Finite Group Databasebeta

← #6400921

Subgroups — #6400921

Order 512.

1747 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61393C211yesmaximal, characteristic
2#60274C211yesmaximal, characteristic, complemented
3#60218C211yesmaximal, characteristic, complemented
4#23180C213yesmaximal, complemented
5#23114C213yesmaximal, complemented
6#23016C213yesmaximal, complemented
7#22782C213yesmaximal, complemented
8#22746C211yesmaximal, characteristic, complemented
9#22550C211yesmaximal, characteristic, complemented
10#22550C211yesmaximal, characteristic, complemented
11#22280C213yesmaximal, complemented
12#22166C213yesmaximal, complemented
13#21518C213yesmaximal, complemented
14#21137C213yesmaximal, complemented
15#20572C211yesmaximal, characteristic, complemented
16#3480C2211yescharacteristic
17#3480C2211yescharacteristic
18#3396C413yescomplemented
19#3395C411yescharacteristic, complemented
20#3394C413yescomplemented
21#3394C413yescomplemented
22#3393C413yescomplemented
23#3392C411yescharacteristic, complemented
24#3392C411yescharacteristic, complemented
25#3391C411yescharacteristic, complemented
26#3147C2213yes—
27#3035C2213yes—
28#3035C2213yes—
29#2994—23no—
30#2993—23no—
31#2991—21no—
32#2988—21no—
33#2971—21no—
34#2969—23no—
35#2966—23no—
36C8×#41—21no—
37Q8×M16C2211yescharacteristic
38Q8×M16C2211yescharacteristic
39C2×C8×Q8C2211yescharacteristic
40C2×C8×Q8C2211yescharacteristic
41C2×#444C2216yes—
42C4×#135C2213yes—
43#2935C2213yes—
44#2935C2213yes—
45#2931C2213yes—
46#2931C2213yes—
47#2925C2213yes—
48#2922C2213yes—
49#2906C2213yes—
50C4×#131C2213yes—
51C42×Q8C2211yescharacteristic
52#1980C2213yes—
53#1980C2213yes—
54#1975C2216yes—
55#1962C2213yes—
56#1962C2213yes—
57#1950C2213yes—
58#1950C2213yes—
59#1938C2216yes—
60#1929C2213yes—
61#1929C2213yes—
62#1922C2211yescharacteristic
63#1922C2213yes—
64#1876C2213yes—
65#1876C2216yes—
66#1876C2216yes—
67#1876C2216yes—
68#1876C2213yes—
69#1866C2213yes—
70#1866C2213yes—
71#1837C2213yes—
72#1806C2213yes—
73#1806C2213yes—
74C4×#108C2213yes—
75#1770C2213yes—
76#1770C2213yes—
77#1767C2213yes—
78#1765C2216yes—
79C4×#104C2213yes—
80C4×#104C2213yes—
81#1744C2213yes—
82C2×#135C2313yes—
83C2×#135C2316yes—
84C2×#131C2313yes—
85C2×#131C2313yes—
86C2×#131C2316yes—
87#559C2×C413yes—
88#559C2×C413yes—
89#557C2×C411yescharacteristic
90#557C2×C411yescharacteristic
91#556C2×C413yes—
92#556C2×C413yes—
93#553—23no—
94#553—23no—
95#550C2×C411yescharacteristic
96#550C2×C411yescharacteristic
97#548—21no—
98#548—21no—
99#532C2313yes—
100#532C2313yes—
101#529—23no—
102#529—23no—
103C4×#41—21no—
104C4×#41—21no—
105C4×#41C2313yes—
106C4×#41C2313yes—
107C2×C4×Q8—21no—
108C2×C4×Q8C2311yescharacteristic
109#495C2×C416yes—
110#495C2×C416yes—
111#494C2×C413yes—
112#493C2×C413yes—
113#492C2×C413yes—
114#492C2×C413yes—
115#464C2×C416yes—
116#464C2×C416yes—
117#463C2×C416yes—
118#463C2×C416yes—
119#462C2×C413yes—
120#462C2×C413yes—
121#461C2×C413yes—
122#461C2×C413yes—
123#460C2×C413yes—
124#460C2×C413yes—
125#459C2×C413yes—
126#459C2×C413yes—
127#446—23no—
