Finite Group Databasebeta

← #205321

Subgroups — #205321

Order 512.

1070 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3182C211yesmaximal, characteristic
2#21894C211yesmaximal, characteristic, complemented
3#21809C211yesmaximal, characteristic
4#20441C211yesmaximal, characteristic, complemented
5C2×#1569C211yesmaximal, characteristic, complemented
6C2×#1552C211yesmaximal, characteristic, complemented
7C2×#1547C211yesmaximal, 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
30C22×C4⋊C8C2211yescharacteristic
31C2×#405—21no—
32C2×C4×M16C2211yescharacteristic
33#2285C2211yescharacteristic
34#1988C2211yescharacteristic
35#1988C2211yescharacteristic
36#1982C2211yescharacteristic
37#1982C2211yescharacteristic
38#1981C2211yescharacteristic
39#1981C2211yescharacteristic
40#1981C2211yescharacteristic
41#1981C2211yescharacteristic
42#1959—22no—
43#1894—22no—
44#1803—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#1552—24no—
59#1547—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×M16D411yescharacteristic
73C22×M16D411yescharacteristic
74C23×C8—21no—
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—
98C2×#117—21no—
99C22×C42C2311yescharacteristic
100#500—22no—
101#500C2×C414yes—
102#492—22no—
103#492C2×C414yes—
104#487—22no—
105#487C2×C414yes—
106#486—22no—
107#486C2×C414yes—
108#468—28no—
109#468—28no—
110#440—28no—
111#440—28no—
112#432—22no—
113#432—22no—
114C2×#107—21no—
115C2×#107—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—21no—
129C2×#103—21no—
130C2×#103—22no—
131C2×#103—22no—
132C2×#103—22no—
133C2×#103—22no—
134C2×#102—21no—
135C2×#102—21no—
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#332—22no—
169#332—22no—
170#332—22no—
171#332—22no—
172#332—22no—
173#332—22no—
174#331—22no—
175#331—22no—
176C2×#41—24no—
177C2×#41—42no—
178C2×#41—82no—
179C2×#41—82no—
180C22×Q8—21no—
181C22×Q8—21no—
182C22×Q8—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—41no—
187C23×C4—41no—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4#4111yescharacteristic
191C2×SD16—24no—
192C2×SD16—24no—
193C2×SD16—42no—
194C2×SD16—42no—
195C2×SD16—42no—
196C2×SD16—42no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—41no—
200C2×M16—41no—
201C2×M16—42no—
202C2×M16—82no—
203C2×M16C22⋊C414yes—
204C2×M16C22⋊C414yes—
205C22×C8—24no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—42no—
213C22×C8—42no—
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8C22×C412yes—
228C4⋊Q8C22×C412yes—
229C4⋊Q8C22×C412yes—
230#127—21no—
231#127—21no—
232#127—21no—
233#127C22×C412yes—
234#125—24no—
235#125—42no—
236#124—24no—
237#124—42no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240C22⋊Q8—42no—
241C22⋊Q8—42no—
242#121—24no—
243#121—24no—
244#121—42no—
245#121—42no—
246C4×Q8—24no—
247C4×Q8—24no—
248C4×Q8—42no—
249C4×Q8—42no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—42no—
253C4×D4—42no—
254#117—24no—
255#117—42no—
256#117—42no—
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—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—41no—
277C2×C42C22×C412yes—
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D412yes—
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D412yes—
282C2×C42#4111yescharacteristic
283C2×C42#4112yes—
284C2×C42C2411yescharacteristic
285#107—24no—
286#107—24no—
287C4⋊C8—24no—
288C4⋊C8—21no—
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—42no—
299C4⋊C8C2×D412yes—
300C4⋊C8C2×D412yes—
301#103—24no—
302#103—24no—
303#103—82no—
304#103—82no—
305#103—82no—
306#103—82no—
307#102—24no—
308#102—24no—
309#102—82no—
310#102—82no—
311C22⋊C8—42no—
312#97—42no—
313#97—42no—
314#97—42no—
315#97C22⋊C414yes—
316#97C22⋊C414yes—
317C4×C8—42no—
318C4×C8—42no—
319C4×C8C22×C412yes—
320C4×C8C22×C412yes—
