Finite Group Databasebeta

← #6513526

Subgroups — #6513526

Order 512.

2380 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3103C211yesmaximal, characteristic, complemented
2C2×#2975C211yesmaximal, characteristic, complemented
3#35165C211yesmaximal, characteristic
4C2×#2538C211yesmaximal, characteristic, complemented
5#28230C214yesmaximal, complemented
6#28150C214yesmaximal, complemented
7#25395C214yesmaximal, complemented
8#25348C214yesmaximal, complemented
9#24557C214yesmaximal, complemented
10#24387C214yesmaximal, complemented
11#21982C211yesmaximal, characteristic, complemented
12#21859C211yesmaximal, characteristic, complemented
13#20734C211yesmaximal, characteristic, complemented
14C2×#535—21no—
15C2×#532C2211yescharacteristic
16C22×C22⋊Q8C2212yes—
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#3103—28no—
20C2×#481C2212yes—
21C2×#475C2212yescomplemented
22C2×#468C2218yescomplemented
23C2×#460C2212yescomplemented
24C2×#448C2212yes—
25#2975—28no—
26C2×C4×SD16C2214yescomplemented
27C2×#436C2214yes—
28C2×#436C2214yes—
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#407C2214yes—
32#2847C2214yes—
33#2617—22no—
34#2547C2212yes—
35#2544—22no—
36#2541C2212yes—
37#2538—22no—
38#2538—22no—
39#2538C2214yes—
40#2538C2214yes—
41#2537C2212yes—
42#2537C2212yes—
43#2535—22no—
44C2×#396—22no—
45#2311—21no—
46C2×#390—21no—
47C2×#389—21no—
48#2285C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C2×#380C2212yes—
51#2077C2218yescomplemented
52#2036C2218yescomplemented
53#2033C2218yescomplemented
54#2000C2218yescomplemented
55#1990C2212yescomplemented
56#1957C2212yescomplemented
57#1947C2214yescomplemented
58#1895C2218yescomplemented
59#1894C2218yescomplemented
60#1874C2214yes—
61#1852C2214yes—
62#1845C2212yescomplemented
63#1834C2212yescomplemented
64#1824C2212yes—
65#1809C2218yescomplemented
66#1803C2212yescomplemented
67#1803C2212yescomplemented
68#1802C2214yescomplemented
69#1787C2212yes—
70C4×#103C2214yescomplemented
71C22×#41—21no—
72C23×Q8C2311yescharacteristic
73C23×D4D411yescharacteristic
74C24×C4C2312yes—
75C22×SD16C2314yes—
76C22×M16C2318yes—
77#535—48no—
78#532—24no—
79#532—24no—
80C2×C4⋊Q8C2311yescharacteristic
81C2×#127—22no—
82C2×#127C2311yescharacteristic
83C2×#124—22no—
84C2×#124—21no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—48no—
87C2×C22⋊Q8C2318yes—
88C2×#121—22no—
89C2×#121—22no—
90C2×#121—21no—
91C2×#121C2312yes—
92C2×#121C2314yes—
93C2×#121C2314yes—
94C2×#120—24no—
95C2×C4×Q8—21no—
96C2×C4×Q8—21no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8C2312yes—
100C2×C4×Q8C2312yes—
101C2×C4×D4—24no—
102C2×C4×D4—21no—
103C2×C4×D4—24no—
104C2×C4×D4D412yes—
105C2×C4×D4D412yes—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C2×#117—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4—22no—
118C22×C22⋊C4—22no—
119C22×C22⋊C4D412yes—
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122#481—28no—
123#475—28no—
124#468—416no—
125#460—28no—
126#448—28no—
127C4×SD16—216no—
128#436—216no—
129#436—216no—
130#431—24no—
131#431D418yescomplemented
132C2×#107C2318yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140C2×#102C2318yes—
141#407—216no—
142C2×C22⋊C8C2318yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2312yes—
147#397—24no—
148#396—28no—
149#396—28no—
150#394—24no—
151#394—24no—
152#392—24no—
153#391—22no—
154#390—24no—
155#390—24no—
156#390—22no—
157#389—24no—
158#389—24no—
159#386—24no—
160#386—24no—
161#386—24no—
162#384—21no—
163#384—21no—
164#384C2312yes—
165#384C2311yescharacteristic
166#384C2311yescharacteristic
167#384C2312yes—
168#382—21no—
169#380—21no—
170#380—24no—
171#380—22no—
172#380—22no—
173#380C2314yes—
174#380C2318yes—
175#380C2314yes—
176#380C2312yes—
177C4×C4⋊C4—21no—
178C4×C4⋊C4C2312yes—
179C2×#95—22no—
180#340C2318yes—
181#340C2314yes—
182#340C2314yes—
183#340C2318yes—
184C25C2×D412yes—
185C2×#41—44no—
186C2×#41—44no—
187C2×#41—44no—
188C22×Q8—24no—
189C22×Q8—21no—
190C22×Q8—21no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—42no—
194C22×Q8C2412yes—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—21no—
202C22×D4—21no—
203C22×D4C2×D412yes—
204C22×D4C2×D412yes—
205C22×D4C2412yes—
206C23×C4—21no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—21no—
211C23×C4—22no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C2×D412yes—
215C23×C4C2×D414yes—
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C2×SD16—28no—
221C2×SD16—28no—
222C2×SD16—44no—
223C2×M16—216no—
224C2×M16—48no—
225C2×M16—48no—
226C22×C8C2414yes—
227C22×C8C2414yes—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230#127—22no—
231#127—22no—
232#127—48no—
233#124—48no—
234#124—44no—
235C22⋊Q8—216no—
236C22⋊Q8—48no—
237C22⋊Q8—416no—
238#121—24no—
239#121—24no—
240#121—48no—
241#121—48no—
242#121—44no—
243#121—48no—
244#121—48no—
245#120—416no—
246C4×Q8—24no—
247C4×Q8—24no—
248C4×Q8—24no—
249C4×Q8—24no—
250C4×Q8—44no—
251C4×Q8—44no—
252C4×D4—28no—
253C4×D4—28no—
254C4×D4—24no—
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—24no—