128#446—23no—
129C8×Q8—21no—
130C8×Q8—21no—
131C8×Q8C2×C411yescharacteristic
132C8×Q8C2×C411yescharacteristic
133C8×Q8C2×C413yes—
134C8×Q8C2×C413yes—
135C8×Q8D411yescharacteristic
136C8×Q8D413yes—
137C8×Q8D413yes—
138C8×Q8D411yescharacteristic
139#444—412no—
140C4×Q16C2×C413yes—
141C4×Q16C2×C413yes—
142C4×SD16C2×C413yes—
143C4×SD16C2×C413yes—
144C4×SD16C2×C413yes—
145C4×SD16C2×C413yes—
146C4×D8C2×C413yes—
147C4×D8C2×C413yes—
148#436—23no—
149#436—23no—
150#436—23no—
151#436—23no—
152#436—23no—
153#436—23no—
154#435—26no—
155#435—26no—
156C8×D4—23no—
157C8×D4—23no—
158C8×D4—23no—
159C8×D4—23no—
160C8×D4—23no—
161C8×D4—23no—
162#433—26no—
163#433—23no—
164#433—23no—
165#432—23no—
166#431—23no—
167#431—23no—
168#431—23no—
169C2×#108C2313yes—
170C2×#108C2313yes—
171#427C2311yescharacteristic
172#424—26no—
173#424C2311yescharacteristic
174#424C2313yes—
175#423—23no—
176#423C2313yes—
177#423C2313yes—
178C2×C4⋊C8—23no—
179C2×C4⋊C8C2313yes—
180C2×C4⋊C8C2313yes—
181C2×#104C2313yes—
182C2×#104C2316yes—
183C2×#104C2316yes—
184C2×#104C2313yes—
185C2×#104C2313yes—
186C2×#104C2313yes—
187#416C2313yes—
188#416C2313yes—
189#408C2313yes—
190#408C2316yes—
191#408C2313yes—
192#405—23no—
193#405—23no—
194#405C2311yescharacteristic
195#405C2313yes—
196C4×M16—23no—
197C4×M16C2313yes—
198C4×M16C2313yes—
199C2×#97C2316yes—
200C2×C4×C8—23no—
201C2×C4×C8C2313yes—
202C2×C4×C8C2313yes—
203C4×C4⋊C4—23no—
204C4×C4⋊C4C2313yes—
205C43C2313yes—
206#341C2313yes—
207#341C2316yes—
208#341C2311yescharacteristic
209#339C2313yes—
210#339C2313yes—
211#339C2311yescharacteristic
212#339C2311yescharacteristic
213#336C2313yes—
214#336C2313yes—
215#336C2311yescharacteristic
216#336C2313yes—
217C2×#41—21no—
218C2×#41—21no—
219C2×#41C2413yes—
220C2×#41C2413yes—
221C22×Q8#4111yescharacteristic
222#135—46no—
223#135—43no—
224#135—43no—
225#135—46no—
226C2×Q16C22×C413yes—
227C2×Q16C22×C416yes—
228C2×SD16C22×C413yes—
229C2×SD16C22×C413yes—
230C2×SD16C22×C416yes—
231C2×SD16C22×C416yes—
232C2×D8C22×C413yes—
233C2×D8C22×C416yes—
234#131—26no—
235#131—43no—
236#131—43no—
237#131—43no—
238#131—46no—
239#131—46no—
240C2×M16—23no—
241C2×M16—23no—
242C2×M16—26no—
243C2×M16C2413yes—
244C2×M16C2413yes—
245C2×M16C2411yescharacteristic
246C2×M16C2411yescharacteristic
247C22×C8—23no—
248C22×C8—23no—
249C22×C8—26no—
250C22×C8C2411yescharacteristic
251C22×C8C2413yes—
252C22×C8C2411yescharacteristic
253C22×C8C2413yes—
254C4⋊Q8C22×C413yes—
255C4⋊Q8C22×C413yes—
256#127C22×C413yes—
257#127C22×C413yes—
258#126—26no—
259#126—26no—
260#125—23no—
261#125—23no—
262#124—23no—
263#124—23no—
264#124C22×C416yes—
265#124C22×C416yes—
266#123—26no—
267#123—26no—
268C22⋊Q8—23no—
269C22⋊Q8—23no—
270#121—23no—
271#121—23no—
272#121—23no—
273#121—23no—
274#121—23no—
275#121—23no—
276C4×Q8—21no—
277C4×Q8—21no—
278C4×Q8—44no—
279C4×Q8C22×C413yes—
280C4×Q8C22×C413yes—
281C4×Q8C22×C413yes—
282C4×Q8C22×C413yes—
283C4×Q8C22×C411yescharacteristic
284C4×Q8C22×C413yes—
285C4×Q8C2×D411yescharacteristic
286C4×Q8C2×D413yes—
287C4×D4—23no—
288C4×D4—23no—
289C4×D4—23no—
