Finite Group Databasebeta

← #6537291

Subgroups — #6537291

Order 512.

1759 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic, complemented
2#34330C211yesmaximal, characteristic
3#29983C211yesmaximal, characteristic, complemented
4#29954C212yesmaximal, complemented
5#29258C211yesmaximal, characteristic, complemented
6#29156C212yesmaximal, complemented
7#29139C211yesmaximal, characteristic, complemented
8#26467C212yesmaximal, complemented
9#26423C214yesmaximal, complemented
10#26027C212yesmaximal, complemented
11#25964C214yesmaximal, complemented
12#24304C212yesmaximal, complemented
13#23777C211yesmaximal, characteristic, complemented
14#23451C212yesmaximal, complemented
15#22708C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic, complemented
17#22418C212yesmaximal
18C2×#550C2211yescharacteristic
19C2×#533C2212yes—
20C2×#478C2214yes—
21C2×C8×D4C2212yes—
22C2×#416C2214yes—
23#2868C2211yescharacteristic
24#2857C2212yes—
25#2552C2212yes—
26#2542—21no—
27#2538—21no—
28#2441C2214yes—
29#2429—22no—
30#2423—22no—
31#2366—21no—
32#2366C2211yescharacteristic
33#2365—21no—
34#2364—22no—
35#2363C2212yes—
36#2357—21no—
37#2355C2211yescharacteristic
38#2354C2211yescharacteristic
39#2354C2212yes—
40Q8×C4⋊C4—21no—
41#2339C2212yes—
42#2333—22no—
43#2328—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#121C2212yes—
47#2297C2212yes—
48#2295C2211yescharacteristic
49#2275—21no—
50C42×D4C2211yescharacteristic
51#2100C2211yescharacteristic
52#2099C2212yes—
53#2096C2212yes—
54#2095C2211yescharacteristic
55#2094C2212yes—
56#2087C2212yes—
57#2068C2212yes—
58#2067C2214yes—
59#2066C2214yes—
60#2058C2212yes—
61#2057C2211yescharacteristic
62#2026C2214yes—
63#2025C2214yes—
64#2024C2212yes—
65#2024C2211yescharacteristic
66#2023C2212yes—
67#2023C2212yes—
68#2023C2212yes—
69#1989C2214yes—
70#1982C2212yes—
71#1957C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1939C2212yes—
75#1920C2214yes—
76#1893C2214yes—
77#1875C2214yes—
78#1864C2214yes—
79#1864C2214yes—
80#1854C2214yes—
81#1854C2214yes—
82#1850C2214yes—
83#1834C2212yes—
84#1831C2211yescharacteristic
85#1807C2214yes—
86#1803C2212yes—
87#1802C2214yes—
88#1797C2212yes—
89#1788C2214yes—
90C4×#107C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93#1735C2214yes—
94C2×#336C2212yes—
95C22×#41C2312yes—
96C23×C8C2314yes—
97#550—24no—
98#550Q818yescomplemented
99#533—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127—21no—
105C2×#127C2311yescharacteristic
106C2×#125C2314yes—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×#121—22no—
111C2×#121C2312yes—
112C2×C4×Q8—21no—
113C2×C4×Q8—22no—
114C2×C4×Q8—22no—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×C4×D4C2311yescharacteristic
126C2×#117C2314yes—
127C2×#117C2312yes—
128C22×C42C2312yes—
129#478—48no—
130C8×D4—28no—
131C2×#107C2312yes—
132C2×#107C2312yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2312yes—
138#416—48no—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2314yes—
147C2×C4×C8C2312yes—
148#399—24no—
149#398—24no—
150#396—24no—
151#391—21no—
152#391—22no—
153#391C2312yes—
154#391C2311yescharacteristic
155#391C2311yescharacteristic
156#390—21no—
157#390—22no—
158#390—21no—
159#390—22no—
160#390—21no—
161#390—21no—
162#390C2312yes—
163#390C2312yes—
164#389—22no—
165#389—21no—
166#389—22no—
167#389—24no—
168#389C2312yes—
169#389C2311yescharacteristic
170#389C2314yes—
171#389C2311yescharacteristic
172#388—22no—
173#388—24no—
174#388—22no—
175#388—22no—
176#387—21no—
177#387—21no—
178#387—22no—
179#387—24no—
180#386—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#384—22no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382—21no—
195#381—22no—
196#380—22no—
197#380C2312yes—
198#380C2314yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217#376—21no—
218C2×#95—22no—
219C43C2311yescharacteristic
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C2×#41—24no—
235C2×#41—44no—
236C2×#41—44no—
237C2×#41C2×Q818yes—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8C2411yescharacteristic
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2×D411yescharacteristic
246C22×D4C2412yes—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2412yes—
253C22×C8—28no—
