Finite Group Databasebeta

← #6564383

Subgroups — #6564383

Order 512.

1865 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3302C212yesmaximal
2#36188C211yesmaximal, characteristic, complemented
3#34316C211yesmaximal, characteristic
4#29864C211yesmaximal, characteristic, complemented
5#29717C212yesmaximal, complemented
6#28717C211yesmaximal, characteristic, complemented
7#28575C212yesmaximal, complemented
8#28161C211yesmaximal, characteristic, complemented
9#27652C212yesmaximal, complemented
10#25323C212yesmaximal, complemented
11#25201C214yesmaximal, complemented
12#24681C212yesmaximal, complemented
13#24534C214yesmaximal, complemented
14#23677C211yesmaximal, characteristic, complemented
15#23627C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic, complemented
17#22490C212yesmaximal, complemented
18C2×#556C2212yes—
19C2×#547C2211yescharacteristic
20#3302—216no—
21C2×#487C2212yes—
22C2×#485C2214yes—
23C2×#474C2212yes—
24C2×#467C2214yes—
25C2×#462C2212yes—
26C2×#452C2214yes—
27C2×C4×SD16C2212yes—
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#2374—22no—
39#2365—21no—
40#2365—21no—
41#2365C2211yescharacteristic
42#2363—21no—
43#2362C2212yes—
44#2354C2211yescharacteristic
45#2354C2211yescharacteristic
46#2345—21no—
47#2341C2212yes—
48#2335—22no—
49#2324—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53#2297C2212yes—
54#2293C2211yescharacteristic
55#2285C2212yes—
56#2274—21no—
57C42×D4C2211yescharacteristic
58#2084C2211yescharacteristic
59#2083C2212yes—
60#2083C2212yes—
61#2074C2214yes—
62#2073C2214yes—
63#2072C2212yes—
64#2072C2212yes—
65#2072C2212yes—
66#2071C2212yes—
67#2071C2211yescharacteristic
68#2071C2212yes—
69#2066C2212yes—
70#2056C2214yes—
71#2055C2214yes—
72#2055C2212yes—
73#2043C2211yescharacteristic
74#2016C2212yes—
75#2013C2211yescharacteristic
76#1990C2212yes—
77#1988C2212yes—
78#1949C2214yes—
79#1939C2212yes—
80#1928C2214yes—
81#1892C2214yes—
82#1878C2214yes—
83#1878C2214yes—
84#1873C2214yes—
85#1850C2214yes—
86#1831C2211yescharacteristic
87#1825C2214yes—
88#1805C2214yes—
89#1803C2212yes—
90#1803C2214yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94#1735C2214yes—
95C2×#336C2212yes—
96C22×#41C2312yes—
97C22×Q16C2314yes—
98C22×SD16C2312yes—
99C23×C8C2314yes—
100#556—28no—
101#556—28no—
102#547—24no—
103#547D418yes—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×#126—24no—
110C2×#124—22no—
111C2×#123—22no—
112C2×C22⋊Q8C2314yes—
113C2×C22⋊Q8C2312yes—
114C2×#121—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—21no—
124C2×#117—21no—
125C2×#117C2314yes—
126C2×#117C2312yes—
127C22×C4⋊C4—21no—
128C22×C42C2312yes—
129#487—28no—
130#485—216no—
131#474—28no—
132#467—216no—
133#462—28no—
134#452—216no—
135C4×SD16—28no—
136C8×D4—28no—
137C2×#107C2314yes—
138C2×#106C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2312yes—
142#416—216no—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×C22⋊C8C2314yes—
153C2×C4×C8C2312yes—
154#400—24no—
155#397—24no—
156#397—24no—
157#391—22no—
158#391—22no—
159#391C2312yes—
160#391C2312yes—
161#391C2314yes—
162#391C2312yes—
163#390—21no—
164#390—22no—
165#390—22no—
166#390C2311yescharacteristic
167#390C2312yes—
168#390C2311yescharacteristic
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389C2311yescharacteristic
177#389C2311yescharacteristic
178#388—22no—
179#388—22no—
180#388—22no—
181#388—21no—
182#388—21no—
183#388—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—24no—
188#387—24no—
189#386—22no—
190#386—22no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#382C2314yes—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—21no—
204#380—22no—
205#380C2312yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4C2312yes—
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
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×Q8C2412yes—
249C22×D4—22no—
250C22×D4C2×D411yescharacteristic
251C22×D4C2411yescharacteristic
252C23×C4—21no—
253C23×C4—22no—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2412yes—
257C2×Q16—28no—
258C2×Q16—44no—
259C2×Q16—44no—
260C2×SD16—24no—
