Finite Group Databasebeta

← #220780

Subgroups — #220780

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3384C211yesmaximal, characteristic
2#22670C211yesmaximal, characteristic
3#22487C211yesmaximal, characteristic, complemented
4D4×#103C211yesmaximal, characteristic, complemented
5C2×C8×D8C211yesmaximal, characteristic
6#9819C211yesmaximal, characteristic, complemented
7#9741C211yesmaximal, characteristic, complemented
8#9219C211yesmaximal, characteristic, complemented
9#9215C211yesmaximal, characteristic, complemented
10#8981C211yesmaximal, characteristic, complemented
11#8973C211yesmaximal, characteristic, complemented
12#8651C211yesmaximal, characteristic, complemented
13#8450C211yesmaximal, characteristic, complemented
14#8447C211yesmaximal, characteristic, complemented
15#8207C211yesmaximal, characteristic, complemented
16C2×#554C2211yescharacteristic
17C2×D4×Q8C2211yescharacteristic
18#3384—24no—
19#3384C418yescomplemented
20C2×#501—21no—
21C2×#498C2211yescharacteristic
22C2×#478—21no—
23C2×#476—21no—
24C2×#476C2211yescharacteristic
25C2×#474C2211yescharacteristic
26C2×#446C2211yescharacteristic
27C2×#442—21no—
28C2×#442—21no—
29C2×C4×D8C2211yescharacteristic
30C2×C8×D4C2211yescharacteristic
31C2×C8×D4C2211yescharacteristic
32C22×#103—21no—
33D4×C4⋊C4C2211yescharacteristic
34D4×C4⋊C4C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1987C2211yescharacteristic
38#1981C2211yescharacteristic
39#1956C2211yescharacteristic
40#1956C2211yescharacteristic
41#1943—21no—
42#1892—21no—
43#1892—21no—
44#1872—21no—
45#1872—21no—
46#1864—21no—
47#1864—21no—
48#1844C2211yescharacteristic
49#1824—21no—
50#1803C2211yescharacteristic
51#1803C2211yescharacteristic
52#1802C2211yescharacteristic
53#1802C2211yescharacteristic
54#1787—21no—
55#1770C2211yescharacteristic
56C4×#103C2211yescharacteristic
57C2×#340—21no—
58C8×D8—24no—
59C2×#335C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65C2×C82C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1279C2211yescharacteristic
69#1277C2211yescharacteristic
70C23×Q8—21no—
71C22×D8D411yescharacteristic
72C23×C8—21no—
73C23×C8—21no—
74#554—22no—
75#554—22no—
76#554—41no—
77#554C2×C414yescomplemented
78D4×Q8—22no—
79D4×Q8—22no—
80D4×Q8—41no—
81D4×Q8C2×C414yescomplemented
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×#125—21no—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8—41no—
89C2×C22⋊Q8—41no—
90C2×C4×Q8C2311yescharacteristic
91C2×C4×D4—21no—
92C2×C4×D4—21no—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4C2311yescharacteristic
96C2×C4×D4C2311yescharacteristic
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100#501—28no—
101#498—22no—
102#498C2×C414yescomplemented
103#478—28no—
104#476—22no—
105#476—28no—
106#476C2×C414yescomplemented
107#474—22no—
108#474C2×C414yescomplemented
109#446—22no—
110#446C2×C414yescomplemented
111#442—28no—
112#442—44no—
113C4×D8C2×C414yes—
114C4×D8D414yes—
115C8×D4—22no—
116C8×D4—22no—
117C8×D4—41no—
118C8×D4—41no—
119C2×#107—21no—
120C2×#107D411yescharacteristic
121C2×#106—21no—
122C2×#106—21no—
123C2×#106—21no—
124C2×#106—21no—
125C2×#106—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×#103—21no—
132C2×#103—21no—
133C2×#103—21no—
134C2×#103—21no—
135C2×#103—21no—
136C2×#103—21no—
137C2×#103—44no—
138C2×#103—41no—
139C2×#103D411yescharacteristic
140C2×#103D411yescharacteristic
141C2×#102—21no—
142C2×#102—21no—
143C2×#102—21no—
144C2×#102—21no—
145C2×#102D411yescharacteristic
146C2×#102D411yescharacteristic
147C2×C22⋊C8—21no—
