Finite Group Databasebeta

← #205333

Subgroups — #205333

Order 512.

1132 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3182C211yesmaximal, characteristic
2#21394C211yesmaximal, characteristic, complemented
3#21258C211yesmaximal, characteristic
4C4⋊C4×M16C211yesmaximal, characteristic, complemented
5C2×#1569C211yesmaximal, characteristic, complemented
6C2×#1561C211yesmaximal, characteristic, complemented
7C2×#1557C211yesmaximal, characteristic
8#9748C211yesmaximal, characteristic, complemented
9#9746C211yesmaximal, characteristic, complemented
10#9744C211yesmaximal, characteristic, complemented
11#9740C211yesmaximal, characteristic, complemented
12#9738C211yesmaximal, characteristic, complemented
13#9736C211yesmaximal, characteristic, complemented
14#9732C211yesmaximal, characteristic, complemented
15#9731C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3182—24no—
19#3182C418yescomplemented
20C2×#500C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#468—21no—
25C2×#468—21no—
26C2×#440—21no—
27C2×#440—21no—
28C2×#432C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C2×#417—22no—
32C2×C4×M16—21no—
33C2×C4×M16C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1988C2211yescharacteristic
36#1988C2211yescharacteristic
37#1983—22no—
38#1982C2211yescharacteristic
39#1982C2211yescharacteristic
40#1981C2211yescharacteristic
41#1981C2211yescharacteristic
42#1981C2211yescharacteristic
43#1981C2211yescharacteristic
44#1872—22no—
45#1771—21no—
46#1771—21no—
47#1770C2211yescharacteristic
48#1770C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1763—22no—
52#1728—21no—
53#1569—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1561—24no—
59#1557—24no—
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65C2×#332C2211yescharacteristic
66C2×#331C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C22×#41—21no—
69C22×SD16—21no—
70C22×SD16—21no—
71C22×M16—21no—
72C22×M16—21no—
73C22×M16D411yescharacteristic
74C22×M16D411yescharacteristic
75#533—22no—
76#533—22no—
77#533—22no—
78#533C2×C414yes—
79#532—22no—
80#532—22no—
81#532—22no—
82#532C2×C414yes—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×#127C2311yescharacteristic
87C2×#125—21no—
88C2×#124—21no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—21no—
91C2×#121—21no—
92C2×#121—21no—
93C2×C4×Q8—21no—
94C2×C4×Q8—21no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×#117—21no—
98C22×C4⋊C4D412yes—
99C22×C42—21no—
100C22×C42C2311yescharacteristic
101#500—22no—
102#500C2×C414yescomplemented
103#492—22no—
104#492C2×C414yescomplemented
105#487—22no—
106#487C2×C414yescomplemented
107#486—22no—
108#486C2×C414yescomplemented
109#468—28no—
110#468—28no—
111#440—28no—
112#440—28no—
113#432—22no—
114#432—22no—
115C2×#107—21no—
116C2×#107—21no—
117#423—22no—
118#423—22no—
119#423—22no—
120#423D414yes—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129#417—48no—
130C2×#103—21no—
131C2×#103—21no—
132C2×#103—22no—
133C2×#103—22no—
134C2×#103—22no—
135C2×#103—22no—
136C2×#102—21no—
137C2×#102—21no—
138C2×#102—22no—
139C2×#102—22no—
140C2×C22⋊C8—21no—
141C4×M16—44no—
142C4×M16C2×C414yes—
143C4×M16D414yes—
144C2×#97—21no—
145C2×#97—21no—
146C2×#97—21no—
147C2×#97D411yescharacteristic
148C2×#97D411yescharacteristic
149C2×C4×C8—21no—
150C2×C4×C8—21no—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4D412yes—
156C4×C4⋊C4C2311yescharacteristic
157C4×C4⋊C4C2311yescharacteristic
158C2×#95—21no—
159#340—22no—
160#340—22no—
161#336—21no—
162#336—21no—
163#336—21no—
164#336—21no—
165#335—22no—
166#335—22no—
167#334—22no—
168#334—22no—
169#332—22no—
170#332—22no—
171#332—22no—
172#332—22no—
173#332—22no—
174#332—22no—
175#331—22no—
176#331—22no—
177C2×#41—24no—
178C2×#41—44no—
179C2×#41—44no—
180C2×#41—42no—
181C22×Q8—21no—
182C22×Q8—21no—
183C22×Q8—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—41no—
188C23×C4—22no—
189C23×C4—41no—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4#4111yescharacteristic
193C2×SD16—24no—
194C2×SD16—24no—
195C2×SD16—42no—
196C2×SD16—42no—
197C2×SD16—42no—
198C2×SD16—42no—
199C2×M16—24no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—41no—
203C2×M16—41no—
204C2×M16—42no—
205C2×M16—42no—
206C2×M16—44no—
207C2×M16—42no—
208C2×M16C22⋊C414yes—
209C2×M16C22⋊C414yes—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C22×C8C2×D411yescharacteristic
219C22×C8C2×D411yescharacteristic
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8C22×C412yes—
230C4⋊Q8C22×C412yes—
231C4⋊Q8C22×C412yes—
232#127—21no—
233#127—21no—
234#127—21no—
235#127C22×C412yes—
236#125—24no—
237#125—42no—
238#124—24no—
239#124—42no—
240C22⋊Q8—24no—
241C22⋊Q8—24no—
242C22⋊Q8—42no—
243C22⋊Q8—42no—
244#121—24no—
245#121—24no—
246#121—42no—
247#121—42no—
248C4×Q8—24no—
249C4×Q8—24no—
250C4×Q8—42no—
251C4×Q8—42no—
252C4×D4—24no—
253C4×D4—24no—
254C4×D4—42no—
255C4×D4—42no—
256#117—24no—
