Finite Group Databasebeta

← #6470781

Subgroups — #6470781

Order 512.

2132 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3200C212yesmaximal
2#37386C211yesmaximal, characteristic, complemented
3#34275C211yesmaximal, characteristic
4#33114C212yesmaximal, complemented
5#32914C212yesmaximal, complemented
6#29766C212yesmaximal, complemented
7#29714C212yesmaximal, complemented
8#25230C214yesmaximal, complemented
9#25211C214yesmaximal, complemented
10#25099C214yesmaximal, complemented
11#25078C214yesmaximal, complemented
12#21936C211yesmaximal, characteristic, complemented
13#20913C212yesmaximal, complemented
14C2×#536C2212yes—
15C2×#535C2211yescharacteristic
16C2×#520C2211yescharacteristic
17C2×#518C2211yescharacteristic
18#3200—216no—
19C2×#485C2214yes—
20C2×#482C2214yes—
21C2×#468C2214yes—
22C2×#465C2214yes—
23C2×#461C2212yes—
24C2×#461C2212yes—
25C2×#448C2214yes—
26C2×#450C2214yes—
27C2×#419C2212yescomplemented
28C22×#103C2212yescomplemented
29C2×#408C2212yes—
30#2857C2212yes—
31#2734—21no—
32#2734—21no—
33#2724—21no—
34#2724—21no—
35#2717—24no—
36#2621C2212yes—
37#2499C2214yescomplemented
38#2434C2214yescomplemented
39#2363C2214yes—
40#2361C2214yes—
41#2358—24no—
42#2328—21no—
43#2326—21no—
44#2317C2211yescharacteristic
45C4×#127—21no—
46C4×#124—21no—
47#2297C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2085C2214yescomplemented
51#2071C2214yes—
52#2068C2214yescomplemented
53#2066C2214yescomplemented
54#2055C2214yes—
55#2055C2214yes—
56#2043C2214yescomplemented
57#2013C2214yes—
58#1981C2212yescomplemented
59#1981C2212yescomplemented
60#1948C2212yescomplemented
61#1947C2212yescomplemented
62#1928C2214yescomplemented
63#1927C2214yescomplemented
64#1877C2214yes—
65#1877C2214yes—
66#1873C2214yes—
67#1873C2214yescomplemented
68#1872C2214yes—
69#1872C2214yescomplemented
70#1851C2214yescomplemented
71#1850C2214yescomplemented
72#1804C2212yes—
73#1804C2212yes—
74#1798C2212yes—
75#1797C2212yes—
76#1736C2212yescomplemented
77#1735C2212yescomplemented
78C22×#41C2311yescharacteristic
79C23×Q8C2312yes—
80C23×D4C2311yescharacteristic
81C22×SD16C2314yes—
82C22×SD16C2314yes—
83C22×M16C2312yes—
84C23×C8C2312yes—
85#536—28no—
86#536—28no—
87#535—24no—
88#535D418yes—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×#127—21no—
93C2×#124—21no—
94C2×#124—22no—
95C2×#123—22no—
96C2×#123—22no—
97C2×C22⋊Q8C2314yes—
98C2×C22⋊Q8C2314yes—
99C2×#121C2314yes—
100C2×#120—22no—
101#520—42no—
102#520—42no—
103#518—42no—
104#518—42no—
105C2×C4×Q8—21no—
106C2×C4×D4—21no—
107C2×C4×D4—22no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4—24no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C2×#117C2312yes—
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—22no—
121C22×C22⋊C4—22no—
122C22×C42C2311yescharacteristic
123#485—216no—
124#482—216no—
125#468—216no—
126#465—216no—
127#461—44no—
128#461—44no—
129#450—216no—
130#448—216no—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137#419—44no—
138C2×#103—44no—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147#408—28no—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2314yes—
150#398—22no—
151#398—22no—
152#397—22no—
153#397—22no—
154#397—24no—
155#397—24no—
156#395—24no—
157#395—24no—
158#391D411yescharacteristic
159#391D411yescharacteristic
160#391C2314yes—
161#391C2314yes—
162#390—24no—
163#390—21no—
