Finite Group Databasebeta

← #205330

Subgroups — #205330

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3184C211yesmaximal, characteristic
2#21395C211yesmaximal, characteristic, complemented
3#21258C211yesmaximal, characteristic, complemented
4C4⋊C4×M16C211yesmaximal, characteristic, complemented
5C2×#1569C211yesmaximal, characteristic, complemented
6C2×#1560C211yesmaximal, characteristic, complemented
7C2×#1557C211yesmaximal, characteristic, complemented
8#9747C211yesmaximal, characteristic, complemented
9#9746C211yesmaximal, characteristic, complemented
10#9743C211yesmaximal, characteristic, complemented
11#9741C211yesmaximal, characteristic, complemented
12#9738C211yesmaximal, characteristic, complemented
13#9735C211yesmaximal, characteristic, complemented
14#9734C211yesmaximal, characteristic, complemented
15#9729C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3184—24no—
19#3184C418yescomplemented
20C2×#498C2211yescharacteristic
21C2×#493C2211yescharacteristic
22C2×#486C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#469—21no—
25C2×#469—21no—
26C2×#442—21no—
27C2×#442—21no—
28C2×#432C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C2×#417—22no—
32C2×C4×M16—21no—
33C2×C4×M16C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1987C2211yescharacteristic
36#1987C2211yescharacteristic
37#1983—22no—
38#1982C2211yescharacteristic
39#1982C2211yescharacteristic
40#1982C2211yescharacteristic
41#1982C2211yescharacteristic
42#1981C2211yescharacteristic
43#1981C2211yescharacteristic
44#1874—22no—
45#1771—21no—
46#1771—21no—
47#1770C2211yescharacteristic
48#1770C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1762—22no—
52#1728—21no—
53#1569—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1560—24no—
59#1557—24no—
60C2×#333C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#331C2211yescharacteristic
65C2×#331C2211yescharacteristic
66C2×#331C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C22×#41—21no—
69C22×#41—21no—
70C22×D8—21no—
71C22×D8—21no—
72C22×M16—21no—
73C22×M16—21no—
74C22×M16D411yescharacteristic
75C22×M16D411yescharacteristic
76#532—22no—
77#532—22no—
78#532—22no—
79#532—22no—
80#532—22no—
81#532—22no—
82#532C2×C414yes—
83#532C2×C414yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×#127C2311yescharacteristic
87C2×#127C2311yescharacteristic
88C2×#124—21no—
89C2×#124—21no—
90C2×#121—21no—
91C2×#121—21no—
92C2×#121—21no—
93C2×#121—21no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C22×C4⋊C4D412yes—
99C22×C42—21no—
100C22×C42C2311yescharacteristic
101#498—22no—
102#498C2×C414yescomplemented
103#493—22no—
104#493C2×C414yescomplemented
105#486—22no—
106#486—22no—
107#486C2×C414yescomplemented
108#486C2×C414yescomplemented
109#469—28no—
110#469—28no—
111#442—28no—
112#442—28no—
113#432—22no—
114#432—22no—
115C2×#106—21no—
116C2×#106—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—22no—
131C2×#103—22no—
132C2×#102—21no—
133C2×#102—21no—
134C2×#102—21no—
135C2×#102—21no—
136C2×#102—22no—
137C2×#102—22no—
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#333—22no—
170#333—22no—
171#332—22no—
172#332—22no—
173#331—22no—
174#331—22no—
175#331—22no—
176#331—22no—
177C2×#41—24no—
178C2×#41—24no—
179C2×#41—44no—
180C2×#41—44no—
181C2×#41—42no—
182C2×#41—42no—
183C2×#41—84no—
184C22×Q8—21no—
185C22×Q8—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4—41no—
193C22×D4—41no—
194C23×C4—22no—
195C23×C4—41no—
196C23×C4—41no—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4#4111yescharacteristic
200C2×D8—24no—
201C2×D8—24no—
202C2×D8—42no—
203C2×D8—42no—
204C2×D8—42no—
205C2×D8—42no—
206C2×M16—24no—
207C2×M16—22no—
208C2×M16—22no—
209C2×M16—41no—
210C2×M16—41no—
211C2×M16—42no—
212C2×M16—42no—
213C2×M16—44no—
214C2×M16—42no—
215C2×M16C22⋊C414yes—
216C2×M16C22⋊C414yes—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8C2×D411yescharacteristic
224C22×C8C2×D411yescharacteristic
225C22×C8C2×D411yescharacteristic
226C22×C8C2×D411yescharacteristic
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8C22×C412yes—
234C4⋊Q8C22×C412yes—
235#127—21no—
236#127—21no—
237#127—21no—
238#127—21no—
239#127—21no—
240#127—21no—
241#127C22×C412yes—
242#127C22×C412yes—
243#124—24no—
244#124—24no—
245#124—42no—
246#124—42no—
247#121—24no—
248#121—24no—
249#121—24no—
250#121—24no—
251#121—42no—
252#121—42no—
253#121—42no—
254#121—42no—
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—42no—
260C4×D4—42no—
261C4×D4—42no—
262C4×D4—42no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C42—22no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42C22×C412yes—
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D412yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D412yes—
286C2×C42#4111yescharacteristic
287C2×C42#4112yes—
