Finite Group Databasebeta

← #6540311

Subgroups — #6540311

Order 512.

1921 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36127C211yesmaximal, characteristic
2#34334C211yesmaximal, characteristic, complemented
3#30176C212yesmaximal, complemented
4#29984C211yesmaximal, characteristic, complemented
5#29141C211yesmaximal, characteristic, complemented
6#29111C212yesmaximal, complemented
7#28157C211yesmaximal, characteristic, complemented
8#26677C214yesmaximal, complemented
9#24965C212yesmaximal, complemented
10#24910C214yesmaximal, complemented
11#24239C212yesmaximal, complemented
12#23931C212yesmaximal, complemented
13#23678C211yesmaximal, characteristic, complemented
14#23576C212yesmaximal, complemented
15#23002C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic, complemented
17#22762C212yesmaximal, complemented
18C2×D4×Q8C2211yescharacteristic
19C2×#547—21no—
20C2×#532C2212yes—
21C2×#461C2212yes—
22C2×#448C2214yes—
23C2×#440C2214yes—
24C2×C4×SD16C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2547—21no—
28#2537—21no—
29#2537C2212yes—
30#2432—22no—
31#2422—22no—
32#2422C2214yes—
33#2369—21no—
34#2368—22no—
35#2365C2211yescharacteristic
36#2360C2212yes—
37#2355C2211yescharacteristic
38#2349—21no—
39D4×C4⋊C4—21no—
40D4×C4⋊C4C2212yes—
41D4×C4⋊C4C2211yescharacteristic
42Q8×C22⋊C4C2212yes—
43#2331—21no—
44#2329—22no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48#2295C2211yescharacteristic
49C2×#391C2212yes—
50#2275—21no—
51C42×D4C2211yescharacteristic
52#2100C2212yes—
53#2095C2212yes—
54#2084C2211yescharacteristic
55#2075C2214yes—
56#2072C2211yescharacteristic
57#2072C2212yes—
58#2065C2212yes—
59#2057C2212yes—
60#2055C2212yes—
61#2041C2211yescharacteristic
62#2035C2214yes—
63#2033C2214yes—
64#2024C2212yes—
65#2024C2212yes—
66#2024C2212yes—
67#2014C2212yes—
68#2014C2211yescharacteristic
69#2000C2214yes—
70#1956C2212yes—
71#1954C2212yes—
72#1947C2212yes—
73#1935C2214yes—
74#1935C2214yes—
75#1924C2214yes—
76#1920C2214yes—
77#1880C2214yes—
78#1877C2214yes—
79#1877C2214yes—
80#1874C2214yes—
81#1863C2214yes—
82#1834C2212yes—
83#1831C2211yescharacteristic
84#1823C2214yes—
85#1817C2212yes—
86#1803C2212yes—
87#1803C2214yes—
88#1802C2212yes—
89#1787C2214yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1762C2214yes—
93C4×#102C2212yes—
94#1749C2212yes—
95C22×#41—22no—
96C23×Q8C2312yes—
97C22×SD16C2312yes—
98C22×SD16C2314yes—
99D4×Q8—24no—
100D4×Q8—24no—
101#547—48no—
102#532—48no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×#127C2311yescharacteristic
109C2×#124—22no—
110C2×#123—24no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2312yes—
114C2×#121—22no—
115C2×#121—22no—
116C2×#121C2314yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—22no—
125C2×C4×D4D411yescharacteristic
126C2×C4×D4D411yescharacteristic
127C2×C4×D4C2312yes—
128C2×C4×D4C2314yes—
129C2×C4×D4C2311yescharacteristic
130C22×C4⋊C4—22no—
131C22×C4⋊C4C2312yes—
132C22×C42C2312yes—
133#461—28no—
134#448—48no—
135#440—48no—
136C4×SD16—28no—
137C2×#107C2314yes—
138C2×#106C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×#97C2314yes—
153C2×C4×C8C2312yes—
154#397—24no—
155#394—24no—
156#394—24no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2312yes—
163#391C2311yescharacteristic
164#391C2311yescharacteristic
165#391C2312yes—
166#391C2312yes—
167#390—21no—
168#390—21no—
169#390C2312yes—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389C2311yescharacteristic
177#389C2311yescharacteristic
178#389C2314yes—
179#387—21no—
180#387—22no—
181#387—21no—
182#387—22no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—22no—
187#385—22no—
188#385—24no—
189#385—24no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#382—21no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380C2312yes—
202#380C2314yes—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376—21no—
221C2×#95—22no—
222C2×#95—22no—
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—48no—
237C2×#41—48no—
238C2×#41—84no—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8C2412yes—
246C22×Q8C2411yescharacteristic
247C22×D4—24no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4C2×D411yescharacteristic
251C22×D4C2412yes—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—24no—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2412yes—
259C2×SD16—24no—
260C2×SD16—28no—
261C2×SD16—28no—
262C2×SD16—24no—
263C2×SD16—44no—
264C2×SD16—42no—
265C22×C8C2412yes—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C4⋊Q8—28no—
