Finite Group Databasebeta

← #6472788

Subgroups — #6472788

Order 512.

3337 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36614C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic, complemented
3#32982C212yesmaximal, complemented
4#32978C212yesmaximal, complemented
5#29895C212yesmaximal, complemented
6#29892C212yesmaximal, complemented
7#27602C214yesmaximal, complemented
8#24559C214yesmaximal, complemented
9#24558C214yesmaximal, complemented
10#24004C214yesmaximal, complemented
11#21944C212yesmaximal, complemented
12#21769C212yesmaximal, complemented
13#20579C211yesmaximal, characteristic, complemented
14C25×C4C2211yescharacteristic
15C2×#525C2211yescharacteristic
16C2×#525C2212yes—
17C23×C4⋊C4C2212yes—
18C23×C22⋊C4—21no—
19C2×#483C2214yescomplemented
20C2×#480C2214yescomplemented
21C2×#407C2212yes—
22C2×#407C2214yes—
23#2849—22no—
24#2849—21no—
25#2848C2211yescharacteristic
26#2848C2211yescharacteristic
27#2848C2212yes—
28#2847C2211yescharacteristic
29#2847C2211yescharacteristic
30#2847C2212yes—
31#2846—21no—
32C4×#120—21no—
33C4×#120—22no—
34C4×#120—21no—
35#2280C2211yescharacteristic
36#2280C2212yes—
37C2×#380—22no—
38C2×#380—21no—
39C2×#380—22no—
40C2×#380—21no—
41C2×#380C2212yes—
42C2×#380C2214yes—
43C2×#380C2212yes—
44C2×#380C2214yes—
45C22×#95—21no—
46#2077C2218yescomplemented
47#2076C2218yescomplemented
48#2036C2218yescomplemented
49#2034C2218yescomplemented
50#1895C2214yescomplemented
51#1894C2214yescomplemented
52#1874C2214yescomplemented
53#1874C2214yescomplemented
54#1872C2214yescomplemented
55#1872C2214yescomplemented
56#1854C2218yescomplemented
57#1853C2214yescomplemented
58#1853C2214yescomplemented
59#1852C2214yescomplemented
60#1852C2214yescomplemented
61#1825C2214yescomplemented
62#1823C2214yescomplemented
63#1823C2214yescomplemented
64#1809C2218yescomplemented
65#1788C2214yescomplemented
66#1779C2214yes—
67#1779C2212yes—
68#1779C2212yes—
69C26D411yescharacteristic
70C22×#41C2311yescharacteristic
71C24×C4—28no—
72C24×C4—22no—
73C24×C4—48no—
74C24×C4D411yescharacteristic
75C24×C4D412yes—
76C24×C4D411yescharacteristic
77C24×C4D411yescharacteristic
78C24×C4D412yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×M16C2314yes—
85#525—28no—
86#525—48no—
87C2×#123—22no—
88C2×#123—21no—
89C2×C22⋊Q8C2312yes—
90C2×C22⋊Q8C2311yescharacteristic
91C2×C22⋊Q8C2311yescharacteristic
92C2×C22⋊Q8C2312yes—
93C2×#121C2312yes—
94C2×#121C2311yescharacteristic
95C2×#121C2311yescharacteristic
96C2×#121C2312yes—
97C2×#120—22no—
98C2×#120—21no—
99C2×#120—21no—
100C2×#120—22no—
101C2×#120—21no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×#117C2311yescharacteristic
107C2×#117C2312yes—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—48no—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C22⋊C4—24no—
116C22×C22⋊C4—24no—
117C22×C22⋊C4—22no—
118C22×C22⋊C4—22no—
119C22×C22⋊C4—48no—
120#483—48no—
121#480—48no—
122C2×#107C2318yescomplemented
123C2×#106C2318yescomplemented
124C2×#103C2318yescomplemented
125C2×#102C2318yescomplemented
126#407—28no—
127#407—48no—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2314yes—
130C2×C22⋊C8C2314yes—
131C2×C22⋊C8C2314yes—
132C2×C22⋊C8C2318yes—
133#386—22no—
134#386—21no—
135#386—22no—
136#386—22no—
137#386—22no—
138#386—21no—
139#386—22no—
140#385—22no—
141#385—21no—
142#385—22no—
143#385—22no—
144#385—22no—
145#385—21no—
146#385—22no—
147#381—22no—
148#381—21no—
149#380—24no—
150#380—24no—
151#380—24no—
152#380—24no—
153#380—24no—
154#380—22no—
155#380—22no—
156#380C2312yes—
157#380C2314yes—
158#380C2312yes—
159#380C2318yes—
160#380C2312yes—
161#380C2312yes—
162#380C2312yes—
163#380C2311yescharacteristic
164#380C2311yescharacteristic
165#380C2314yes—
166#379—21no—
