Finite Group Databasebeta

← #397978

Subgroups — #397978

Order 512.

1836 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic, complemented
3C2×#1807C212yesmaximal
4C22×#336C211yesmaximal, characteristic
5#14697C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic, complemented
7#11327C212yesmaximal, complemented
8#11323C211yesmaximal, characteristic, complemented
9#11220C212yesmaximal, complemented
10#11207C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal
12C24×C8C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8—21no—
15C23×C42C2211yescharacteristic
16C22×#106C2212yes—
17C22×#106C2212yes—
18#2857—28no—
19C2×#391—21no—
20C2×#389—21no—
21C2×#389C2212yes—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#385—21no—
25C2×#384C2211yescharacteristic
26C2×#380—21no—
27C2×#380C2211yescharacteristic
28C2×#380C2212yes—
29C2×#380C2211yescharacteristic
30#2058—24no—
31#1807—28no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336—24no—
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1436—21no—
43#1436—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C22⋊#95—21no—
47C23×Q8—21no—
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8—24no—
52C23×C8—24no—
53C23×C8—24no—
54C23×C8—24no—
55C23×C8—42no—
56C23×C8—42no—
57C23×C8Q818yescomplemented
58C23×C8C2312yes—
59C23×C8C2312yes—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—42no—
64C2×C22⋊Q8—28no—
65C2×C22⋊Q8—44no—
66C2×C22⋊Q8—44no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—22no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C42—24no—
83C22×C42—22no—
84C22×C42—22no—
85C22×C42—42no—
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—24no—
94C2×#106—42no—
95C2×#106—42no—
96#391—28no—
97#389—28no—
98#389—24no—
99#389—24no—
100#389—22no—
101#389—22no—
102#389—42no—
103#385—44no—
104#384—24no—
105#380—28no—
106#380—24no—
107#380—22no—
108#380—24no—
109#380D418yes—
110#380Q814yescomplemented
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—42no—
126#340—42no—
127#336—24no—
128#336—22no—
129#336D414yes—
130C25C2×D411yescharacteristic
131C22×Q8—24no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—42no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—41no—
159C23×C4—41no—
160C23×C4—41no—
161C23×C4—41no—
162C23×C4SD1614yes—
163C23×C4SD1614yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D414yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×Q814yes—
171C23×C4C2×Q812yes—
172C23×C4#4112yes—
173C23×C4C2411yescharacteristic
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—216no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—216no—
186C22×C8—48no—
187C22×C8—48no—
188C22×C8—48no—
189C22×C8—48no—
190C22×C8—48no—
191C22×C8C2×D414yes—
192C22×C8C2×Q814yes—
193C22×C8C2×Q814yes—
194C22×C8#4114yes—
195C4⋊Q8—48no—
196C4⋊Q8—48no—
197C22⋊Q8—432no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4C2×Q812yes—
245C2×C4⋊C4C2×Q812yes—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—21no—
264C2×C42—22no—
265C2×C42—48no—
266C2×C42—48no—
267C2×C42—41no—
268C2×C42—41no—
269C2×C42C2×D412yes—
270C2×C42#4112yes—
271#106—48no—
272#106—48no—
273#95—28no—
274#95—28no—
275#95—24no—
276#95—42no—
277#95—42no—
278#95—42no—
279#95—42no—
280#95—42no—
281#95—42no—
282#95—42no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24#12012yes—
292C24C22⋊Q812yes—
293C24C4⋊Q812yes—
294C24C2×SD1612yes—
295C24C2×SD1612yes—
296C24AGL(1,ℤ/8)12yes—
297C24#13712yes—
298C24C22×D411yescharacteristic
299C2×Q8—48no—
300C2×Q8—48no—
301C2×Q8—48no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×Q8—84no—
305C2×Q8—84no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4#12012yes—
401C22×C4#12112yes—
402C22×C4#12112yes—
403C22×C4C22⋊Q818yes—
404C22×C4C22⋊Q812yes—
405C22×C4C4⋊Q818yes—
406C22×C4C4⋊Q812yes—
407C22×C4C2×SD1612yes—
408C22×C4C2×SD1612yes—
409C22×C4C2×SD1612yes—
410C22×C4C2×SD1612yes—
411C22×C4#13512yes—
412C22×C4#13512yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×Q812yes—
416C22×C4C22×Q811yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4112yes—
419C22×C4C2×#4111yescharacteristic
420C22×C4H(5,𝔽2)11yescharacteristic
421C22×C4H(5,𝔽2)12yes—
422C22×C42−1+411yescharacteristic
