Finite Group Databasebeta

← #401714

Subgroups — #401714

Order 512.

1592 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2349C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#1802C211yesmaximal, characteristic
4#15152C211yesmaximal, characteristic, complemented
5#11293C212yesmaximal, complemented
6#11268C212yesmaximal, complemented
7#11228C211yesmaximal, characteristic, complemented
8#11202C211yesmaximal, characteristic, complemented
9#10788C211yesmaximal, characteristic
10#10788C212yesmaximal
11#10788C212yesmaximal
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2212yes—
18C2×#400—21no—
19#2349—24no—
20#2349—24no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#385—21no—
24C2×#385—21no—
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#380—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C22×#95—21no—
30#2099—24no—
31#1802—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1445—21no—
42#1442—21no—
43#1441—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C24×C4—21no—
50C24×C4—21no—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—24no—
60C2×C4×D4—24no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—42no—
64C2×C4×D4—42no—
65C2×C4×D4D414yes—
66C2×C4×D4Q814yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×C4⋊C8—22no—
85C2×C4⋊C8—42no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92#400—44no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389D414yes—
97#385—28no—
98#385—28no—
99#384—22no—
100#384—22no—
101#382—22no—
102#382—22no—
103#380—28no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C2×#95—28no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D411yescharacteristic
118C2×#95D411yescharacteristic
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—42no—
124#340—42no—
125#336—22no—
126#336—24no—
127#336—42no—
128C25—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4C2×D412yes—
135C22×D4C2×Q812yes—
136C22×D4#4112yes—
137C22×D4#4114yes—
138C22×D4#4114yes—
139C23×C4—21no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—82no—
156C23×C4—82no—
157C23×C4—82no—
158C23×C4—82no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2411yescharacteristic
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C4×D4—28no—
181C4×D4—28no—
182C4×D4—28no—
183C4×D4—28no—
184C4×D4—28no—
185C4×D4—44no—
186C4×D4—48no—
187C4×D4—48no—
188C4×D4—44no—
189C4×D4—44no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
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—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×Q812yes—
230C2×C4⋊C4#4112yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×Q812yes—
251C2×C42#4112yes—
252C4⋊C8—44no—
253#102—48no—
254#102—48no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—22no—
259#95—24no—
260#95—22no—
261#95—42no—
262#95—44no—
263#95—44no—
264#95—42no—
265#95—82no—
266#95—82no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—24no—
271C24—24no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—42no—
277C24—42no—
278C24C22×D411yescharacteristic
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4C22⋊Q814yes—
293C2×D4#12314yes—
294C2×D4#12414yes—
295C2×D4#12518yes—
296C2×D4#12618yes—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—216no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—48no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
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—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—84no—
357C22×C4—88no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4—84no—
365C22×C4—88no—
366C22×C4#12012yes—
367C22×C4#12012yes—
368C22×C4C22⋊Q812yes—
369C22×C4C22⋊Q812yes—
370C22×C4#13512yes—
371C22×C4AGL(1,ℤ/8)12yes—
372C22×C4#13712yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×Q811yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4112yes—
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4112yes—
381C22×C4H(5,𝔽2)12yes—
382C22×C42−1+412yes—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—44no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—84no—
417C4⋊C4—84no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4C22⋊Q814yes—
421C22⋊C4—216no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—24no—
431C42—41no—
432C42—41no—
433C42—42no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—28no—
438C23—28no—
439C23—42no—
440C23—42no—
441C23—44no—
442C23—44no—
443C23—42no—
444C23—44no—
445C23—42no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23C2×#12011yescharacteristic
458C23C2×C22⋊Q811yescharacteristic
459C23#52511yescharacteristic
460C23#54612yes—
461C23#54812yes—
462C23C2×#13511yescharacteristic
463C23C2×AGL(1,ℤ/8)11yescharacteristic
464C23C2×#13711yescharacteristic
465C23#57511yescharacteristic
466C23#57612yes—
467C23#57712yes—
468D4—24no—
469D4—24no—
470D4—28no—
471D4—28no—
472D4—24no—
473D4#40018yescomplemented
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—44no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—816no—
517C2×C4—816no—
518C2×C4—816no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4#44914yes—
538C2×C4C2×C22⋊Q811yescharacteristic
539C2×C4C2×C22⋊Q811yescharacteristic
540C2×C4C2×#12311yescharacteristic
541C2×C4#52511yescharacteristic
542C2×C4#52511yescharacteristic
543C2×C4C2×#12411yescharacteristic
544C2×C4C2×#12512yes—
545C2×C4C2×#12612yes—
546C2×C4#53411yescharacteristic
547C2×C4#53611yescharacteristic
548C2×C4#53912yes—
549C2×C4#54412yes—
550C2×C4#54612yes—
551C2×C4#54812yes—
552C2×C4#55212yes—
553C2×C4#55212yes—
554C2×C4#55312yes—
555C2×C4#55312yes—
556C8—82no—
557C8—82no—
558C22—28no—
559C22—48no—
560C22—48no—
561C22—44no—
562C22—44no—
563C22—48no—
564C22—48no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22#240511yescharacteristic
570C22#243611yescharacteristic
571C22#244212yes—
572C22#248112yes—
573C22#284811yescharacteristic
574C22C2×#44911yescharacteristic
575C22#300711yescharacteristic
576C22#300811yescharacteristic
577C22#300911yescharacteristic
578C22#307611yescharacteristic
579C22#307911yescharacteristic
580C22#308112yes—
581C22#310711yescharacteristic
582C22#311511yescharacteristic
583C22#311812yes—
584C22#329612yes—
585C22#330612yes—
586C22#331412yes—
587C22#332212yes—
588C22#336112yes—
589C22#336612yes—
590C22#337212yes—
591C22#337612yes—
592C4—24no—
593C4—42no—
594C4—42no—
595C4—44no—
596C4—42no—
597C4—42no—
598C4—81no—
599C4—82no—
600C4—82no—
601C4—162no—
602C4—162no—
603C4C2×#40011yescharacteristic
604C4#244212yes—
605C4#248112yes—
606C4#248411yescharacteristic
607C4#259111yescharacteristic
608C4#262711yescharacteristic
609C2—48no—
610C2#1482211yescharacteristic
611C2#2403811yescharacteristic
612C2#2404911yescharacteristic
613C2#2593011yescharacteristic
614C2#2594511yescharacteristic
615C2#2611512yes—
616C2#2612412yes—
617C2#2756512yes—
618C2#2757412yes—
619C2#3295511yescharacteristic
620C2#3296211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2099C2semidirect
4. #15152C2semidirect
5. #11293C2semidirect
6. #11268C2semidirect
7. #11228C2semidirect
8. #11202C2semidirect
473. D4#400semidirect

Maximal subgroups

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

subgroupindex
C2×#23492
C2×#20992
C2×#18022
#151522
#112932
#112682
#112282
#112022
#107882