Finite Group Databasebeta

← #401712

Subgroups — #401712

Order 512.

1722 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#1802C211yesmaximal, characteristic
4#15152C211yesmaximal, characteristic, complemented
5#11292C212yesmaximal, complemented
6#11267C212yesmaximal, complemented
7#11227C211yesmaximal, characteristic, complemented
8#11202C211yesmaximal, characteristic, complemented
9#10788C211yesmaximal, characteristic
10#10788C212yesmaximal
11#10788C212yesmaximal
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C23×C4⋊C4—21no—
16C22×C4⋊C8C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2212yes—
19C2×#400—21no—
20D4×C4⋊C4—24no—
21D4×C4⋊C4—24no—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#385—21no—
25C2×#385—21no—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#380—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
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—28no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—44no—
75C22×C4⋊C4—44no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×C4⋊C8—22no—
90C2×C4⋊C8—42no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97#400—44no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389D414yes—
102#385—28no—
103#385—28no—
104#384—22no—
105#384—22no—
106#384—22no—
107#384—22no—
108#380—28no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95D411yescharacteristic
120C2×#95D411yescharacteristic
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—42no—
126#340—42no—
127#336—22no—
128#336—24no—
129#336—42no—
130C25—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4C2×D412yes—
137C22×D4C2×Q812yes—
138C22×D4#4112yes—
139C22×D4#4114yes—
140C22×D4#4114yes—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—82no—
162C23×C4—82no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
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—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C4×D4—28no—
185C4×D4—28no—
186C4×D4—28no—
187C4×D4—28no—
188C4×D4—28no—
189C4×D4—44no—
190C4×D4—48no—
191C4×D4—48no—
192C4×D4—44no—
193C4×D4—44no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
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—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—432no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4—82no—
237C2×C4⋊C4D814yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×Q812yes—
242C2×C4⋊C4#4112yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×Q812yes—
263C2×C42#4112yes—
264C4⋊C8—44no—
265#102—48no—
266#102—48no—
267#95—24no—
268#95—24no—
269#95—22no—
270#95—22no—
271#95—42no—
272#95—42no—
273#95—44no—
274#95—44no—
275#95—82no—
276#95—82no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—24no—
281C24—24no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—42no—
287C24—42no—
288C24C22×D411yescharacteristic
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4C22⋊Q814yes—
303C2×D4#12314yes—
304C2×D4#12414yes—
305C2×D4#12518yes—
306C2×D4#12618yes—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—216no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—48no—
337C22×C4—48no—
338C22×C4—48no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—48no—
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—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—84no—
377C22×C4—88no—
378C22×C4#12012yes—
379C22×C4#12012yes—
380C22×C4C22⋊Q812yes—
381C22×C4C22⋊Q812yes—
382C22×C4C2×D812yes—
383C22×C4C2×D812yes—
384C22×C4C2×D812yes—
385C22×C4AGL(1,ℤ/8)12yes—
386C22×C4#13712yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×Q811yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4112yes—
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4112yes—
395C22×C4H(5,𝔽2)12yes—
396C22×C42−1+412yes—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—28no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—44no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—42no—
424C4⋊C4—48no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C4⋊C4C22⋊Q814yes—
437C22⋊C4—216no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—24no—
447C42—41no—
448C42—41no—
449C42—42no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—28no—
454C23—28no—
455C23—44no—
456C23—42no—
457C23—44no—
458C23—42no—
459C23—44no—
460C23—42no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23C2×#12011yescharacteristic
474C23C2×C22⋊Q811yescharacteristic
475C23#52511yescharacteristic
476C23#54612yes—
477C23#54812yes—
478C23C22×D811yescharacteristic
479C23C2×AGL(1,ℤ/8)11yescharacteristic
480C23C2×#13711yescharacteristic
481C23#57511yescharacteristic
482C23#57712yes—
483C23#57712yes—
484D4—24no—
485D4—24no—
486D4—28no—
487D4—28no—
488D4—24no—
489D4#40018yescomplemented
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—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—432no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—816no—
535C2×C4—816no—
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—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4#44714yes—
555C2×C4#47414yes—
556C2×C4C2×C22⋊Q811yescharacteristic
557C2×C4C2×C22⋊Q811yescharacteristic
558C2×C4C2×#12311yescharacteristic
559C2×C4#52511yescharacteristic
560C2×C4#52511yescharacteristic
561C2×C4C2×#12411yescharacteristic
562C2×C4C2×#12512yes—
563C2×C4C2×#12612yes—
564C2×C4#53411yescharacteristic
565C2×C4#53611yescharacteristic
566C2×C4#53912yes—
567C2×C4#54412yes—
568C2×C4#54612yes—
569C2×C4#54812yes—
570C2×C4#55212yes—
571C2×C4#55212yes—
572C2×C4#55312yes—
573C2×C4#55312yes—
574C8—82no—
575C8—82no—
576C22—28no—
577C22—48no—
578C22—44no—
579C22—44no—
580C22—48no—
581C22—48no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—48no—
587C22#240511yescharacteristic
588C22#243611yescharacteristic
589C22#244212yes—
590C22#248112yes—
591C22#284811yescharacteristic
592C22C2×#44711yescharacteristic
593C22#300711yescharacteristic
594C22#300911yescharacteristic
595C22#301011yescharacteristic
596C22C2×#47411yescharacteristic
597C22#307711yescharacteristic
598C22#308412yes—
599C22#310011yescharacteristic
600C22#311011yescharacteristic
601C22#312512yes—
602C22#329412yes—
603C22#330712yes—
604C22#331012yes—
605C22#333112yes—
606C22#335812yes—
607C22#337212yes—
608C22#337312yes—
609C22#337612yes—
610C4—24no—
611C4—24no—
612C4—44no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—82no—
617C4—82no—
618C4—81no—
619C4—162no—
620C4—162no—
621C4C2×#40011yescharacteristic
622C4#244212yes—
623C4#248112yes—
624C4#248411yescharacteristic
625C4#259111yescharacteristic
626C4#262711yescharacteristic
627C2—48no—
628C2#1482211yescharacteristic
629C2#2403211yescharacteristic
630C2#2405511yescharacteristic
631C2#2593311yescharacteristic
632C2#2594511yescharacteristic
633C2#2610612yes—
634C2#2611812yes—
635C2#2755012yes—
636C2#2757312yes—
637C2#3295311yescharacteristic
638C2#3296511yescharacteristic
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. #11292C2semidirect
6. #11267C2semidirect
7. #11227C2semidirect
8. #11202C2semidirect
489. D4#400semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
C2×#20992
C2×#18022
#151522
#112922
#112672
#112272
#112022
#107882