Finite Group Databasebeta

← #6489194

Subgroups — #6489194

Order 512.

1818 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#35322C211yesmaximal, characteristic, complemented
2#34276C211yesmaximal, characteristic
3#32916C212yesmaximal
4#32915C212yesmaximal, complemented
5#30046C212yesmaximal, complemented
6#30042C212yesmaximal, complemented
7#29147C211yesmaximal, characteristic, complemented
8#29142C211yesmaximal, characteristic, complemented
9#28449C211yesmaximal, characteristic, complemented
10#28440C211yesmaximal, characteristic, complemented
11#28170C212yesmaximal, complemented
12#28168C212yesmaximal, complemented
13#27985C212yesmaximal, complemented
14#25735C212yesmaximal
15#23935C212yesmaximal
16#23934C212yesmaximal, complemented
17#22278C212yesmaximal, complemented
18#22277C212yesmaximal, complemented
19#20781C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#533C2212yes—
22C2×#520C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#497C2212yes—
25C2×#494C2212yes—
26C2×#462C2212yes—
27C2×#462C2212yes—
28C2×#462C2212yes—
29C2×#461C2212yes—
30C2×#452C2214yes—
31C2×#451C2214yes—
32#2868C2211yescharacteristic
33#2862C2212yes—
34#2862C2211yescharacteristic
35#2621C2212yes—
36#2621C2212yes—
37#2565—22no—
38#2565—22no—
39#2499—22no—
40#2499—22no—
41#2362—22no—
42#2360—22no—
43#2359—22no—
44#2359—22no—
45#2327C2211yescharacteristic
46#2327C2212yes—
47#2327C2212yes—
48#2327C2211yescharacteristic
49#2326C2211yescharacteristic
50#2326C2211yescharacteristic
51#2317C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×#124C2211yescharacteristic
55C4×#124C2211yescharacteristic
56#2297C2211yescharacteristic
57C2×#391C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2074C2214yes—
60#2072C2214yes—
61#2024C2212yes—
62#2024C2212yes—
63#2024C2212yes—
64#2023C2212yes—
65#2016C2214yes—
66#2016C2212yes—
67#2016C2212yes—
68#2016C2212yes—
69#2015C2214yes—
70#2013C2212yes—
71#1960C2212yes—
72#1960C2212yes—
73#1958C2212yes—
74#1956C2212yes—
75#1880C2214yes—
76#1878C2214yes—
77#1878C2214yes—
78#1878C2214yes—
79#1878C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1877C2214yes—
83#1873C2214yes—
84#1872C2214yes—
85#1839C2212yes—
86#1836C2212yes—
87#1835C2212yes—
88#1832C2212yes—
89#1817C2212yes—
90#1805C2212yes—
91#1805C2212yes—
92#1805C2212yes—
93#1804C2212yes—
94#1798C2212yes—
95#1767C2211yescharacteristic
96C4×C4⋊C8C2211yescharacteristic
97#1750C2212yes—
98#1746C2212yes—
99C22×#41C2311yescharacteristic
100C23×Q8C2311yescharacteristic
101C22×Q16C2314yes—
102C22×Q16C2314yes—
103C22×Q16C2314yes—
104C22×SD16C2314yes—
105#536—24no—
106#536—24no—
107#533—48no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2312yes—
114C2×#125—22no—
115C2×#124C2311yescharacteristic
116C2×#124C2311yescharacteristic
117C2×#123—22no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8C2314yes—
121#520—44no—
122#519—44no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—22no—
126C2×C4×Q8C2311yescharacteristic
127C2×C4×Q8C2311yescharacteristic
128C2×C4×Q8C2311yescharacteristic
129C2×C4×Q8C2311yescharacteristic
130C2×C4×Q8C2311yescharacteristic
131C2×C4×Q8C2314yes—
132C2×C4×Q8C2311yescharacteristic
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4C2311yescharacteristic
136C2×C4×D4C2311yescharacteristic
137C2×#117—22no—
138C2×#117—21no—
139C2×#117—21no—
140C2×#117—21no—
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C2×#117C2311yescharacteristic
144C22×C4⋊C4—21no—
145C22×C4⋊C4C2311yescharacteristic
