Finite Group Databasebeta

← #164942

Subgroups — #164942

Order 512.

1366 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36260C211yesmaximal, characteristic
2#22835C211yesmaximal, characteristic, complemented
3#22700C211yesmaximal, characteristic, complemented
4#7724C212yesmaximal, complemented
5#7684C212yesmaximal, complemented
6#7659C211yesmaximal, characteristic, complemented
7#7659C211yesmaximal, characteristic, complemented
8#7651C211yesmaximal, characteristic, complemented
9#7606C212yesmaximal, complemented
10#7572C212yesmaximal, complemented
11#7569C211yesmaximal, characteristic, complemented
12C2×#547C411yescharacteristic
13C2×#546—21no—
14C2×#546—21no—
15C2×D42C411yescharacteristic
16C2×#436—21no—
17#2858C2211yescharacteristic
18#2856C2212yes—
19#2542C411yescharacteristic
20#2541C411yescharacteristic
21#2540—21no—
22#2540—21no—
23#2539—22no—
24#2539—22no—
25#2501C412yes—
26#2433C412yes—
27C4×#127C2211yescharacteristic
28C2×#390C2212yes—
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1923—22no—
33#1851—22no—
34#1831C2211yescharacteristic, complemented
35#1831C2211yescharacteristic, complemented
36#1817C2212yescomplemented
37#1797C2212yescomplemented
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2211yescharacteristic, complemented
40C4×C4⋊C8C2211yescharacteristic, complemented
41C4×C22⋊C8C2212yescomplemented
42#1728C2212yescomplemented
43#1270C2211yescharacteristic
44#1270C2211yescharacteristic
45#1270C2212yes—
46#1270C2211yescharacteristic, complemented
47#1270C2212yescomplemented
48#1270C2212yescomplemented
49#1270C2212yescomplemented
50#1270C2211yescharacteristic, complemented
51#1270C2212yescomplemented
52#1270C2212yescomplemented
53C22×#41—21no—
54C23×D4—21no—
55C23×D4C2×C412yes—
56C22×M16—22no—
57#547—44no—
58#546—44no—
59#546—44no—
60#546—44no—
61#546—44no—
62D42—44no—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×#127C2×C411yescharacteristic
65C2×#127C2×C411yescharacteristic
66C2×#127C2×C412yes—
67C2×#124—21no—
68C2×#124—21no—
69C2×#123—22no—
70C2×#123—22no—
71C2×#123—22no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×#121—22no—
76C2×#121—22no—
77C2×#121—21no—
78C2×#121—21no—
79C2×#121—21no—
80C2×#121C2×C412yes—
81C2×#121C2×C412yes—
82C2×#120—22no—
83C2×#120—22no—
84C2×#120—22no—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4C2×C411yescharacteristic
88C2×C4×D4C2×C411yescharacteristic
89C2×C4×D4C2×C411yescharacteristic
90C2×C4×D4D411yescharacteristic
91C2×C4×D4D411yescharacteristic
92C22×C4⋊C4C2×C412yes—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—22no—
95C22×C42C2312yes—
96#436—44no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—22no—
108#399—22no—
109#399—22no—
110#399—22no—
111#392—22no—
112#392—22no—
113#392—22no—
114#390—21no—
115#390—21no—
116#390—22no—
117#390—22no—
118#390—21no—
119#390—21no—
120#390—22no—
121#390—22no—
122#390C2×C412yes—
123#390C2×C412yes—
124#390C2×C412yes—
125#390C2311yescharacteristic
126#390C2311yescharacteristic
127#390C2312yes—
128#384C2×C411yescharacteristic
129#384C2×C412yes—
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D411yescharacteristic
133C4×C22⋊C4C2×C412yes—
134C4×C22⋊C4D412yes—
135C4×C22⋊C4D412yes—
136C43C2311yescharacteristic
137#336—22no—
138#336—21no—
139#336—21no—
140#336—22no—
141#336C2311yescharacteristic
142#336C2312yes—
143#336C2311yescharacteristic
144#336C2312yes—
145#336C2311yescharacteristic
146#336C2311yescharacteristic
147C25—22no—
148C2×#41—44no—
149C2×#41—44no—
150C2×#41—44no—
151C22×Q8—21no—
152C22×D4—24no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—21no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—42no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—82no—
168C22×D4C22⋊C411yescharacteristic
169C22×D4C22×C411yescharacteristic
170C22×D4C22×C412yes—
171C22×D4C22×C411yescharacteristic
172C23×C4—22no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4C22×C412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C2×M16—44no—
179C2×M16—44no—
180C2×M16—82no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—21no—
187C4⋊Q8—42no—
188#127—22no—
189#127—41no—
190#127—42no—
191#127—82no—
192#124—42no—
193#124—42no—
194#124—42no—
195#124—42no—
196#123—48no—
197#123—48no—
198#123—48no—
199C22⋊Q8—44no—
200C22⋊Q8—44no—
201C22⋊Q8—44no—
202#121—44no—
203#121—44no—
204#121—44no—
205#121—44no—
206#121—44no—
207#121—44no—
208#121—44no—
209#121—44no—
210#121—44no—
211#120—48no—
212#120—48no—
213#120—48no—
214C4×D4—24no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223C4×D4D814yes—
224C4×D4SD1614yes—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4C22⋊C412yes—
259C2×C22⋊C4C22⋊C412yes—
260C2×C22⋊C4C22⋊C412yes—
