Finite Group Databasebeta

← #205319

Subgroups — #205319

Order 512.

1140 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3184C211yesmaximal, characteristic
2#21893C211yesmaximal, characteristic, complemented
3#21809C211yesmaximal, characteristic, complemented
4#20441C211yesmaximal, characteristic, complemented
5C2×#1569C211yesmaximal, characteristic, complemented
6C2×#1551C211yesmaximal, characteristic, complemented
7C2×#1547C211yesmaximal, 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
30C22×C4⋊C8C2211yescharacteristic
31C2×#405—21no—
32C2×C4×M16C2211yescharacteristic
33#2285C2211yescharacteristic
34#1987C2211yescharacteristic
35#1987C2211yescharacteristic
36#1982C2211yescharacteristic
37#1982C2211yescharacteristic
38#1982C2211yescharacteristic
39#1982C2211yescharacteristic
40#1981C2211yescharacteristic
41#1981C2211yescharacteristic
42#1959—22no—
43#1895—22no—
44#1802—22no—
45#1788—22no—
46#1771—21no—
47#1771—21no—
48#1770C2211yescharacteristic
49#1770C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1729—21no—
53#1569—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1551—24no—
59#1547—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×M16D411yescharacteristic
74C22×M16D411yescharacteristic
75C23×C8—21no—
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—
98C2×#117—21no—
99C22×C42C2311yescharacteristic
100#498—22no—
101#498C2×C414yes—
102#493—22no—
103#493C2×C414yes—
104#486—22no—
105#486—22no—
106#486C2×C414yes—
107#486C2×C414yes—
108#469—28no—
109#469—28no—
110#442—28no—
111#442—28no—
112#432—22no—
113#432—22no—
114C2×#106—21no—
115C2×#106—21no—
116#423—22no—
117#423—22no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—22no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8D414yes—
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×#103—22no—
129C2×#103—22no—
130C2×#102—21no—
131C2×#102—21no—
132C2×#102—21no—
133C2×#102—21no—
134C2×#102—22no—
135C2×#102—22no—
136C2×#102—22no—
137C2×#102—22no—
138C2×C22⋊C8—21no—
139#405—44no—
140C4×M16C2×C414yes—
141C4×M16D414yes—
142C2×#97—21no—
143C2×#97—21no—
144C2×#97—21no—
145C2×#97D411yescharacteristic
146C2×#97D411yescharacteristic
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151#384D412yes—
152#382—21no—
153#380—21no—
154#380D412yes—
155C4×C4⋊C4—21no—
156C4×C4⋊C4C2311yescharacteristic
157C4×C4⋊C4C2311yescharacteristic
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—42no—
179C2×#41—42no—
180C2×#41—82no—
181C2×#41—82no—
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×M16—42no—
209C2×M16—82no—
210C2×M16C22⋊C414yes—
211C2×M16C22⋊C414yes—
212C22×C8—24no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—42no—
220C22×C8—42no—
221C22×C8C2×D411yescharacteristic
222C22×C8C2×D411yescharacteristic
223C22×C8C2×D411yescharacteristic
224C22×C8C2×D411yescharacteristic
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8C22×C412yes—
232C4⋊Q8C22×C412yes—
233#127—21no—
234#127—21no—
235#127—21no—
236#127—21no—
237#127—21no—
238#127—21no—
239#127C22×C412yes—
240#127C22×C412yes—
241#124—24no—
242#124—24no—
243#124—42no—
244#124—42no—
245#121—24no—
246#121—24no—
247#121—24no—
248#121—24no—
249#121—42no—
250#121—42no—
251#121—42no—
252#121—42no—
253C4×D4—24no—
254C4×D4—24no—
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—42no—
258C4×D4—42no—
259C4×D4—42no—
260C4×D4—42no—
261#117—42no—
262#117—42no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42C22×C412yes—
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D412yes—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D412yes—
283C2×C42#4111yescharacteristic
284C2×C42#4112yes—
285C2×C42C2411yescharacteristic
286#106—24no—
287#106—24no—
288C4⋊C8—24no—
289C4⋊C8—21no—
290C4⋊C8—21no—
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—42no—
300C4⋊C8C2×D412yes—
301C4⋊C8C2×D412yes—
302#103—82no—
303#103—82no—
304#102—24no—
305#102—24no—
306#102—24no—
307#102—24no—
308#102—82no—
309#102—82no—
310#102—82no—
311#102—82no—
312C22⋊C8—42no—
313#97—42no—
314#97—42no—
315#97—42no—
316#97C22⋊C414yes—
317#97C22⋊C414yes—
