Finite Group Databasebeta

← #6477734

Subgroups — #6477734

Order 512.

1862 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C211yesmaximal, characteristic
2C2×#2924C211yesmaximal, characteristic
3#35311C211yesmaximal, characteristic
4C2×#2625C211yesmaximal, characteristic
5#33141C214yesmaximal, complemented
6#29786C214yesmaximal, complemented
7#24616C214yesmaximal, complemented
8#24609C212yesmaximal, complemented
9#24601C212yesmaximal, complemented
10#24514C214yesmaximal, complemented
11#24508C212yesmaximal, complemented
12#24502C212yesmaximal, complemented
13#21944C211yesmaximal, characteristic
14#21762C211yesmaximal, characteristic
15#21759C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#3195—28no—
20C2×#483C2212yes—
21C2×#483C2212yes—
22C2×#470C2214yes—
23C2×#451C2212yes—
24C2×#451C2212yes—
25#2924—28no—
26C2×#433C2214yes—
27C2×#432C2212yes—
28C2×#431C2212yes—
29C2×#424C2212yes—
30C2×#407C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C2×#405C2212yes—
33#2855C2212yes—
34#2855C2212yes—
35#2625—21no—
36#2625—21no—
37#2625—22no—
38#2625C2214yes—
39#2625C2212yes—
40#2625C2212yes—
41#2625C2212yes—
42#2625C2212yes—
43#2503—21no—
44#2502—21no—
45#2499—22no—
46#2484—21no—
47#2484—21no—
48#2483—22no—
49C2×#400—21no—
50C2×#398—21no—
51C2×#397—22no—
52#2317C2212yes—
53#2310C2212yes—
54C2×#391C2212yes—
55C2×#389C2212yes—
56#2280C2211yescharacteristic
57C2×#380C2211yescharacteristic
58#2088C2214yescomplemented
59#2088C2214yescomplemented
60#2068C2214yescomplemented
61#2068C2214yescomplemented
62#2066C2214yescomplemented
63#2066C2214yescomplemented
64#2056C2214yescomplemented
65#2056C2214yescomplemented
66#1971C2214yescomplemented
67#1969C2214yescomplemented
68#1943C2214yescomplemented
69#1941C2214yescomplemented
70#1935C2214yescomplemented
71#1935C2214yescomplemented
72#1933C2214yescomplemented
73#1933C2214yescomplemented
74#1892C2214yes—
75#1892C2214yes—
76#1873C2214yes—
77#1872C2212yes—
78#1872C2212yes—
79#1854C2214yes—
80#1853C2212yes—
81#1852C2212yes—
82#1824C2212yes—
83#1823C2211yescharacteristic
84#1823C2211yescharacteristic
85#1809C2214yes—
86#1787C2211yescharacteristic
87#1787C2211yescharacteristic
88#1787C2212yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×Q16C2314yes—
93C22×M16C2312yes—
94C23×C8C2312yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×#117—24no—
108C2×#117—24no—
109C2×#117D411yescharacteristic
110C2×#117D411yescharacteristic
111C2×#117D412yes—
112C2×#117C2312yes—
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4D412yes—
128C22×C42C2312yes—
129#483—28no—
130#483—28no—
131#470—216no—
132#451—28no—
133#451—28no—
134#433—216no—
135#432—28no—
136#431—28no—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140#424—28no—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149#407D418yes—
150C2×C22⋊C8—24no—
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2311yescharacteristic
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156#405—28no—
157C2×#97C2314yes—
158C2×C4×C8C2314yes—
159#400—24no—
160#400—22no—
161#400—22no—
162#400—22no—
163#400—22no—
164#398—24no—
165#398—22no—
166#398—22no—
167#398—22no—
168#398—22no—
169#397—24no—
170#397—24no—
171#397—24no—
172#397—24no—
173#397—22no—
174#397—22no—
175#397—22no—
176#397—22no—
177#391—22no—
178#391—22no—
179#391—22no—
180#391—22no—
181#391C2312yes—
182#391C2312yes—
183#391C2312yes—
184#391C2312yes—
185#391C2314yes—
186#391C2314yes—
187#389—22no—
188#389—22no—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2314yes—
194#389C2314yes—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#385—22no—
200#385—22no—
201#385—22no—
202#385—22no—
203#384—22no—
204#382—22no—
205#381—22no—
206#381—22no—
207#381—21no—
208#381—22no—
209#381—22no—
210#380—21no—
211#380—21no—
212#380C2312yes—
213#380C2312yes—
214#380C2311yescharacteristic
215#380C2311yescharacteristic
216#380C2312yes—
