Finite Group Databasebeta

← #205248

Subgroups — #205248

Order 512.

1166 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3184C211yesmaximal, characteristic
2#21390C211yesmaximal, characteristic, complemented
3#21257C211yesmaximal, characteristic, complemented
4#20480C211yesmaximal, characteristic, complemented
5C2×#1565C211yesmaximal, characteristic, complemented
6C2×#1553C211yesmaximal, characteristic, complemented
7C2×#1551C211yesmaximal, 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×#431C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C2×#416—22no—
32C2×C4×M16C2211yescharacteristic
33C4×#117C2211yescharacteristic
34#1987C2211yescharacteristic
35#1987C2211yescharacteristic
36#1983—22no—
37#1982C2211yescharacteristic
38#1982C2211yescharacteristic
39#1982C2211yescharacteristic
40#1982C2211yescharacteristic
41#1981C2211yescharacteristic
42#1981C2211yescharacteristic
43#1875—22no—
44#1770C2211yescharacteristic
45#1770C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48C4×#102—22no—
49#1753—21no—
50#1753—21no—
51#1745—21no—
52C4×#97—21no—
53#1565—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1553—24no—
59#1551—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×M16D411yescharacteristic
73C22×M16D411yescharacteristic
74C23×C8—21no—
75#532—22no—
76#532—22no—
77#532—22no—
78#532—22no—
79#532—22no—
80#532—22no—
81#532C2×C414yes—
82#532C2×C414yes—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×#127C2311yescharacteristic
86C2×#127C2311yescharacteristic
87C2×#124—21no—
88C2×#124—21no—
89C2×#121—21no—
90C2×#121—21no—
91C2×#121—21no—
92C2×#121—21no—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×#117D412yes—
98C22×C42C2311yescharacteristic
99#498—22no—
100#498C2×C414yes—
101#493—22no—
102#493C2×C414yes—
103#486—22no—
104#486—22no—
105#486C2×C414yes—
106#486C2×C414yes—
107#469—28no—
108#469—28no—
109#442—28no—
110#442—28no—
111#432—22no—
112#432D414yes—
113#431—22no—
114#431—22no—
115C2×#106—21no—
116C2×#106—21no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126#416—48no—
127C2×#103—22no—
128C2×#103—22no—
129C2×#102—21no—
130C2×#102—21no—
131C2×#102—21no—
132C2×#102—21no—
133C2×#102—22no—
134C2×#102—22no—
135C2×#102—22no—
136C2×#102—22no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C4×M16C2×C414yes—
140C4×M16D414yes—
141C2×#97—21no—
142C2×#97—21no—
143C2×#97—21no—
144C2×#97D411yescharacteristic
145C2×#97D411yescharacteristic
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C4×C4⋊C4D412yes—
152C4×C4⋊C4C2311yescharacteristic
153C4×C4⋊C4C2311yescharacteristic
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156#376—21no—
157C43—21no—
158#340—22no—
159#340—22no—
160#336—21no—
161#336—21no—
162#336—21no—
163#336—21no—
164#335—22no—
165#335—22no—
166#334—22no—
167#334—22no—
168#333—22no—
169#333—22no—
170#332—22no—
171#332—22no—
172#331—22no—
173#331—22no—
174#331—22no—
175#331—22no—
176C2×#41—24no—
177C2×#41—24no—
178C2×#41—44no—
179C2×#41—44no—
180C2×#41—42no—
181C2×#41—42no—
182C2×#41—84no—
183C22×Q8—21no—
184C22×Q8—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4—41no—
192C22×D4—41no—
193C23×C4—41no—
194C23×C4—41no—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4#4111yescharacteristic
198C2×D8—24no—
199C2×D8—24no—
200C2×D8—42no—
201C2×D8—42no—
202C2×D8—42no—
203C2×D8—42no—
204C2×M16—22no—
205C2×M16—22no—
206C2×M16—41no—
207C2×M16—41no—
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×C8—44no—
217C22×C8—42no—
218C22×C8C2×D411yescharacteristic
219C22×C8C2×D411yescharacteristic
220C22×C8C2×D411yescharacteristic
221C22×C8C2×D411yescharacteristic
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8C22×C412yes—
229C4⋊Q8C22×C412yes—
230#127—21no—
231#127—21no—
232#127—21no—
233#127—21no—
234#127—21no—
235#127—21no—
236#127C22×C412yes—
237#127C22×C412yes—
238#124—24no—
239#124—24no—
240#124—42no—
241#124—42no—
242#121—24no—
243#121—24no—
244#121—24no—
245#121—24no—
246#121—42no—
247#121—42no—
248#121—42no—
249#121—42no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—24no—
253C4×D4—24no—
254C4×D4—42no—
255C4×D4—42no—
256C4×D4—42no—
257C4×D4—42no—
258#117—24no—
259#117—42no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C42—22no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—22no—
275C2×C42C22×C412yes—
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D412yes—
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D412yes—
280C2×C42#4111yescharacteristic
281C2×C42#4112yes—
282C2×C42C2411yescharacteristic
283#106—24no—