261C2×SD16—42no—
262C2×SD16—42no—
263C22×C8—28no—
264C22×C8—44no—
265C22×C8—44no—
266C22×C8C2412yes—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C4⋊Q8—24no—
270C4⋊Q8—22no—
271C4⋊Q8—24no—
272C4⋊Q8—42no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8C2×D414yes—
276#126—416no—
277#124—48no—
278#123—216no—
279C22⋊Q8—24no—
280C22⋊Q8—28no—
281C22⋊Q8—28no—
282C22⋊Q8C2×D418yes—
283#121—28no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×Q8—48no—
289C4×D4—24no—
290C4×D4—22no—
291C4×D4—42no—
292C4×D4C2×D414yes—
293C4×D4C2×D414yes—
294C4×D4#4114yes—
295#117—28no—
296#117—28no—
297#117—44no—
298#117—44no—
299#117—44no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—24no—
348C2×C42—22no—
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—28no—
364#106—24no—
365C4⋊C8—24no—
366C4⋊C8—44no—
367C4⋊C8—42no—
368#103—28no—
369#103—28no—
370#103—28no—
371#103—24no—
372#103—42no—
373#103—44no—
374#103—42no—
375#102—24no—
376#102—28no—
377C22⋊C8—28no—
378C4×C8—24no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—21no—
392#95—24no—
393#95—24no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—21no—
398C24C22×D412yes—
399#41—416no—
400#41—88no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8C22×D414yes—
414C2×D4—28no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4#12114yes—
423C2×D4C22×D412yes—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—28no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D414yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C2×#4114yes—
457C22×C4C2511yescharacteristic
458Q16—88no—
459SD16—48no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—88no—
470C4⋊C4—24no—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4#12114yes—
501C4⋊C4C22×D412yes—
502C22⋊C4—28no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—42no—
506C22⋊C4—44no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—81no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4#12118yes—
517C42—22no—
518C42—22no—
519C42—24no—
520C42—22no—
521C42—22no—
522C42—42no—
523C42—42no—
524C42—41no—
525C42—44no—
526C42—48no—
527C42—44no—
528C42—42no—
529C42—82no—
530C42—82no—
531C42—82no—
532C42#13514yes—
533C42C22×D412yes—
534C42C2×#4112yes—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C2×#12114yes—
549C23C23×D411yescharacteristic
550Q8—28no—
551Q8—44no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—88no—
557D4—24no—
558D4—28no—
559D4—42no—
560D4—88no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—48no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—88no—
610C2×C4C2×#12114yes—
611C2×C4C2×#12112yes—
612C2×C4C2×#12112yes—
613C2×C4C2×#13512yes—
614C2×C4C2×#13512yes—
615C2×C4C2×#13512yes—
616C2×C4#57512yes—
617C2×C4#57812yes—
618C2×C4C23×D411yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4#58512yes—
621C2×C4#58512yes—
622C8—42no—
623C8—44no—
624C8—42no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22C22×#12111yescharacteristic
640C22#348511yescharacteristic
641C22#350011yescharacteristic
642C22C22×#13511yescharacteristic
643C22C2×#57511yescharacteristic
644C22C2×#57811yescharacteristic
645C22#358611yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—24no—
651C4—42no—
652C4—42no—
653C4—41no—
654C4—44no—
655C4—42no—
656C4—81no—
657C4—82no—
658C4—81no—
659C4—82no—
660C4—82no—
661C4—84no—
662C4#303514yes—
663C4#348512yes—
664C4#352112yes—
665C2—44no—
666C2—82no—
667C2#3600911yescharacteristic
668C2C2×#303511yescharacteristic
669C2#6043211yescharacteristic
670C2#6170111yescharacteristic
671C2#6173011yescharacteristic
672C2#6252811yescharacteristic
673C2#6258711yescharacteristic
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. #29717C2semidirect
6. #28717C2semidirect
7. #28575C2semidirect
8. #28161C2semidirect
9. #27652C2semidirect
10. #25323C2semidirect
11. #25201C2semidirect
12. #24681C2semidirect
13. #24534C2semidirect
14. #23677C2semidirect
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×#33022
#361882
#343162
#298642
#297172
#287172
#285752
#281612
#276522
#253232
#252012
#246812
#245342
#236772
#236272
#226962
#224902