164#390—21no—
165#390D411yescharacteristic
166#390D411yescharacteristic
167#390C2314yes—
168#389—22no—
169#389—22no—
170#389C2314yes—
171#388—24no—
172#388—24no—
173#387—24no—
174#387—24no—
175#386—24no—
176#386—24no—
177#386D414yes—
178#385—22no—
179#385—24no—
180#385—24no—
181#385—24no—
182#384—21no—
183#384C2312yes—
184#382—21no—
185#382C2312yes—
186#380—22no—
187#380—22no—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2312yes—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—24no—
192C4×C22⋊C4C2314yes—
193#376—21no—
194C43—21no—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C2×#41—41no—
203C2×#41—41no—
204C2×#41—41no—
205C2×#41—41no—
206C2×#41C2×D414yes—
207C2×#41#4114yes—
208C22×Q8—22no—
209C22×Q8—24no—
210C22×Q8—24no—
211C22×Q8—42no—
212C22×Q8—42no—
213C22×Q8—42no—
214C22×Q8—42no—
215C22×Q8C2×D411yescharacteristic
216C22×Q8C2414yes—
217C22×D4—24no—
218C22×D4—21no—
219C22×D4—22no—
220C22×D4—41no—
221C22×D4—41no—
222C22×D4—41no—
223C22×D4—41no—
224C22×D4C2×D414yes—
225C22×D4C2×D412yes—
226C22×D4C2×D411yescharacteristic
227C22×D4C2412yes—
228C23×C4—21no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4C2×D412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C2×SD16—28no—
237C2×SD16—28no—
238C2×SD16—44no—
239C2×SD16—44no—
240C2×SD16—44no—
241C2×SD16—44no—
242C2×M16—24no—
243C2×M16—42no—
244C2×M16—42no—
245C22×C8—24no—
246C22×C8—42no—
247C22×C8—42no—
248C22×C8C2414yes—
249C22×C8C2414yes—
250C4⋊Q8—42no—
251C4⋊Q8—42no—
252C4⋊Q8—42no—
253C4⋊Q8—42no—
254C4⋊Q8—48no—
255#127—28no—
256#124—28no—
257#124—48no—
258#123—48no—
259#123—48no—
260C22⋊Q8—28no—
261C22⋊Q8—28no—
262C22⋊Q8—28no—
263C22⋊Q8—28no—
264#121—28no—
265#121C2×D4116yes—
266#120—216no—
267C4×Q8—44no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—42no—
271C4×D4—42no—
272C4×D4—44no—
273C4×D4—44no—
274C4×D4—44no—
275C4×D4—48no—
276C4×D4—88no—
277#117—22no—
278#117—22no—
279#117—22no—
280#117—24no—
281#117—24no—
282#117—24no—
283#117—41no—
284#117—41no—
285#117—42no—
286#117C2×D414yes—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—41no—
301C2×C4⋊C4—41no—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—48no—
321C2×C22⋊C4—84no—
322C2×C22⋊C4C2×D414yes—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D414yes—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—41no—
332C2×C42—41no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2412yes—
336C2×C42C2412yes—
337#107—28no—
338#106—28no—
339C4⋊C8—42no—
340C4⋊C8—42no—
341C4⋊C8—42no—
342C4⋊C8—42no—
343#103—28no—
344#103—28no—
345#103—28no—
346#103—28no—
347#103—44no—
348#103—44no—
349#102—28no—
350#102—28no—
351C22⋊C8—28no—
352C22⋊C8—28no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—22no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95C2×D414yes—
363#95C2×D414yes—
364C24—22no—
365C24—24no—
366C24—42no—
367C24—42no—
368C24—82no—
369C24—82no—
370C24C22×D411yescharacteristic
371C24C22×D412yes—
372#41—44no—
373#41—44no—
374#41—44no—
375#41—44no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—41no—
383C2×Q8—41no—
384C2×Q8—48no—
385C2×Q8—48no—
386C2×Q8—44no—
387C2×Q8—82no—
388C2×Q8—82no—
389C2×Q8—84no—
390C2×Q8—84no—
391C2×Q8#12114yes—
392C2×D4—216no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—41no—
402C2×D4—42no—
403C2×D4—41no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—84no—
410C2×D4—84no—