290C4×D4—23no—
291C4×D4—23no—
292C4×D4—23no—
293C4×D4—23no—
294C4×D4—23no—
295C4×D4—23no—
296C4×D4—23no—
297C4×D4C22×C413yes—
298C4×D4C22×C413yes—
299C4×D4C22×C413yes—
300C4×D4C22×C413yes—
301#117—23no—
302#117—23no—
303#117—23no—
304#117—23no—
305#117C2411yescharacteristic
306#117C2411yescharacteristic
307C2×C4⋊C4—23no—
308C2×C4⋊C4#4113yes—
309C2×C42—23no—
310C2×C42—23no—
311C2×C42—23no—
312C2×C42—23no—
313C2×C42C2411yescharacteristic
314C2×C42C2413yes—
315C2×C42C2413yes—
316#108—26no—
317#108—43no—
318#107C22×C411yescharacteristic
319#107C22×C411yescharacteristic
320#106C22×C411yescharacteristic
321#106C22×C411yescharacteristic
322C4⋊C8—23no—
323C4⋊C8—23no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—23no—
327C4⋊C8—23no—
328C4⋊C8—23no—
329C4⋊C8—23no—
330C4⋊C8C22×C413yes—
331C4⋊C8C22×C413yes—
332C4⋊C8C22×C416yes—
333C4⋊C8C2×D413yes—
334C4⋊C8C2×D413yes—
335C4⋊C8C2×D416yes—
336#104—26no—
337#104—43no—
338#104—46no—
339#104—43no—
340#104—43no—
341#104—46no—
342#103C22×C413yes—
343#103C22×C413yes—
344#103C22×C413yes—
345#103C22×C413yes—
346#102C22×C413yes—
347#102C22×C413yes—
348#102C22×C413yes—
349#102C22×C413yes—
350C22⋊C8—23no—
351C22⋊C8—26no—
352C22⋊C8—23no—
353C22⋊C8—23no—
354C22⋊C8—23no—
355C22⋊C8—26no—
356#97—21no—
357#97—23no—
358#97—23no—
359#97—23no—
360#97—46no—
361C4×C8—21no—
362C4×C8—23no—
363C4×C8—23no—
364C4×C8—23no—
365C4×C8C22×C413yes—
366C4×C8C22×C413yes—
367C4×C8C22×C413yes—
368C4×C8C22×C413yes—
369C4×C8C2×D413yes—
370C4×C8C2×D413yes—
371C4×C8C2×D413yes—
372C4×C8C2×D413yes—
373#95—23no—
374#41—23no—
375#41—23no—
376#41—23no—
377#41—23no—
378#41—23no—
379#41—23no—
380#41—23no—
381#41—23no—
382#41—44no—
383#41—44no—
384C2×Q8—24no—
385C2×Q8C4×D413yes—
386C2×Q8C4×D411yescharacteristic
387C2×Q8C23×C413yes—
388C2×Q8C23×C413yes—
389C2×D4—23no—
390C2×D4—23no—
391C2×D4—23no—
392C2×D4—23no—
393C2×D4—23no—
394C2×D4—23no—
395C2×D4C23×C413yes—
396C2×D4C23×C413yes—
397C22×C4—21no—
398C22×C4—23no—
399C22×C4—23no—
400C22×C4—23no—
401C22×C4—23no—
402C22×C4—23no—
403C22×C4—23no—
404C22×C4—23no—
405C22×C4C2×#4113yes—
406C22×C4C2×#4113yes—
407C22×C4C2511yescharacteristic
408Q16—43no—
409Q16—46no—
410SD16—43no—
411SD16—43no—
412SD16—46no—
413SD16—46no—
414D8—43no—
415D8—46no—
416M16—21no—
417M16—23no—
418M16—21no—
419M16—23no—
420M16—23no—
421M16—23no—
422M16—21no—
423M16—46no—
424M16—46no—
425M16—86no—
426M16H(5,𝔽2)12yes—
427C2×C8—21no—
428C2×C8—23no—
429C2×C8—23no—
430C2×C8—23no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—23no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—23no—
438C2×C8—23no—
439C2×C8—46no—
440C2×C8—46no—
441C2×C8—86no—
442C2×C8C23×C411yescharacteristic
443C2×C8C23×C411yescharacteristic
444C2×C8C23×C413yes—
445C2×C8C23×C413yes—
446C2×C8C22×D411yescharacteristic
447C2×C8C22×D413yes—
448C2×C8C22×D411yescharacteristic
449C2×C8C22×D413yes—
450C4⋊C4—23no—
451C4⋊C4—23no—
452C4⋊C4—23no—
453C4⋊C4—23no—
454C4⋊C4—23no—
455C4⋊C4—23no—
456C4⋊C4—23no—
457C4⋊C4—23no—
458C4⋊C4—46no—
459C4⋊C4C4×D413yes—
460C4⋊C4C4×D413yes—
461C4⋊C4C4×D416yes—