284#106—24no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—42no—
296C4⋊C8C2×D412yes—
297C4⋊C8C2×D412yes—
298#103—44no—
299#103—44no—
300#102—24no—
301#102—24no—
302#102—24no—
303#102—24no—
304#102—44no—
305#102—44no—
306#102—44no—
307#102—44no—
308C22⋊C8—24no—
309C22⋊C8—42no—
310#97—24no—
311#97—42no—
312#97—42no—
313#97C22⋊C414yes—
314#97C22⋊C414yes—
315C4×C8—42no—
316C4×C8—42no—
317C4×C8—42no—
318C4×C8C22×C412yes—
319C4×C8C22×C412yes—
320C4×C8C2×D412yes—
321C4×C8C2×D412yes—
322#95—22no—
323#95—21no—
324#95—21no—
325C24—41no—
326C24—41no—
327C24#12111yescharacteristic
328#41—416no—
329#41—84no—
330#41—84no—
331#41—88no—
332#41—88no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—82no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4—82no—
367C2×D4—82no—
368C2×D4—84no—
369C2×D4—84no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4C2×C22⋊C412yes—
386C22×C4C4×D412yes—
387C22×C4C4×D412yes—
388C22×C4#12112yes—
389C22×C4#12111yescharacteristic
390C22×C4#12111yescharacteristic
391C22×C4#12111yescharacteristic
392C22×C4#12412yes—
393C22×C4#12712yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C2×#4111yescharacteristic
397D8—48no—
398D8—48no—
399M16—28no—
400M16—44no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—84no—
422C2×C8C2×C22⋊C412yes—
423C2×C8C2×C22⋊C412yes—
424C2×C8C2×C22⋊C412yes—
425C2×C8C2×C22⋊C412yes—
426C2×C8#13512yes—
427C2×C8#13512yes—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4AGL(1,ℤ/8)14yes—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C42—44no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—82no—
451C42C2×C22⋊C412yes—
452C42C4×D412yes—
453C42C4×D412yes—
454C42#12112yes—
455C42#12412yes—
456C42#12712yes—
457C42C23×C411yescharacteristic
458C42C22×D411yescharacteristic
459C42C22×D411yescharacteristic
460C42C2×#4111yescharacteristic
461C23—21no—
462C23—21no—
463C23—22no—
464C23—22no—
465C23—42no—
466C23—42no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23—81no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23#39012yes—
476C23C2×#12111yescharacteristic
477Q8—44no—
478Q8—82no—
479Q8—82no—
480Q8—81no—
481Q8—81no—
482D4—42no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—44no—
487D4—44no—
488D4—82no—
489D4—82no—
490D4—82no—
491D4—82no—
492D4—88no—
493D4—88no—
494D4—81no—
495D4—81no—
496D4—81no—
497D4—81no—
498D4—81no—
499D4—81no—
500D4—81no—
501D4—81no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—88no—
525C2×C4—88no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4#39012yes—
529C2×C4#39012yes—
530C2×C4#39012yes—
531C2×C4C22×C22⋊C411yescharacteristic
532C2×C4C2×C4×D411yescharacteristic
533C2×C4C2×C4×D411yescharacteristic
534C2×C4C2×#12111yescharacteristic
535C2×C4C2×#12411yescharacteristic
536C2×C4C2×#12711yescharacteristic
537C2×C4C2×#13511yescharacteristic
538C2×C4C2×#13511yescharacteristic
539C2×C4C2×AGL(1,ℤ/8)12yes—
540C2×C4#57511yescharacteristic
541C2×C4#57511yescharacteristic
542C2×C4#57512yes—
543C8—22no—
544C8—22no—
545C8—22no—
546C8—42no—
547C8—42no—
548C8—81no—
549C8—81no—
550C8—81no—
551C8—81no—
552C8#41614yescomplemented
553C22—22no—
554C22—22no—
555C22—22no—
556C22—42no—
557C22—42no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—162no—
565C22—162no—
566C22—162no—
567C22—162no—
568C22C2×#39011yescharacteristic
569C22#303511yescharacteristic
570C22#303511yescharacteristic
571C22#305512yes—
572C22#306812yes—
573C4—22no—
574C4—21no—
575C4—21no—
576C4—22no—
577C4—22no—
578C4—44no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4C2×#41611yescharacteristic
583C4#289611yescharacteristic
584C4C4×AGL(1,ℤ/8)12yes—
585C4#303511yescharacteristic
586C4#303511yescharacteristic
587C4#315111yescharacteristic
588C4#315311yescharacteristic
589C2—42no—
590C2—162no—
591C2—162no—
592C2#1918012yes—
593C2#1918412yes—
594C2#2123611yescharacteristic
595C2#2124911yescharacteristic
596C2#2125711yescharacteristic
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. #21390C2semidirect
3. #21257C2semidirect
4. #20480C2semidirect
5. C2×#1565C2semidirect
6. C2×#1553C2semidirect
7. C2×#1551C2semidirect
8. #9747C2semidirect
9. #9746C2semidirect
10. #9743C2semidirect
11. #9741C2semidirect
12. #9738C2semidirect
13. #9735C2semidirect
14. #9734C2semidirect
15. #9729C2semidirect
19. #3184C4semidirect
552. C8#416semidirect

Maximal subgroups

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

subgroupindex
C2×#31842
#213902
#212572
#204802
C2×#15652
C2×#15532
C2×#15512
#97472
#97462
#97432
#97412
#97382
#97352
#97342
#97292