148C2×C22⋊C8—21no—
149C2×#97—21no—
150C2×#97—21no—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154#389—21no—
155#389—21no—
156#385—41no—
157#385—41no—
158#384C2311yescharacteristic
159#384C2311yescharacteristic
160#380—21no—
161#380—21no—
162C4×C4⋊C4C2311yescharacteristic
163#340—21no—
164#340—21no—
165#340—21no—
166#340—21no—
167#340—21no—
168#340—21no—
169#340—21no—
170#340—21no—
171#340—41no—
172#340D411yescharacteristic
173#340D411yescharacteristic
174#336—21no—
175#335—22no—
176#333—41no—
177#332—41no—
178Q8⋊C8—22no—
179D4⋊C8—22no—
180D4⋊C8—22no—
181C82—22no—
182C22×Q8—24no—
183C22×Q8—21no—
184C22×Q8—21no—
185C22×Q8—41no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8—81no—
190C22×Q8—81no—
191C22×Q8C2×D411yescharacteristic
192C22×D4C2×D411yescharacteristic
193C22×D4C2×D411yescharacteristic
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—41no—
197C23×C4—41no—
198C23×C4—41no—
199C23×C4—41no—
200C2×D8C22⋊C414yes—
201C2×D8SD1614yes—
202C2×D8Q1614yes—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—42no—
209C22×C8—42no—
210C22×C8—42no—
211C22×C8—42no—
212C22×C8—81no—
213C22×C8—81no—
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C4⋊Q8—24no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—42no—
226C4⋊Q8C22×C412yes—
227C4⋊Q8C22×C412yes—
228#125—24no—
229#125—42no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C22⋊Q8—44no—
233C22⋊Q8—44no—
234C22⋊Q8—44no—
235C22⋊Q8—44no—
236C22⋊Q8—42no—
237C22⋊Q8—42no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8C22×C412yes—
241C4×Q8#4112yes—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—42no—
249C4×D4—42no—
250C4×D4—42no—
251C4×D4—42no—
252C4×D4—81no—
253C4×D4—81no—
254C4×D4C22×C412yes—
255C4×D4C22×C412yes—
256C4×D4C2×D412yes—
257C4×D4C2×D412yes—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—41no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4—41no—
277C2×C4⋊C4—41no—
278C2×C4⋊C4—81no—
279C2×C4⋊C4—81no—
280C2×C4⋊C4C2×D411yescharacteristic
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2×D411yescharacteristic
283C2×C4⋊C4C2×D411yescharacteristic
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42C2411yescharacteristic
294#107—24no—
295#107C22⋊C414yes—
296#106—24no—
297#106—24no—
298#106—24no—
299#106—42no—
300#106—81no—
301C4⋊C8—24no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8C22×C412yes—
310#103—22no—
311#103—41no—
312#103—42no—
313#103—42no—
314#103—42no—
315#103—81no—
316#103—81no—
317#103—81no—
318#103—82no—
319#102—24no—
320#102—24no—
321#102—42no—
322#102—42no—
323#102C22⋊C414yes—
324#102C22⋊C414yes—
325C22⋊C8—42no—
326C22⋊C8—42no—
327#97—24no—
328#97—42no—
329C4×C8—21no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—21no—
333C4×C8C22×C412yes—
334C4×C8C2×D412yes—
335#95—21no—
336#95—41no—
337#95—41no—
338C24—21no—
339C24—21no—
340C2×Q8—21no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—44no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—48no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—81no—
356C2×Q8—81no—
357C2×Q8—82no—
358C2×Q8—82no—
359C2×Q8—84no—
360C2×Q8—84no—
361C2×Q8—84no—
362C2×Q8C4×D412yes—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4C2×C22⋊C412yes—
370C2×D4C2×C22⋊C412yes—
371C2×D4C2×SD1612yes—
372C2×D4C2×SD1612yes—
373C2×D4C2×Q1612yes—
374C2×D4C2×Q1612yes—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4—161no—
406C22×C4—161no—
407C22×C4—161no—
408C22×C4—161no—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4H(5,𝔽2)11yescharacteristic
