Finite Group Databasebeta

← #6564386

Subgroups — #6564386

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
1C2×#3304C212yesmaximal
2#36188C211yesmaximal, characteristic, complemented
3#34330C211yesmaximal, characteristic
4#29866C211yesmaximal, characteristic, complemented
5#29717C212yesmaximal, complemented
6#28716C211yesmaximal, characteristic, complemented
7#28571C212yesmaximal, complemented
8#28160C211yesmaximal, characteristic, complemented
9#27650C212yesmaximal, complemented
10#25325C212yesmaximal, complemented
11#25205C214yesmaximal, complemented
12#24683C212yesmaximal, complemented
13#24536C214yesmaximal, complemented
14#23679C211yesmaximal, characteristic, complemented
15#23627C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic, complemented
17#22490C212yesmaximal, complemented
18C2×#556C2212yes—
19C2×#550C2211yescharacteristic
20#3304—216no—
21C2×#486C2212yes—
22C2×#482C2214yes—
23C2×#475C2212yes—
24C2×#465C2214yes—
25C2×#460C2212yes—
26C2×#452C2214yes—
27C2×C4×Q16C2212yes—
28C2×C8×D4C2212yes—
29C2×#416C2214yes—
30#2868C2211yescharacteristic
31#2857C2212yes—
32#2568C2212yes—
33#2547—21no—
34#2547—21no—
35#2499C2214yes—
36#2496—22no—
37#2496—22no—
38#2366—21no—
39#2366C2211yescharacteristic
40#2365—21no—
41#2365—21no—
42#2364—22no—
43#2363C2212yes—
44#2357—21no—
45#2355C2211yescharacteristic
46#2354C2211yescharacteristic
47#2339C2212yes—
48#2333—22no—
49#2328—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×#121C2212yes—
53#2297C2212yes—
54#2295C2211yescharacteristic
55#2285C2212yes—
56#2275—21no—
57C42×D4C2211yescharacteristic
58#2084C2212yes—
59#2084C2212yes—
60#2083C2211yescharacteristic
61#2074C2214yes—
62#2073C2214yes—
63#2072C2212yes—
64#2072C2211yescharacteristic
65#2072C2212yes—
66#2071C2212yes—
67#2071C2212yes—
68#2071C2212yes—
69#2068C2212yes—
70#2056C2214yes—
71#2056C2212yes—
72#2055C2214yes—
73#2043C2211yescharacteristic
74#2016C2211yescharacteristic
75#2013C2212yes—
76#1990C2212yes—
77#1988C2212yes—
78#1949C2214yes—
79#1939C2212yes—
80#1927C2214yes—
81#1892C2214yes—
82#1877C2214yes—
83#1877C2214yes—
84#1873C2214yes—
85#1850C2214yes—
86#1831C2211yescharacteristic
87#1825C2214yes—
88#1804C2214yes—
89#1803C2212yes—
90#1802C2214yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94#1735C2214yes—
95C2×#336C2212yes—
96C22×#41C2312yes—
97C22×Q16C2312yes—
98C22×SD16C2314yes—
99C23×C8C2314yes—
100#556—28no—
101#556—28no—
102#550—24no—
103#550D418yes—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×#127—21no—
109C2×#127C2311yescharacteristic
110C2×#126—24no—
111C2×#124—22no—
112C2×C22⋊Q8C2314yes—
113C2×#121—22no—
114C2×#121C2312yes—
115C2×C4×Q8—22no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×C4×D4C2311yescharacteristic
128C2×#117C2314yes—
129C2×#117C2312yes—
130C22×C42C2312yes—
131#486—28no—
132#482—216no—
133#475—28no—
134#465—216no—
135#460—28no—
136#452—216no—
137C4×Q16—28no—
138C8×D4—28no—
139C2×#107C2312yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2312yes—
144#416—216no—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C4×C8C2312yes—
156#400—24no—
157#397—24no—
158#397—24no—
159#391—22no—
160#391—22no—
161#391C2312yes—
162#391C2314yes—
163#391C2311yescharacteristic
164#391C2312yes—
165#391C2311yescharacteristic
166#390—21no—
167#390—21no—
168#390—22no—
169#390—21no—
170#390—21no—
171#390C2312yes—
172#390C2312yes—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389C2311yescharacteristic
178#389C2311yescharacteristic
179#388—22no—
180#388—22no—
181#388—22no—
182#387—21no—
183#387—21no—
184#387—22no—
185#387—24no—
186#387—24no—
187#386—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#386—22no—
194#384—21no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#382—21no—
201#382C2314yes—
202#381—22no—
203#380—22no—
204#380—22no—
205#380—22no—
206#380—22no—
207#380C2312yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4C2312yes—
227#376—21no—
228C2×#95—22no—
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2311yescharacteristic
240#336C2312yes—
241#336C2314yes—
242C2×#41—24no—
243C2×#41—44no—
244C2×#41—44no—
245C2×#41C2×D418yes—
246C22×Q8—22no—
247C22×Q8C2412yes—
248C22×Q8C2411yescharacteristic
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4C2×D411yescharacteristic
254C22×D4C2412yes—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2412yes—