462C4⋊C4C23×C411yescharacteristic
463C4⋊C4C23×C411yescharacteristic
464C4⋊C4C23×C411yescharacteristic
465C4⋊C4C23×C411yescharacteristic
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—26no—
469C22⋊C4—26no—
470C22⋊C4—23no—
471C22⋊C4—23no—
472C22⋊C4—23no—
473C22⋊C4—23no—
474C22⋊C4—26no—
475C22⋊C4—26no—
476C42—23no—
477C42—23no—
478C42—23no—
479C42—23no—
480C42—21no—
481C42—21no—
482C42—23no—
483C42—21no—
484C42—21no—
485C42—23no—
486C42—46no—
487C42—46no—
488C42C23×C413yes—
489C42C23×C413yes—
490C42C22×D413yes—
491C42C22×D413yes—
492C23—23no—
493C23—23no—
494C23C22×#4111yescharacteristic
495Q8—26no—
496Q8—23no—
497Q8—23no—
498Q8—23no—
499Q8—23no—
500Q8#44414yescomplemented
501D4—23no—
502D4—23no—
503D4—23no—
504D4—23no—
505D4—43no—
506D4—43no—
507D4—43no—
508D4—43no—
509D4—46no—
510D4—46no—
511D4—83no—
512D4—83no—
513C2×C4—26no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—23no—
520C2×C4—23no—
521C2×C4—43no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—46no—
525C2×C4—46no—
526C2×C4—43no—
527C2×C4—43no—
528C2×C4—43no—
529C2×C4—43no—
530C2×C4—43no—
531C2×C4—43no—
532C2×C4—43no—
533C2×C4—83no—
534C2×C4—83no—
535C2×C4C2×C4×D413yes—
536C2×C4C2×C4×D413yes—
537C2×C4C2×C4×D413yes—
538C2×C4C2×C4×D413yes—
539C2×C4C24×C411yescharacteristic
540C2×C4C23×D411yescharacteristic
541C2×C4C2×H(5,𝔽2)11yescharacteristic
542C2×C4#58511yescharacteristic
543C2×C4#58511yescharacteristic
544C2×C4#58511yescharacteristic
545C8—21no—
546C8—21no—
547C8—23no—
548C8—23no—
549C8—21no—
550C8—21no—
551C8—23no—
552C8—23no—
553C8—22no—
554C8—43no—
555C8—41no—
556C8—43no—
557C22—21no—
558C22—21no—
559C22—43no—
560C22—43no—
561C22—43no—
562C22—43no—
563C22—43no—
564C22—43no—
565C22C22×C4×D411yescharacteristic
566C22#351711yescharacteristic
567C22#352411yescharacteristic
568C4—23no—
569C4—23no—
570C4—23no—
571C4—42no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—43no—
577C4—43no—
578C4—83no—
579C4C2×#44416yes—
580C4#342811yescharacteristic
581C4C4×H(5,𝔽2)11yescharacteristic
582C4#348711yescharacteristic
583C4#348811yescharacteristic
584C2—21no—
585C2—21no—
586C2—83no—
587C2—83no—
588C2#3429511yescharacteristic
589C2C22×#44411yescharacteristic
590C2#6018311yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C22×C4×D4
prime radical#61393C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60274C2semidirect
3. #60218C2semidirect
4. #23180C2semidirect
5. #23114C2semidirect
6. #23016C2semidirect
7. #22782C2semidirect
8. #22746C2semidirect
9. #22550C2semidirect
10. #22550C2semidirect
11. #22280C2semidirect
12. #22166C2semidirect
13. #21518C2semidirect
14. #21137C2semidirect
15. #20572C2semidirect
18. #3396C4semidirect
19. #3395C4semidirect
20. #3394C4semidirect
21. #3394C4semidirect
22. #3393C4semidirect
23. #3392C4semidirect
24. #3392C4semidirect
25. #3391C4semidirect
500. Q8#444semidirect

Maximal subgroups

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

subgroupindex
#613932
#602742
#602182
#231802
#231142
#230162
#227822
#227462
#225502
#222802
#221662
#215182
#211372
#205722