288C2×C42C2411yescharacteristic
289#106—24no—
290#106—24no—
291C4⋊C8—24no—
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—21no—
302C4⋊C8—42no—
303C4⋊C8C2×D412yes—
304C4⋊C8C2×D412yes—
305#103—44no—
306#103—44no—
307#102—24no—
308#102—24no—
309#102—24no—
310#102—24no—
311#102—44no—
312#102—44no—
313#102—44no—
314#102—44no—
315C22⋊C8—42no—
316#97—42no—
317#97—42no—
318#97—42no—
319#97C22⋊C414yes—
320#97C22⋊C414yes—
321C4×C8—42no—
322C4×C8—42no—
323C4×C8C22×C412yes—
324C4×C8C22×C412yes—
325C4×C8C2×D412yes—
326C4×C8C2×D412yes—
327#95—22no—
328#95—21no—
329#95—21no—
330C24—41no—
331C24—41no—
332C24#12111yescharacteristic
333#41—416no—
334#41—84no—
335#41—84no—
336#41—88no—
337#41—88no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—82no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—42no—
371C2×D4—82no—
372C2×D4—82no—
373C2×D4—84no—
374C2×D4—84no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4—82no—
393C22×C4C2×C22⋊C412yes—
394C22×C4C4×D412yes—
395C22×C4C4×D412yes—
396C22×C4#12112yes—
397C22×C4#12111yescharacteristic
398C22×C4#12111yescharacteristic
399C22×C4#12111yescharacteristic
400C22×C4#12412yes—
401C22×C4#12712yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C2×#4111yescharacteristic
405D8—48no—
406D8—48no—
407M16—28no—
408M16—44no—
409M16—48no—
410M16—84no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8C2×C22⋊C412yes—
432C2×C8C2×C22⋊C412yes—
433C2×C8C2×C22⋊C412yes—
434C2×C8C2×C22⋊C412yes—
435C2×C8AGL(1,ℤ/8)12yes—
436C2×C8#13712yes—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—44no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4AGL(1,ℤ/8)14yes—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C42—44no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42C2×C22⋊C412yes—
460C42C4×D412yes—
461C42C4×D412yes—
462C42#12112yes—
463C42#12412yes—
464C42#12712yes—
465C42C23×C411yescharacteristic
466C42C22×D411yescharacteristic
467C42C22×D411yescharacteristic
468C42C2×#4111yescharacteristic
469C23—21no—
470C23—21no—
471C23—22no—
472C23—22no—
473C23—42no—
474C23—42no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23#39012yes—
484C23C2×#12111yescharacteristic
485Q8—44no—
486Q8—82no—
487Q8—82no—
488Q8—81no—
489Q8—81no—
490D4—42no—
491D4—42no—
492D4—42no—
493D4—42no—
494D4—44no—
495D4—44no—
496D4—82no—
497D4—82no—
498D4—82no—
499D4—82no—
500D4—88no—
501D4—88no—
502D4—81no—
503D4—81no—
504D4—81no—
505D4—81no—
506D4—81no—
507D4—81no—
508D4—81no—
509D4—81no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—88no—
533C2×C4—88no—
534C2×C4—88no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4#39012yes—
538C2×C4#39012yes—
539C2×C4#39012yes—
540C2×C4C22×C22⋊C411yescharacteristic
541C2×C4C2×C4×D411yescharacteristic
542C2×C4C2×C4×D411yescharacteristic
543C2×C4C2×#12111yescharacteristic
544C2×C4C2×#12411yescharacteristic
545C2×C4C2×#12711yescharacteristic
546C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
547C2×C4C2×AGL(1,ℤ/8)12yes—
548C2×C4C2×#13711yescharacteristic
549C2×C4#57511yescharacteristic
550C2×C4#57511yescharacteristic
551C2×C4#57512yes—
552C8—22no—
553C8—22no—
554C8—22no—
555C8—42no—
556C8—42no—
557C8—81no—
558C8—81no—
559C8—81no—
560C8—81no—
561C8#41714yescomplemented
562C22—22no—
563C22—22no—
564C22—22no—
565C22—42no—
566C22—42no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—162no—
574C22—162no—
575C22—162no—
576C22—162no—
577C22C2×#39011yescharacteristic
578C22#303611yescharacteristic
579C22#303711yescharacteristic
580C22#305512yes—
581C22#306812yes—
582C4—22no—
583C4—21no—
584C4—21no—
585C4—22no—
586C4—22no—
587C4—44no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4C2×#41711yescharacteristic
592C4#289611yescharacteristic
593C4C4×AGL(1,ℤ/8)12yes—
594C4#303611yescharacteristic
595C4#303711yescharacteristic
596C4#315111yescharacteristic
597C4#315311yescharacteristic
598C2—42no—
599C2—162no—
600C2—162no—
601C2#1930612yes—
602C2#1931012yes—
603C2#2123611yescharacteristic
604C2#2124911yescharacteristic
605C2#2125811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3184C2
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. #21395C2semidirect
3. #21258C2semidirect
4. C4⋊C4×M16C2semidirect
5. C2×#1569C2semidirect
6. C2×#1560C2semidirect
7. C2×#1557C2semidirect
8. #9747C2semidirect
9. #9746C2semidirect
10. #9743C2semidirect
11. #9741C2semidirect
12. #9738C2semidirect
13. #9735C2semidirect
14. #9734C2semidirect
15. #9729C2semidirect
19. #3184C4semidirect
102. #498C2×C4semidirect
104. #493C2×C4semidirect
107. #486C2×C4semidirect
108. #486C2×C4semidirect
561. C8#417semidirect

Maximal subgroups

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

subgroupindex
C2×#31842
#213952
#212582
C4⋊C4×M162
C2×#15692
C2×#15602
C2×#15572
#97472
#97462
#97432
#97412
#97382
#97352
#97342
#97292