Finite Group Databasebeta

← #6540310

Subgroups — #6540310

Order 512.

1933 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
1#36127C211yesmaximal, characteristic, complemented
2#34316C211yesmaximal, characteristic, complemented
3#30169C212yesmaximal, complemented
4#29980C211yesmaximal, characteristic, complemented
5#29141C211yesmaximal, characteristic, complemented
6#29113C212yesmaximal, complemented
7#28157C211yesmaximal, characteristic, complemented
8#26680C214yesmaximal, complemented
9#24968C212yesmaximal, complemented
10#24916C214yesmaximal, complemented
11#24237C212yesmaximal, complemented
12#23932C212yesmaximal, complemented
13#23676C211yesmaximal, characteristic, complemented
14#23576C212yesmaximal, complemented
15#23002C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic, complemented
17#22758C212yesmaximal, complemented
18C2×#547—21no—
19C2×#547C2211yescharacteristic
20C2×#532C2212yes—
21C2×#461C2212yes—
22C2×#449C2214yes—
23C2×#440C2214yes—
24C2×C4×D8C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2547—21no—
28#2537—21no—
29#2537C2212yes—
30#2432—22no—
31#2422—22no—
32#2422C2214yes—
33#2374—22no—
34#2365C2211yescharacteristic
35#2363—21no—
36#2362C2212yes—
37#2354C2211yescharacteristic
38#2354C2211yescharacteristic
39D4×C4⋊C4—21no—
40D4×C4⋊C4C2212yes—
41#2345—21no—
42#2341C2212yes—
43#2335—22no—
44#2324—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48#2297C2212yes—
49#2293C2211yescharacteristic
50#2274—21no—
51C42×D4C2211yescharacteristic
52#2099C2212yes—
53#2093C2212yes—
54#2083C2211yescharacteristic
55#2076C2214yes—
56#2072C2212yes—
57#2071C2211yescharacteristic
58#2067C2212yes—
59#2057C2212yes—
60#2054C2212yes—
61#2041C2211yescharacteristic
62#2036C2214yes—
63#2033C2214yes—
64#2024C2212yes—
65#2024C2212yes—
66#2023C2212yes—
67#2014C2212yes—
68#2011C2211yescharacteristic
69#2000C2214yes—
70#1956C2212yes—
71#1954C2212yes—
72#1947C2212yes—
73#1936C2214yes—
74#1935C2214yes—
75#1924C2214yes—
76#1921C2214yes—
77#1880C2214yes—
78#1879C2214yes—
79#1877C2214yes—
80#1875C2214yes—
81#1863C2214yes—
82#1834C2212yes—
83#1831C2211yescharacteristic
84#1825C2214yes—
85#1817C2212yes—
86#1803C2214yes—
87#1802C2212yes—
88#1802C2212yes—
89#1788C2214yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1762C2214yes—
93C4×#102C2212yes—
94#1749C2212yes—
95C22×#41—22no—
96C22×#41C2312yes—
97C22×SD16C2314yes—
98C22×D8C2312yes—
99#547—24no—
100#547—24no—
101#547—48no—
102#532—48no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×#127C2311yescharacteristic
108C2×#124—22no—
109C2×#123—22no—
110C2×#123—24no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2312yes—
113C2×#121—21no—
114C2×#121—22no—
115C2×#121—22no—
116C2×#121C2314yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—22no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2312yes—
125C2×C4×D4C2314yes—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117—21no—
130C2×#117C2312yes—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—22no—
133C22×C42C2312yes—
134#461—28no—
135#449—48no—
136#440—48no—
137C4×D8—28no—
138C2×#107C2312yes—
139C2×#107C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153C2×#97C2314yes—
154C2×C4×C8C2312yes—
155#397—24no—
156#394—24no—
157#394—24no—
158#391—22no—
159#391C2312yes—
160#391C2312yes—
161#390—21no—
162#390—22no—
163#390—22no—
164#390—21no—
165#390—21no—
166#390C2312yes—
167#390C2311yescharacteristic
168#390C2311yescharacteristic
169#390C2312yes—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389C2311yescharacteristic
175#389C2311yescharacteristic
176#389C2314yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—21no—
181#388—21no—
182#388—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#386—22no—
187#386—22no—
188#385—22no—
189#385—24no—
190#385—24no—
191#384—21no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#380—22no—
198#380—21no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380C2312yes—
203#380C2314yes—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223C43C2311yescharacteristic
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2314yes—
235#336C2312yes—
236C2×#41—24no—
237C2×#41—24no—
238C2×#41—48no—
239C2×#41—44no—
240C2×#41—44no—
241C2×#41—48no—
242C2×#41—84no—
243C22×Q8—22no—
244C22×Q8C2412yes—
245C22×D4—22no—
246C22×D4—24no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4C2×D411yescharacteristic
250C22×D4C2411yescharacteristic
251C22×D4C2412yes—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—24no—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2412yes—
259C2×SD16—28no—
260C2×SD16—28no—
261C2×SD16—44no—
262C2×D8—24no—
263C2×D8—24no—
264C2×D8—42no—
265C22×C8C2412yes—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C4⋊Q8—22no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—82no—
