Finite Group Databasebeta

← #6476713

Subgroups — #6476713

Order 512.

2063 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2621C211yesmaximal, characteristic, complemented
2#34204C211yesmaximal, characteristic
3#33117C212yesmaximal, complemented
4#32914C212yesmaximal, complemented
5#30120C214yesmaximal, complemented
6#29341C212yesmaximal, complemented
7#28320C212yesmaximal, complemented
8#27684C212yesmaximal, complemented
9#24666C214yesmaximal, complemented
10#24209C212yesmaximal, complemented
11#24010C212yesmaximal, complemented
12#23919C212yesmaximal
13#22047C212yesmaximal, complemented
14#22041C212yesmaximal, complemented
15#20716C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic, complemented
17C22×C22⋊Q8C2212yes—
18C22×#121C2211yescharacteristic
19C2×#518C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C2×#485C2214yes—
22C2×#461C2214yescomplemented
23C2×#448C2212yes—
24C2×#448C2212yes—
25C2×#450C2214yes—
26C2×#433C2212yescomplemented
27#2621—22no—
28#2621—22no—
29#2621C2212yes—
30#2621C2212yes—
31#2621C2212yescomplemented
32#2621C2212yescomplemented
33C2×#398—22no—
34C2×#397—22no—
35#2363—22no—
36#2361—22no—
37#2342—22no—
38D4×C22⋊C4—22no—
39#2324C2212yes—
40#2324C2212yes—
41#2316C2211yescharacteristic
42#2309C2211yescharacteristic
43C4×#121C2212yes—
44C4×#121C2212yes—
45C2×#391C2212yescomplemented
46C2×#382C2211yescharacteristic, complemented
47#2280C2211yescharacteristic
48#2274C2211yescharacteristic
49#2101C2214yescomplemented
50#2092C2214yescomplemented
51#2043C2214yescomplemented
52#2036C2214yescomplemented
53#2035C2214yescomplemented
54#2013C2214yescomplemented
55#2000C2214yes—
56#2000C2214yes—
57#1981C2214yescomplemented
58#1973C2214yescomplemented
59#1968C2214yescomplemented
60#1949C2214yescomplemented
61#1935C2214yescomplemented
62#1933C2214yescomplemented
63#1928C2214yescomplemented
64#1927C2214yescomplemented
65#1877C2214yescomplemented
66#1877C2214yescomplemented
67#1874C2212yes—
68#1874C2212yes—
69#1873C2214yes—
70#1872C2214yes—
71#1853C2212yescomplemented
72#1852C2212yescomplemented
73#1818C2212yescomplemented
74#1804C2214yescomplemented
75#1798C2212yes—
76#1797C2212yescomplemented
77#1787C2212yescomplemented
78#1787C2212yescomplemented
79#1779C2212yes—
80#1750C2212yes—
81#1749C2211yescharacteristic, complemented
82C4×C22⋊C8C2211yescharacteristic, complemented
83C23×Q8C2312yes—
84C23×D4—21no—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C22×SD16C2314yes—
88C22×SD16C2314yes—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—44no—
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2312yes—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—48no—
96C2×C22⋊Q8C2314yes—
97C2×C22⋊Q8C2314yes—
98C2×#121—21no—
99C2×#121—21no—
100C2×#121—28no—
101C2×#121C2312yes—
102C2×#121C2312yes—
103#518—44no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C2×#117—44no—
110C2×#117D412yes—
111C2×#117D412yes—
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C22×C4⋊C4—22no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4D412yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—22no—
125C22×C22⋊C4—22no—
126C22×C22⋊C4D412yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#485—48no—
130#461—48no—
131#450—48no—
132#448—28no—
133#448—28no—
134#433—44no—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2314yes—
150C2×#97C2312yes—
151C2×C4×C8C2312yes—
152#398—24no—
153#398—24no—
154#398—24no—
155#398—24no—
156#397—24no—
157#397—24no—
158#397—24no—
159#397—24no—
160#391—24no—
161#391C2314yes—
162#391C2314yes—
163#390—22no—
164#390—22no—
165#390—22no—
166#390C2312yes—
167#390C2312yes—
168#389—22no—
169#389C2312yes—
170#389C2312yes—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#386—22no—
176#386—22no—
177#386—22no—
178#386—22no—
179#385—22no—
180#385—22no—
181#382—22no—
182#382C2312yes—
183#382C2312yes—
184#381—21no—
185#380—22no—
186#380C2312yes—
187#380C2312yes—
188#379—21no—
189#379—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C22⋊C4—21no—
194C4×C22⋊C4—21no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C2×#95—22no—
199C2×#95—24no—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2314yes—
205#336C2312yes—
206#336C2312yes—
207C25—21no—
208C25C2×D411yescharacteristic
209C22×Q8—22no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—44no—
213C22×Q8—44no—
214C22×Q8C2414yes—
215C22×D4—22no—
216C22×D4—28no—
217C22×D4—22no—
218C22×D4—24no—
219C22×D4—21no—
220C22×D4—24no—
221C22×D4—42no—
222C22×D4—84no—
223C22×D4C2412yes—
224C23×C4—21no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—42no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D412yes—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C2×SD16—28no—
241C2×SD16—28no—
242C2×SD16—44no—
243C2×SD16—44no—
244C2×SD16—44no—
245C2×SD16—44no—
246C22×C8C2414yes—
247C22×C8C2414yes—
248C4⋊Q8—28no—
249C4⋊Q8—42no—
250C4⋊Q8—42no—
251C4⋊Q8—48no—