260C4×D4—44no—
261#117—44no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—48no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—82no—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4—84no—
303C2×C22⋊C4C2×D414yes—
304C2×C22⋊C4C2×D414yes—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—44no—
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D414yes—
312C2×C42#4114yes—
313C2×C42C2411yescharacteristic
314C2×C42C2411yescharacteristic
315#107—48no—
316#106—28no—
317C4⋊C8—24no—
318C4⋊C8C2×D418yes—
319#103—28no—
320#103—28no—
321#103—44no—
322#102—28no—
323#102—48no—
324C22⋊C8—24no—
325C22⋊C8—216no—
326C22⋊C8C2×D418yes—
327C4×C8—24no—
328C4×C8C2×D418yes—
329#95—28no—
330#95—22no—
331#95—22no—
332#95—24no—
333#95—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—22no—
338C24—42no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24C22×D414yes—
343C24C22×D411yescharacteristic
344#41—816no—
345C2×Q8—24no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8H(5,𝔽2)14yes—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—416no—
379C2×D4—81no—
380C2×D4—81no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—84no—
384C2×D4—84no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—88no—
418C22×C4—88no—
419C22×C4#12114yes—
420C22×C4#12114yes—
421C22×C4AGL(1,ℤ/8)14yes—
422C22×C4#13714yes—
423C22×C4C22×D414yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C2×#4112yes—
431C22×C4C2511yescharacteristic
432SD16—416no—
433M16—816no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8C22×D418yes—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—28no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—416no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C22⋊C4—28no—
464C22⋊C4—28no—
465C22⋊C4—416no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—84no—
472C22⋊C4—84no—
473C22⋊C4—88no—
474C42—44no—
475C42—44no—
476C42—44no—
477C42#12114yes—
478C42#12114yes—
479C42C22×D412yes—
480C42C22×D412yes—
481C42C2×#4112yes—
482C42C2×#4112yes—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—48no—
495C23—48no—
496C23—48no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—84no—
505C23—84no—
506C23—84no—
507C23—84no—
508C23C2×#12112yes—
509C23C2×AGL(1,ℤ/8)12yes—
510C23C2×#13712yes—
511C23C23×D411yescharacteristic
512Q8—24no—
513Q8—24no—
514Q8—48no—
515Q8—42no—
516Q8—42no—
517Q8—84no—
518D4—44no—
519D4—44no—
520D4—48no—
521D4—48no—
522D4—84no—
523D4—88no—
524D4—84no—
525D4—168no—
526D4—168no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—44no—
540C2×C4—48no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—48no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—88no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—88no—
564C2×C4—88no—
565C2×C4#468116yescomplemented
566C2×C4C2×#12112yes—
567C2×C4C2×#12112yes—
568C2×C4C2×#12112yes—
569C2×C4C2×#12112yes—
570C2×C4C2×#12112yes—
571C2×C4C2×AGL(1,ℤ/8)12yes—
572C2×C4C2×AGL(1,ℤ/8)12yes—
573C2×C4C2×#13712yes—
574C2×C4C2×#13712yes—
575C2×C4C23×D411yescharacteristic
576C2×C4C22×#4111yescharacteristic
577C2×C4C2×H(5,𝔽2)12yes—
578C2×C4#58512yes—
579C8—28no—
580C8—44no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—816no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22C2×#46814yes—
603C22C22×#12111yescharacteristic
604C22#348411yescharacteristic
605C22#348511yescharacteristic
606C22C22×AGL(1,ℤ/8)11yescharacteristic
607C22C22×#13711yescharacteristic
608C22#358511yescharacteristic
609C22#358611yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—44no—
614C4—44no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—84no—
619C4—82no—
620C4C2×#46814yes—
621C4C2×#46814yes—
622C4#348812yes—
623C4#349412yes—
624C2—24no—
625C2—84no—
626C2—162no—
627C2#3551411yescharacteristic
628C2C22×#46811yescharacteristic
629C2#6045611yescharacteristic
630C2#6158411yescharacteristic
631C2#6158711yescharacteristic
632C2#6171911yescharacteristic
633C2#6172211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35165C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3103C2semidirect
2. C2×#2975C2semidirect
4. C2×#2538C2semidirect
5. #28230C2semidirect
6. #28150C2semidirect
7. #25395C2semidirect
8. #25348C2semidirect
9. #24557C2semidirect
10. #24387C2semidirect
11. #21982C2semidirect
12. #21859C2semidirect
13. #20734C2semidirect
21. C2×#475C22semidirect
22. C2×#468C22semidirect
23. C2×#460C22semidirect
26. C2×C4×SD16C22semidirect
51. #2077C22semidirect
52. #2036C22semidirect
53. #2033C22semidirect
54. #2000C22semidirect
55. #1990C22semidirect
56. #1957C22semidirect
57. #1947C22semidirect
58. #1895C22semidirect
59. #1894C22semidirect
62. #1845C22semidirect
63. #1834C22semidirect
65. #1809C22semidirect
66. #1803C22semidirect
67. #1803C22semidirect
68. #1802C22semidirect
70. C4×#103C22semidirect
131. #431D4semidirect
565. C2×C4#468semidirect

Maximal subgroups

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

subgroupindex
C2×#31032
C2×#29752
#351652
C2×#25382
#282302
#281502
#253952
#253482
#245572
#243872
#219822
#218592
#207342