Finite Group Databasebeta

← #6433146

Subgroups — #6433146

Order 512.

3353 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36614C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic, complemented
3#29895C212yesmaximal, complemented
4#29894C214yesmaximal, complemented
5#29893C212yesmaximal, complemented
6#29892C212yesmaximal, complemented
7#29891C214yesmaximal, complemented
8#29890C212yesmaximal, complemented
9#23642C214yesmaximal, complemented
10#23482C214yesmaximal, complemented
11#20909C212yesmaximal, complemented
12#20844C212yesmaximal, complemented
13#20579C211yesmaximal, characteristic, complemented
14C25×C4C2211yescharacteristic
15C2×#525C2211yescharacteristic
16C2×#525C2212yes—
17C23×C4⋊C4C2212yes—
18C23×C22⋊C4—21no—
19C2×#428C2214yescomplemented
20C2×#417C2214yescomplemented
21C2×#407C2212yes—
22C2×#407C2214yes—
23#2849—21no—
24#2849—22no—
25#2848C2211yescharacteristic
26#2848C2212yes—
27#2848C2211yescharacteristic
28#2847C2211yescharacteristic
29#2847C2212yes—
30#2847C2211yescharacteristic
31#2846—21no—
32C4×#120—22no—
33C4×#120—21no—
34C4×#120—21no—
35#2280C2212yes—
36#2280C2211yescharacteristic
37C2×#380—22no—
38C2×#380—22no—
39C2×#380—21no—
40C2×#380—21no—
41C2×#380C2214yes—
42C2×#380C2212yes—
43C2×#380C2214yes—
44C2×#380C2212yes—
45C22×#95—21no—
46#2077C2218yescomplemented
47#2077C2218yescomplemented
48#2076C2218yescomplemented
49#2075C2218yescomplemented
50#2036C2218yescomplemented
51#2035C2218yescomplemented
52#2034C2218yescomplemented
53#2033C2218yescomplemented
54#1874C2214yescomplemented
55#1874C2214yescomplemented
56#1872C2214yescomplemented
57#1872C2214yescomplemented
58#1853C2214yescomplemented
59#1853C2214yescomplemented
60#1852C2214yescomplemented
61#1852C2214yescomplemented
62#1779C2212yes—
63#1779C2214yes—
64#1779C2212yes—
65#1736C2214yescomplemented
66#1736C2214yescomplemented
67C26D411yescharacteristic
68C22×#41C2311yescharacteristic
69C24×C4—28no—
70C24×C4—22no—
71C24×C4—48no—
72C24×C4D411yescharacteristic
73C24×C4D412yes—
74C24×C4D412yes—
75C24×C4D411yescharacteristic
76C24×C4D411yescharacteristic
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C22×M16C2314yes—
83#525—28no—
84#525—48no—
85C2×#123—22no—
86C2×#123—21no—
87C2×C22⋊Q8C2311yescharacteristic
88C2×C22⋊Q8C2311yescharacteristic
89C2×C22⋊Q8C2312yes—
90C2×C22⋊Q8C2312yes—
91C2×#121C2311yescharacteristic
92C2×#121C2311yescharacteristic
93C2×#121C2312yes—
94C2×#121C2312yes—
95C2×#120—22no—
96C2×#120—21no—
97C2×#120—21no—
98C2×#120—22no—
99C2×#120—21no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×#117C2312yes—
105C2×#117C2311yescharacteristic
106C22×C4⋊C4—22no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—48no—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—24no—
115C22×C22⋊C4—22no—
116C22×C22⋊C4—22no—
117C22×C22⋊C4—48no—
118#428—48no—
119C2×#107C2318yescomplemented
120C2×#106C2318yescomplemented
121#417—48no—
122C2×#103C2318yescomplemented
123C2×#102C2318yescomplemented
124#407—28no—
125#407—48no—
126C2×C22⋊C8C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2318yes—
129C2×C22⋊C8C2318yes—
130#386—21no—
131#386—22no—
132#386—21no—
133#386—22no—
134#386—22no—
135#386—22no—
136#386—22no—
137#385—21no—
138#385—22no—
139#385—21no—
140#385—22no—
141#385—22no—
142#385—22no—
143#385—22no—
144#381—22no—
145#381—21no—
146#380—24no—
147#380—24no—
148#380—24no—
149#380—24no—
150#380—24no—
151#380—24no—
152#380C2312yes—
153#380C2318yes—
154#380C2312yes—
155#380C2318yes—
156#380C2312yes—
157#380C2312yes—
158#380C2311yescharacteristic
159#380C2312yes—
160#380C2311yescharacteristic
161#379—22no—
