Finite Group Databasebeta

← #6566104

Subgroups — #6566104

Order 512.

1971 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3353C212yesmaximal
2#36188C211yesmaximal, characteristic, complemented
3#34308C211yesmaximal, characteristic
4#29864C211yesmaximal, characteristic, complemented
5#29757C212yesmaximal, complemented
6#28689C211yesmaximal, characteristic, complemented
7#28577C212yesmaximal, complemented
8#28205C211yesmaximal, characteristic, complemented
9#27650C212yesmaximal, complemented
10#25456C212yesmaximal, complemented
11#25230C214yesmaximal, complemented
12#24816C212yesmaximal, complemented
13#24588C214yesmaximal, complemented
14#23677C211yesmaximal, characteristic, complemented
15#23657C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22504C212yesmaximal, complemented
18C2×#556C2212yes—
19C2×#547C2211yescharacteristic
20C2×#520—21no—
21#3353—216no—
22C2×#500C2212yes—
23C2×#484C2214yes—
24C2×#482C2214yes—
25C2×#475C2212yes—
26C2×#474C2212yes—
27C2×#468C2214yes—
28C2×C4×SD16C2212yes—
29C2×#436C2212yes—
30C2×#428C2214yes—
31#2868C2211yescharacteristic
32#2857C2212yes—
33#2568C2212yes—
34#2547—21no—
35#2547—21no—
36#2499C2214yes—
37#2496—22no—
38#2496—22no—
39#2365—21no—
40#2365—21no—
41#2365C2211yescharacteristic
42#2361C2212yes—
43#2360—21no—
44#2359—22no—
45D4×C4⋊C4C2211yescharacteristic
46D4×C4⋊C4C2211yescharacteristic
47#2344C2212yes—
48#2339—21no—
49#2323—21no—
50#2322—22no—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×#121C2212yes—
54#2293C2211yescharacteristic
55C2×#389C2212yes—
56#2285C2212yes—
57C42×D4C2211yescharacteristic
58#2085C2214yes—
59#2085C2214yes—
60#2084C2212yes—
61#2084C2211yescharacteristic
62#2083C2212yes—
63#2083C2212yes—
64#2083C2212yes—
65#2083C2212yes—
66#2071C2211yescharacteristic
67#2071C2212yes—
68#2071C2212yes—
69#2068C2214yes—
70#2068C2214yes—
71#2066C2212yes—
72#2055C2212yes—
73#2043C2212yes—
74#2043C2211yescharacteristic
75#2013C2211yescharacteristic
76#1990C2212yes—
77#1988C2212yes—
78#1949C2214yes—
79#1947C2214yes—
80#1939C2212yes—
81#1928C2214yes—
82#1927C2214yes—
83#1894C2214yes—
84#1877C2214yes—
85#1872C2214yes—
86#1851C2214yes—
87#1831C2211yescharacteristic
88#1825C2214yes—
89#1804C2214yes—
90#1803C2212yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94#1736C2214yes—
95#1728C2212yes—
96C22×#41—21no—
97C22×SD16C2312yes—
98C22×M16C2314yes—
99#556—28no—
100#556—28no—
101#547—24no—
102#547D418yes—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#126—24no—
109C2×#124—22no—
110C2×#123—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8C2314yes—
113C2×#121C2312yes—
114#520—48no—
115C2×C4×Q8—21no—
116C2×C4×Q8—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C2×#117—21no—
129C2×#117C2314yes—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C42C2312yes—
135#500—28no—
136#484—216no—
137#482—216no—
138#475—28no—
139#474—28no—
140#468—216no—
141C4×SD16—28no—
142#436—28no—
143#428—216no—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#106C2314yes—
148C2×#106C2312yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2314yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#102C2312yes—
158C2×#102C2314yes—
159C2×C22⋊C8C2314yes—
160C2×C4×C8C2312yes—
161#400—24no—
162#397—24no—
163#397—24no—
164#391—22no—
165#391—22no—
166#391—21no—
167#391—21no—
168#391C2312yes—
169#391C2314yes—
170#390C2311yescharacteristic
171#390C2311yescharacteristic
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389C2312yes—
181#389C2311yescharacteristic
182#389C2312yes—
183#389C2312yes—
184#389C2311yescharacteristic
185#388—21no—
186#388—22no—
187#388—21no—
188#388—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—24no—
193#387—24no—
194#386—22no—
195#386—22no—
196#385—22no—
197#385—22no—
198#385—22no—
199#384—21no—
200#384—21no—
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#384C2312yes—
204#384C2312yes—
205#382C2314yes—
206#381—22no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380—22no—
211#380—21no—
212#380C2312yes—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—21no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4C2312yes—
229C2×#95—22no—
230C43C2311yescharacteristic
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#336C2311yescharacteristic
239#336C2312yes—
240#336C2312yes—
241#336C2311yescharacteristic
242#336C2314yes—
243C2×#41—28no—
244C2×#41—44no—
245C2×#41—44no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8C2412yes—
249C22×D4—22no—
250C22×D4C2×D411yescharacteristic
251C22×D4C2411yescharacteristic
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2412yes—
259C2×SD16—24no—
260C2×SD16—24no—
261C2×SD16—42no—
262C2×M16—28no—
263C2×M16—28no—
264C2×M16—44no—