321C4×C8C2×D412yes—
322C4×C8C2×D412yes—
323#95—22no—
324#95—21no—
325#95—21no—
326C24—41no—
327C24#12111yescharacteristic
328#41—88no—
329#41—84no—
330#41—84no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—44no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—82no—
359C2×D4—84no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—44no—
373C22×C4—84no—
374C22×C4C2×C22⋊C412yes—
375C22×C4C4×D412yes—
376C22×C4C4×D412yes—
377C22×C4#12112yes—
378C22×C4#12111yescharacteristic
379C22×C4#12111yescharacteristic
380C22×C4#12111yescharacteristic
381C22×C4#12412yes—
382C22×C4#12712yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C2×#4111yescharacteristic
386SD16—48no—
387SD16—48no—
388M16—28no—
389M16—44no—
390M16—84no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—48no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8C2×C22⋊C412yes—
413C2×C8C2×C22⋊C412yes—
414C2×C8C2×C22⋊C412yes—
415C2×C8C2×C22⋊C412yes—
416C2×C8AGL(1,ℤ/8)12yes—
417C2×C8#13712yes—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—44no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—82no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—44no—
447C42C2×C22⋊C412yes—
448C42C4×D412yes—
449C42C4×D412yes—
450C42#12112yes—
451C42#12412yes—
452C42#12712yes—
453C42C23×C411yescharacteristic
454C42C22×D411yescharacteristic
455C42C22×D411yescharacteristic
456C42C2×#4111yescharacteristic
457C23—21no—
458C23—21no—
459C23—22no—
460C23—22no—
461C23—42no—
462C23—81no—
463C23—81no—
464C23—82no—
465C23—82no—
466C23#39012yes—
467C23C2×#12111yescharacteristic
468Q8—42no—
469Q8—42no—
470Q8—44no—
471Q8—44no—
472Q8—81no—
473Q8—81no—
474Q8—81no—
475Q8—81no—
476Q8—81no—
477Q8—81no—
478D4—42no—
479D4—42no—
480D4—44no—
481D4—82no—
482D4—82no—
483D4—88no—
484D4—81no—
485D4—81no—
486D4—81no—
487D4—81no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—88no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4#39012yes—
517C2×C4#39012yes—
518C2×C4#39012yes—
519C2×C4C22×C22⋊C411yescharacteristic
520C2×C4C2×C4×D411yescharacteristic
521C2×C4C2×C4×D411yescharacteristic
522C2×C4C2×#12111yescharacteristic
523C2×C4C2×#12411yescharacteristic
524C2×C4C2×#12711yescharacteristic
525C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
526C2×C4C2×#13711yescharacteristic
527C2×C4#57511yescharacteristic
528C2×C4#57511yescharacteristic
529C2×C4#57612yes—
530C2×C4#57712yes—
531C8—22no—
532C8—22no—
533C8—22no—
534C8—42no—
535C8—42no—
536C8—81no—
537C8—81no—
538C8—81no—
539C8—81no—
540C8#41714yes—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—42no—
545C22—42no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—162no—
550C22—162no—
551C22C2×#39011yescharacteristic
552C22#303611yescharacteristic
553C22#303711yescharacteristic
554C22#306212yes—
555C22#306912yes—
556C4—21no—
557C4—21no—
558C4—22no—
559C4—22no—
560C4—22no—
561C4—82no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4—82no—
566C4C2×#41711yescharacteristic
567C4#289611yescharacteristic
568C4#294612yes—
569C4#303611yescharacteristic
570C4#303711yescharacteristic
571C4#315411yescharacteristic
572C4#315511yescharacteristic
573C2—42no—
574C2—162no—
575C2#1936812yes—
576C2#1937212yes—
577C2#2123511yescharacteristic
578C2#2125011yescharacteristic
579C2#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. #21894C2semidirect
4. #20441C2semidirect
5. C2×#1569C2semidirect
6. C2×#1552C2semidirect
8. #9748C2semidirect
9. #9746C2semidirect
10. #9744C2semidirect
11. #9740C2semidirect
12. #9738C2semidirect
13. #9736C2semidirect
14. #9732C2semidirect
15. #9731C2semidirect
19. #3182C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31822
#218942
#218092
#204412
C2×#15692
C2×#15522
C2×#15472
#97482
#97462
#97442
#97402
#97382
#97362
#97322
#97312