217#379—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C2×#95—22no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95—21no—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230C25C2×D411yescharacteristic
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—44no—
238C22×Q8—44no—
239C22×Q8C2412yes—
240C22×Q8C2412yes—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—44no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D414yes—
253C23×C4#4114yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C2×Q16—28no—
258C2×Q16—44no—
259C2×Q16—44no—
260C2×M16—44no—
261C2×M16C2×D418yes—
262C22×C8—24no—
263C22×C8—44no—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2414yes—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278C22⋊Q8—44no—
279#117—24no—
280#117—24no—
281#117—28no—
282#117—22no—
283#117—22no—
284#117—24no—
285#117—24no—
286#117—22no—
287#117—22no—
288#117—44no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—84no—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—82no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348#107—44no—
349#106—28no—
350#106—44no—
351C4⋊C8—28no—
352C4⋊C8—28no—
353#103—28no—
354#103—28no—
355#103—44no—
356#103—44no—
357#103—44no—
358#103—44no—
359C22⋊C8—22no—
360C22⋊C8—22no—
361C22⋊C8—24no—
362C22⋊C8C2×D418yes—
363C22⋊C8C2×D418yes—
364#97—28no—
365C4×C8—28no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—21no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389C24—22no—
390C24—22no—
391C24—42no—
392C24—42no—
393C24#12114yes—
394C24C22×D411yescharacteristic
395C24C22×D411yescharacteristic
396C24C22×D411yescharacteristic
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—84no—
410C2×Q8—84no—
411C2×Q8—84no—
412C2×Q8—84no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—82no—
446C22×C4#12114yes—
447C22×C4#12114yes—
448C22×C4#12114yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2511yescharacteristic
463Q16—88no—
464M16—48no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—48no—
472C2×C8C22×D418yes—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C42—24no—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—44no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—28no—
532C23—44no—
533C23—44no—
534C23—84no—
535C23—84no—
536C23C2×#12112yes—
537C23C2×#12112yes—
538C23C2×#12112yes—
539C23#57712yes—
540C23#57812yes—
541C23C23×D411yescharacteristic
542Q8—44no—
543Q8—44no—
544Q8—42no—
545Q8—42no—
546Q8—44no—
547Q8—84no—
548Q8—84no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—84no—
595C2×C4—88no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4C2×#12112yes—
601C2×C4C2×#12112yes—
602C2×C4C2×#12112yes—
603C2×C4C23×D411yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4#58514yes—
606C8—28no—
607C8—42no—
608C8—42no—
609C22—22no—
610C22—22no—
611C22—24no—
612C22—24no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—88no—
624C22#306314yes—
625C22C22×#12111yescharacteristic
626C22#344912yes—
627C22C2×#57711yescharacteristic
628C22C2×#57811yescharacteristic
629C22#358612yes—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—84no—
643C4—82no—
644C4#348512yes—
645C4#349112yes—
646C2—24no—
647C2—42no—
648C2—42no—
649C2#3508911yescharacteristic
650C2C2×#306311yescharacteristic
651C2#6045511yescharacteristic
652C2#6097712yes—
653C2#6098312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
5. #33141C2semidirect
6. #29786C2semidirect
7. #24616C2semidirect
8. #24609C2semidirect
9. #24601C2semidirect
10. #24514C2semidirect
11. #24508C2semidirect
12. #24502C2semidirect
58. #2088C22semidirect
59. #2088C22semidirect
60. #2068C22semidirect
61. #2068C22semidirect
62. #2066C22semidirect
63. #2066C22semidirect
64. #2056C22semidirect
65. #2056C22semidirect
66. #1971C22semidirect
67. #1969C22semidirect
68. #1943C22semidirect
69. #1941C22semidirect
70. #1935C22semidirect
71. #1935C22semidirect
72. #1933C22semidirect
73. #1933C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31952
C2×#29242
#353112
C2×#26252
#331412
#297862
#246162
#246092
#246012
#245142
#245082
#245022
#219442
#217622
#217592