Finite Group Databasebeta

← #165065

Subgroups — #165065

Order 512.

1336 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36260C211yesmaximal, characteristic
2#22787C212yesmaximal, complemented
3#7704C212yesmaximal, complemented
4#7699C212yesmaximal, complemented
5#7664C212yesmaximal, complemented
6#7640C211yesmaximal, characteristic, complemented
7#7637C211yesmaximal, characteristic, complemented
8#7626C211yesmaximal, characteristic, complemented
9#7603C211yesmaximal, characteristic, complemented
10#7592C211yesmaximal, characteristic, complemented
11C2×#1270C211yesmaximal, characteristic, complemented
12C2×#547C411yescharacteristic
13C2×#546—22no—
14C2×D42C411yescharacteristic
15#2858C2211yescharacteristic
16#2856C2211yescharacteristic
17#2856C2211yescharacteristic
18#2542C411yescharacteristic
19#2541C411yescharacteristic
20#2540—22no—
21#2539—22no—
22#2539—22no—
23#2501C411yescharacteristic
24#2501C411yescharacteristic
25#2433C411yescharacteristic
26#2433C411yescharacteristic
27C4×#127C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C42×D4C2211yescharacteristic
31#1924—22no—
32#1917—22no—
33#1869—22no—
34#1839C2212yescomplemented
35#1817C2212yescomplemented
36#1797C2212yescomplemented
37#1752—22no—
38#1749C2212yescomplemented
39#1745C2212yescomplemented
40C2×#336C2212yescomplemented
41#1270C2211yescharacteristic
42#1270C2211yescharacteristic
43#1270C2211yescharacteristic
44#1270C2211yescharacteristic
45#1270C2212yescomplemented
46#1270C2212yescomplemented
47#1270C2212yescomplemented
48#1270C2212yescomplemented
49#1270C2212yescomplemented
50#1270C2212yescomplemented
51C22×#41—21no—
52C23×D4—21no—
53C23×D4C2×C411yescharacteristic
54C23×D4C2×C411yescharacteristic
55C23×C8—22no—
56#547—44no—
57#546—48no—
58#546—48no—
59D42—44no—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×#127C2×C411yescharacteristic
62C2×#127C2×C411yescharacteristic
63C2×#127C2×C411yescharacteristic
64C2×#127C2×C411yescharacteristic
65C2×#124—22no—
66C2×#123—22no—
67C2×#123—22no—
68C2×#123—21no—
69C2×#123—21no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—21no—
72C2×#121—22no—
73C2×#121—22no—
74C2×#121—21no—
75C2×#121—22no—
76C2×#121C2×C411yescharacteristic
77C2×#121C2×C411yescharacteristic
78C2×#121C2×C411yescharacteristic
79C2×#121C2×C411yescharacteristic
80C2×#120—22no—
81C2×#120—22no—
82C2×#120—21no—
83C2×#120—21no—
84C2×C4×D4—22no—
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2×C411yescharacteristic
87C2×C4×D4C2×C411yescharacteristic
88C2×C4×D4D412yes—
89C22×C4⋊C4C2×C411yescharacteristic
90C22×C4⋊C4C2×C411yescharacteristic
91C22×C22⋊C4—22no—
92C22×C22⋊C4—22no—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C2×C4⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×#97—22no—
100C2×#97—22no—
101#399—22no—
102#399—22no—
103#399—22no—
104#392—22no—
105#392—22no—
106#392—22no—
107#390—22no—
108#390—22no—
109#390—22no—
110#390—22no—
111#390—22no—
112#390—22no—
113#390C2×C411yescharacteristic
114#390C2×C411yescharacteristic
115#390C2×C411yescharacteristic
116#390C2×C411yescharacteristic
117#390C2×C411yescharacteristic
118#390C2×C411yescharacteristic
119#390C2311yescharacteristic
120#390C2311yescharacteristic
121#390C2311yescharacteristic
122#390C2311yescharacteristic
123#384C2×C411yescharacteristic
124#384C2×C411yescharacteristic
125#384C2×C411yescharacteristic
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4D412yes—
128C4×C22⋊C4C2×C411yescharacteristic
129C4×C22⋊C4C2×C411yescharacteristic
130C4×C22⋊C4D412yes—
131C4×C22⋊C4D412yes—
132C43C2311yescharacteristic
133#336—22no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336C2312yes—
139#336C2312yes—
140#336C2312yes—
141#336C2312yes—
142C25—21no—
143C25—21no—
144C2×#41—48no—
145C2×#41—44no—
146C22×Q8—21no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—41no—
161C22×D4—41no—
162C22×D4—42no—
163C22×D4—42no—
164C22×D4—48no—
165C22×D4—44no—
166C22×D4—81no—
167C22×D4—81no—
168C22×D4C22⋊C411yescharacteristic
169C22×D4C22×C411yescharacteristic
170C22×D4C22×C411yescharacteristic
171C22×D4C22×C411yescharacteristic
172C22×D4C22×C411yescharacteristic
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—44no—
186C22×C8—44no—
187C22×C8—82no—
188C4⋊Q8—42no—
189#127—22no—
190#127—41no—
191#127—42no—
192#127—81no—
193#127—81no—
194#124—44no—
195#124—44no—
196#123—48no—
197#123—48no—
198#123—44no—
199#123—44no—
200C22⋊Q8—48no—