254C22×C8—44no—
255C22×C8—44no—
256C22×C8C2412yes—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262C4⋊Q8—82no—
263#127—28no—
264#127—22no—
265#127C2×Q814yes—
266#125—28no—
267#125—44no—
268C22⋊Q8—48no—
269C22⋊Q8—48no—
270C22⋊Q8—84no—
271#121—24no—
272#121—216no—
273#121C2×Q818yes—
274C4×Q8—22no—
275C4×Q8—44no—
276C4×Q8—48no—
277C4×Q8—48no—
278C4×Q8C2×Q814yes—
279C4×D4—28no—
280C4×D4—24no—
281C4×D4—42no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4—82no—
287C4×D4—84no—
288C4×D4C2×Q814yes—
289C4×D4#4114yes—
290#117—28no—
291#117—44no—
292#117—44no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—44no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—24no—
352#107—42no—
353#106—44no—
354#106—44no—
355C4⋊C8—24no—
356C4⋊C8—42no—
357C4⋊C8—44no—
358#103—24no—
359#103—42no—
360#103—44no—
361#102—28no—
362#102—44no—
363#102—44no—
364#102—44no—
365C22⋊C8—28no—
366C4×C8—24no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—24no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—24no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—22no—
386C24C22×D412yes—
387#41—416no—
388#41—88no—
389C2×Q8—24no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—42no—
398C2×Q8C22×Q812yes—
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—81no—
411C2×D4—81no—
412C2×D4—82no—
413C2×D4—82no—
414C2×D4—82no—
415C2×D4—82no—
416C2×D4C22⋊Q814yes—
417C2×D4C22×Q814yes—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×Q814yes—
453C22×C4C2×#4114yes—
454C22×C4C2511yescharacteristic
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—88no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—84no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4C22⋊Q814yes—
494C4⋊C4C22×Q812yes—
495C4⋊C4H(5,𝔽2)14yes—
496C22⋊C4—28no—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—48no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4—84no—
509C22⋊C4C22⋊Q818yes—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—42no—
517C42—42no—
518C42—41no—
519C42—48no—
520C42—42no—
521C42—44no—
522C42—82no—
523C42—82no—
524C42#13514yes—
525C42C22×Q812yes—
526C42C2×#4112yes—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23C2×C22⋊Q814yes—
541C23C23×D411yescharacteristic
542Q8—24no—
543Q8—48no—
544Q8—42no—
545Q8—88no—
546D4—28no—
547D4—28no—
548D4—48no—
549D4—44no—
550D4—84no—
551D4—88no—
552D4—88no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—88no—
601C2×C4—164no—
602C2×C4—164no—
603C2×C4C2×C22⋊Q812yes—
604C2×C4C2×C22⋊Q814yes—
605C2×C4C2×C22⋊Q812yes—
606C2×C4C2×#13512yes—
607C2×C4C2×#13512yes—
608C2×C4C2×#13512yes—
609C2×C4#57512yes—
610C2×C4#57712yes—
611C2×C4C23×Q811yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C2×H(5,𝔽2)12yes—
614C2×C4#58512yes—
615C8—42no—
616C8—44no—
617C8—42no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22C22×C22⋊Q811yescharacteristic
633C22#348411yescharacteristic
634C22#350011yescharacteristic
635C22C22×#13511yescharacteristic
636C22C2×#57511yescharacteristic
637C22C2×#57711yescharacteristic
638C22#358511yescharacteristic
639C4—22no—
640C4—22no—
641C4—24no—
642C4—22no—
643C4—22no—
644C4—42no—
645C4—42no—
646C4—41no—
647C4—42no—
648C4—42no—
649C4—44no—
650C4—82no—
651C4—82no—
652C4—81no—
653C4—84no—
654C4—81no—
655C4#307614yes—
656C4#350112yes—
657C4#351412yes—
658C2—44no—
659C2—84no—
660C2#3570211yescharacteristic
661C2C2×#307611yescharacteristic
662C2#6049511yescharacteristic
663C2#6156611yescharacteristic
664C2#6160411yescharacteristic
665C2#6253111yescharacteristic
666C2#6258611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
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. #36134C2semidirect
3. #29983C2semidirect
4. #29954C2semidirect
5. #29258C2semidirect
6. #29156C2semidirect
7. #29139C2semidirect
8. #26467C2semidirect
9. #26423C2semidirect
10. #26027C2semidirect
11. #25964C2semidirect
12. #24304C2semidirect
13. #23777C2semidirect
14. #23451C2semidirect
15. #22708C2semidirect
16. #22696C2semidirect
98. #550Q8semidirect

Maximal subgroups

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

subgroupindex
#361342
#343302
#299832
#299542
#292582
#291562
#291392
#264672
#264232
#260272
#259642
#243042
#237772
#234512
#227082
#226962
#224182