#1797C2211yescharacteristic
86#1797C2212yes—
87#1788C2214yes—
88C22×#41C2312yes—
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C22×M16C2314yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#124C2312yes—
95#525—28no—
96#525—28no—
97C2×#123—22no—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×#121—21no—
101C2×#121—21no—
102C2×#121C2312yes—
103C2×#121C2312yes—
104C2×#120—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×#117C2312yes—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4D418yescomplemented
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4D411yescharacteristic
121C22×C22⋊C4D411yescharacteristic
122C22×C42—21no—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#484—28no—
127#484—28no—
128#483—216no—
129#481—28no—
130#481—28no—
131#480—216no—
132#428—216no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2312yes—
140#417—216no—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149#407—28no—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2314yes—
153#397—22no—
154#397—22no—
155#394—22no—
156#394—22no—
157#394—22no—
158#394—22no—
159#392—22no—
160#392—22no—
161#391C2312yes—
162#391C2312yes—
163#390—21no—
164#390—21no—
165#390C2312yes—
166#390C2312yes—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389C2312yes—
173#389C2314yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—22no—
180#387—22no—
181#387—24no—
182#387—24no—
183#387—24no—
184#387—24no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#385—22no—
190#385—22no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#382C2312yes—
194#380—21no—
195#380—21no—
196#380—22no—
197#380—24no—
198#380—24no—
199#380C2314yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—24no—
207C4×C22⋊C4—24no—
208C4×C22⋊C4D412yes—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4C2312yes—
211C4×C22⋊C4C2312yes—
212C2×#95—21no—
213C2×#95—21no—
214C2×#95—21no—
215C2×#95—21no—
216C2×#95—21no—
217C2×#95D411yescharacteristic
218C2×#95D411yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#336C2312yes—
224#336C2312yes—
225#336C2314yes—
226C25C2×D411yescharacteristic
227C2×#41—24no—
228C2×#41—24no—
229C2×#41—44no—
230C2×#41—44no—
231C22×Q8C2412yes—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2412yes—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—28no—
241C23×C4—24no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—42no—
246C23×C4—42no—
247C23×C4—82no—
248C23×C4—82no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D414yes—
255C23×C4C2×D411yescharacteristic
256C23×C4#4114yes—
257C23×C4C2411yescharacteristic
258C23×C4C2412yes—
259C2×M16—28no—
260C2×M16—44no—
261C2×M16—44no—
262C22×C8C2414yes—
263C22×C8C2414yes—
264C4⋊Q8—42no—
265#127—42no—
266#124—44no—
267#123—48no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272#121—24no—
273#121—24no—
274#121—24no—
275#121—24no—
276#121—44no—
277#121—44no—
278#120—48no—
279C4×D4—48no—
280C4×D4—48no—
281#117—24no—
282#117—24no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—216no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4C2×D4116yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—42no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—82no—
343C2×C42—82no—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—28no—
351#107—28no—
352#106—28no—
353#106—28no—
354C4⋊C8—44no—
355C4⋊C8—42no—
356C4⋊C8—42no—
357#103—28no—
358#103—28no—
359#103—44no—
360#103—44no—
361#102—28no—
362#102—28no—
363#102—44no—
364#102—44no—
365C22⋊C8—28no—
366C22⋊C8—24no—
367C22⋊C8—24no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—41no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24#12114yes—
393C24C22×D412yes—
394C24C22×D411yescharacteristic
395#41—416no—
396#41—88no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—42no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—81no—
413C2×D4—81no—
414C2×D4—81no—
415C2×D4—81no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4—82no—
419C2×D4—82no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—216no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4—84no—
465C22×C4—84no—
466C22×C4—816no—
467C22×C4#12114yes—
468C22×C4#12118yes—
469C22×C4#13714yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D414yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C2×#4114yes—
481C22×C4C2×#4112yes—
482C22×C4C2511yescharacteristic
483M16—416no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C4⋊C4—216no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—81no—
509C4⋊C4—81no—
510C4⋊C4—84no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—81no—
514C4⋊C4—81no—
515C4⋊C4C22×D414yes—
516C4⋊C4C22×D414yes—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—41no—
537C42—41no—
538C42—42no—
539C42—44no—
540C42—82no—
541C42—82no—
542C42—84no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—28no—
552C23—22no—
553C23—22no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23C2×#12114yes—
568C23C2×#12112yes—
569C23C2×#13712yes—
570C23#57512yes—
571C23C23×D411yescharacteristic
572Q8—44no—
573Q8—44no—
574D4—44no—
575D4—44no—
576D4—88no—
577D4—88no—
578D4—88no—
579D4—162no—
580D4—162no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—88no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—816no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4C2×#12114yes—
639C2×C4C2×#12112yes—
640C2×C4C2×#13712yes—
641C2×C4C2×#13712yes—
642C2×C4C23×D411yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4#58512yes—
645C2×C4#58512yes—
646C8—44no—
647C8—44no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—416no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22#303714yes—
670C22C22×#12111yescharacteristic
671C22#348511yescharacteristic
672C22#348511yescharacteristic
673C22C22×#13711yescharacteristic
674C22C2×#57511yescharacteristic
675C22#358511yescharacteristic
676C22#358511yescharacteristic
677C4—22no—
678C4—22no—
679C4—24no—
680C4—22no—
681C4—44no—
682C4—44no—
683C4—42no—
684C4—41no—
685C4—41no—
686C4—42no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4#344912yes—
693C4#349112yes—
694C2—24no—
695C2—44no—
696C2—162no—
697C2#3508911yescharacteristic
698C2C2×#303711yescharacteristic
699C2#6042311yescharacteristic
700C2#6170911yescharacteristic
701C2#6171311yescharacteristic
702C2#6171811yescharacteristic
703C2#6171911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35032C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#3092C2semidirect
3. C2×#2345C2semidirect
4. #33110C2semidirect
5. #33109C2semidirect
6. #33107C2semidirect
7. #29761C2semidirect
8. #29757C2semidirect
9. #29755C2semidirect
10. #24591C2semidirect
11. #24590C2semidirect
12. #24491C2semidirect
13. #24490C2semidirect
14. #21934C2semidirect
15. #21769C2semidirect
55. #2085C22semidirect
56. #2084C22semidirect
57. #2083C22semidirect
58. #2068C22semidirect
59. #2067C22semidirect
60. #2066C22semidirect
61. #2065C22semidirect
62. #2043C22semidirect
63. #2042C22semidirect
64. #2041C22semidirect
65. #1983C22semidirect
66. #1982C22semidirect
67. #1981C22semidirect
68. #1949C22semidirect
69. #1948C22semidirect
70. #1947C22semidirect
71. #1928C22semidirect
72. #1927C22semidirect
73. #1926C22semidirect
74. #1925C22semidirect
113. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30922
#350322
C2×#23452
#331102
#331092
#331072
#297612
#297572
#297552
#245912
#245902
#244912
#244902
#219342
#217692