201C22⋊Q8—44no—
202#121—44no—
203#121—48no—
204#121—42no—
205#121—42no—
206#121—42no—
207#121—42no—
208#121—44no—
209#121—44no—
210#121—44no—
211#121—44no—
212#120—44no—
213#120—44no—
214#120—48no—
215#120—48no—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—44no—
220C4×D4—44no—
221C4×D4—42no—
222C4×D4—42no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C412yes—
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4C22⋊C411yescharacteristic
258C2×C22⋊C4C22⋊C411yescharacteristic
259C2×C22⋊C4C22⋊C412yes—
260C2×C22⋊C4C22⋊C412yes—
261C2×C42—81no—
262C2×C42—81no—
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2411yescharacteristic
272C4⋊C8—82no—
273C22⋊C8—44no—
274C22⋊C8—44no—
275C22⋊C8—82no—
276#97—44no—
277#97—82no—
278#95C22⋊C412yes—
279#95C22⋊C412yes—
280#95C22⋊C412yes—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—42no—
286C24—42no—
287C24—44no—
288C24—42no—
289C24—42no—
290C24—44no—
291C24—81no—
292C24—81no—
293C24—82no—
294C24—82no—
295C24C2×C22⋊C411yescharacteristic
296C24C2×C22⋊C411yescharacteristic
297#41—816no—
298C2×Q8—44no—
299C2×Q8—42no—
300C2×D4—24no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—44no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—44no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—44no—
323C2×D4—42no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—42no—
334C2×D4—44no—
335C2×D4—82no—
336C2×D4—81no—
337C2×D4—81no—
338C2×D4—82no—
339C2×D4—82no—
340C2×D4—82no—
341C2×D4—82no—
342C2×D4—82no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—816no—
348C2×D4#10014yes—
349C2×D4#13112yes—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4C2×C22⋊C412yes—
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C412yes—
383C22×C4AGL(1,ℤ/8)14yes—
384C22×C4AGL(1,ℤ/8)14yes—
385C22×C4C23×C411yescharacteristic
386C22×C4C22×D412yes—
387C2×C8—24no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—88no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C42—21no—
437C42—21no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—41no—
445C42—41no—
446C42—41no—
447C23—21no—
448C23—21no—
449C23—22no—
450C23—21no—
451C23—21no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—84no—
475C23—84no—
476C23—88no—
477C23—84no—
478C23—88no—
479C23—84no—
480C23—162no—
481C23—164no—
482C23—162no—
483C23—164no—
484C23C2×#10012yes—
485C23C2×#10012yes—
486C23#41814yes—
487C23C22×C22⋊C411yescharacteristic
488Q8—84no—
489D4—28no—
490D4—42no—
491D4—42no—
492D4—81no—
493D4—81no—
494D4—81no—
495D4—81no—
496D4—81no—
497D4—81no—
498D4—84no—
499D4—88no—
500D4—84no—
501D4—84no—
502D4—84no—
503D4—84no—
504D4—84no—
505D4—84no—
506D4—164no—
507D4—162no—
508D4—164no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4C2×#10012yes—
551C2×C4#41814yes—
552C2×C4C2×#13111yescharacteristic
553C2×C4#56811yescharacteristic
554C2×C4C2×AGL(1,ℤ/8)12yes—
555C2×C4C2×AGL(1,ℤ/8)12yes—
556C8—44no—
557C8—82no—
558C8—82no—
559C8—82no—
560C22—22no—
561C22—22no—
562C22—24no—
563C22—24no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—44no—
571C22—81no—
572C22—81no—
573C22—82no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—168no—
590C22—168no—
591C22#155514yes—
592C22#288011yescharacteristic
593C22#288311yescharacteristic
594C22C22×#10011yescharacteristic
595C22C2×#41812yes—
596C22#290112yes—
597C4—21no—
598C4—21no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—24no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—42no—
607C4—42no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—81no—
612C2—24no—
613C2—42no—
614C2—82no—
615C2—161no—
616C2—161no—
617C2—161no—
618C2#1742111yescharacteristic
619C2#1751112yes—
620C2#1762012yes—
621C2#1768511yescharacteristic
622C2C2×#155511yescharacteristic
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. #22787C2semidirect
3. #7704C2semidirect
4. #7699C2semidirect
5. #7664C2semidirect
6. #7640C2semidirect
7. #7637C2semidirect
8. #7626C2semidirect
9. #7603C2semidirect
10. #7592C2semidirect
11. C2×#1270C2semidirect
34. #1839C22semidirect
35. #1817C22semidirect
36. #1797C22semidirect
38. #1749C22semidirect
39. #1745C22semidirect
40. C2×#336C22semidirect
45. #1270C22semidirect
46. #1270C22semidirect
47. #1270C22semidirect
48. #1270C22semidirect
49. #1270C22semidirect
50. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#362602
#227872
#77042
#76992
#76642
#76402
#76372
#76262
#76032
#75922
C2×#12702