Finite Group Databasebeta

← #202833

Subgroups — #202833

Order 512.

894 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21410C211yesmaximal, characteristic
2#21365C211yesmaximal, characteristic, complemented
3#21258C211yesmaximal, characteristic
4C4⋊C4×M16C211yesmaximal, characteristic, complemented
5C2×#1557C211yesmaximal, characteristic
6#9574C211yesmaximal, characteristic, complemented
7#9573C211yesmaximal, characteristic, complemented
8#9572C211yesmaximal, characteristic, complemented
9#9571C211yesmaximal, characteristic, complemented
10#9566C211yesmaximal, characteristic, complemented
11#9565C211yesmaximal, characteristic, complemented
12#9564C211yesmaximal, characteristic, complemented
13#9563C211yesmaximal, characteristic, complemented
14#8041C211yesmaximal, characteristic, complemented
15#7876C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×#417—21no—
21C2×#417—21no—
22C2×C4×M16—21no—
23C2×C4×M16C2211yescharacteristic
24#2285C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1983—21no—
29#1983—21no—
30#1982C2211yescharacteristic
31#1982C2211yescharacteristic
32#1981C2211yescharacteristic
33#1981C2211yescharacteristic
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1948C2211yescharacteristic
37#1948C2211yescharacteristic
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1895—21no—
41#1895—21no—
42#1853—21no—
43#1852—21no—
44#1776—21no—
45#1776—21no—
46#1771—21no—
47#1771—21no—
48#1770C2211yescharacteristic
49#1770C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1762—21no—
53#1762—21no—
54#1728—21no—
55#1557C418yescomplemented
56C2×#332C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#331C2211yescharacteristic
59C2×#331C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×#41—21no—
69C22×M16—21no—
70C22×M16—21no—
71C22×M16D411yescharacteristic
72C22×M16D411yescharacteristic
73#532—22no—
74#532—22no—
75#532—22no—
76#532C2×C414yes—
77C2×C4⋊Q8C2311yescharacteristic
78C2×#127C2311yescharacteristic
79C2×#124—21no—
80C2×#121—21no—
81C2×#121—21no—
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×#117—21no—
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4D411yescharacteristic
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#432—22no—
90#432C2×C414yes—
91C2×#107—21no—
92C2×#107—21no—
93C2×#106—21no—
94C2×#106—21no—
95#423—22no—
96#423—22no—
97#423—22no—
98#423C2×C414yes—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107#417—44no—
108#417—44no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×#103—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×#102—21no—
117C2×#102—21no—
118C2×#102—21no—
119C2×#102—21no—
120C2×#102—21no—
121C2×C22⋊C8—21no—
122C4×M16—22no—
123C4×M16—22no—
124C4×M16—44no—
125C2×#97—21no—
126C2×#97—21no—
127C2×#97—21no—
128C2×#97D411yescharacteristic
129C2×#97D411yescharacteristic
130C2×C4×C8—21no—
131C2×C4×C8—21no—
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134#384C2311yescharacteristic
135#384C2311yescharacteristic
136#382—21no—
137#380—21no—
138#380—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4D411yescharacteristic
144C4×C4⋊C4D411yescharacteristic
145C4×C4⋊C4C2311yescharacteristic
146C4×C4⋊C4C2311yescharacteristic
147C2×#95—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340—21no—
155#340—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#332C2×C414yescomplemented
161#332C2×C414yescomplemented
162#331C2×C414yescomplemented
163#331C2×C414yescomplemented
164C2×#41—24no—
165C2×#41—42no—
166C2×#41—42no—
167C2×#41—42no—
168C2×#41—42no—
169C2×#41—42no—
170C22×Q8—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—41no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—41no—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4#4111yescharacteristic
181C2×M16—22no—
182C2×M16—24no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—41no—
187C2×M16—41no—
188C2×M16—42no—
189C2×M16—42no—
190C2×M16—42no—
191C2×M16—42no—
192C2×M16—42no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8C22×C412yes—
207#127—21no—
208#127—21no—
209#127—21no—
210#127C22×C412yes—
211#124—24no—
212#124—42no—
213#121—24no—
214#121—24no—
215#121—42no—
216#121—42no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—42no—
220C4×D4—42no—
221#117—82no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—41no—
250C2×C42C22×C412yes—
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42#4111yescharacteristic
254C2×C42#4112yes—
255C2×C42#4112yes—
256C2×C42#4112yes—
257C2×C42C2411yescharacteristic
258#107—42no—
259#107—42no—
260#106—42no—
261#106—42no—
262C4⋊C8—24no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—42no—
