Finite Group Databasebeta

← #406283

Subgroups — #406283

Order 512.

1652 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic
2C2×#1892C211yesmaximal, characteristic
3C2×#1872C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5C22×#340C211yesmaximal, characteristic
6#13907C211yesmaximal, characteristic
7#11268C212yesmaximal, complemented
8#11208C212yesmaximal, 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#2503—24no—
21#2503—24no—
22C2×#400—22no—
23C2×#391C2212yes—
24C2×#389C2212yes—
25C2×#382C2212yes—
26C2×#381—21no—
27C2×#381—21no—
28C2×#380—21no—
29C2×#380C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1892—24no—
32#1872—24no—
33#1823—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—42no—
77C22×C4⋊C4D414yes—
78C22×C4⋊C4D414yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4D411yescharacteristic
87C22×C22⋊C4D411yescharacteristic
88C22×C42C2312yes—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—24no—
94C2×#103—42no—
95C2×#103—42no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99#400—216no—
100#391—24no—
101#391—24no—
102#389—24no—
103#389—42no—
104#382—24no—
105#382—24no—
106#381—28no—
107#381—28no—
108#380—22no—
109#380—22no—
110#380—44no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C2×#95—22no—
114C2×#95—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—24no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—44no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—82no—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4#4112yes—
171C23×C4#4112yes—
172C23×C4#4112yes—
173C23×C4C2411yescharacteristic
174C22×C8—28no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—48no—
192C22×C8—48no—
193C22×C8—44no—
194C22×C8—44no—
195C22×C8#4114yes—
196C22⋊Q8—28no—
197C22⋊Q8—28no—
198C22⋊Q8—44no—
199C22⋊Q8—44no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
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⋊C4C2×D412yes—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4#4114yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262#103—44no—
263#103—48no—
264#103—44no—
265C22⋊C8—44no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—24no—
271#95—44no—
272#95—44no—
273#95—42no—
274#95—82no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24#12012yes—
280C24#12112yes—
281C24#12412yes—
282C24#13512yes—
283C24#13512yes—
284C24AGL(1,ℤ/8)12yes—
285C24#13712yes—
286C24C22×D411yescharacteristic
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—42no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
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—22no—
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—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—84no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—84no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4#12012yes—
382C22×C4#12114yes—
383C22×C4#12114yes—
384C22×C4#12112yes—
385C22×C4#12314yes—
386C22×C4#12314yes—
387C22×C4#12412yes—
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#12314yes—
438C22⋊C4—48no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—84no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—42no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—22no—
458C23—22no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23#39414yes—
463C23#44914yes—
464C23#45214yes—
465C23#46314yes—
466C23#46414yes—
467C23#48814yes—
468C23#48914yes—
469C23C2×#12011yescharacteristic
470C23C2×#12111yescharacteristic
471C23C2×#12411yescharacteristic
472C23#54612yes—
473C23#54712yes—
474C23C2×#13511yescharacteristic
475C23C2×#13511yescharacteristic
476C23C2×AGL(1,ℤ/8)11yescharacteristic
477C23C2×#13711yescharacteristic
478C23#57712yes—
479C23#57812yes—
480Q8—44no—
481Q8—42no—
482Q8—42no—
483Q8—81no—
484Q8—81no—
485Q8—82no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—816no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—82no—
538C2×C4—816no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4#39414yes—
547C2×C4C2×#12111yescharacteristic
548C2×C4C2×#12111yescharacteristic
549C2×C4C2×#12311yescharacteristic
550C2×C4C2×#12311yescharacteristic
551C2×C4C2×#12311yescharacteristic
552C2×C4#52511yescharacteristic
553C2×C4#52912yes—
554C2×C4#53212yes—
555C2×C4#53411yescharacteristic
556C2×C4#53611yescharacteristic
557C2×C4#53712yes—
558C2×C4#53912yes—
559C2×C4#54612yes—
560C2×C4#54712yes—
561C2×C4#55112yes—
562C2×C4#55112yes—
563C2×C4#55212yes—
564C2×C4#55212yes—
565C8—44no—
566C8—81no—
567C8—81no—
568C22—24no—
569C22—24no—
570C22—28no—
571C22—28no—
572C22—24no—
573C22—28no—
574C22—28no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22#204218yes—
579C22C2×#39411yescharacteristic
580C22#241211yescharacteristic
581C22#241311yescharacteristic
582C22#243012yes—
583C22#248812yes—
584C22C2×#44911yescharacteristic
585C22C2×#45211yescharacteristic
586C22#301312yes—
587C22C2×#46311yescharacteristic
588C22C2×#46411yescharacteristic
589C22#304211yescharacteristic
590C22#304411yescharacteristic
591C22C2×#48811yescharacteristic
592C22C2×#48911yescharacteristic
593C22#313712yes—
594C22#329612yes—
595C22#330012yes—
596C22#331012yes—
597C22#331212yes—
598C22#333612yes—
599C22#333712yes—
600C22#334312yes—
601C22#334412yes—
602C4—24no—
603C4—44no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4—84no—
608C4—82no—
609C4—82no—
610C4—162no—
611C4#241912yes—
612C4#248611yescharacteristic
613C4#249812yes—
614C4#253411yescharacteristic
615C4#261811yescharacteristic
616C4#285411yescharacteristic
617C2—28no—
618C2#1492711yescharacteristic
619C2C2×#204211yescharacteristic
620C2#2351211yescharacteristic
621C2#2452711yescharacteristic
622C2#2452911yescharacteristic
623C2#2457711yescharacteristic
624C2#2457911yescharacteristic
625C2#2553912yes—
626C2#2554112yes—
627C2#2769912yes—
628C2#2771312yes—
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. #11268C2semidirect
8. #11208C2semidirect
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×#25032
C2×#18922
C2×#18722
C2×#18232
C22×#3402
#139072
#112682
#112082
#111902
#110672
#107142