Finite Group Databasebeta

← #6557142

Subgroups — #6557142

Order 512.

1838 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36290C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic
3#30220C211yesmaximal, characteristic, complemented
4#30202C212yesmaximal, complemented
5#29329C212yesmaximal, complemented
6#29130C211yesmaximal, characteristic, complemented
7#28755C211yesmaximal, characteristic, complemented
8#27768C212yesmaximal, complemented
9#27748C212yesmaximal, complemented
10#27284C214yesmaximal, complemented
11#27272C214yesmaximal, complemented
12#25858C212yesmaximal, complemented
13#23819C211yesmaximal, characteristic, complemented
14#23508C212yesmaximal, complemented
15#22431C212yesmaximal, complemented
16#22431C212yesmaximal, complemented
17#20567C211yesmaximal, characteristic, complemented
18C2×#547C2212yes—
19C2×#532C2212yes—
20C2×#491C2212yes—
21C2×#488C2214yes—
22C2×#436C2212yes—
23C2×#417C2214yes—
24C2×#417C2214yes—
25#2868C2211yescharacteristic
26#2859C2211yescharacteristic
27#2857C2212yes—
28#2851C2212yes—
29#2571C2211yescharacteristic
30#2552C2211yescharacteristic
31#2547C2212yes—
32#2537—21no—
33#2537—21no—
34#2509—21no—
35#2507—21no—
36#2505—22no—
37#2473—22no—
38#2442C2212yes—
39#2442C2212yes—
40#2432C2214yes—
41#2422—22no—
42#2422—22no—
43#2349—21no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4—21no—
46D4×C4⋊C4—21no—
47D4×C4⋊C4C2211yescharacteristic
48D4×C4⋊C4C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C4×#125C2211yescharacteristic
51C2×#389C2212yes—
52C42×D4C2211yescharacteristic
53#2100C2211yescharacteristic
54#2100C2211yescharacteristic
55#2100C2212yes—
56#2074C2214yes—
57#2073C2212yes—
58#2043C2212yes—
59#2043C2212yes—
60#2043C2214yes—
61#2043C2212yes—
62#2043C2212yes—
63#2042C2214yes—
64#2042C2214yes—
65#2041C2211yescharacteristic
66#2041C2212yes—
67#2041C2211yescharacteristic
68#2026C2214yes—
69#2025C2212yes—
70#1981C2212yes—
71#1981C2212yes—
72#1960C2212yes—
73#1958C2214yes—
74#1956C2212yes—
75#1956C2212yes—
76#1936C2214yes—
77#1933C2214yes—
78#1933C2214yes—
79#1894C2214yes—
80#1894C2214yes—
81#1872C2214yes—
82#1872C2214yes—
83#1802C2212yes—
84#1802C2212yes—
85#1788C2214yes—
86#1788C2214yes—
87C8×C4⋊C4C2212yes—
88#1767C2211yescharacteristic
89C4×#102C2212yes—
90C4×#102C2212yes—
91#1753C2214yes—
92C4×C22⋊C8C2212yes—
93#1744C2211yescharacteristic
94#1728C2212yes—
95C22×#41C2314yes—
96C22×M16C2314yes—
97#547—28no—
98#547—28no—
99#532—48no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#125—21no—
105C2×#125C2311yescharacteristic
106C2×#125C2312yes—
107C2×#124C2314yes—
108C2×#123—24no—
109C2×C22⋊Q8C2314yes—
110C2×#121—22no—
111C2×#121—22no—
112C2×#121—22no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C22×C4⋊C4—22no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C42C2312yes—
128#491—44no—
129#488—48no—
130#436—28no—
131C2×C4⋊C8C2312yes—
132#417—48no—
133#417—48no—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×C22⋊C8C2314yes—
149C2×#97C2312yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2314yes—
152#400—22no—
153#400—22no—
154#400—22no—
155#398—22no—
156#398—22no—
157#398—22no—
158#397—24no—
159#394—24no—
160#394—24no—
161#391C2314yes—
162#390—21no—
163#390—21no—
164#390C2312yes—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389C2311yescharacteristic
176#389C2311yescharacteristic
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#387—21no—
182#387—21no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—24no—
192#385—24no—
193#384—21no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#382—21no—
202#382—21no—
203#382C2311yescharacteristic
204#382C2312yes—
205#380—22no—
206#380—22no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380—22no—
211#380C2312yes—
212#380C2312yes—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223C2×#95—22no—
224C43C2311yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2314yes—
233#336C2311yescharacteristic
234#336C2312yes—
235#336C2311yescharacteristic
236C2×#41—28no—
237C2×#41—28no—
238C2×#41—28no—
239C2×#41—44no—
