Finite Group Databasebeta

← #6470366

Subgroups — #6470366

Order 512.

3462 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C211yesmaximal, characteristic, complemented
2C22×#407C211yesmaximal, characteristic
3C2×#2848C212yesmaximal
4#32937C214yesmaximal, complemented
5#29743C214yesmaximal, complemented
6#24559C214yesmaximal, complemented
7#24551C214yesmaximal, complemented
8#24386C214yesmaximal, complemented
9#24380C214yesmaximal, complemented
10#21942C211yesmaximal, characteristic, complemented
11#21765C212yesmaximal, complemented
12C25×C4C2211yescharacteristic
13C23×M16C2212yes—
14C22×C22⋊Q8C2212yes—
15C22×C22⋊Q8C2212yes—
16C22×C22⋊Q8C2212yes—
17#3195—28no—
18C2×#483C2214yescomplemented
19C2×#470C2214yescomplemented
20C2×#470C2214yescomplemented
21C2×#467C2214yescomplemented
22C2×#451C2214yescomplemented
23C2×#407—28no—
24C2×#407C2211yescharacteristic
25C2×#407C2212yes—
26C2×#407C2214yes—
27C2×#407C2211yescharacteristic
28C22×C22⋊C8C2214yes—
29#2848—22no—
30#2848—22no—
31#2848—22no—
32#2848—22no—
33#2848C2214yes—
34#2848C2214yes—
35#2848C2214yes—
36#2848C2214yes—
37C2×#385—22no—
38C2×#385—22no—
39C2×#385—22no—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#379—22no—
44#2077C2218yescomplemented
45#2036C2218yescomplemented
46#2035C2218yescomplemented
47#2001C2218yescomplemented
48#1894C2218yescomplemented
49#1894C2218yescomplemented
50#1892C2218yescomplemented
51#1872C2214yescomplemented
52#1872C2214yescomplemented
53#1853C2214yescomplemented
54#1852C2214yescomplemented
55#1823C2214yescomplemented
56#1809C2214yescomplemented
57#1809C2214yescomplemented
58#1807C2214yescomplemented
59#1787C2214yescomplemented
60C26D411yescharacteristic
61C23×Q8C2312yes—
62C24×C4—24no—
63C24×C4—24no—
64C24×C4—44no—
65C24×C4—44no—
66C24×C4D411yescharacteristic
67C24×C4D411yescharacteristic
68C24×C4D412yes—
69C24×C4D412yes—
70C24×C4D411yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2312yes—
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C22×Q16C2314yescomplemented
77C22×M16—28no—
78C22×M16—44no—
79C22×M16—44no—
80C22×M16C2314yes—
81C22×M16C2312yes—
82C22×M16C2312yes—
83C23×C8C2314yes—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—48no—
93C2×C22⋊Q8—48no—
94C2×C22⋊Q8C2314yes—
95C2×C22⋊Q8C2314yes—
96C2×C22⋊Q8C2314yes—
97C2×C22⋊Q8C2314yes—
98C2×C22⋊Q8C2314yes—
99C2×C22⋊Q8C2314yes—
100C22×C4⋊C4—22no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—22no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4—22no—
107C22×C22⋊C4—22no—
108C22×C22⋊C4—22no—
109C22×C22⋊C4—22no—
110C22×C22⋊C4—22no—
111C22×C22⋊C4—22no—
112#483—216no—
113#470—216no—
114#470—48no—
115#467—48no—
116#451—216no—
117C2×#107C2314yescomplemented
118C2×#106C2314yescomplemented
119C2×#106C2314yescomplemented
120C2×#103C2318yescomplemented
121C2×#103C2318yescomplemented
122C2×#103C2318yescomplemented
123#407—24no—
124#407—28no—
125#407—216no—
126#407D418yes—
127C2×C22⋊C8—216no—
128C2×C22⋊C8C2318yes—
129C2×C22⋊C8C2314yes—
130C2×C22⋊C8C2314yes—
131#385—24no—
132#385—24no—
133#385—24no—
134#385—24no—
135#385—24no—
136#385—24no—
137#385—24no—
138#385—24no—
139#385—24no—
140#385—24no—
141#385—24no—
142#385—24no—
143#380—22no—
144#380—22no—
145#380—22no—
146#380—22no—
147#380—22no—
148#380—22no—
149#380C2314yes—
150#380C2314yes—
151#380C2314yes—
152#380C2314yes—
153#380C2314yes—
154#380C2314yes—
155#379—24no—
156#379—24no—
157#379—24no—
158#379—24no—