412D8#103116yescomplemented
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—84no—
436C2×C8—84no—
437C2×C8C2×C22⋊C412yes—
438C2×C8C2×C22⋊C412yes—
439C2×C8C2×SD1612yes—
440C2×C8C2×Q1612yes—
441C2×C8AGL(1,ℤ/8)12yes—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4C2×C22⋊C412yes—
475C4⋊C4C2×C22⋊C412yes—
476C4⋊C4C4×D412yes—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C42—22no—
486C42—22no—
487C42—41no—
488C42—41no—
489C42C23×C411yescharacteristic
490C42C22×D411yescharacteristic
491C42C2×#4111yescharacteristic
492C42H(5,𝔽2)11yescharacteristic
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23D4211yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—41no—
511Q8—41no—
512Q8—42no—
513Q8—42no—
514Q8—82no—
515Q8—88no—
516Q8—81no—
517Q8—81no—
518Q8—161no—
519D4—22no—
520D4—22no—
521D4—22no—
522D4—22no—
523D4—22no—
524D4—22no—
525D4C2×#10314yes—
526D4C2×#10314yescomplemented
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
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—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—88no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—164no—
569C2×C4—164no—
570C2×C4—164no—
571C2×C4C22×C22⋊C411yescharacteristic
572C2×C4C2×C4×D411yescharacteristic
573C2×C4#51811yescharacteristic
574C2×C4D4211yescharacteristic
575C2×C4#54611yescharacteristic
576C2×C4#54611yescharacteristic
577C2×C4C22×SD1611yescharacteristic
578C2×C4C22×Q1611yescharacteristic
579C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
580C2×C4#57511yescharacteristic
581C2×C4#57611yescharacteristic
582C2×C4#57711yescharacteristic
583C2×C4#57711yescharacteristic
584C2×C4#57811yescharacteristic
585C8—22no—
586C8—22no—
587C8—42no—
588C8—42no—
589C8—41no—
590C8—81no—
591C8—81no—
592C8—81no—
593C8C2×#10314yescomplemented
594C22—24no—
595C22—24no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22D4×C22⋊C411yescharacteristic
609C22#327211yescharacteristic
610C22#327311yescharacteristic
611C22D4×SD1611yescharacteristic
612C22D4×Q1611yescharacteristic
613C22#329011yescharacteristic
614C22#329111yescharacteristic
615C4—21no—
616C4—21no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—82no—
622C4—81no—
623C4—81no—
624C4—82no—
625C4—161no—
626C4—161no—
627C4C22×#10311yescharacteristic
628C4#289811yescharacteristic
629C4C4×AGL(1,ℤ/8)11yescharacteristic
630C4#294611yescharacteristic
631C4#326411yescharacteristic
632C4#326711yescharacteristic
633C4#329811yescharacteristic
634C4#329911yescharacteristic
635C2—44no—
636C2—44no—
637C2#1989011yescharacteristic
638C2#1989111yescharacteristic
639C2#1992611yescharacteristic
640C2#1992711yescharacteristic
641C2D4×#10311yescharacteristic
642C2#2244211yescharacteristic
643C2#2246711yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D8D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22487C2semidirect
4. D4×#103C2semidirect
6. #9819C2semidirect
7. #9741C2semidirect
8. #9219C2semidirect
9. #9215C2semidirect
10. #8981C2semidirect
11. #8973C2semidirect
12. #8651C2semidirect
13. #8450C2semidirect
14. #8447C2semidirect
15. #8207C2semidirect
19. #3384C4semidirect
77. #554C2×C4semidirect
81. D4×Q8C2×C4semidirect
102. #498C2×C4semidirect
106. #476C2×C4semidirect
108. #474C2×C4semidirect
110. #446C2×C4semidirect
412. D8#103semidirect
526. D4C2×#103semidirect
593. C8C2×#103semidirect

Maximal subgroups

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

subgroupindex
C2×#33842
#226702
#224872
D4×#1032
C2×C8×D82
#98192
#97412
#92192
#92152
#89812
#89732
#86512
#84502
#84472
#82072