Finite Group Databasebeta

← #377802

Subgroups — #377802

Order 512.

1010 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23638C211yesmaximal, characteristic
2C2×#1922C211yesmaximal, characteristic, complemented
3#20835C211yesmaximal, characteristic
4#20386C211yesmaximal, characteristic
5C2×#1753C211yesmaximal, characteristic
6#11201C211yesmaximal, characteristic
7#11201C211yesmaximal, characteristic
8#11097C211yesmaximal, characteristic
9#10910C211yesmaximal, characteristic, complemented
10#10887C211yesmaximal, characteristic, complemented
11#10883C211yesmaximal, characteristic, complemented
12#10882C211yesmaximal, characteristic, complemented
13#10860C211yesmaximal, characteristic, complemented
14#10818C211yesmaximal, characteristic, complemented
15#10809C211yesmaximal, characteristic, complemented
16C23×M16C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×#407C2211yescharacteristic
22C2×#407C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2280C2211yescharacteristic
29#2280C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#2077C412yescomplemented
32#2077C412yescomplemented
33#2076C412yescomplemented
34#2075C412yescomplemented
35#1922—24no—
36#1851—22no—
37#1851—22no—
38#1753—24no—
39#1753—22no—
40#1753—22no—
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43C2×#341C2211yescharacteristic
44#1735C2211yescharacteristic
45#1735C2211yescharacteristic
46#1735C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55#1729C2211yescharacteristic
56#1729C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C24×C4C2311yescharacteristic
62C22×M16—22no—
63C22×M16—22no—
64C22×M16—22no—
65C22×M16—22no—
66C22×M16—41no—
67C22×M16—41no—
68C22×M16D412yes—
69C22×M16D412yes—
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C22×C22⋊C4—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#108—22no—
86C2×#108—22no—
87C2×#108—41no—
88#427—22no—
89#427—22no—
90C2×#107C2×C411yescharacteristic
91C2×#107C2×C411yescharacteristic
92C2×#106C2×C411yescharacteristic
93C2×#106C2×C411yescharacteristic
94#424—22no—
95#424—22no—
96#424—22no—
97#424—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—22no—
101#407—22no—
102#407—22no—
103#407—41no—
104#407—41no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8C2×C412yes—
110C2×C22⋊C8C2×C412yes—
111#405—22no—
112#405—22no—
113#405—41no—
114#405—41no—
115C2×#97—22no—
116C2×#97—21no—
117C2×#97—21no—
118C2×#97—22no—
119C2×#97—41no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8—22no—
123C2×C4×C8—22no—
124C2×C4×C8—41no—
125#381—21no—
126#381—21no—
127#380C2×C412yes—
128#380C2×C412yes—
129#380C2×C412yes—
130#380C2×C412yes—
131#379—21no—
132#379—21no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C2×#95—21no—
138#341—22no—
139#341—22no—
140#341—41no—
141#340C2×C412yes—
142#340C2×C412yes—
143#340C2×C412yes—
144#340C2×C412yes—
145#340C2×C412yes—
146#340C2×C412yes—
147#339—22no—
148#339—22no—
149#339—22no—
150#339—22no—
151#339—22no—
152#339—22no—
153#339—41no—
154#339—41no—
155#336—21no—
156#336—21no—
157#336—22no—
158#336—22no—
159#336—22no—
160#336—22no—
161#336D414yes—
162C25#4111yescharacteristic
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4C22×C412yes—
178C23×C4C2×D412yes—
179C23×C4#4111yescharacteristic
180C23×C4C2411yescharacteristic
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—24no—
184C2×M16—24no—
185C2×M16—24no—
186C2×M16—24no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—42no—
208C2×M16—42no—
209C2×M16—44no—
210C2×M16—44no—
211C2×M16—44no—
212C2×M16—44no—
213C2×M16#4112yes—
214C2×M16#4112yes—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8C22×C412yes—
238C22×C8C22×C411yescharacteristic
239C22×C8C22×C411yescharacteristic
240C22×C8C2×D412yes—
241C22×C8C2×D412yes—
242C22×C8C2×D411yescharacteristic
243C22×C8C2×D411yescharacteristic
244C22×C8C2×D412yes—
245C22×C8#4112yes—
246C22×C8#4112yes—
247C22×C8#4112yes—
248C22×C8#4112yes—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265C2×C4⋊C4C22×C412yes—
266C2×C4⋊C4C22×C412yes—