162#379—21no—
163#379—21no—
164#379—21no—
165#379—22no—
166C4×C22⋊C4—22no—
167C4×C22⋊C4—22no—
168C4×C22⋊C4—21no—
169C4×C22⋊C4—21no—
170C2×#95—24no—
171C2×#95—22no—
172C2×#95—24no—
173C2×#95—22no—
174#340C2318yescomplemented
175#340C2318yescomplemented
176#340C2318yescomplemented
177#340C2318yescomplemented
178C25—24no—
179C25—24no—
180C25—24no—
181C25—44no—
182C25—44no—
183C25C2×D411yescharacteristic
184C25C2×D412yes—
185C25C2×D412yes—
186C25C2×D411yescharacteristic
187C25C2×D411yescharacteristic
188C2×#41—24no—
189C2×#41—28no—
190C22×Q8C2411yescharacteristic
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4C2411yescharacteristic
197C23×C4—28no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—28no—
202C23×C4—28no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—28no—
208C23×C4—28no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—21no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—464no—
225C23×C4—48no—
226C23×C4—48no—
227C23×C4—48no—
228C23×C4C2×D412yes—
229C23×C4C2×D412yes—
230C23×C4C2×D412yes—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D412yes—
237C23×C4C2×D412yes—
238C23×C4C2×D412yes—
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4#4118yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×M16—28no—
248C2×M16—28no—
249C2×M16—28no—
250C22×C8C2418yes—
251#123—48no—
252#123—44no—
253C22⋊Q8—24no—
254C22⋊Q8—24no—
255C22⋊Q8—44no—
256C22⋊Q8—44no—
257#121—24no—
258#121—24no—
259#121—44no—
260#121—44no—
261#120—48no—
262#120—48no—
263#120—44no—
264#120—44no—
265#120—44no—
266C4×D4—48no—
267C4×D4—48no—
268C4×D4—44no—
269C4×D4—44no—
270#117—24no—
271#117—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—48no—
282C2×C4⋊C4—48no—
283C2×C4⋊C4—48no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4—48no—
286C2×C4⋊C4C2418yes—
287C2×C4⋊C4C2411yescharacteristic
288C2×C4⋊C4C2411yescharacteristic
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—48no—
311C2×C22⋊C4—416no—
312C2×C22⋊C4—416no—
313C2×C22⋊C4—48no—
314C2×C42—21no—
315C2×C42—22no—
316#107—48no—
317#106—48no—
318#103—48no—
319#102—48no—
320C22⋊C8—28no—
321C22⋊C8—28no—
322C22⋊C8—48no—
323C22⋊C8—48no—
324#95—28no—
325#95—28no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—21no—
331#95—22no—
332#95—21no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—232no—
338C24—24no—
339C24—24no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—21no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—416no—
362C24—432no—
363C24—432no—
364C24—416no—
365C24#12314yes—
366C24AGL(1,ℤ/8)14yes—
367C24#13714yes—
368C24C22×D412yes—
369C24C22×D412yes—
370C24C22×D411yescharacteristic
371C24C22×D411yescharacteristic
372C24C22×D411yescharacteristic
373#41—416no—
374C2×Q8—22no—
375C2×Q8—24no—
376C2×D4—22no—
377C2×D4—24no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—44no—
389C2×D4—42no—
390C2×D4—44no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—232no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
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—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—21no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—432no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—432no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—432no—
444C22×C4—416no—
445C22×C4—432no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—832no—
458C22×C4#12314yes—
459C22×C4#12314yes—
460C22×C4#12314yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2×#4114yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4114yes—
469C22×C4H(5,𝔽2)18yes—
470C22×C4C2511yescharacteristic
