Finite Group Databasebeta

← #190478

Subgroups — #190478

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2273C211yesmaximal, characteristic
2#21906C212yesmaximal, complemented
3#8700C212yesmaximal, complemented
4#8559C212yesmaximal, complemented
5#8557C212yesmaximal, complemented
6#8298C212yesmaximal
7#8152C212yesmaximal, complemented
8#8152C212yesmaximal, complemented
9C23×C22⋊C4C411yescharacteristic
10C23×C22⋊C4C411yescharacteristic
11C23×C22⋊C4C2211yescharacteristic
12#2273—28no—
13C2×C4×C22⋊C4C411yescharacteristic
14C2×C4×C22⋊C4C411yescharacteristic
15C2×C4×C22⋊C4C411yescharacteristic
16C2×C4×C22⋊C4C411yescharacteristic
17C2×C4×C22⋊C4C411yescharacteristic
18C2×C4×C22⋊C4C411yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20C2×C4×C22⋊C4C2211yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25#1924—28no—
26#1797C2214yescomplemented
27#1779C2214yes—
28#1749C2214yescomplemented
29C2×#323C2214yes—
30C2×#323C2214yes—
31#1280C2214yescomplemented
32#1280C2214yescomplemented
33C26C2×C411yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4C2×C411yescharacteristic
38C24×C4C2×C411yescharacteristic
39C24×C4C2311yescharacteristic
40C22×C22⋊C4—24no—
41C22×C22⋊C4—24no—
42C22×C22⋊C4—24no—
43C22×C22⋊C4—24no—
44C22×C22⋊C4—24no—
45C22×C22⋊C4—24no—
46C22×C22⋊C4—24no—
47C22×C22⋊C4—24no—
48C22×C22⋊C4—24no—
49C22×C22⋊C4C2×C411yescharacteristic
50C22×C22⋊C4C2×C411yescharacteristic
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2311yescharacteristic
60C22×C22⋊C4C2311yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—24no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8C2318yes—
73C2×#97—24no—
74C4×C22⋊C4—24no—
75C4×C22⋊C4—24no—
76C4×C22⋊C4—24no—
77C4×C22⋊C4—24no—
78C4×C22⋊C4—24no—
79C4×C22⋊C4—24no—
80C4×C22⋊C4—24no—
81C4×C22⋊C4—24no—
82C4×C22⋊C4—24no—
83C4×C22⋊C4—24no—
84C4×C22⋊C4—24no—
85C4×C22⋊C4D418yes—
86C2×#95C2×C411yescharacteristic
87C2×#95C2×C411yescharacteristic
88C2×#95C2×C411yescharacteristic
89C2×#95C2×C411yescharacteristic
90C2×#95C2×C411yescharacteristic
91C2×#95C2×C411yescharacteristic
92C2×#95C2311yescharacteristic
93C2×#95C2311yescharacteristic
94#336—24no—
95#336—24no—
96#323—28no—
97#323—28no—
98C25—22no—
99C25—22no—
100C25—22no—
101C25—22no—
102C25—22no—
103C25—22no—
104C25—81no—
105C25—81no—
106C25—81no—
107C25—81no—
108C25M1614yes—
109C25C22×C411yescharacteristic
110C25C22×C411yescharacteristic
111C25C22×C411yescharacteristic
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—81no—
129C23×C4—81no—
130C23×C4—81no—
131C23×C4—81no—
132C23×C4—81no—
133C23×C4—81no—
134C23×C4—81no—
135C23×C4—81no—
136C23×C4—81no—
137C23×C4—81no—
138C23×C4—81no—
139C23×C4—81no—
140C23×C4M1614yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D414yes—
153C23×C4C2411yescharacteristic
154C22×C8—28no—
155C2×C22⋊C4—232no—
156C2×C22⋊C4—232no—
157C2×C22⋊C4—232no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4C22⋊C414yes—
189C2×C22⋊C4C22⋊C414yes—
190C2×C22⋊C4M1614yes—
191C2×C22⋊C4M1614yes—
192C2×C22⋊C4#4114yes—
193C2×C22⋊C4#4114yes—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—81no—
201C2×C42—81no—
202C2×C42—81no—
203C2×C42—81no—
204C2×C42—81no—
205C2×C42—81no—
206C2×C42—81no—
207C2×C42—81no—
208C2×C42C2×D414yes—
209C2×C42C2×D414yes—
210C4⋊C8—84no—
211C22⋊C8—28no—
212C22⋊C8—28no—
213C22⋊C8—84no—
214C22⋊C8—84no—
215#97—84no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95C22⋊C414yes—
223#95C22⋊C414yes—
224C24—28no—
225C24—28no—
226C24—28no—
227C24—28no—
228C24—21no—
229C24—21no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—28no—