261C2×Q16—24no—
262C2×Q16—42no—
263C2×Q16—42no—
264C2×SD16—28no—
265C2×SD16—44no—
266C2×SD16—44no—
267C22×C8—28no—
268C22×C8—44no—
269C22×C8—44no—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—42no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278#127—28no—
279#127—22no—
280#127C2×D414yes—
281#126—416no—
282#124—48no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285#121—24no—
286#121—216no—
287#121C2×D418yes—
288C4×Q8—22no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—48no—
292C4×Q8C2×D414yes—
293C4×D4—28no—
294C4×D4—24no—
295C4×D4—42no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—82no—
301C4×D4—84no—
302C4×D4C2×D414yes—
303C4×D4#4114yes—
304#117—28no—
305#117—28no—
306#117—44no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C42—21no—
347C2×C42—24no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—44no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—24no—
364#106—28no—
365C4⋊C8—24no—
366C4⋊C8—44no—
367C4⋊C8—42no—
368#103—24no—
369#103—28no—
370#103—24no—
371#103—42no—
372#103—44no—
373#103—42no—
374#102—28no—
375#102—28no—
376#102—28no—
377C22⋊C8—28no—
378C4×C8—24no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—21no—
394#95—24no—
395#95—24no—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—22no—
400C24C22×D412yes—
401#41—416no—
402#41—88no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8C22×D412yes—
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—81no—
428C2×D4—81no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4#12114yes—
434C2×D4C22×D414yes—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D414yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4114yes—
471C22×C4C2511yescharacteristic
472Q16—48no—
473SD16—88no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—88no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—81no—
509C4⋊C4—81no—
510C4⋊C4#12114yes—
511C4⋊C4C22×D412yes—
512C22⋊C4—28no—
513C22⋊C4—24no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4#12118yes—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—42no—
533C42—42no—
534C42—41no—
535C42—48no—
536C42—44no—
537C42—42no—
538C42—82no—
539C42—82no—
540C42#13514yes—
541C42C22×D412yes—
542C42C2×#4112yes—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—24no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23C2×#12114yes—
557C23C23×D411yescharacteristic
558Q8—24no—
559Q8—42no—
560Q8—42no—
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8—88no—
565D4—28no—
566D4—28no—
567D4—44no—
568D4—48no—
569D4—84no—
570D4—88no—
571D4—88no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—88no—
620C2×C4—164no—
621C2×C4—164no—
622C2×C4C2×#12114yes—
623C2×C4C2×#12112yes—
624C2×C4C2×#12112yes—
625C2×C4C2×#13512yes—
626C2×C4C2×#13512yes—
627C2×C4C2×#13512yes—
628C2×C4#57512yes—
629C2×C4#57712yes—
630C2×C4C23×D411yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4#58512yes—
633C2×C4#58512yes—
634C8—42no—
635C8—44no—
636C8—42no—
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—
651C22C22×#12111yescharacteristic
652C22#348511yescharacteristic
653C22#350011yescharacteristic
654C22C22×#13511yescharacteristic
655C22C2×#57511yescharacteristic
656C22C2×#57711yescharacteristic
657C22#358611yescharacteristic
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—24no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—41no—
667C4—42no—
668C4—82no—
669C4—81no—
670C4—82no—
671C4—81no—
672C4—82no—
673C4—84no—
674C4#303514yes—
675C4#348512yes—
676C4#352112yes—
677C2—44no—
678C2—84no—
679C2#3600911yescharacteristic
680C2C2×#303511yescharacteristic
681C2#6042911yescharacteristic
682C2#6170111yescharacteristic
683C2#6173911yescharacteristic
684C2#6253111yescharacteristic
685C2#6258711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
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. #29866C2semidirect
5. #29717C2semidirect
6. #28716C2semidirect
7. #28571C2semidirect
8. #28160C2semidirect
9. #27650C2semidirect
10. #25325C2semidirect
11. #25205C2semidirect
12. #24683C2semidirect
13. #24536C2semidirect
14. #23679C2semidirect
15. #23627C2semidirect
16. #22696C2semidirect
17. #22490C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33042
#361882
#343302
#298662
#297172
#287162
#285712
#281602
#276502
#253252
#252052
#246832
#245362
#236792
#236272
#226962
#224902