167#379—22no—
168#379—21no—
169#379—22no—
170#379—21no—
171C4×C22⋊C4—22no—
172C4×C22⋊C4—22no—
173C4×C22⋊C4—21no—
174C4×C22⋊C4—21no—
175C2×#95—24no—
176C2×#95—24no—
177C2×#95—22no—
178C2×#95—22no—
179#340C2318yescomplemented
180#340C2318yescomplemented
181#340C2318yescomplemented
182#340C2318yescomplemented
183C25—24no—
184C25—24no—
185C25—24no—
186C25—44no—
187C25—44no—
188C25C2×D411yescharacteristic
189C25C2×D412yes—
190C25C2×D412yes—
191C25C2×D411yescharacteristic
192C25C2×D411yescharacteristic
193C2×#41—24no—
194C2×#41—44no—
195C22×Q8C2411yescharacteristic
196C22×D4—22no—
197C22×D4—21no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—21no—
201C22×D4C2411yescharacteristic
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—28no—
207C23×C4—28no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—21no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—464no—
230C23×C4—48no—
231C23×C4—48no—
232C23×C4—48no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D412yes—
235C23×C4C2×D412yes—
236C23×C4C2×D412yes—
237C23×C4C2×D412yes—
238C23×C4C2×D412yes—
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D412yes—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4#4118yes—
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2412yes—
252C2×M16—28no—
253C2×M16—48no—
254C22×C8C2418yes—
255#123—48no—
256#123—44no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259C22⋊Q8—28no—
260C22⋊Q8—44no—
261#121—24no—
262#121—24no—
263#121—28no—
264#121—44no—
265#120—48no—
266#120—48no—
267#120—44no—
268#120—44no—
269#120—44no—
270C4×D4—48no—
271C4×D4—48no—
272C4×D4—44no—
273C4×D4—44no—
274#117—24no—
275#117—44no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—48no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4—48no—
288C2×C4⋊C4—48no—
289C2×C4⋊C4—48no—
290C2×C4⋊C4C2418yes—
291C2×C4⋊C4C2411yescharacteristic
292C2×C4⋊C4C2411yescharacteristic
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—48no—
316C2×C22⋊C4—48no—
317C2×C22⋊C4—416no—
318C2×C22⋊C4—416no—
319C2×C42—22no—
320C2×C42—21no—
321#107—48no—
322#106—48no—
323#103—48no—
324#102—48no—
325C22⋊C8—28no—
326C22⋊C8—28no—
327C22⋊C8—28no—
328C22⋊C8—28no—
329C22⋊C8—48no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—21no—
337#95—21no—
338#95—22no—
339#95—22no—
340C24—22no—
341C24—24no—
342C24—22no—
343C24—24no—
344C24—232no—
345C24—24no—
346C24—24no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—21no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—416no—
369C24—416no—
370C24—432no—
371C24—432no—
372C24#12314yes—
373C24AGL(1,ℤ/8)14yes—
374C24#13714yes—
375C24C22×D411yescharacteristic
376C24C22×D411yescharacteristic
377C24C22×D412yes—
378C24C22×D411yescharacteristic
379C24C22×D412yes—
380#41—88no—
381C2×Q8—22no—
382C2×Q8—24no—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—42no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—42no—
397C2×D4—44no—
398C22×C4—232no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—28no—
431C22×C4—21no—
432C22×C4—416no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—432no—
446C22×C4—432no—
447C22×C4—416no—
448C22×C4—432no—
449C22×C4—432no—
450C22×C4—416no—
451C22×C4—416no—
452C22×C4—44no—
453C22×C4—416no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—42no—
464C22×C4—832no—
465C22×C4#12314yes—
466C22×C4#12314yes—
467C22×C4#12314yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4114yes—
476C22×C4H(5,𝔽2)18yes—
477C22×C4C2511yescharacteristic
478M16—88no—
479C2×C8—28no—
480C2×C8—28no—
481C2×C8—28no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—416no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—44no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C42—44no—
516C42—42no—