159C2×#95—22no—
160C2×#95—22no—
161C2×#95—22no—
162#340C2318yescomplemented
163#340C2318yescomplemented
164#340C2318yescomplemented
165C25—24no—
166C25—28no—
167C25—42no—
168C25—42no—
169C25—44no—
170C25C2×D412yes—
171C25C2×D411yescharacteristic
172C25C2×D411yescharacteristic
173C25C2×D412yes—
174C25C2×D411yescharacteristic
175C22×Q8—22no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—44no—
179C22×Q8—44no—
180C22×Q8C2414yes—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—28no—
188C23×C4—28no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—432no—
219C23×C4—432no—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D412yes—
224C23×C4C2×D412yes—
225C23×C4C2×D412yes—
226C23×C4C2×D412yes—
227C23×C4C2×D412yes—
228C23×C4C2×D412yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D412yes—
231C23×C4C2×D412yes—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D414yes—
234C23×C4#4114yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C2×Q16—28no—
241C2×Q16—44no—
242C2×Q16—44no—
243C2×M16—24no—
244C2×M16—28no—
245C2×M16—28no—
246C2×M16—42no—
247C2×M16—42no—
248C2×M16—44no—
249C2×M16—42no—
250C2×M16—42no—
251C2×M16—432no—
252C2×M16C2×D418yes—
253C22×C8—28no—
254C22×C8—28no—
255C22×C8—44no—
256C22×C8C2414yes—
257C22×C8C2414yes—
258C22⋊Q8—28no—
259C22⋊Q8—28no—
260C22⋊Q8—28no—
261C22⋊Q8—28no—
262C22⋊Q8—216no—
263C22⋊Q8—216no—
264C22⋊Q8—48no—
265C22⋊Q8—48no—
266C22⋊Q8—48no—
267C22⋊Q8—48no—
268C22⋊Q8—48no—
269C22⋊Q8—48no—
270C22⋊Q8—48no—
271C22⋊Q8—48no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—48no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4—48no—
288C2×C4⋊C4C2414yes—
289C2×C4⋊C4C2414yes—
290C2×C4⋊C4C2414yes—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4—48no—
305C2×C22⋊C4—48no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4—48no—
309#107—44no—
310#106—28no—
311#106—44no—
312#103—216no—
313#103—48no—
314#103—48no—
315C22⋊C8—28no—
316C22⋊C8—216no—
317C22⋊C8C2×D4116yes—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—22no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—41no—
342C24—41no—
343C24—42no—
344C24—42no—
345C24—41no—
346C24—41no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24—416no—
352C24—432no—
353C24—416no—
354C24—432no—
355C24—416no—
356C24#12114yes—
357C24#13718yes—
358C24C22×D411yescharacteristic
359C24C22×D411yescharacteristic
360C24C22×D412yes—
361C24C22×D412yes—
362C24C22×D411yescharacteristic
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—28no—
366C2×Q8—28no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—88no—
371C2×Q8—88no—
372C22×C4—216no—
373C22×C4—216no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
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—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—432no—
437C22×C4—432no—
438C22×C4—416no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—416no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4#12118yes—
455C22×C4#12114yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D414yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2×#4114yes—
464C22×C4C2×#4112yes—
465C22×C4H(5,𝔽2)14yes—
466C22×C4H(5,𝔽2)14yes—
467C22×C4C2511yescharacteristic
468Q16—88no—
469M16—48no—
470M16—416no—
471M16—48no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—416no—
478C2×C8C22×D418yes—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C23—21no—
505C23—22no—
506C23—22no—
