Finite Group Databasebeta

← #6481846

Subgroups — #6481846

Order 512.

2216 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×#3246C211yesmaximal, characteristic, complemented
2C22×#432C211yesmaximal, characteristic, complemented
3#36625C211yesmaximal, characteristic, complemented
4#36624C211yesmaximal, characteristic
5#33134C214yesmaximal, complemented
6#29735C214yesmaximal, complemented
7#24710C214yesmaximal, complemented
8#24708C214yesmaximal, complemented
9#24661C214yesmaximal, complemented
10#24659C214yesmaximal, complemented
11#21961C211yesmaximal, characteristic, complemented
12#21885C211yesmaximal, characteristic, complemented
13#21884C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C2×#532C2211yescharacteristic
16C2×#525—21no—
17C23×C42C2211yescharacteristic
18#3246—28no—
19C2×#475C2212yescomplemented
20C2×#476C2212yescomplemented
21C2×#465C2214yes—
22C2×#460C2212yescomplemented
23C2×#461C2212yescomplemented
24C2×#432—28no—
25C2×#432C2214yes—
26C2×#432C2214yes—
27C22×C4⋊C8C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C22×C22⋊C8C2212yes—
30C22×#97C2212yes—
31#2857C2212yes—
32#2857C2212yes—
33#2857C2212yes—
34#2856C2212yes—
35#2855—21no—
36#2855C2212yes—
37#2854—21no—
38#2854C2212yes—
39#2851—22no—
40#2850—22no—
41#2848—21no—
42#2848C2212yes—
43#2847—21no—
44#2847C2212yes—
45C4×C22⋊Q8—22no—
46C4×C22⋊Q8C2214yes—
47C4×#121—22no—
48C4×#121C2214yes—
49#2297—21no—
50#2296—21no—
51#2285C2211yescharacteristic
52#2285C2211yescharacteristic
53#2280—21no—
54#2077C2218yes—
55#2036C2218yes—
56#2033C2218yes—
57#2000C2218yes—
58#1988C2212yescomplemented
59#1987C2212yescomplemented
60#1982C2212yescomplemented
61#1981C2212yescomplemented
62#1973C2218yescomplemented
63#1947C2214yescomplemented
64#1947C2214yescomplemented
65#1943C2218yescomplemented
66#1936C2218yescomplemented
67#1935C2218yescomplemented
68#1893C2214yes—
69#1892C2214yes—
70#1808C2214yes—
71#1803C2214yescomplemented
72#1802C2214yescomplemented
73C22×#41—21no—
74C24×C4D411yescharacteristic
75C24×C4D411yescharacteristic
76C24×C4C2311yescharacteristic
77C22×SD16C2314yes—
78C23×C8C2312yes—
79C23×C8C2312yes—
80#533—24no—
81#533—24no—
82#532—24no—
83#532—24no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2311yescharacteristic
87C2×#127C2311yescharacteristic
88C2×#125—21no—
89C2×#124—21no—
90#525—48no—
91C2×#123—22no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8C2312yes—
96C2×C22⋊Q8C2314yes—
97C2×#121—21no—
98C2×#121—22no—
99C2×#121—22no—
100C2×#121C2312yes—
101C2×#121C2314yes—
102C2×#120—22no—
103C2×C4×Q8—21no—
104C2×C4×Q8C2312yes—
105C2×C4×D4—24no—
106C2×C4×D4—21no—
107C2×C4×D4C2312yes—
108C2×#117—21no—
109C2×#117—21no—
110C2×#117—21no—
111C22×C42—24no—
112C22×C42—24no—
113C22×C42D412yes—
114C22×C42D412yes—
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121#476—28no—
122#475—28no—
123#465—216no—
124#461—28no—
125#460—28no—
126#432—216no—
127#432—216no—
128C2×#107C2318yes—
129C2×#106C2314yes—
130C2×C4⋊C8—24no—
131C2×C4⋊C8D418yescomplemented
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×#103C2314yes—
137C2×#103C2318yes—
138C2×#102C2314yes—
139C2×#102C2318yes—
140C2×C22⋊C8—28no—
141C2×C22⋊C8C2318yes—
142C2×#97—28no—
143C2×#97C2318yes—
144#391—22no—
145#391—22no—
146#391C2314yes—
147#390—22no—
148#390—22no—
149#390C2314yes—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389C2314yes—
155#389C2314yes—
156#388—24no—
157#388—24no—
158#387—24no—
159#387—24no—
160#386—24no—
161#386—22no—
162#386—24no—
163#386—22no—
164#385—24no—
165#385—22no—
166#385—24no—
167#385—22no—
168#384C2312yes—
169#384C2312yes—
170#382—21no—
171#382—21no—
172#381—22no—
173#380—21no—
174#380—22no—
175#380—21no—
176#380—22no—
177#380—22no—
178#380—22no—
179#380C2312yes—
180#380C2312yes—
181#380C2314yes—
182#380C2314yes—
183#379—22no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—24no—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C22⋊C4—24no—
190C4×C22⋊C4—24no—
191#340C2318yes—
192#340C2318yes—
193#340C2318yes—
194C25C2×D411yescharacteristic
195C2×#41—44no—
196C2×#41—44no—
197C2×#41—44no—
198C22×Q8—21no—
199C22×Q8—21no—
200C22×Q8C2412yes—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—21no—
204C22×D4—21no—
205C22×D4C2412yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4C2×D412yes—
217C23×C4C2×D412yes—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D412yes—
220C23×C4C2×D412yes—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D412yes—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C2×SD16—28no—
229C2×SD16—48no—
230C22×C8—24no—
231C22×C8—44no—
232C22×C8—44no—
233C22×C8C2×D418yes—
234C22×C8C2414yes—
235C22×C8C2414yes—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238C4⋊Q8—22no—
239C4⋊Q8—22no—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242#127—22no—
243#127—22no—