261C2×C42—82no—
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2411yescharacteristic
272C4⋊C8—42no—
273C4⋊C8—42no—
274C4⋊C8—42no—
275C4⋊C8—42no—
276C4⋊C8—82no—
277C22⋊C8—44no—
278C22⋊C8—44no—
279C22⋊C8—82no—
280C4×C8—42no—
281C4×C8—42no—
282C4×C8—82no—
283#95C22⋊C412yes—
284#95C22⋊C412yes—
285#95C22⋊C412yes—
286C24—21no—
287C24—22no—
288C24—21no—
289C24—44no—
290C24—44no—
291C24—44no—
292C24—44no—
293C24—82no—
294C24—82no—
295C24—82no—
296C24C2×C22⋊C412yes—
297#41—816no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×D4—21no—
302C2×D4—22no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—44no—
313C2×D4—41no—
314C2×D4—42no—
315C2×D4—41no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—44no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—81no—
333C2×D4—81no—
334C2×D4—82no—
335C2×D4—82no—
336C2×D4—82no—
337C2×D4—82no—
338C2×D4—81no—
339C2×D4—81no—
340C2×D4—82no—
341C2×D4—88no—
342C2×D4—88no—
343C2×D4—816no—
344C2×D4#10214yes—
345C2×D4#13112yes—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×D814yes—
376C22×C4C2×SD1614yes—
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380M16—88no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—41no—
396C2×C8—42no—
397C2×C8—41no—
398C2×C8—42no—
399C4⋊C4—22no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—44no—
404C4⋊C4—42no—
405C4⋊C4—44no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C4⋊C4#10014yes—
410C4⋊C4#10214yes—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—41no—
418C22⋊C4—42no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—42no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4#10218yes—
436C42—21no—
437C42—21no—
438C42—22no—
439C42—42no—
440C42—41no—
441C42—41no—
442C42—42no—
443C42—41no—
444C42—42no—
445C42C2×D812yes—
446C42C2×SD1612yes—
447C42AGL(1,ℤ/8)12yes—
448C42AGL(1,ℤ/8)12yes—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—42no—
464C23—41no—
465C23—42no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23—41no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—88no—
477C23—88no—
478C23—88no—
479C23—88no—
480C23—164no—
481C23—164no—
482C23—164no—
483C23C2×#10214yes—
484C23C22×C22⋊C411yescharacteristic
485Q8—84no—
486D4—28no—
487D4—42no—
488D4—42no—
489D4—84no—
490D4—84no—
491D4—88no—
492D4—81no—
493D4—81no—
494D4—82no—
495D4—82no—
496D4—88no—
497D4—88no—
498D4—84no—
499D4—164no—
500D4—162no—
501D4—164no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—88no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4C2×#10014yes—
541C2×C4C2×#10012yes—
542C2×C4C2×#10212yes—
543C2×C4C2×#10214yes—
544C2×C4C2×#10212yes—
545C2×C4#41812yes—
546C2×C4#41812yes—
547C2×C4C2×#13111yescharacteristic
548C2×C4#56811yescharacteristic
549C2×C4C22×D811yescharacteristic
550C2×C4C22×SD1611yescharacteristic
551C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
552C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
553C8—42no—
554C8—42no—
555C8—81no—
556C8—81no—
557C8—82no—
558C8—82no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—82no—
574C22—81no—
575C22—82no—
576C22—81no—
577C22—81no—
578C22—82no—
579C22—81no—
580C22—82no—
581C22—81no—
582C22—82no—
583C22—81no—
584C22—82no—
585C22—81no—
586C22—82no—
587C22—1616no—
588C22#288011yescharacteristic
589C22#288311yescharacteristic
590C22C22×#10011yescharacteristic
591C22C22×#10211yescharacteristic
592C22C2×#41811yescharacteristic
593C22#290111yescharacteristic
594C22#290111yescharacteristic
595C4—21no—
596C4—21no—
597C4—24no—
598C4—22no—
599C4—22no—
600C4—21no—
601C4—21no—
602C4—24no—
603C4—21no—
604C4—21no—
605C4—41no—
606C4—42no—
607C4—81no—
608C4—81no—
609C4—82no—
610C4#154514yes—
611C2—44no—
612C2—82no—
613C2—161no—
614C2—162no—
615C2#1742111yescharacteristic
616C2#1750711yescharacteristic
617C2#1751111yescharacteristic
618C2#1761311yescharacteristic
619C2#1762611yescharacteristic
620C2#1768811yescharacteristic
621C2C2×#154511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36260C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22835C2semidirect
3. #22700C2semidirect
4. #7724C2semidirect
5. #7684C2semidirect
6. #7659C2semidirect
7. #7659C2semidirect
8. #7651C2semidirect
9. #7606C2semidirect
10. #7572C2semidirect
11. #7569C2semidirect
34. #1831C22semidirect
35. #1831C22semidirect
36. #1817C22semidirect
37. #1797C22semidirect
39. C4×C4⋊C8C22semidirect
40. C4×C4⋊C8C22semidirect
41. C4×C22⋊C8C22semidirect
42. #1728C22semidirect
46. #1270C22semidirect
47. #1270C22semidirect
48. #1270C22semidirect
49. #1270C22semidirect
50. #1270C22semidirect
51. #1270C22semidirect
52. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#362602
#228352
#227002
#77242
#76842
#76592
#76512
#76062
#75722
#75692