146C22×C42C2311yescharacteristic
147#497—28no—
148#494—28no—
149#462—44no—
150#462—44no—
151#462—44no—
152#461—44no—
153#452—48no—
154#451—48no—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#102C2314yes—
165C2×C22⋊C8C2314yes—
166C2×#97C2312yes—
167C2×C4×C8C2312yes—
168#398—22no—
169#398—22no—
170#398—24no—
171#397—22no—
172#397—22no—
173#397—24no—
174#391—22no—
175#391—22no—
176#391—22no—
177#391—22no—
178#391—22no—
179#391C2311yescharacteristic
180#391C2311yescharacteristic
181#391C2311yescharacteristic
182#391C2311yescharacteristic
183#391C2311yescharacteristic
184#391C2311yescharacteristic
185#391C2314yes—
186#391C2314yes—
187#390C2311yescharacteristic
188#390C2311yescharacteristic
189#389—22no—
190#389—22no—
191#389—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#387—24no—
197#387—24no—
198#387—24no—
199#387—24no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#386—22no—
205#385—22no—
206#385—22no—
207#385—22no—
208#384—22no—
209#384—22no—
210#384C2311yescharacteristic
211#384C2311yescharacteristic
212#382—22no—
213#382—22no—
214#382C2312yes—
215#382C2311yescharacteristic
216#382C2311yescharacteristic
217#382C2312yes—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4D412yes—
221C4×C4⋊C4D412yes—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4D412yes—
227#376C2311yescharacteristic
228#376C2311yescharacteristic
229C2×#95—22no—
230C43C2311yescharacteristic
231C43C2311yescharacteristic
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#336C2314yes—
237#336C2314yes—
238#336C2314yes—
239#336C2312yes—
240#336C2312yes—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—24no—
244C2×#41—42no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—21no—
248C22×Q8—22no—
249C22×Q8—24no—
250C22×Q8—42no—
251C22×Q8C2414yes—
252C22×Q8C2411yescharacteristic
253C22×Q8C2411yescharacteristic
254C22×Q8C2411yescharacteristic
255C22×D4—21no—
256C22×D4C2411yescharacteristic
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4C2×D412yes—
263C23×C4C2411yescharacteristic
264C2×Q16—28no—
265C2×Q16—28no—
266C2×Q16—28no—
267C2×Q16—44no—
268C2×Q16—44no—
269C2×Q16—44no—
270C2×Q16—44no—
271C2×Q16—44no—
272C2×Q16—44no—
273C2×SD16—28no—
274C2×SD16—44no—
275C2×SD16—44no—
276C22×C8C2414yes—
277C22×C8C2414yes—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8—24no—
282C4⋊Q8—24no—
283C4⋊Q8—24no—
284C4⋊Q8—24no—
285C4⋊Q8—44no—
286C4⋊Q8—44no—
287C4⋊Q8#4114yes—
288#125—48no—
289#124—22no—
290#124—22no—
291#124—22no—
292#124#4114yes—
293#123—216no—
294C22⋊Q8—216no—
295C22⋊Q8—28no—
296C22⋊Q8—28no—
297C22⋊Q8—48no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—42no—
302C4×Q8—42no—
303C4×Q8—42no—
304C4×Q8—44no—
305C4×Q8—44no—
306C4×Q8—48no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×D4—24no—
310C4×D4—42no—
311C4×D4—44no—
312C4×D4—44no—
313#117—22no—
314#117—22no—
315#117—24no—
316#117—44no—
317#117—44no—
318#117—42no—
319#117—44no—
320#117—48no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—42no—
341C2×C4⋊C4—44no—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2414yes—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4C2×D411yescharacteristic
359C2×C22⋊C4C2×D411yescharacteristic
360C2×C22⋊C4C2×D412yes—
361C2×C42—24no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—24no—
369C2×C42—21no—
370C2×C42C2×D412yes—
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2×D412yes—
374C2×C42C2×D412yes—
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381C4⋊C8—24no—
382C4⋊C8—24no—
383#103—44no—