257#117—42no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C42—22no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42—41no—
280C2×C42C22×C412yes—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D412yes—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D412yes—
285C2×C42#4111yescharacteristic
286C2×C42#4112yes—
287C2×C42C2411yescharacteristic
288#107—24no—
289#107—24no—
290C4⋊C8—24no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—42no—
302C4⋊C8C2×D412yes—
303C4⋊C8C2×D412yes—
304#103—24no—
305#103—24no—
306#103—44no—
307#103—44no—
308#103—44no—
309#103—44no—
310#102—24no—
311#102—24no—
312#102—44no—
313#102—44no—
314C22⋊C8—42no—
315#97—42no—
316#97—42no—
317#97—42no—
318#97C22⋊C414yes—
319#97C22⋊C414yes—
320C4×C8—42no—
321C4×C8—42no—
322C4×C8C22×C412yes—
323C4×C8C22×C412yes—
324C4×C8C2×D412yes—
325C4×C8C2×D412yes—
326#95—22no—
327#95—21no—
328#95—21no—
329C24—41no—
330C24#12111yescharacteristic
331#41—416no—
332#41—84no—
333#41—84no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—42no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—42no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—44no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—42no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—42no—
361C2×D4—82no—
362C2×D4—84no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—44no—
378C22×C4—84no—
379C22×C4—82no—
380C22×C4C2×C22⋊C412yes—
381C22×C4C4×D412yes—
382C22×C4C4×D412yes—
383C22×C4#12112yes—
384C22×C4#12111yescharacteristic
385C22×C4#12111yescharacteristic
386C22×C4#12111yescharacteristic
387C22×C4#12412yes—
388C22×C4#12712yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C2×#4111yescharacteristic
392SD16—48no—
393SD16—48no—
394M16—28no—
395M16—44no—
396M16—48no—
397M16—84no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8C2×C22⋊C412yes—
419C2×C8C2×C22⋊C412yes—
420C2×C8C2×C22⋊C412yes—
421C2×C8C2×C22⋊C412yes—
422C2×C8AGL(1,ℤ/8)12yes—
423C2×C8#13712yes—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—44no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4AGL(1,ℤ/8)14yes—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C42—44no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—44no—
452C42C2×C22⋊C412yes—
453C42C4×D412yes—
454C42C4×D412yes—
455C42#12112yes—
456C42#12412yes—
457C42#12712yes—
458C42C23×C411yescharacteristic
459C42C22×D411yescharacteristic
460C42C22×D411yescharacteristic
461C42C2×#4111yescharacteristic
462C23—21no—
463C23—21no—
464C23—22no—
465C23—22no—
466C23—42no—
467C23—81no—
468C23—81no—
469C23—82no—
470C23—82no—
471C23#39012yes—
472C23C2×#12111yescharacteristic
473Q8—42no—
474Q8—42no—
475Q8—44no—
476Q8—44no—
477Q8—81no—
478Q8—81no—
479Q8—81no—
480Q8—81no—
481Q8—81no—
482Q8—81no—
483D4—42no—
484D4—42no—
485D4—44no—
486D4—82no—
487D4—82no—
488D4—88no—
489D4—81no—
490D4—81no—
491D4—81no—
492D4—81no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—88no—
517C2×C4—88no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4#39012yes—
523C2×C4#39012yes—
524C2×C4#39012yes—
525C2×C4C22×C22⋊C411yescharacteristic
526C2×C4C2×C4×D411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×#12111yescharacteristic
529C2×C4C2×#12411yescharacteristic
530C2×C4C2×#12711yescharacteristic
531C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
532C2×C4C2×AGL(1,ℤ/8)12yes—
533C2×C4C2×#13711yescharacteristic
534C2×C4#57511yescharacteristic
535C2×C4#57511yescharacteristic
536C2×C4#57512yes—
537C8—22no—
538C8—22no—
539C8—22no—
540C8—42no—
541C8—42no—
542C8—81no—
543C8—81no—
544C8—81no—
545C8—81no—
546C8#41714yescomplemented
547C22—22no—
548C22—22no—
549C22—22no—
550C22—42no—
551C22—42no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—162no—
556C22—162no—
557C22C2×#39011yescharacteristic
558C22#303611yescharacteristic
559C22#303711yescharacteristic
560C22#305512yes—
561C22#306912yes—
562C4—21no—
563C4—21no—
564C4—22no—
565C4—22no—
566C4—22no—
567C4—44no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4C2×#41711yescharacteristic
573C4#289611yescharacteristic
574C4C4×AGL(1,ℤ/8)12yes—
575C4#303611yescharacteristic
576C4#303711yescharacteristic
577C4#315111yescharacteristic
578C4#315311yescharacteristic
579C2—42no—
580C2—162no—
581C2#1930412yes—
582C2#1930812yes—
583C2#2123611yescharacteristic
584C2#2125011yescharacteristic
585C2#2125811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21394C2semidirect
4. C4⋊C4×M16C2semidirect
5. C2×#1569C2semidirect
6. C2×#1561C2semidirect
8. #9748C2semidirect
9. #9746C2semidirect
10. #9744C2semidirect
11. #9740C2semidirect
12. #9738C2semidirect
13. #9736C2semidirect
14. #9732C2semidirect
15. #9731C2semidirect
19. #3182C4semidirect
102. #500C2×C4semidirect
104. #492C2×C4semidirect
106. #487C2×C4semidirect
108. #486C2×C4semidirect
546. C8#417semidirect

Maximal subgroups

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

subgroupindex
C2×#31822
#213942
#212582
C4⋊C4×M162
C2×#15692
C2×#15612
C2×#15572
#97482
#97462
#97442
#97402
#97382
#97362
#97322
#97312