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C22⋊C4—24no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—44no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42C2×D412yes—
296C2×C42#4112yes—
297#108—44no—
298#107—22no—
299#107—22no—
300#106—22no—
301#106—22no—
302C4⋊C8—42no—
303C4⋊C8—42no—
304C4⋊C8—44no—
305C22⋊C8—28no—
306C22⋊C8—24no—
307C22⋊C8—24no—
308C22⋊C8—44no—
309#97—44no—
310#97—42no—
311#97—42no—
312C4×C8—44no—
313C4×C8—42no—
314C4×C8—42no—
315#95—24no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—42no—
327C24—42no—
328C24C4×D412yes—
329C24C2×#4111yescharacteristic
330C22×C4—24no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4C4×D412yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C42−1+411yescharacteristic
394C22×C42−1+411yescharacteristic
395M16—22no—
396M16—22no—
397M16—22no—
398M16—22no—
399M16—48no—
400M16—48no—
401M16—41no—
402M16—41no—
403M16—41no—
404M16—41no—
405M16—41no—
406M16—41no—
407M16—41no—
408M16—41no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—41no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—22no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8C4×D412yes—
455C2×C8C4×D412yes—
456C2×C8#12112yes—
457C2×C8#12112yes—
458C2×C8#12314yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C22⋊C4—48no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—82no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23C2×C4×D411yescharacteristic
522C23#52511yescharacteristic
523C23#52911yescharacteristic
524C23#52911yescharacteristic
525C23#52911yescharacteristic
526C23#53911yescharacteristic
527C23#53911yescharacteristic
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4C2×C4×D411yescharacteristic
568C2×C4C4×#4111yescharacteristic
569C2×C4C4×#4111yescharacteristic
570C2×C4C4×#4111yescharacteristic
571C2×C4#52011yescharacteristic
572C2×C4#52011yescharacteristic
573C2×C4C2×#12111yescharacteristic
574C2×C4C2×#12311yescharacteristic
575C2×C4#52511yescharacteristic
576C2×C4#53211yescharacteristic
577C2×C4#53411yescharacteristic
578C2×C4#53611yescharacteristic
579C2×C4#54611yescharacteristic
580C2×C4#54611yescharacteristic
581C2×C4#54711yescharacteristic
582C2×C4#54811yescharacteristic
583C2×C4#55111yescharacteristic
584C2×C4#55111yescharacteristic
585C2×C4#55111yescharacteristic
586C2×C4#55211yescharacteristic
587C2×C4#55211yescharacteristic
588C2×C4#55211yescharacteristic
589C2×C4#55311yescharacteristic
590C2×C4#55311yescharacteristic
591C2×C4#55611yescharacteristic
592C2×C4#55611yescharacteristic
593C2×C4#55611yescharacteristic
594C2×C4#55911yescharacteristic
595C8—22no—
596C8—22no—
597C8—22no—
598C8—22no—
599C8—24no—
600C8—42no—
601C8—42no—
602C8—42no—
603C8—81no—
604C8—81no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22C4×#12111yescharacteristic
625C22C4×#12311yescharacteristic
626C22#232211yescharacteristic
627C22#246011yescharacteristic
628C22#246611yescharacteristic
629C22#251711yescharacteristic
630C22#252311yescharacteristic
631C22#295612yes—
632C22#295712yes—
633C4—44no—
634C4—42no—
635C4—82no—
636C4—81no—
637C4—81no—
638C4—81no—
639C4—81no—
640C4—82no—
641C4—82no—
642C4#235911yescharacteristic
643C4#236711yescharacteristic
644C4#237311yescharacteristic
645C4#237311yescharacteristic
646C4#241211yescharacteristic
647C4#241511yescharacteristic
648C4#249911yescharacteristic
649C4#260211yescharacteristic
650C2—21no—
651C2—21no—
652C2—21no—
653C2—21no—
654C2—82no—
655C2#1420211yescharacteristic
656C2#2154711yescharacteristic
657C2#2154911yescharacteristic
658C2#2159412yes—
659C2#2209712yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×M16 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1922C2semidirect
9. #10910C2semidirect
10. #10887C2semidirect
11. #10883C2semidirect
12. #10882C2semidirect
13. #10860C2semidirect
14. #10818C2semidirect
15. #10809C2semidirect
31. #2077C4semidirect
32. #2077C4semidirect
33. #2076C4semidirect
34. #2075C4semidirect

Maximal subgroups

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

subgroupindex
#236382
C2×#19222
#208352
#203862
C2×#17532
#112012
#110972
#109102
#108872
#108832
#108822
#108602
#108182
#108092