471M16—416no—
472C2×C8—28no—
473C2×C8—28no—
474C2×C8—28no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—28no—
478C4⋊C4—24no—
479C4⋊C4—28no—
480C4⋊C4—24no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—48no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—416no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C22⋊C4—416no—
500C22⋊C4—44no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C42—42no—
508C42—44no—
509C23—216no—
510C23—216no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—22no—
515C23—22no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—22no—
522C23—216no—
523C23—216no—
524C23—28no—
525C23—28no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—48no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—48no—
544C23—48no—
545C23—48no—
546C23—48no—
547C23—48no—
548C23—44no—
549C23—44no—
550C23—48no—
551C23—4128no—
552C23—48no—
553C23—48no—
554C23—44no—
555C23—44no—
556C23—48no—
557C23—48no—
558C23—48no—
559C23—41no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23C2×#12312yes—
565C23C2×#12312yes—
566C23C2×#12312yes—
567C23#53414yes—
568C23C2×AGL(1,ℤ/8)14yes—
569C23C2×AGL(1,ℤ/8)12yes—
570C23C2×#13714yes—
571C23C2×#13712yes—
572C23#57514yes—
573C23C23×D411yescharacteristic
574Q8—44no—
575D4—44no—
576D4—88no—
577D4—88no—
578D4—84no—
579D4—84no—
580D4—84no—
581C2×C4—28no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—48no—
622C2×C4—816no—
623C2×C4—816no—
624C2×C4—816no—
625C2×C4—84no—
626C2×C4C2×#12312yes—
627C2×C4C2×#12312yes—
628C2×C4C2×#12312yes—
629C2×C4#53414yes—
630C2×C4#538116yes—
631C2×C4C22×#4112yes—
632C2×C4C2×H(5,𝔽2)14yes—
633C8—48no—
634C22—232no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—416no—
654C22—416no—
655C22—432no—
656C22—432no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—82no—
666C22—82no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22#262214yes—
671C22#309214yes—
672C22#309314yes—
673C22C22×#12311yescharacteristic
674C22C2×#53412yes—
675C22C22×AGL(1,ℤ/8)11yescharacteristic
676C22C22×#13711yescharacteristic
677C22C2×#57512yes—
678C4—22no—
679C4—22no—
680C4—42no—
681C4—42no—
682C4—88no—
683C4—82no—
684C4—82no—
685C4—81no—
686C4—81no—
687C4—81no—
688C4#262214yes—
689C4C2×#53814yes—
690C2—24no—
691C2—24no—
692C2—24no—
693C2—44no—
694C2—44no—
695C2—161no—
696C2C2×#262211yescharacteristic
697C2C2×#309211yescharacteristic
698C2C2×#309311yescharacteristic
699C2#6075012yes—
700C2#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. #29895C2semidirect
4. #29894C2semidirect
5. #29893C2semidirect
6. #29892C2semidirect
7. #29891C2semidirect
8. #29890C2semidirect
9. #23642C2semidirect
10. #23482C2semidirect
11. #20909C2semidirect
12. #20844C2semidirect
13. #20579C2semidirect
19. C2×#428C22semidirect
20. C2×#417C22semidirect
46. #2077C22semidirect
47. #2077C22semidirect
48. #2076C22semidirect
49. #2075C22semidirect
50. #2036C22semidirect
51. #2035C22semidirect
52. #2034C22semidirect
53. #2033C22semidirect
54. #1874C22semidirect
55. #1874C22semidirect
56. #1872C22semidirect
57. #1872C22semidirect
58. #1853C22semidirect
59. #1853C22semidirect
60. #1852C22semidirect
61. #1852C22semidirect
65. #1736C22semidirect
66. #1736C22semidirect
119. C2×#107C23semidirect
120. C2×#106C23semidirect
122. C2×#103C23semidirect
123. C2×#102C23semidirect
174. #340C23semidirect
175. #340C23semidirect
176. #340C23semidirect
177. #340C23semidirect

Maximal subgroups

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

subgroupindex
#366142
C22×#3802
#298952
#298942
#298932
#298922
#298912
#298902
#236422
#234822
#209092
#208442
#205792