Finite Group Databasebeta

← #394450

Subgroups — #394450

Order 512.

2222 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic, complemented
2C2×Q8×C22⋊C4C211yesmaximal, characteristic
3C2×#2024C211yesmaximal, characteristic, complemented
4C2×#1787C212yesmaximal
5#15125C211yesmaximal, characteristic
6#11122C212yesmaximal, complemented
7#11095C212yesmaximal
8#11077C211yesmaximal, characteristic, complemented
9#11042C211yesmaximal, characteristic
10#10789C211yesmaximal, characteristic, complemented
11#10789C212yesmaximal, complemented
12C24×Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C2×#475—28no—
17C2×#475—24no—
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×C22⋊C8C2212yes—
25C2×#393—21no—
26Q8×C22⋊C4—24no—
27Q8×C22⋊C4—24no—
28C2×#391—21no—
29C2×#391—21no—
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#380—21no—
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×C4×C22⋊C4—21no—
36C2×C4×C22⋊C4C2211yescharacteristic
37#2024—24no—
38#1787—28no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1446—21no—
45#1442—21no—
46#1439—21no—
47C4⋊#95C2211yescharacteristic
48C4⋊#95C2212yes—
49C23×Q8—22no—
50C23×Q8—22no—
51C23×Q8—22no—
52C23×Q8—22no—
53C23×Q8—21no—
54C23×Q8—21no—
55C23×Q8—22no—
56C23×Q8—24no—
57C23×Q8—24no—
58C23×Q8—42no—
59C23×Q8—42no—
60C23×Q8Q814yescomplemented
61C23×Q8C2311yescharacteristic
62C23×Q8C2311yescharacteristic
63C24×C4—21no—
64C24×C4C2311yescharacteristic
65C23×C8C2312yes—
66C23×C8C2312yes—
67C2×C4⋊Q8—24no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—42no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—24no—
75C2×C4×Q8—24no—
76C2×C4×Q8—42no—
77C2×C4×Q8—42no—
78C2×C4×Q8D414yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4D411yescharacteristic
89C22×C22⋊C4D411yescharacteristic
90C22×C42—21no—
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#475—232no—
95C2×#106—24no—
96C2×#106—24no—
97C2×#106—24no—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—22no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—42no—
110C2×#103—42no—
111C2×#103D418yes—
112C2×C22⋊C8—24no—
113C2×C22⋊C8—24no—
114C2×C22⋊C8—42no—
115#393—44no—
116#391—28no—
117#391—28no—
118#391—22no—
119#391—22no—
120#391—22no—
121#391D414yes—
122#380—28no—
123#380—22no—
124#380—22no—
125#380—24no—
126#380—42no—
127C4×C22⋊C4—28no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95D411yescharacteristic
141C2×#95D411yescharacteristic
142#340—24no—
143#340—42no—
144#340—42no—
145#336—22no—
146#336—42no—
147C25C2×D411yescharacteristic
148C22×Q8—24no—
149C22×Q8—28no—
150C22×Q8—28no—
151C22×Q8—24no—
152C22×Q8—28no—
153C22×Q8—28no—
154C22×Q8—24no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—216no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—44no—
169C22×Q8—44no—
170C22×Q8—44no—
171C22×Q8—48no—
172C22×Q8—48no—
173C22×Q8—48no—
174C22×Q8—48no—
175C22×Q8—48no—
176C22×Q8—48no—
177C22×Q8—48no—
178C22×Q8—44no—
179C22×Q8—44no—
180C22×Q8—48no—
181C22×Q8—44no—
182C22×Q8—48no—
183C22×Q8—44no—
184C22×Q8—42no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8—82no—
188C22×Q8C2×D412yes—
189C22×Q8C2×D412yes—
190C22×Q8C2×D414yes—
191C22×Q8C2×Q812yes—
192C22×Q8C2×Q812yes—
193C22×Q8#4112yes—
194C22×Q8#4114yes—
195C22×Q8#4112yes—
196C23×C4—21no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—22no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—22no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×Q812yes—
221C23×C4#4112yes—