244#125—44no—
245#124—44no—
246#123—48no—
247C22⋊Q8—24no—
248C22⋊Q8—24no—
249C22⋊Q8—44no—
250C22⋊Q8—44no—
251C22⋊Q8—48no—
252C22⋊Q8—48no—
253C22⋊Q8—44no—
254#121—24no—
255#121—24no—
256#121—44no—
257#121—44no—
258#121—48no—
259#121—48no—
260#121—44no—
261#120—48no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—44no—
265C4×D4—24no—
266C4×D4—24no—
267C4×D4—416no—
268C4×D4—44no—
269#117—44no—
270#117—44no—
271#117—44no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2414yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—28no—
305C2×C42—28no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D414yes—
321C2×C42#4114yes—
322C2×C42C2412yes—
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325#107—48no—
326#106—28no—
327C4⋊C8—24no—
328C4⋊C8—24no—
329C4⋊C8C2×D418yes—
330C4⋊C8C2×D418yes—
331#103—28no—
332#103—48no—
333#102—28no—
334#102—48no—
335C22⋊C8—216no—
336#97—216no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—22no—
343#95—22no—
344#95—24no—
345#95—22no—
346#95—22no—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—42no—
353C24C22×D412yes—
354C24C22×D411yescharacteristic
355#41—816no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—24no—
359C2×Q8—24no—
360C2×Q8—44no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—44no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—48no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—48no—
419C22×C4—416no—
420C22×C4—416no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4#12114yes—
425C22×C4#12114yes—
426C22×C4AGL(1,ℤ/8)14yes—
427C22×C4#13714yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4112yes—
438C22×C4C2511yescharacteristic
439SD16—88no—
440C2×C8—28no—
441C2×C8—24no—
442C2×C8—28no—
443C2×C8—48no—
444C2×C8—48no—
445C2×C8C22×D418yes—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—48no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C42—24no—
479C42—24no—
480C42—24no—
481C42—24no—
482C42—24no—
483C42—22no—
484C42—22no—
485C42—44no—
486C42—44no—
487C42—44no—
488C42#12114yes—
489C42#12114yes—
490C42C22×D412yes—
491C42C22×D412yes—
492C42C2×#4112yes—
493C42C2×#4112yes—
494C23—22no—
495C23—22no—
496C23—28no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—48no—
504C23—48no—
505C23—44no—
506C23—44no—
507C23—42no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23C2×#12112yes—
513C23C2×AGL(1,ℤ/8)12yes—
514C23C2×#13712yes—
515C23#57514yes—
516C23C23×D411yescharacteristic
517Q8—42no—
518Q8—42no—
519Q8—44no—
520Q8—84no—
521Q8—84no—
522D4—42no—
523D4—42no—
524D4—44no—
525D4—84no—
526D4—816no—
527D4—88no—
528D4—84no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—44no—
545C2×C4—48no—
546C2×C4—48no—
547C2×C4—48no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—48no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—88no—
571C2×C4#468116yes—
572C2×C4C2×#12112yes—
573C2×C4C2×#12112yes—
574C2×C4C2×#12112yes—
575C2×C4C2×#12112yes—
576C2×C4C2×#12112yes—
577C2×C4C2×AGL(1,ℤ/8)12yes—
578C2×C4C2×AGL(1,ℤ/8)12yes—
579C2×C4C2×#13712yes—
580C2×C4C2×#13712yes—
581C2×C4C23×D411yescharacteristic
582C2×C4C22×#4111yescharacteristic
583C2×C4#58512yes—
584C2×C4#58512yes—
585C8—28no—
586C8—44no—
587C22—22no—
588C22—24no—
589C22—24no—
590C22—22no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—416no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—82no—
600C22—82no—
601C22—84no—
602C22—84no—
603C22—82no—
604C22C2×#46814yes—
605C22C22×#12111yescharacteristic
606C22#344912yes—
607C22C22×AGL(1,ℤ/8)11yescharacteristic
608C22C22×#13711yescharacteristic
609C22C2×#57512yes—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—82no—
617C4—82no—
618C4—84no—
619C4—84no—
620C4—82no—
621C4C2×#46814yes—
622C4C2×#46814yes—
623C4#348511yescharacteristic
624C4#348711yescharacteristic
625C4#349312yes—
626C2—24no—
627C2—44no—
628C2—162no—
629C2#3515011yescharacteristic
630C2C22×#46811yescharacteristic
631C2#6044411yescharacteristic
632C2#6096212yes—
633C2#6096312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3246C2semidirect
2. C22×#432C2semidirect
3. #36625C2semidirect
5. #33134C2semidirect
6. #29735C2semidirect
7. #24710C2semidirect
8. #24708C2semidirect
9. #24661C2semidirect
10. #24659C2semidirect
11. #21961C2semidirect
12. #21885C2semidirect
13. #21884C2semidirect
19. C2×#475C22semidirect
20. C2×#476C22semidirect
22. C2×#460C22semidirect
23. C2×#461C22semidirect
58. #1988C22semidirect
59. #1987C22semidirect
60. #1982C22semidirect
61. #1981C22semidirect
62. #1973C22semidirect
63. #1947C22semidirect
64. #1947C22semidirect
65. #1943C22semidirect
66. #1936C22semidirect
67. #1935C22semidirect
71. #1803C22semidirect
72. #1802C22semidirect
131. C2×C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32462
C22×#4322
#366252
#366242
#331342
#297352
#247102
#247082
#246612
#246592
#219612
#218852
#218842