269C4⋊Q8—22no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8—82no—
273C4⋊Q8#4114yescomplemented
274#127—24no—
275#124—48no—
276#123—416no—
277C22⋊Q8—216no—
278C22⋊Q8—24no—
279C22⋊Q8—24no—
280C22⋊Q8—48no—
281#121—28no—
282#121—44no—
283#121—48no—
284#121—48no—
285C4×Q8—22no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×Q8#4114yescomplemented
289C4×D4—28no—
290C4×D4—24no—
291C4×D4—28no—
292C4×D4—24no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—48no—
300C4×D4—44no—
301C4×D4—44no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—48no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—44no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—44no—
360#106—24no—
361C4⋊C8—24no—
362C4⋊C8—42no—
363#103—24no—
364#103—28no—
365#103—44no—
366#103—44no—
367#103—44no—
368#102—24no—
369#102—24no—
370#102—42no—
371#102—44no—
372C22⋊C8—44no—
373C22⋊C8—44no—
374#97—44no—
375C4×C8—24no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—24no—
392#95—24no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—22no—
397C24C22×D412yes—
398#41—832no—
399C2×Q8—28no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8—88no—
412C2×Q8—88no—
413C2×Q8C2×#4112yes—
414C2×Q8C2×#4114yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
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—84no—
431C2×D4AGL(1,ℤ/8)14yes—
432C2×D4H(5,𝔽2)14yes—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—84no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2511yescharacteristic
473SD16—48no—
474SD16—416no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C4⋊C4—28no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
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—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—81no—
508C4⋊C4—81no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4C2×#4112yes—
515C22⋊C4—28no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—48no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—42no—
533C42—42no—
534C42—41no—
535C42—42no—
536C42—82no—
537C42—82no—
538C42—82no—
539C42#13714yes—
540C42C2×#4112yes—
541C42C2×#4112yes—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—24no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23C2×AGL(1,ℤ/8)14yes—
556C23C23×D411yescharacteristic
557Q8—24no—
558Q8—28no—
559Q8—44no—
560Q8—42no—
561Q8—88no—
562Q8—88no—
563D4—28no—
564D4—24no—
565D4—24no—
566D4—48no—
567D4—48no—
568D4—42no—
569D4—42no—
570D4—816no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—48no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—44no—
618C2×C4—84no—
619C2×C4—88no—
620C2×C4—816no—
621C2×C4—88no—
622C2×C4—164no—
623C2×C4#52512yes—
624C2×C4#52512yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4C2×#13712yes—
627C2×C4C2×#13712yes—
628C2×C4C2×#13712yes—
629C2×C4#57712yes—
630C2×C4C22×#4111yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×H(5,𝔽2)12yes—
633C2×C4#58512yes—
634C8—42no—
635C8—42no—
636C8—44no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22C2×#52511yescharacteristic
652C22#348411yescharacteristic
653C22#350011yescharacteristic
654C22C22×AGL(1,ℤ/8)11yescharacteristic
655C22C22×#13711yescharacteristic
656C22C2×#57711yescharacteristic
657C22#358611yescharacteristic
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—41no—
663C4—42no—
664C4—42no—
665C4—44no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—82no—
670C4—81no—
671C4—82no—
672C4—81no—
673C4—84no—
674C4#311214yes—
675C4#349412yes—
676C4#351312yes—
677C2—44no—
678C2—84no—
679C2#3573011yescharacteristic
680C2C2×#311211yescharacteristic
681C2#6059911yescharacteristic
682C2#6158611yescharacteristic
683C2#6160011yescharacteristic
684C2#6254011yescharacteristic
685C2#6258311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
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
2. #34334C2semidirect
3. #30176C2semidirect
4. #29984C2semidirect
5. #29141C2semidirect
6. #29111C2semidirect
7. #28157C2semidirect
8. #26677C2semidirect
9. #24965C2semidirect
10. #24910C2semidirect
11. #24239C2semidirect
12. #23931C2semidirect
13. #23678C2semidirect
14. #23576C2semidirect
15. #23002C2semidirect
16. #22839C2semidirect
17. #22762C2semidirect
273. C4⋊Q8#41semidirect
288. C4×Q8#41semidirect

Maximal subgroups

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

subgroupindex
#361272
#343342
#301762
#299842
#291412
#291112
#281572
#266772
#249652
#249102
#242392
#239312
#236782
#235762
#230022
#228392
#227622