241C24—28no—
242C24—28no—
243C24—41no—
244C24—41no—
245C24—41no—
246C24—41no—
247C24—41no—
248C24—41no—
249C24—41no—
250C24—41no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24—41no—
255C24—82no—
256C24—82no—
257C24—82no—
258C24—82no—
259C24—84no—
260C24—84no—
261C24—84no—
262C24—84no—
263C24—84no—
264C24—84no—
265C24—84no—
266C24—84no—
267C24—84no—
268C24—84no—
269C24—84no—
270C24—84no—
271C24C2×C22⋊C412yes—
272C24#11712yes—
273C24#11712yes—
274C24C2×M1612yes—
275C24C2×M1612yes—
276C24C2×M1612yes—
277C24C2×M1612yes—
278C24C2×M1612yes—
279C24C2×M1612yes—
280C24C23×C411yescharacteristic
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—28no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—84no—
344C22×C4—84no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4—84no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C412yes—
360C22×C4#11712yes—
361C22×C4#11712yes—
362C22×C4C2×M1612yes—
363C22×C4C2×M1612yes—
364C22×C4C2×M1612yes—
365C22×C4C2×M1612yes—
366C22×C4C2×M1612yes—
367C22×C4C2×M1612yes—
368C22×C4C22×D412yes—
369C22×C4C2×#4112yes—
370C22×C4C2×#4112yes—
371C2×C8—48no—
372C2×C8—48no—
373C2×C8—48no—
374C22⋊C4—28no—
375C22⋊C4—28no—
376C22⋊C4—28no—
377C22⋊C4—28no—
378C22⋊C4—28no—
379C22⋊C4—28no—
380C22⋊C4—28no—
381C22⋊C4—28no—
382C22⋊C4—28no—
383C22⋊C4—28no—
384C22⋊C4—28no—
385C22⋊C4—28no—
386C42—44no—
387C42—44no—
388C42—82no—
389C42—82no—
390C42—82no—
391C42—82no—
392C42—82no—
393C42—82no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—232no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—88no—
466C23—88no—
467C23—816no—
468C23—816no—
469C23—816no—
470C23—816no—
471C23—88no—
472C23—88no—
473C23—88no—
474C23—88no—
475C23#40714yes—
476C23#43214yes—
477C23#43214yes—
478C23C22×C22⋊C411yescharacteristic
479C23C2×#11711yescharacteristic
480C23C2×#11711yescharacteristic
481C23C22×M1611yescharacteristic
482C23C22×M1611yescharacteristic
483C23C22×M1611yescharacteristic
484C23C22×M1611yescharacteristic
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
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—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
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—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—88no—
544C2×C4—88no—
545C2×C4—88no—
546C2×C4—88no—
547C2×C4—88no—
548C2×C4—88no—
549C2×C4#40714yes—
550C2×C4#52514yes—
551C8—88no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—832no—
598C22#228011yescharacteristic
599C22C2×#40711yescharacteristic
600C22C2×#40711yescharacteristic
601C22#288311yescharacteristic
602C22#288311yescharacteristic
603C22#288811yescharacteristic
604C22#288811yescharacteristic
605C22C2×#43211yescharacteristic
606C22C2×#43211yescharacteristic
607C22#292011yescharacteristic
608C22#292011yescharacteristic
609C4—24no—
610C4—24no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—42no—
621C2—21no—
622C2—21no—
623C2—21no—
624C2—21no—
625C2—44no—
626C2—44no—
627C2—44no—
628C2#1736711yescharacteristic
629C2#1736711yescharacteristic
630C2#1736711yescharacteristic
631C2#1736911yescharacteristic
632C2#2071511yescharacteristic
633C2#2072411yescharacteristic
634C2#2072911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21906C2semidirect
3. #8700C2semidirect
4. #8559C2semidirect
5. #8557C2semidirect
7. #8152C2semidirect
8. #8152C2semidirect
26. #1797C22semidirect
28. #1749C22semidirect
31. #1280C22semidirect
32. #1280C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22732
#219062
#87002
#85592
#85572
#82982
#81522