252C22⋊Q8—28no—
253C22⋊Q8—28no—
254C22⋊Q8—44no—
255C22⋊Q8—44no—
256C22⋊Q8—48no—
257C22⋊Q8—48no—
258#121—28no—
259#121—28no—
260#121—24no—
261#121—24no—
262#121—24no—
263#121#4118yes—
264C4×D4—28no—
265C4×D4—44no—
266C4×D4—48no—
267C4×D4—84no—
268C4×D4—84no—
269#117—24no—
270#117—24no—
271#117—24no—
272#117—42no—
273#117—42no—
274#117—44no—
275#117—44no—
276#117—42no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—48no—
289C2×C4⋊C4—48no—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—48no—
315C2×C22⋊C4—48no—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C42—24no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—42no—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332#107—44no—
333#106—44no—
334C4⋊C8—28no—
335#103—28no—
336#103—28no—
337#103—44no—
338#103—44no—
339#103—44no—
340#103—44no—
341#102—44no—
342#102—44no—
343C22⋊C8—24no—
344C22⋊C8—24no—
345C22⋊C8—28no—
346C22⋊C8—44no—
347#97—42no—
348C4×C8—42no—
349#95—24no—
350#95—24no—
351#95—28no—
352#95—28no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—24no—
364C24—41no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—82no—
369C24—82no—
370C24C22×D411yescharacteristic
371C24C22×D411yescharacteristic
372C24C22×D411yescharacteristic
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—88no—
383C2×Q8—88no—
384C2×D4—24no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—416no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—42no—
395C2×D4—82no—
396C2×D4—82no—
397C2×D4—816no—
398C2×D4C2×#4114yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—416no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C2×#4114yes—
449C22×C4H(5,𝔽2)14yes—
450C22×C4C2511yescharacteristic
451SD16—88no—
452SD16—88no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4C2×#4114yes—
477C22⋊C4—28no—
478C22⋊C4—28no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—42no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—42no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—42no—
500C42—42no—
501C42—42no—
502C42—82no—
503C42—82no—
504C42H(5,𝔽2)14yes—
505C23—22no—
506C23—21no—
507C23—21no—
508C23—22no—
509C23—21no—
510C23—21no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—44no—
519C23—48no—
520C23—48no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—84no—
529C23—82no—
530C23—84no—
531C23—84no—
532C23—164no—
533C23—164no—
534C23#52512yes—
535C23#53512yes—
536C23#57714yes—
537C23C23×D411yescharacteristic
538Q8—44no—
539Q8—44no—
540Q8—44no—
541Q8—44no—
542Q8—88no—
543D4—28no—
544D4—48no—
545D4—416no—
546D4—44no—
547D4—88no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—88no—
591C2×C4—88no—
592C2×C4—164no—
593C2×C4—164no—
594C2×C4#52512yes—
595C2×C4#53414yes—
596C2×C4#53512yes—
597C2×C4C22×#4112yes—
598C2×C4C2×H(5,𝔽2)12yes—
599C2×C4C2×H(5,𝔽2)12yes—
600C8—44no—
601C8—44no—
602C22—22no—
603C22—24no—
604C22—24no—
605C22—22no—
606C22—22no—
607C22—24no—
608C22—42no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—44no—
613C22—42no—
614C22—44no—
615C22—41no—
616C22—42no—
617C22—41no—
618C22—44no—
619C22—88no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—168no—
625C22#312314yes—
626C22C2×#52511yescharacteristic
627C22C2×#53411yescharacteristic
628C22C2×#53511yescharacteristic
629C22C2×#57712yes—
630C22#358612yes—
631C4—22no—
632C4—22no—
633C4—24no—
634C4—41no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—44no—
639C4—84no—
640C4—82no—
641C4—82no—
642C4—84no—
643C4#351914yes—
644C2—24no—
645C2—42no—
646C2—81no—
647C2—84no—
648C2#3507611yescharacteristic
649C2C2×#312311yescharacteristic
650C2#6056911yescharacteristic
651C2#6075812yes—
652C2#6081312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2621C2semidirect
3. #33117C2semidirect
4. #32914C2semidirect
5. #30120C2semidirect
6. #29341C2semidirect
7. #28320C2semidirect
8. #27684C2semidirect
9. #24666C2semidirect
10. #24209C2semidirect
11. #24010C2semidirect
13. #22047C2semidirect
14. #22041C2semidirect
15. #20716C2semidirect
16. C22×C4⋊Q8C22semidirect
22. C2×#461C22semidirect
26. C2×#433C22semidirect
31. #2621C22semidirect
32. #2621C22semidirect
45. C2×#391C22semidirect
46. C2×#382C22semidirect
49. #2101C22semidirect
50. #2092C22semidirect
51. #2043C22semidirect
52. #2036C22semidirect
53. #2035C22semidirect
54. #2013C22semidirect
57. #1981C22semidirect
58. #1973C22semidirect
59. #1968C22semidirect
60. #1949C22semidirect
61. #1935C22semidirect
62. #1933C22semidirect
63. #1928C22semidirect
64. #1927C22semidirect
65. #1877C22semidirect
66. #1877C22semidirect
71. #1853C22semidirect
72. #1852C22semidirect
73. #1818C22semidirect
74. #1804C22semidirect
76. #1797C22semidirect
77. #1787C22semidirect
78. #1787C22semidirect
81. #1749C22semidirect
82. C4×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26212
#342042
#331172
#329142
#301202
#293412
#283202
#276842
#246662
#242092
#240102
#239192
#220472
#220412
#207162