272C4⋊Q8#4114yescomplemented
273#127—24no—
274#124—48no—
275#123—216no—
276#123—416no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—48no—
280#121—28no—
281#121—28no—
282#121—44no—
283#121—48no—
284#121—48no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×D4—28no—
288C4×D4—24no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—42no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—48no—
298C4×D4—44no—
299C4×D4#4114yescomplemented
300#117—44no—
301#117—44no—
302#117—44no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—48no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—44no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—24no—
367#107—44no—
368C4⋊C8—24no—
369C4⋊C8—42no—
370#103—28no—
371#103—44no—
372#103—44no—
373#102—24no—
374#102—24no—
375#102—24no—
376#102—42no—
377#102—44no—
378#102—44no—
379C22⋊C8—44no—
380C22⋊C8—44no—
381#97—44no—
382C4×C8—24no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—21no—
392#95—24no—
393#95—24no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—21no—
398C24—22no—
399C24C22×D412yes—
400#41—88no—
401#41—88no—
402#41—832no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8C2×#4114yes—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—28no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4—84no—
433C2×D4C2×#4112yes—
434C2×D4H(5,𝔽2)14yes—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—84no—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C2511yescharacteristic
474SD16—416no—
475D8—48no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C4⋊C4—28no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—81no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4C2×#4112yes—
514C22⋊C4—28no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—48no—
524C22⋊C4—44no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—81no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C42—22no—
534C42—22no—
535C42—24no—
536C42—22no—
537C42—22no—
538C42—42no—
539C42—42no—
540C42—41no—
541C42—44no—
542C42—42no—
543C42—82no—
544C42—82no—
545C42—82no—
546C42AGL(1,ℤ/8)14yes—
547C42C2×#4112yes—
548C42C2×#4112yes—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—41no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23—41no—
565C23—42no—
566C23—41no—
567C23—42no—
568C23—41no—
569C23—42no—
570C23C23×D411yescharacteristic
571Q8—28no—
572Q8—44no—
573Q8—88no—
574D4—24no—
575D4—24no—
576D4—24no—
577D4—48no—
578D4—48no—
579D4—44no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—88no—
584D4—816no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—48no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—84no—
631C2×C4—88no—
632C2×C4—816no—
633C2×C4—88no—
634C2×C4#52512yes—
635C2×C4#52512yes—
636C2×C4C2×AGL(1,ℤ/8)12yes—
637C2×C4C2×AGL(1,ℤ/8)12yes—
638C2×C4C2×AGL(1,ℤ/8)12yes—
639C2×C4#57512yes—
640C2×C4#57812yes—
641C2×C4C22×#4111yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C2×H(5,𝔽2)12yes—
644C2×C4#58512yes—
645C8—42no—
646C8—42no—
647C8—44no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—44no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—42no—
658C22—44no—
659C22—42no—
660C22—42no—
661C22—81no—
662C22—82no—
663C22—82no—
664C22—81no—
665C22—82no—
666C22—81no—
667C22—81no—
668C22—82no—
669C22C2×#52511yescharacteristic
670C22#348411yescharacteristic
671C22#350011yescharacteristic
672C22C22×AGL(1,ℤ/8)11yescharacteristic
673C22C2×#57511yescharacteristic
674C22C2×#57811yescharacteristic
675C22#358511yescharacteristic
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—42no—
681C4—41no—
682C4—42no—
683C4—42no—
684C4—44no—
685C4—42no—
686C4—81no—
687C4—82no—
688C4—82no—
689C4—81no—
690C4—84no—
691C4#311514yes—
692C4#349412yes—
693C4#351312yes—
694C2—82no—
695C2—82no—
696C2—84no—
697C2#3573011yescharacteristic
698C2C2×#311511yescharacteristic
699C2#6060011yescharacteristic
700C2#6158211yescharacteristic
701C2#6159511yescharacteristic
702C2#6253811yescharacteristic
703C2#6258611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36127C2semidirect
2. #34316C2semidirect
3. #30169C2semidirect
4. #29980C2semidirect
5. #29141C2semidirect
6. #29113C2semidirect
7. #28157C2semidirect
8. #26680C2semidirect
9. #24968C2semidirect
10. #24916C2semidirect
11. #24237C2semidirect
12. #23932C2semidirect
13. #23676C2semidirect
14. #23576C2semidirect
15. #23002C2semidirect
16. #22839C2semidirect
17. #22758C2semidirect
272. C4⋊Q8#41semidirect
299. C4×D4#41semidirect

Maximal subgroups

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

subgroupindex
#361272
#343162
#301692
#299802
#291412
#291132
#281572
#266802
#249682
#249162
#242372
#239322
#236762
#235762
#230022
#228392
#227582