507C23—21no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—28no—
512C23—216no—
513C23—22no—
514C23—21no—
515C23—22no—
516C23—22no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—216no—
521C23—216no—
522C23—216no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—44no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—48no—
535C23—48no—
536C23—48no—
537C23—44no—
538C23—48no—
539C23—44no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—48no—
544C23—48no—
545C23—48no—
546C23—44no—
547C23—44no—
548C23—48no—
549C23—44no—
550C23—48no—
551C23—4128no—
552C23—44no—
553C23—44no—
554C23—48no—
555C23C2×#12112yes—
556C23C2×#12114yes—
557C23#53412yes—
558C23#53412yes—
559C23C2×#13714yes—
560C23C2×#13714yes—
561C23C2×#13714yes—
562C23#57514yes—
563C23C23×D411yescharacteristic
564Q8—48no—
565Q8—48no—
566Q8—88no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—48no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—88no—
614C2×C4—88no—
615C2×C4—88no—
616C2×C4—88no—
617C2×C4C2×#12112yes—
618C2×C4C2×#12114yes—
619C2×C4#53412yes—
620C2×C4#53412yes—
621C2×C4C23×D411yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C2×H(5,𝔽2)12yes—
624C2×C4C2×H(5,𝔽2)12yes—
625C8—28no—
626C8—44no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—22no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—22no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—22no—
643C22—22no—
644C22—432no—
645C22—416no—
646C22—416no—
647C22—432no—
648C22—416no—
649C22—42no—
650C22—42no—
651C22—41no—
652C22—41no—
653C22—42no—
654C22—41no—
655C22—41no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22#305618yes—
660C22#319514yescomplemented
661C22C22×#12111yescharacteristic
662C22C2×#53411yescharacteristic
663C22C2×#53411yescharacteristic
664C22C22×#13712yes—
665C22C2×#57512yes—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—24no—
670C4—42no—
671C4—84no—
672C4—84no—
673C4—84no—
674C4—84no—
675C4#348412yes—
676C4#349012yes—
677C2—24no—
678C2—28no—
679C2—42no—
680C2—44no—
681C2—42no—
682C2#3498111yescharacteristic
683C2C2×#305612yes—
684C2C2×#319511yescharacteristic
685C2#6074111yescharacteristic
686C2#6075112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC26D4
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
1. C2×#3195C2semidirect
4. #32937C2semidirect
5. #29743C2semidirect
6. #24559C2semidirect
7. #24551C2semidirect
8. #24386C2semidirect
9. #24380C2semidirect
10. #21942C2semidirect
11. #21765C2semidirect
18. C2×#483C22semidirect
19. C2×#470C22semidirect
20. C2×#470C22semidirect
21. C2×#467C22semidirect
22. C2×#451C22semidirect
44. #2077C22semidirect
45. #2036C22semidirect
46. #2035C22semidirect
47. #2001C22semidirect
48. #1894C22semidirect
49. #1894C22semidirect
50. #1892C22semidirect
51. #1872C22semidirect
52. #1872C22semidirect
53. #1853C22semidirect
54. #1852C22semidirect
55. #1823C22semidirect
56. #1809C22semidirect
57. #1809C22semidirect
58. #1807C22semidirect
59. #1787C22semidirect
76. C22×Q16C23semidirect
117. C2×#107C23semidirect
118. C2×#106C23semidirect
119. C2×#106C23semidirect
120. C2×#103C23semidirect
121. C2×#103C23semidirect
122. C2×#103C23semidirect
162. #340C23semidirect
163. #340C23semidirect
164. #340C23semidirect
660. C22#3195semidirect

Maximal subgroups

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

subgroupindex
C2×#31952
C22×#4072
C2×#28482
#329372
#297432
#245592
#245512
#243862
#243802
#219422
#217652