318C4×C8—42no—
319C4×C8—42no—
320C4×C8C22×C412yes—
321C4×C8C22×C412yes—
322C4×C8C2×D412yes—
323C4×C8C2×D412yes—
324#95—22no—
325#95—21no—
326#95—21no—
327C24—41no—
328C24—41no—
329C24#12111yescharacteristic
330#41—88no—
331#41—84no—
332#41—84no—
333#41—88no—
334#41—88no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—82no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—44no—
349C2×D4—44no—
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—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—82no—
369C2×D4—82no—
370C2×D4—84no—
371C2×D4—84no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4C2×C22⋊C412yes—
388C22×C4C4×D412yes—
389C22×C4C4×D412yes—
390C22×C4#12112yes—
391C22×C4#12111yescharacteristic
392C22×C4#12111yescharacteristic
393C22×C4#12111yescharacteristic
394C22×C4#12412yes—
395C22×C4#12712yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C2×#4111yescharacteristic
399D8—48no—
400D8—48no—
401M16—28no—
402M16—44no—
403M16—84no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—48no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8C2×C22⋊C412yes—
426C2×C8C2×C22⋊C412yes—
427C2×C8C2×C22⋊C412yes—
428C2×C8C2×C22⋊C412yes—
429C2×C8AGL(1,ℤ/8)12yes—
430C2×C8#13712yes—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—44no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—82no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42C2×C22⋊C412yes—
455C42C4×D412yes—
456C42C4×D412yes—
457C42#12112yes—
458C42#12412yes—
459C42#12712yes—
460C42C23×C411yescharacteristic
461C42C22×D411yescharacteristic
462C42C22×D411yescharacteristic
463C42C2×#4111yescharacteristic
464C23—21no—
465C23—21no—
466C23—22no—
467C23—22no—
468C23—42no—
469C23—42no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23#39012yes—
479C23C2×#12111yescharacteristic
480Q8—44no—
481Q8—82no—
482Q8—82no—
483Q8—81no—
484Q8—81no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—44no—
490D4—44no—
491D4—82no—
492D4—82no—
493D4—82no—
494D4—82no—
495D4—88no—
496D4—88no—
497D4—81no—
498D4—81no—
499D4—81no—
500D4—81no—
501D4—81no—
502D4—81no—
503D4—81no—
504D4—81no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—88no—
528C2×C4—88no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4#39012yes—
532C2×C4#39012yes—
533C2×C4#39012yes—
534C2×C4C22×C22⋊C411yescharacteristic
535C2×C4C2×C4×D411yescharacteristic
536C2×C4C2×C4×D411yescharacteristic
537C2×C4C2×#12111yescharacteristic
538C2×C4C2×#12411yescharacteristic
539C2×C4C2×#12711yescharacteristic
540C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
541C2×C4C2×#13711yescharacteristic
542C2×C4#57511yescharacteristic
543C2×C4#57511yescharacteristic
544C2×C4#57612yes—
545C2×C4#57712yes—
546C8—22no—
547C8—22no—
548C8—22no—
549C8—42no—
550C8—42no—
551C8—81no—
552C8—81no—
553C8—81no—
554C8—81no—
555C8#41714yes—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—162no—
568C22—162no—
569C22—162no—
570C22—162no—
571C22C2×#39011yescharacteristic
572C22#303611yescharacteristic
573C22#303711yescharacteristic
574C22#306212yes—
575C22#306812yes—
576C4—22no—
577C4—21no—
578C4—21no—
579C4—22no—
580C4—22no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4C2×#41711yescharacteristic
586C4#289611yescharacteristic
587C4#294612yes—
588C4#303611yescharacteristic
589C4#303711yescharacteristic
590C4#315411yescharacteristic
591C4#315511yescharacteristic
592C2—42no—
593C2—162no—
594C2—162no—
595C2#1937012yes—
596C2#1937412yes—
597C2#2123511yescharacteristic
598C2#2124911yescharacteristic
599C2#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. #21893C2semidirect
3. #21809C2semidirect
4. #20441C2semidirect
5. C2×#1569C2semidirect
6. C2×#1551C2semidirect
7. C2×#1547C2semidirect
8. #9747C2semidirect
9. #9746C2semidirect
10. #9743C2semidirect
11. #9741C2semidirect
12. #9738C2semidirect
13. #9735C2semidirect
14. #9734C2semidirect
15. #9729C2semidirect
19. #3184C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31842
#218932
#218092
#204412
C2×#15692
C2×#15512
C2×#15472
#97472
#97462
#97432
#97412
#97382
#97352
#97342
#97292