Finite Group Databasebeta

← #405523

Subgroups — #405523

Order 512.

1760 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C211yesmaximal, characteristic
2C2×#1853C211yesmaximal, characteristic
3C22×#340C211yesmaximal, characteristic
4#14130C211yesmaximal, characteristic
5#11384C212yesmaximal, complemented
6#11327C212yesmaximal, complemented
7#11220C212yesmaximal, complemented
8#11207C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10721C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8—21no—
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#106C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19#2440—24no—
20#2440—24no—
21C2×#398—21no—
22C2×#395—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2212yes—
26C2×#385—21no—
27C2×#385—21no—
28C2×#384C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#1853—24no—
33C2×#340—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1444—21no—
44#1442—21no—
45C4⋊#95C2212yes—
46C22⋊#95—21no—
47C23×Q8—21no—
48C24×C4—21no—
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—42no—
54C23×C8—42no—
55C23×C8Q814yescomplemented
56C23×C8C2312yes—
57C23×C8C2312yes—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—42no—
62C2×C22⋊Q8—28no—
63C2×C22⋊Q8—44no—
64C2×C22⋊Q8—44no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—42no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4D414yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4D411yescharacteristic
90C22×C22⋊C4D411yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C2×#106—24no—
94C2×#106—24no—
95C2×#106—42no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8Q814yescomplemented
99#398—28no—
100#395—28no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—24no—
106#389—42no—
107#385—28no—
108#385—44no—
109#384—22no—
110#384—22no—
111#380—22no—
112#380—22no—
113#380D414yes—
114#380Q814yescomplemented
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95D411yescharacteristic
125C2×#95D411yescharacteristic
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—42no—
131#340—42no—
132#340—42no—
133#336—42no—
134#336—42no—
135C25C2×D411yescharacteristic
136C22×Q8—24no—
137C22×Q8—44no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—82no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—82no—
160C23×C4—82no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×Q812yes—
169C23×C4C2×Q812yes—
170C23×C4C2×Q812yes—
171C23×C4#4112yes—
172C23×C4C2411yescharacteristic
173C22×C8—28no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—48no—
187C22×C8—44no—
188C22×C8—44no—
189C22×C8—48no—
190C22×C8C2×Q814yes—
191C22×C8C2×Q814yes—
192C4⋊Q8—48no—
193C4⋊Q8—44no—
194C4⋊Q8—44no—
195C22⋊Q8—432no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—48no—
227C2×C4⋊C4—48no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—82no—
242C2×C4⋊C4—82no—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×Q814yes—
248C2×C4⋊C4#4114yes—
249C2×C4⋊C4#4114yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4SD1614yes—
261C2×C22⋊C4SD1614yes—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270#106—48no—
271C22⋊C8—28no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—22no—
276#95—22no—
277#95—24no—
278#95—44no—
279#95—44no—
280#95—82no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24#12012yes—
286C24C22⋊Q812yes—
287C24C4⋊Q812yes—
288C24C2×SD1612yes—
289C24C2×SD1612yes—
290C24AGL(1,ℤ/8)12yes—
291C24#13712yes—
292C24C22×D411yescharacteristic
293C2×Q8—48no—
294C2×Q8—48no—
295C2×Q8—88no—
296C2×Q8—84no—
297C2×Q8—88no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—48no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—48no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—48no—