384#103—44no—
385#103—44no—
386#103—44no—
387#103—44no—
388#103—44no—
389#103—44no—
390#102—44no—
391C22⋊C8—28no—
392#97—24no—
393C4×C8—24no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—22no—
408#95—22no—
409#95C2×D412yes—
410#95C2×D412yes—
411#95C2×D412yes—
412#95C2×D412yes—
413C24—21no—
414C24C22×D411yescharacteristic
415#41—48no—
416#41—48no—
417C2×Q8—24no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—24no—
433C2×Q8—48no—
434C2×Q8—48no—
435C2×Q8—42no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8—44no—
440C2×Q8C2×#4112yes—
441C2×Q8C2×#4112yes—
442C2×Q8C2×#4112yes—
443C2×D4—24no—
444C2×D4—21no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4C2×#4112yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—24no—
465C22×C4—21no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4H(5,𝔽2)14yes—
487C22×C4C2511yescharacteristic
488Q16—88no—
489Q16—88no—
490Q16—88no—
491SD16—88no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—48no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—24no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—48no—
534C22⋊C4—42no—
535C22⋊C4—44no—
536C22⋊C4—82no—
537C42—24no—
538C42—24no—
539C42—24no—
540C42—21no—
541C42—21no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—42no—
546C42—42no—
547C42—42no—
548C42—42no—
549C42—48no—
550C42—42no—
551C42—42no—
552C42—42no—
553C42—42no—
554C42—42no—
555C42C2×#4112yes—
556C42C2×#4112yes—
557C42H(5,𝔽2)14yes—
558C23—22no—
559C23—21no—
560C23—21no—
561C23—22no—
562C23—22no—
563C23—42no—
564C23—41no—
565C23—41no—
566C23—42no—
567C23#53512yes—
568C23C23×D411yescharacteristic
569Q8—24no—
570Q8—24no—
571Q8—24no—
572Q8—48no—
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—44no—
577Q8—44no—
578Q8—44no—
579Q8—44no—
580Q8—88no—
581D4—24no—
582D4—48no—
583D4—42no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—48no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—162no—
631C2×C4—162no—
632C2×C4—162no—
633C2×C4#52514yes—
634C2×C4#53512yes—
635C2×C4#53512yes—
636C2×C4#53512yes—
637C2×C4#57814yes—
638C2×C4C22×#4111yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C2×H(5,𝔽2)12yes—
641C2×C4C2×H(5,𝔽2)12yes—
642C8—44no—
643C8—44no—
644C22—24no—
645C22—22no—
646C22—41no—
647C22—41no—
648C22—42no—
649C22—42no—
650C22—44no—
651C22—82no—
652C22—81no—
653C22—81no—
654C22C2×#52511yescharacteristic
655C22C2×#53511yescharacteristic
656C22C2×#53511yescharacteristic
657C22C2×#57812yes—
658C22#358612yes—
659C4—24no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—24no—
664C4—42no—
665C4—41no—
666C4—41no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—84no—
672C4—81no—
673C4—81no—
674C4—81no—
675C4—81no—
676C4—84no—
677C4#351912yes—
678C4#351912yes—
679C2—42no—
680C2—82no—
681C2#3523711yescharacteristic
682C2#6059412yes—
683C2#6081412yes—
684C2#6082111yescharacteristic
685C2#6082611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35322C2semidirect
4. #32915C2semidirect
5. #30046C2semidirect
6. #30042C2semidirect
7. #29147C2semidirect
8. #29142C2semidirect
9. #28449C2semidirect
10. #28440C2semidirect
11. #28170C2semidirect
12. #28168C2semidirect
13. #27985C2semidirect
16. #23934C2semidirect
17. #22278C2semidirect
18. #22277C2semidirect
19. #20781C2semidirect

Maximal subgroups

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

subgroupindex
#353222
#342762
#329162
#329152
#300462
#300422
#291472
#291422
#284492
#284402
#281702
#281682
#279852
#257352
#239352
#239342
#222782
#222772
#207812