Finite Group Databasebeta

← #416152

Subgroups — #416152

Order 512.

1752 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#1956C211yesmaximal, characteristic, complemented
3C2×#1956C211yesmaximal, characteristic, complemented
4C2×#1834C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊C8C211yesmaximal, characteristic
6#14686C211yesmaximal, characteristic
7#11190C214yesmaximal, complemented
8#10970C214yesmaximal
9#10841C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C23×C42C2211yescharacteristic
13C22×#103C2214yes—
14C22×#103C2214yes—
15C22×C22⋊C8C2212yes—
16C22×C4×C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2857—24no—
19#2857—24no—
20C2×#391C2212yes—
21C2×#389—21no—
22C2×#389—21no—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#380—21no—
27C2×#380—21no—
28C2×#380—21no—
29C2×#380—21no—
30C2×#380C2212yes—
31#1956—24no—
32#1956—24no—
33#1834—24no—
34C4×C22⋊C8—24no—
35C2×#340C2214yes—
36C2×#340C2214yes—
37C2×#336C2212yes—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C23×Q8C2312yes—
42C24×C4D411yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8D414yes—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—28no—
56C2×C22⋊Q8—44no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C42—22no—
72C22×C42—22no—
73C22×C42—22no—
74C22×C42—22no—
75C22×C42—22no—
76C22×C42—22no—
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#103—28no—
82C2×#103—28no—
83C2×#103—28no—
84C2×#103—44no—
85C2×#103—44no—
86C2×#103—44no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8D414yes—
96#391—24no—
97#391—24no—
98#389—28no—
99#389—28no—
100#384—22no—
101#384—22no—
102#384—22no—
103#384—22no—
104#384—22no—
105#384D414yes—
106#380—28no—
107#380—28no—
108#380—44no—
109#380—44no—
110#380—44no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116#340—28no—
117#340—44no—
118#336—24no—
119C25C2×D411yescharacteristic
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—24no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—24no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8C2×D414yes—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4SD1614yes—
156C23×C4Q1614yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4#4114yes—
163C23×C4C2411yescharacteristic
164C22×C8—22no—
165C22×C8—22no—
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×C8C2×D414yes—
177C22×C8#4114yes—
178C4⋊Q8—28no—
179C4⋊Q8—28no—
180C4⋊Q8—44no—
181C4⋊Q8—44no—
182C22⋊Q8—416no—
183C22⋊Q8—416no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C2×D414yes—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251#103—416no—
252#103—416no—
253C22⋊C8—216no—
254C4×C8—28no—
255C4×C8—44no—
256#95—28no—
257#95—28no—
258#95—44no—
259#95—44no—
260#95—44no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24#12412yes—
274C24C2×SD1612yes—
275C24C2×Q1612yes—
276C24C22×D411yescharacteristic
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—28no—
280C2×Q8—28no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—44no—
284C2×Q8—44no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—44no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—28no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
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—42no—
357C22×C4—42no—
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#12012yes—
370C22×C4#12012yes—
371C22×C4#12112yes—
372C22×C4#12112yes—
373C22×C4#12312yes—
374C22×C4#12312yes—
375C22×C4#12412yes—
376C22×C4C2×SD1612yes—
377C22×C4C2×SD1612yes—
378C22×C4C2×Q1612yes—
379C22×C4C2×Q1612yes—
380C22×C4#13512yes—
381C22×C4#13512yes—
382C22×C4AGL(1,ℤ/8)12yes—
383C22×C4#13712yes—
384C22×C4#13712yes—
385C22×C4#13712yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4112yes—
393C22×C4C2×#4112yes—
394C22×C4H(5,𝔽2)12yes—
395C22×C4H(5,𝔽2)12yes—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
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—28no—