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—432no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8C22⋊Q818yescomplemented
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—416no—
448C4⋊C4—48no—
449C4⋊C4—42no—
450C4⋊C4—41no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—41no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C22⋊C4—416no—
463C22⋊C4—48no—
464C22⋊C4—84no—
465C42—24no—
466C42—24no—
467C42—44no—
468C42—44no—
469C42—42no—
470C42—42no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23#39514yes—
500C23#44814yes—
501C23#45014yes—
502C23#47514yes—
503C23#47614yes—
504C23#49814yes—
505C23#50114yes—
506C23C2×#12011yescharacteristic
507C23C2×#12111yescharacteristic
508C23C2×C22⋊Q811yescharacteristic
509C23C2×C4⋊Q811yescharacteristic
510C23#53211yescharacteristic
511C23#53411yescharacteristic
512C23#53711yescharacteristic
513C23C22×SD1611yescharacteristic
514C23C22×SD1611yescharacteristic
515C23C2×#13511yescharacteristic
516C23C2×#13511yescharacteristic
517C23C2×AGL(1,ℤ/8)11yescharacteristic
518C23C2×#13711yescharacteristic
519C23#57511yescharacteristic
520C23#57511yescharacteristic
521Q8—816no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—416no—
551C2×C4—416no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—82no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—81no—
579C2×C4—82no—
580C2×C4—81no—
581C2×C4—82no—
582C2×C4—81no—
583C2×C4—82no—
584C2×C4—81no—
585C2×C4—82no—
586C2×C4—81no—
587C2×C4—82no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4#39514yes—
603C2×C4#46514yes—
604C2×C4#46514yes—
605C2×C4C2×C22⋊Q812yes—
606C2×C4C2×C22⋊Q812yes—
607C2×C4C2×C4⋊Q812yes—
608C2×C4#53312yes—
609C2×C4#54312yes—
610C2×C4#54412yes—
611C2×C4D4212yes—
612C2×C4#54611yescharacteristic
613C2×C4#54612yes—
614C2×C4#54612yes—
615C2×C4#54712yes—
616C2×C4D4×Q811yescharacteristic
617C2×C4#55112yes—
618C2×C4#55211yescharacteristic
619C2×C4#55412yes—
620C2×C4#55411yescharacteristic
621C8—28no—
622C8—44no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22#205818yescomplemented
638C22C2×#39511yescharacteristic
639C22#241011yescharacteristic
640C22#241711yescharacteristic
641C22#243011yescharacteristic
642C22#256411yescharacteristic
643C22#265911yescharacteristic
644C22#285711yescharacteristic
645C22C2×#45011yescharacteristic
646C22C2×#44811yescharacteristic
647C22#300211yescharacteristic
648C22#300211yescharacteristic
649C22C2×#46511yescharacteristic
650C22C2×#46511yescharacteristic
651C22#305511yescharacteristic
652C22#305611yescharacteristic
653C22C2×#47611yescharacteristic
654C22C2×#47511yescharacteristic
655C22#307611yescharacteristic
656C22#307611yescharacteristic
657C22C2×#49811yescharacteristic
658C22#316011yescharacteristic
659C22C2×#50111yescharacteristic
660C22#316211yescharacteristic
661C22#316711yescharacteristic
662C22#316711yescharacteristic
663C22#318011yescharacteristic
664C22#318111yescharacteristic
665C22#318711yescharacteristic
666C22#318711yescharacteristic
667C22#318811yescharacteristic
668C22#319011yescharacteristic
669C22#324611yescharacteristic
670C22#324611yescharacteristic
671C22#324711yescharacteristic
672C22#324811yescharacteristic
673C4—24no—
674C4—24no—
675C4—24no—
676C4—42no—
677C4—42no—
678C4—82no—
679C4—84no—
680C4—84no—
681C4—82no—
682C4—161no—
683C4—161no—
684C4#243712yes—
685C4#243812yes—
686C4#244012yes—
687C4#254912yes—
688C2—28no—
689C2#1475811yescharacteristic
690C2C2×#205811yescharacteristic
691C2#2355111yescharacteristic
692C2#2437911yescharacteristic
693C2#2438111yescharacteristic
694C2#2483611yescharacteristic
695C2#2483811yescharacteristic
696C2#2549511yescharacteristic
697C2#2549711yescharacteristic
698C2#2915611yescharacteristic
699C2#2915911yescharacteristic
700C2#3029311yescharacteristic
701C2#3029411yescharacteristic
702C2#3320711yescharacteristic
703C2#3321711yescharacteristic
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×#2058C2semidirect
6. #11384C2semidirect
7. #11327C2semidirect
8. #11323C2semidirect
9. #11220C2semidirect
10. #11207C2semidirect
57. C23×C8Q8semidirect
110. #380Q8semidirect
437. C2×C8C22⋊Q8semidirect
637. C22#2058semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#20582
C2×#18072
C22×#3362
#146972
#113842
#113272
#113232
#112202
#112072
#107212