Finite Group Databasebeta

← #406284

Subgroups — #406284

Order 512.

1696 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic
2C2×#1892C211yesmaximal, characteristic
3C2×#1872C211yesmaximal, characteristic
4C2×#1824C211yesmaximal, characteristic
5C22×#340C211yesmaximal, characteristic
6#13907C211yesmaximal, characteristic
7#11267C212yesmaximal, complemented
8#11207C212yesmaximal, complemented
9#11190C212yesmaximal, complemented
10#11067C212yesmaximal, complemented
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2502—24no—
21#2502—24no—
22C2×#398—22no—
23C2×#391C2212yes—
24C2×#389C2212yes—
25C2×#385—21no—
26C2×#385—21no—
27C2×#384C2212yes—
28C2×#380—21no—
29C2×#380C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1892—24no—
32#1872—24no—
33#1824—24no—
34C2×#340—24no—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1441—22no—
45C4⋊#95C2212yes—
46C22⋊#95—21no—
47C23×Q8C2311yescharacteristic
48C24×C4—21no—
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8—22no—
52C23×C8—24no—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—42no—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2312yes—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8D414yes—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—42no—
79C22×C4⋊C4D414yes—
80C22×C4⋊C4D414yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4D411yescharacteristic
89C22×C22⋊C4D411yescharacteristic
90C22×C42C2312yes—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—24no—
96C2×#103—42no—
97C2×#103—42no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101#398—216no—
102#391—24no—
103#391—24no—
104#389—24no—
105#389—42no—
106#385—28no—
107#385—28no—
108#384—24no—
109#384—24no—
110#380—22no—
111#380—22no—
112#380—44no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95D412yes—
122#340—24no—
123#340—24no—
124#340—22no—
125#340—22no—
126#340—24no—
127#340—24no—
128#340—24no—
129#336—24no—
130#336—42no—
131C25C2×D411yescharacteristic
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8C2×D412yes—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—44no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—82no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4#4112yes—
169C23×C4#4112yes—
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C22×C8—28no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—48no—
190C22×C8—48no—
191C22×C8—44no—
192C22×C8—44no—
193C22×C8#4114yes—
194C22⋊Q8—28no—
195C22⋊Q8—28no—
196C22⋊Q8—44no—
197C22⋊Q8—44no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—48no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4—82no—
241C2×C4⋊C4—82no—
242C2×C4⋊C4—82no—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4#4114yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4SD1614yes—
259C2×C22⋊C4Q1614yes—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266#103—44no—
267#103—48no—
268#103—44no—
269C22⋊C8—44no—
270#95—28no—
271#95—28no—
272#95—24no—
273#95—44no—
274#95—42no—
275#95—44no—
276#95—82no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24#12012yes—
282C24#12112yes—
283C24#12412yes—
284C24C2×SD1612yes—
285C24C2×Q1612yes—
286C24#13712yes—
287C24#13712yes—
288C24C22×D411yescharacteristic
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—42no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—48no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—84no—
373C22×C4—88no—
374C22×C4—88no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—84no—
378C22×C4—84no—
379C22×C4#12012yes—
380C22×C4#12114yes—
381C22×C4#12114yes—
382C22×C4#12112yes—
383C22×C4#12314yes—
384C22×C4#12314yes—
385C22×C4#12412yes—
386C22×C4C2×SD1614yes—
387C22×C4C2×Q1614yes—
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4112yes—
395C22×C4C2×#4112yes—
396C22×C4H(5,𝔽2)12yes—
397C22×C42−1+412yes—
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—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—42no—
415C2×C8—42no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—48no—
425C4⋊C4—42no—
426C4⋊C4—41no—
427C4⋊C4—42no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—88no—
437C4⋊C4—88no—
438C4⋊C4—88no—
439C4⋊C4#12314yes—
440C22⋊C4—28no—
441C22⋊C4—28no—
442C22⋊C4—48no—
443C22⋊C4—84no—
444C42—42no—
445C42—42no—
446C42—42no—
447C42—42no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23#39414yes—
465C23#44814yes—
466C23#45114yes—
467C23#46114yes—
468C23#46214yes—
469C23#48714yes—
470C23#49114yes—
471C23C2×#12011yescharacteristic
472C23C2×#12111yescharacteristic
473C23C2×#12411yescharacteristic
474C23#54612yes—
475C23#54712yes—
476C23C22×SD1611yescharacteristic
477C23C22×Q1611yescharacteristic
478C23C2×#13711yescharacteristic
479C23C2×#13711yescharacteristic
480C23#57612yes—
481C23#57712yes—
482Q8—44no—
483Q8—42no—
484Q8—42no—
485Q8—81no—
486Q8—81no—
487Q8—82no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—24no—
496C2×C4—28no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—816no—
528C2×C4—816no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—81no—
537C2×C4—82no—
538C2×C4—81no—
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#39414yes—
549C2×C4C2×#12111yescharacteristic
550C2×C4C2×#12111yescharacteristic
551C2×C4C2×#12311yescharacteristic
552C2×C4C2×#12311yescharacteristic
553C2×C4C2×#12311yescharacteristic
554C2×C4#52511yescharacteristic
555C2×C4#52912yes—
556C2×C4#53212yes—
557C2×C4#53411yescharacteristic
558C2×C4#53611yescharacteristic
559C2×C4#53712yes—
560C2×C4#53912yes—
561C2×C4#54612yes—
562C2×C4#54712yes—
563C2×C4#55112yes—
564C2×C4#55112yes—
565C2×C4#55212yes—
566C2×C4#55212yes—
567C8—44no—
568C8—81no—
569C8—81no—
570C22—24no—
571C22—24no—
572C22—28no—
573C22—28no—
574C22—24no—
575C22—28no—
576C22—28no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22#204318yes—
581C22C2×#39411yescharacteristic
582C22#241211yescharacteristic
583C22#241311yescharacteristic
584C22#243012yes—
585C22#248812yes—
586C22C2×#44811yescharacteristic
587C22C2×#45111yescharacteristic
588C22#301212yes—
589C22C2×#46111yescharacteristic
590C22C2×#46211yescharacteristic
591C22#304111yescharacteristic
592C22#304311yescharacteristic
593C22C2×#48711yescharacteristic
594C22C2×#49111yescharacteristic
595C22#313612yes—
596C22#329512yes—
597C22#329912yes—
598C22#331412yes—
599C22#331612yes—
600C22#333512yes—
601C22#333812yes—
602C22#334412yes—
603C22#334612yes—
604C4—24no—
605C4—44no—
606C4—44no—
607C4—44no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4—84no—
612C4—162no—
613C4#241912yes—
614C4#248611yescharacteristic
615C4#249812yes—
616C4#253411yescharacteristic
617C4#261811yescharacteristic
618C4#285411yescharacteristic
619C2—28no—
620C2#1492711yescharacteristic
621C2C2×#204311yescharacteristic
622C2#2351111yescharacteristic
623C2#2453011yescharacteristic
624C2#2453211yescharacteristic
625C2#2458011yescharacteristic
626C2#2458211yescharacteristic
627C2#2554312yes—
628C2#2554512yes—
629C2#2769712yes—
630C2#2771512yes—
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
7. #11267C2semidirect
8. #11207C2semidirect
9. #11190C2semidirect
10. #11067C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
C2×#18922
C2×#18722
C2×#18242
C22×#3402
#139072
#112672
#112072
#111902
#110672
#107142