240C2×#41—44no—
241C2×#41—44no—
242C22×Q8C2414yes—
243C22×D4—24no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C2×D411yescharacteristic
247C22×D4C2412yes—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—24no—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2412yes—
255C2×M16—28no—
256C2×M16—44no—
257C2×M16—44no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—44no—
264C4⋊Q8—42no—
265C4⋊Q8—42no—
266#127—24no—
267#125—22no—
268#125—41no—
269#125—44no—
270#125—82no—
271#124—28no—
272#124—44no—
273#123—416no—
274C22⋊Q8—28no—
275C22⋊Q8—28no—
276#121—48no—
277#121—48no—
278#121—48no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—42no—
282C4×D4—42no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—48no—
288C4×D4—48no—
289C4×D4#4118yes—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—48no—
319C2×C4⋊C4—84no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—44no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C4⋊C8—24no—
355#103—28no—
356#103—44no—
357#103—44no—
358#103—44no—
359#103—44no—
360#103—44no—
361#103—44no—
362#103—44no—
363#102—24no—
364#102—28no—
365#102—24no—
366#102—24no—
367#102—42no—
368#102—44no—
369C22⋊C8—28no—
370#97—42no—
371C4×C8—24no—
372C4×C8—44no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—22no—
393C24C22×D412yes—
394#41—416no—
395#41—416no—
396#41—88no—
397#41—88no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—84no—
420C2×D4#12414yes—
421C2×D4H(5,𝔽2)14yes—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—84no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4118yes—
461C22×C4C2511yescharacteristic
462M16—88no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4#12414yes—
510C4⋊C4H(5,𝔽2)12yes—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C22⋊C4#12418yes—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—24no—
527C42—21no—
528C42—22no—
529C42—21no—
530C42—42no—
531C42—42no—
532C42—42no—
533C42—42no—
534C42C2×#4114yes—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23C2×#12414yes—
549C23C23×D411yescharacteristic
550Q8—44no—
551Q8—44no—
552Q8—44no—
553Q8—44no—
554D4—28no—
555D4—24no—
556D4—24no—
557D4—48no—
558D4—48no—
559D4—42no—
560D4—42no—
561D4—816no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—816no—
611C2×C4—88no—
612C2×C4—88no—
613C2×C4—164no—
614C2×C4C2×#12412yes—
615C2×C4C2×#12414yes—
616C2×C4C2×#12412yes—
617C2×C4#57512yes—
618C2×C4#57512yes—
619C2×C4#57712yes—
620C2×C4#57812yes—
621C2×C4C22×#4112yes—
622C2×C4C2×H(5,𝔽2)12yes—
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4#58511yescharacteristic
625C8—42no—
626C8—44no—
627C8—42no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22C22×#12411yescharacteristic
643C22#348411yescharacteristic
644C22#348811yescharacteristic
645C22C2×#57511yescharacteristic
646C22C2×#57511yescharacteristic
647C22C2×#57711yescharacteristic
648C22C2×#57811yescharacteristic
649C4—24no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—41no—
654C4—42no—
655C4—41no—
656C4—42no—
657C4—42no—
658C4—82no—
659C4—81no—
660C4—84no—
661C4—81no—
662C4—82no—
663C4—81no—
664C4—82no—
665C4#314214yes—
666C4#351312yes—
667C4#351912yes—
668C2—44no—
669C2—84no—
670C2#3590311yescharacteristic
671C2C2×#314211yescharacteristic
672C2#6062311yescharacteristic
673C2#6159511yescharacteristic
674C2#6160411yescharacteristic
675C2#6199611yescharacteristic
676C2#6199811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
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
1. #36290C2semidirect
3. #30220C2semidirect
4. #30202C2semidirect
5. #29329C2semidirect
6. #29130C2semidirect
7. #28755C2semidirect
8. #27768C2semidirect
9. #27748C2semidirect
10. #27284C2semidirect
11. #27272C2semidirect
12. #25858C2semidirect
13. #23819C2semidirect
14. #23508C2semidirect
15. #22431C2semidirect
16. #22431C2semidirect
17. #20567C2semidirect

Maximal subgroups

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

subgroupindex
#362902
#361272
#302202
#302022
#293292
#291302
#287552
#277682
#277482
#272842
#272722
#258582
#238192
#235082
#224312
#205672