265C22×C8C2414yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C4⋊Q8—24no—
269C4⋊Q8—22no—
270C4⋊Q8—24no—
271C4⋊Q8—42no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8C2×D414yes—
275#126—416no—
276#124—48no—
277#123—216no—
278C22⋊Q8—28no—
279C22⋊Q8—28no—
280C22⋊Q8—28no—
281#121—24no—
282#121C2×D418yes—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—44no—
286C4×Q8—48no—
287C4×Q8—44no—
288C4×D4—24no—
289C4×D4—22no—
290C4×D4—42no—
291C4×D4—44no—
292C4×D4—48no—
293C4×D4—48no—
294C4×D4—44no—
295C4×D4C2×D414yes—
296C4×D4C2×D414yes—
297C4×D4#4114yes—
298#117—28no—
299#117—28no—
300#117—48no—
301#117—44no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
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—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—48no—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D418yes—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2414yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2412yes—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42—44no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—28no—
368#107—28no—
369#107—28no—
370#106—24no—
371#106—28no—
372C4⋊C8—24no—
373C4⋊C8—24no—
374C4⋊C8—44no—
375#103—24no—
376#103—24no—
377#103—28no—
378#103—42no—
379#103—44no—
380#102—24no—
381#102—28no—
382C22⋊C8—28no—
383C4×C8—24no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—24no—
391#95—24no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—24no—
398#95—24no—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402C24—21no—
403C24C22×D412yes—
404#41—432no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—44no—
413C2×Q8—42no—
414C2×Q8—44no—
415C2×Q8—42no—
416C2×Q8—44no—
417C2×D4—28no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4#12114yes—
426C2×D4C22×D412yes—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—28no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D414yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C2×#4114yes—
466C22×C4C2511yescharacteristic
467SD16—48no—
468M16—416no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C82−1+414yes—
478C4⋊C4—28no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—81no—
509C4⋊C4—82no—
510C4⋊C4#12114yes—
511C4⋊C4C22×D412yes—
512C4⋊C4C22×D414yes—
513C22⋊C4—24no—
514C22⋊C4—28no—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—81no—
524C22⋊C4—82no—
525C22⋊C4#12118yes—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—24no—
530C42—22no—
531C42—42no—
532C42—42no—
533C42—41no—
534C42—44no—
535C42—48no—
536C42—44no—
537C42—44no—
538C42—42no—
539C42AGL(1,ℤ/8)14yes—
540C42C22×D412yes—
541C42C2×#4112yes—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23C2×#12114yes—
556C23C23×D411yescharacteristic
557Q8—48no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—42no—
562Q8—88no—
563D4—24no—
564D4—28no—
565D4—416no—
566D4—42no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—416no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—48no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—164no—
617C2×C4—164no—
618C2×C4C2×#12114yes—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4C2×AGL(1,ℤ/8)12yes—
622C2×C4C2×AGL(1,ℤ/8)12yes—
623C2×C4C2×AGL(1,ℤ/8)12yes—
624C2×C4#57512yes—
625C2×C4#57812yes—
626C2×C4C23×D411yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×2−1+412yes—
629C2×C4#58512yes—
630C8—24no—
631C8—42no—
632C8—44no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22C22×#12111yescharacteristic
648C22#348511yescharacteristic
649C22#349911yescharacteristic
650C22C22×AGL(1,ℤ/8)11yescharacteristic
651C22C2×#57511yescharacteristic
652C22C2×#57811yescharacteristic
653C22#358611yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—24no—
659C4—42no—
660C4—42no—
661C4—41no—
662C4—42no—
663C4—44no—
664C4—81no—
665C4—82no—
666C4—81no—
667C4—82no—
668C4—82no—
669C4—84no—
670C4#303614yes—
671C4#348912yes—
672C4#351512yes—
673C2—44no—
674C2—82no—
675C2#3602011yescharacteristic
676C2C2×#303611yescharacteristic
677C2#6043211yescharacteristic
678C2#6172211yescharacteristic
679C2#6173011yescharacteristic
680C2#6247911yescharacteristic
681C2#6251111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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. #36188C2semidirect
4. #29864C2semidirect
5. #29757C2semidirect
6. #28689C2semidirect
7. #28577C2semidirect
8. #28205C2semidirect
9. #27650C2semidirect
10. #25456C2semidirect
11. #25230C2semidirect
12. #24816C2semidirect
13. #24588C2semidirect
14. #23677C2semidirect
15. #23657C2semidirect
16. #22700C2semidirect
17. #22504C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33532
#361882
#343082
#298642
#297572
#286892
#285772
#282052
#276502
#254562
#252302
#248162
#245882
#236772
#236572
#227002
#225042