272C4⋊C8C22×C412yes—
273C4⋊C8C22×C412yes—
274C4⋊C8C22×C412yes—
275C4⋊C8C22×C412yes—
276#103—42no—
277#103—42no—
278#103—42no—
279#103—42no—
280#102—42no—
281#102—42no—
282#102—42no—
283#102—42no—
284#102—42no—
285#102—42no—
286#102—42no—
287#102—42no—
288C22⋊C8—24no—
289#97—24no—
290#97—22no—
291#97—22no—
292#97—42no—
293#97—42no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—42no—
299C4×C8—42no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—41no—
305#95—41no—
306C24—41no—
307C24#12111yescharacteristic
308#41—48no—
309#41—48no—
310#41—84no—
311#41—84no—
312C2×Q8—22no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—44no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—81no—
334C2×D4—81no—
335C2×D4—82no—
336C2×D4—82no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—44no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—82no—
360C22×C4—82no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4#11712yes—
364C22×C4C4×D412yes—
365C22×C4C4×D412yes—
366C22×C4#12111yescharacteristic
367C22×C4#12111yescharacteristic
368C22×C4#12111yescharacteristic
369C22×C4#12312yes—
370C22×C4#12412yes—
371C22×C4#12612yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375M16—28no—
376M16—44no—
377M16—48no—
378M16—84no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8AGL(1,ℤ/8)12yes—
408C2×C8#13712yes—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4AGL(1,ℤ/8)12yes—
432C4⋊C4AGL(1,ℤ/8)12yes—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—82no—
436C42—42no—
437C42—42no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—82no—
443C42#11712yes—
444C42C4×D412yes—
445C42C4×D412yes—
446C42#12312yes—
447C42#12412yes—
448C42#12612yes—
449C42C23×C411yescharacteristic
450C42C2×#4111yescharacteristic
451C42C2×#4111yescharacteristic
452C42C2×#4111yescharacteristic
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—42no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23—81no—
466C23#38812yes—
467C23C2×#12111yescharacteristic
468Q8—42no—
469Q8—42no—
470Q8—81no—
471Q8—81no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—42no—
476D4—82no—
477D4—82no—
478D4—88no—
479D4—81no—
480D4—81no—
481D4—81no—
482D4—81no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—88no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4#38812yes—
526C2×C4#38812yes—
527C2×C4#38812yes—
528C2×C4C2×#11711yescharacteristic
529C2×C4C2×C4×D411yescharacteristic
530C2×C4C2×C4×D411yescharacteristic
531C2×C4C2×#12311yescharacteristic
532C2×C4C2×#12411yescharacteristic
533C2×C4C2×#12611yescharacteristic
534C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
535C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
536C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
537C2×C4C2×#13711yescharacteristic
538C2×C4#57511yescharacteristic
539C2×C4#57511yescharacteristic
540C2×C4#57511yescharacteristic
541C2×C4#57511yescharacteristic
542C8—22no—
543C8—22no—
544C8—41no—
545C8—41no—
546C8—42no—
547C8—42no—
548C8—42no—
549C8—42no—
550C8—81no—
551C8—81no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—161no—
563C22—161no—
564C22—161no—
565C22—161no—
566C22C2×#38811yescharacteristic
567C22#303611yescharacteristic
568C22#303711yescharacteristic
569C22#305511yescharacteristic
570C22#305511yescharacteristic
571C22#306811yescharacteristic
572C22#306911yescharacteristic
573C4—22no—
574C4—21no—
575C4—21no—
576C4—22no—
577C4—22no—
578C4—42no—
579C4—42no—
580C4—82no—
581C4—81no—
582C4—81no—
583C4—161no—
584C4—161no—
585C4C4×AGL(1,ℤ/8)11yescharacteristic
586C4C4×AGL(1,ℤ/8)11yescharacteristic
587C4C4×AGL(1,ℤ/8)11yescharacteristic
588C4C4×#13711yescharacteristic
589C4#309211yescharacteristic
590C4#309311yescharacteristic
591C4#313811yescharacteristic
592C4#314011yescharacteristic
593C2—42no—
594C2—162no—
595C2#1930411yescharacteristic
596C2#1930611yescharacteristic
597C2#1930811yescharacteristic
598C2#1931011yescharacteristic
599C2#2105911yescharacteristic
600C2#2107611yescharacteristic
601C2#2109511yescharacteristic
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. #21365C2semidirect
4. C4⋊C4×M16C2semidirect
6. #9574C2semidirect
7. #9573C2semidirect
8. #9572C2semidirect
9. #9571C2semidirect
10. #9566C2semidirect
11. #9565C2semidirect
12. #9564C2semidirect
13. #9563C2semidirect
14. #8041C2semidirect
15. #7876C2semidirect
55. #1557C4semidirect
160. #332C2×C4semidirect
161. #332C2×C4semidirect
162. #331C2×C4semidirect
163. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
#214102
#213652
#212582
C4⋊C4×M162
C2×#15572
#95742
#95732
#95722
#95712
#95662
#95652
#95642
#95632
#80412
#78762