411C4⋊C4—416no—
412C4⋊C4—44no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C22⋊C4—416no—
426C22⋊C4—88no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—44no—
432C42—44no—
433C42#12114yes—
434C42#12114yes—
435C42#12314yes—
436C42#12414yes—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23#48714yes—
472C23C2×#12011yescharacteristic
473C23C2×#12111yescharacteristic
474C23C2×#12311yescharacteristic
475C23C2×#12411yescharacteristic
476C23#53211yescharacteristic
477C23#53511yescharacteristic
478C23#53511yescharacteristic
479C23C22×SD1611yescharacteristic
480C23C22×Q1611yescharacteristic
481C23C2×#13511yescharacteristic
482C23C2×#13511yescharacteristic
483C23C2×AGL(1,ℤ/8)11yescharacteristic
484C23C2×#13711yescharacteristic
485C23C2×#13711yescharacteristic
486C23C2×#13711yescharacteristic
487Q8—48no—
488Q8—48no—
489Q8—84no—
490Q8—84no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—416no—
517C2×C4—416no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
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—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#39414yes—
547C2×C4#39414yes—
548C2×C4#44814yes—
549C2×C4#44914yes—
550C2×C4#45114yes—
551C2×C4#45214yes—
552C2×C4#46514yes—
553C2×C4#46714yes—
554C2×C4#46814yes—
555C2×C4#47014yes—
556C2×C4#48314yes—
557C2×C4#48414yes—
558C2×C4#48514yes—
559C2×C4#48514yes—
560C2×C4C2×#12111yescharacteristic
561C2×C4C2×#12111yescharacteristic
562C2×C4C2×#12311yescharacteristic
563C2×C4#52512yes—
564C2×C4C2×#12411yescharacteristic
565C2×C4#52912yes—
566C2×C4#53512yes—
567C2×C4#54012yes—
568C2×C4#54612yes—
569C2×C4#54612yes—
570C2×C4#54612yes—
571C2×C4#55012yes—
572C2×C4#55312yes—
573C2×C4#55312yes—
574C2×C4#55312yes—
575C2×C4#55512yes—
576C8—44no—
577C8—44no—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22C2×#39411yescharacteristic
594C22#243111yescharacteristic
595C22#243511yescharacteristic
596C22#250011yescharacteristic
597C22#250411yescharacteristic
598C22#261911yescharacteristic
599C22#285011yescharacteristic
600C22C2×#44811yescharacteristic
601C22C2×#45111yescharacteristic
602C22C2×#44911yescharacteristic
603C22C2×#45211yescharacteristic
604C22C2×#46511yescharacteristic
605C22C2×#46711yescharacteristic
606C22C2×#46811yescharacteristic
607C22C2×#47011yescharacteristic
608C22C2×#48311yescharacteristic
609C22C2×#48511yescharacteristic
610C22C2×#48511yescharacteristic
611C22C2×#48411yescharacteristic
612C22C2×#48711yescharacteristic
613C22#313111yescharacteristic
614C22#313911yescharacteristic
615C22#314111yescharacteristic
616C22#316711yescharacteristic
617C22#317011yescharacteristic
618C22#318211yescharacteristic
619C22#318511yescharacteristic
620C22#320611yescharacteristic
621C22#320711yescharacteristic
622C22#320811yescharacteristic
623C22#320911yescharacteristic
624C22#320911yescharacteristic
625C22#321011yescharacteristic
626C22#321011yescharacteristic
627C22#321011yescharacteristic
628C4—42no—
629C4—42no—
630C4—42no—
631C4—84no—
632C4—84no—
633C4—162no—
634C4—162no—
635C4#203518yes—
636C4#203618yes—
637C4#241412yes—
638C4#242012yes—
639C4#249012yes—
640C4#253512yes—
641C2—44no—
642C2#1521911yescharacteristic
643C2C2×#203511yescharacteristic
644C2C2×#203611yescharacteristic
645C2#2559811yescharacteristic
646C2#2560011yescharacteristic
647C2#2585611yescharacteristic
648C2#2585711yescharacteristic
649C2#2788911yescharacteristic
650C2#2789211yescharacteristic
651C2#2797211yescharacteristic
652C2#2797511yescharacteristic
653C2#2980711yescharacteristic
654C2#2980811yescharacteristic
655C2#3304311yescharacteristic
656C2#3304911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1956C2semidirect
3. C2×#1956C2semidirect
4. C2×#1834C2semidirect
7. #11190C2semidirect
9. #10841C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#19562
C2×#18342
C2×C4×C22⋊C82
#146862
#111902
#109702
#108412