367C22×C4—84no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4—88no—
371C22×C4—88no—
372C22×C4—84no—
373C22×C4—84no—
374C22×C4#12012yes—
375C22×C4C22⋊Q814yes—
376C22×C4C22⋊Q814yes—
377C22×C4C22⋊Q812yes—
378C22×C4C4⋊Q814yes—
379C22×C4C4⋊Q814yes—
380C22×C4C4⋊Q812yes—
381C22×C4C2×SD1612yes—
382C22×C4C2×SD1612yes—
383C22×C4C2×SD1612yes—
384C22×C4C2×SD1612yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×Q811yescharacteristic
388C22×C4C22×Q811yescharacteristic
389C22×C4C22×Q811yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4112yes—
392C22×C4C2×#4112yes—
393C22×C4H(5,𝔽2)12yes—
394C22×C4H(5,𝔽2)12yes—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—416no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—42no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—88no—
436C4⋊C4—84no—
437C4⋊C4#12314yes—
438C22⋊C4—28no—
439C22⋊C4—28no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—42no—
447C42—42no—
448C23—24no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23#39514yes—
465C23#44814yes—
466C23#45014yes—
467C23#47514yes—
468C23#47614yes—
469C23#49814yes—
470C23#50114yes—
471C23C2×#12011yescharacteristic
472C23C2×C22⋊Q811yescharacteristic
473C23C2×C4⋊Q811yescharacteristic
474C23#54612yes—
475C23#54612yes—
476C23C22×SD1611yescharacteristic
477C23C22×SD1611yescharacteristic
478C23C2×AGL(1,ℤ/8)11yescharacteristic
479C23C2×#13711yescharacteristic
480C23#57612yes—
481C23#57812yes—
482Q8—816no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—816no—
534C2×C4—816no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4#39514yes—
550C2×C4C2×C22⋊Q811yescharacteristic
551C2×C4C2×C22⋊Q811yescharacteristic
552C2×C4C2×#12311yescharacteristic
553C2×C4#52511yescharacteristic
554C2×C4#52912yes—
555C2×C4C2×C4⋊Q811yescharacteristic
556C2×C4C2×C4⋊Q811yescharacteristic
557C2×C4#53312yes—
558C2×C4#53411yescharacteristic
559C2×C4#53511yescharacteristic
560C2×C4#53812yes—
561C2×C4#54312yes—
562C2×C4#54612yes—
563C2×C4#54612yes—
564C2×C4#55412yes—
565C2×C4#55412yes—
566C2×C4#55412yes—
567C2×C4#55412yes—
568C8—44no—
569C8—44no—
570C22—24no—
571C22—28no—
572C22—28no—
573C22—24no—
574C22—28no—
575C22—28no—
576C22—24no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22#205818yes—
581C22C2×#39511yescharacteristic
582C22#240411yescharacteristic
583C22#240511yescharacteristic
584C22#245012yes—
585C22#254912yes—
586C22C2×#45011yescharacteristic
587C22C2×#44811yescharacteristic
588C22#300811yescharacteristic
589C22#301111yescharacteristic
590C22C2×#47611yescharacteristic
591C22C2×#47511yescharacteristic
592C22#308312yes—
593C22C2×#49811yescharacteristic
594C22C2×#50111yescharacteristic
595C22#316412yes—
596C22#329512yes—
597C22#329512yes—
598C22#329812yes—
599C22#329812yes—
600C22#330812yes—
601C22#331212yes—
602C22#331412yes—
603C22#331812yes—
604C4—24no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—44no—
609C4—42no—
610C4—84no—
611C4—82no—
612C4—84no—
613C4—82no—
614C4—82no—
615C4#245012yes—
616C4#250211yescharacteristic
617C4#254912yes—
618C4#258311yescharacteristic
619C4#262611yescharacteristic
620C4#285511yescharacteristic
621C2—28no—
622C2#1491411yescharacteristic
623C2C2×#205811yescharacteristic
624C2#2355511yescharacteristic
625C2#2396111yescharacteristic
626C2#2396911yescharacteristic
627C2#2403311yescharacteristic
628C2#2404111yescharacteristic
629C2#2641012yes—
630C2#2641212yes—
631C2#2877112yes—
632C2#2877512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11384C2semidirect
6. #11327C2semidirect
7. #11220C2semidirect
8. #11207C2semidirect
55. C23×C8Q8semidirect
98. C2×C22⋊C8Q8semidirect
114. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
C2×#18532
C22×#3402
#141302
#113842
#113272
#112202
#112072
#107302
#107212
#107142