411C2×D4#12118yes—
412C2×D4#12114yes—
413C2×D4C22×D414yes—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—84no—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4112yes—
452C22×C4C2511yescharacteristic
453SD16—88no—
454SD16—88no—
455M16—84no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—84no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—84no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4C22×D414yes—
484C4⋊C4H(5,𝔽2)12yes—
485C4⋊C4H(5,𝔽2)12yes—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—216no—
489C22⋊C4—28no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—84no—
500C42—24no—
501C42—24no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—41no—
507C42—41no—
508C42—44no—
509C42—44no—
510C42—44no—
511C42—82no—
512C23—24no—
513C23—22no—
514C23—21no—
515C23—22no—
516C23—21no—
517C23—22no—
518C23—24no—
519C23—22no—
520C23—24no—
521C23—48no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—88no—
527C23—164no—
528C23—162no—
529C23—162no—
530C23C2×#12112yes—
531C23C2×#12114yes—
532C23C23×D411yescharacteristic
533Q8—44no—
534Q8—44no—
535Q8—44no—
536Q8—44no—
537Q8—44no—
538Q8—88no—
539D4—24no—
540D4—24no—
541D4—44no—
542D4—48no—
543D4—48no—
544D4—42no—
545D4—42no—
546D4—816no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—84no—
584C2×C4—88no—
585C2×C4—164no—
586C2×C4—164no—
587C2×C4—164no—
588C2×C4—162no—
589C2×C4C2×#12112yes—
590C2×C4C2×#12114yes—
591C2×C4#57714yes—
592C2×C4C23×D411yescharacteristic
593C2×C4C22×#4111yescharacteristic
594C2×C4C2×H(5,𝔽2)11yescharacteristic
595C2×C4C2×H(5,𝔽2)11yescharacteristic
596C2×C4#58511yescharacteristic
597C2×C4#58511yescharacteristic
598C8—44no—
599C8—44no—
600C22—28no—
601C22—24no—
602C22—24no—
603C22—22no—
604C22—42no—
605C22—44no—
606C22—42no—
607C22—41no—
608C22—42no—
609C22—41no—
610C22—48no—
611C22—44no—
612C22—84no—
613C22—82no—
614C22—82no—
615C22—168no—
616C22C22×#12111yescharacteristic
617C22#348411yescharacteristic
618C22#348511yescharacteristic
619C22C2×#57712yes—
620C22#358612yes—
621C4—24no—
622C4—22no—
623C4—24no—
624C4—44no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—84no—
631C4—84no—
632C4—82no—
633C4#345911yescharacteristic
634C4#345911yescharacteristic
635C4#349611yescharacteristic
636C4#349611yescharacteristic
637C2—44no—
638C2—42no—
639C2—84no—
640C2#3498411yescharacteristic
641C2#6042912yes—
642C2#6166211yescharacteristic
643C2#6167411yescharacteristic
644C2#6175312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #37386C2semidirect
4. #33114C2semidirect
5. #32914C2semidirect
6. #29766C2semidirect
7. #29714C2semidirect
8. #25230C2semidirect
9. #25211C2semidirect
10. #25099C2semidirect
11. #25078C2semidirect
12. #21936C2semidirect
13. #20913C2semidirect
27. C2×#419C22semidirect
28. C22×#103C22semidirect
37. #2499C22semidirect
38. #2434C22semidirect
50. #2085C22semidirect
52. #2068C22semidirect
53. #2066C22semidirect
56. #2043C22semidirect
58. #1981C22semidirect
59. #1981C22semidirect
60. #1948C22semidirect
61. #1947C22semidirect
62. #1928C22semidirect
63. #1927C22semidirect
67. #1873C22semidirect
69. #1872C22semidirect
70. #1851C22semidirect
71. #1850C22semidirect
76. #1736C22semidirect
77. #1735C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32002
#373862
#342752
#331142
#329142
#297662
#297142
#252302
#252112
#250992
#250782
#219362
#209132