517C23—28no—
518C23—216no—
519C23—28no—
520C23—28no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—216no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—216no—
529C23—216no—
530C23—28no—
531C23—28no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—44no—
547C23—44no—
548C23—48no—
549C23—48no—
550C23—48no—
551C23—48no—
552C23—48no—
553C23—48no—
554C23—48no—
555C23—48no—
556C23—48no—
557C23—44no—
558C23—4128no—
559C23—48no—
560C23—48no—
561C23—48no—
562C23—48no—
563C23—48no—
564C23—44no—
565C23—48no—
566C23—48no—
567C23—41no—
568C23—42no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23C2×#12312yes—
573C23C2×#12312yes—
574C23C2×#12312yes—
575C23#53414yes—
576C23C2×AGL(1,ℤ/8)12yes—
577C23C2×AGL(1,ℤ/8)14yes—
578C23C2×#13714yes—
579C23C2×#13712yes—
580C23#57514yes—
581C23C23×D411yescharacteristic
582Q8—44no—
583D4—44no—
584D4—88no—
585D4—84no—
586D4—84no—
587D4—84no—
588D4—88no—
589C2×C4—216no—
590C2×C4—28no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—216no—
599C2×C4—28no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4—48no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—44no—
614C2×C4—48no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—48no—
622C2×C4—48no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—48no—
631C2×C4—84no—
632C2×C4—816no—
633C2×C4—816no—
634C2×C4—816no—
635C2×C4C2×#12312yes—
636C2×C4C2×#12312yes—
637C2×C4C2×#12312yes—
638C2×C4#53414yes—
639C2×C4C22×#4112yes—
640C2×C4C2×H(5,𝔽2)14yes—
641C8—48no—
642C22—232no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—22no—
650C22—24no—
651C22—22no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—22no—
656C22—22no—
657C22—24no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—416no—
663C22—416no—
664C22—432no—
665C22—432no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—82no—
675C22—82no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22#309214yes—
680C22#319418yes—
681C22C22×#12311yescharacteristic
682C22C2×#53412yes—
683C22C22×AGL(1,ℤ/8)11yescharacteristic
684C22C22×#13711yescharacteristic
685C22C2×#57512yes—
686C4—22no—
687C4—24no—
688C4—22no—
689C4—22no—
690C4—42no—
691C4—88no—
692C4—82no—
693C4—82no—
694C4—81no—
695C4—81no—
696C4—81no—
697C4#349014yes—
698C2—24no—
699C2—28no—
700C2—44no—
701C2—42no—
702C2—42no—
703C2—161no—
704C2#3499211yescharacteristic
705C2C2×#309211yescharacteristic
706C2#6051111yescharacteristic
707C2C2×#319412yes—
708C2#6075112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36614C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#380C2semidirect
3. #32982C2semidirect
4. #32978C2semidirect
5. #29895C2semidirect
6. #29892C2semidirect
7. #27602C2semidirect
8. #24559C2semidirect
9. #24558C2semidirect
10. #24004C2semidirect
11. #21944C2semidirect
12. #21769C2semidirect
13. #20579C2semidirect
19. C2×#483C22semidirect
20. C2×#480C22semidirect
46. #2077C22semidirect
47. #2076C22semidirect
48. #2036C22semidirect
49. #2034C22semidirect
50. #1895C22semidirect
51. #1894C22semidirect
52. #1874C22semidirect
53. #1874C22semidirect
54. #1872C22semidirect
55. #1872C22semidirect
56. #1854C22semidirect
57. #1853C22semidirect
58. #1853C22semidirect
59. #1852C22semidirect
60. #1852C22semidirect
61. #1825C22semidirect
62. #1823C22semidirect
63. #1823C22semidirect
64. #1809C22semidirect
65. #1788C22semidirect
122. C2×#107C23semidirect
123. C2×#106C23semidirect
124. C2×#103C23semidirect
125. C2×#102C23semidirect
179. #340C23semidirect
180. #340C23semidirect
181. #340C23semidirect
182. #340C23semidirect

Maximal subgroups

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

subgroupindex
#366142
C22×#3802
#329822
#329782
#298952
#298922
#276022
#245592
#245582
#240042
#219442
#217692
#205792