222C23×C4C2411yescharacteristic
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8C2×D414yes—
237C4⋊Q8—28no—
238C4⋊Q8—48no—
239C4⋊Q8—44no—
240C4×Q8—28no—
241C4×Q8—44no—
242C4×Q8—48no—
243C4×Q8—48no—
244C4×Q8—44no—
245C4×Q8—44no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D414yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—82no—
283C2×C22⋊C4SD1614yes—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296#106—216no—
297C4⋊C8—28no—
298#103—216no—
299#103—48no—
300#103—48no—
301#103—48no—
302C22⋊C8—48no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—22no—
309#95—22no—
310#95—44no—
311#95—42no—
312#95—44no—
313#95—42no—
314#95—82no—
315C24—21no—
316C24—21no—
317C24—22no—
318C24—22no—
319C24—21no—
320C24—21no—
321C24—22no—
322C24—22no—
323C24C22⋊Q812yes—
324C24C2×SD1612yes—
325C24#13712yes—
326C24C22×D411yescharacteristic
327C2×Q8—216no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—416no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—416no—
354C2×Q8—48no—
355C2×Q8—432no—
356C2×Q8—416no—
357C2×Q8—416no—
358C2×Q8—416no—
359C2×Q8—432no—
360C2×Q8—416no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8—42no—
364C2×Q8—41no—
365C2×Q8—41no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—88no—
370C2×Q8—88no—
371C2×Q8—84no—
372C2×Q8—84no—
373C2×Q8#12014yes—
374C2×Q8C22⋊Q814yes—
375C2×Q8C22⋊Q814yes—
376C2×Q8C22⋊Q818yes—
377C2×Q8#12418yes—
378C2×Q8#12414yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—216no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—416no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—416no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4#12012yes—
456C22×C4#12012yes—
457C22×C4C22⋊Q812yes—
458C22×C4C2×SD1612yes—
459C22×C4C2×SD1612yes—
460C22×C4#13712yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×Q811yescharacteristic
465C22×C4C2×#4111yescharacteristic
466C22×C4C2×#4111yescharacteristic
467C22×C4C2×#4112yes—
468C22×C4C2×#4111yescharacteristic
469C22×C4H(5,𝔽2)12yes—
470C22×C4H(5,𝔽2)12yes—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8—42no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—41no—
494C4⋊C4—42no—
495C4⋊C4—41no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C22⋊C4—28no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—88no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—48no—
510C42—48no—
511C42—41no—
512C42—41no—
513C42—42no—
514C23—22no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23#47514yescomplemented
537C23C2×#12011yescharacteristic
538C23C2×C22⋊Q811yescharacteristic
539C23#52511yescharacteristic
540C23D4212yes—
541C23#54612yes—
542C23C22×SD1611yescharacteristic
543C23C2×#13711yescharacteristic
544C23C2×#13711yescharacteristic
545C23#57511yescharacteristic
546C23#57712yes—
547C23#57812yes—
548Q8—24no—
549Q8—24no—
550Q8—24no—
551Q8—28no—
552Q8—28no—
553Q8—28no—
554Q8—432no—
555Q8—42no—
556Q8—42no—
557Q8—42no—
558Q8—42no—
559Q8—44no—
560Q8—816no—
561Q8#39318yescomplemented
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—432no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4#44814yes—
628C2×C4C2×#12011yescharacteristic
629C2×C4C2×C22⋊Q811yescharacteristic
630C2×C4C2×C22⋊Q812yes—
631C2×C4C2×C22⋊Q811yescharacteristic
632C2×C4#52511yescharacteristic
633C2×C4#52511yescharacteristic
634C2×C4C2×#12411yescharacteristic
635C2×C4C2×#12412yes—
636C2×C4#53411yescharacteristic
637C2×C4#53511yescharacteristic
638C2×C4#53812yes—
639C2×C4#54312yes—
640C2×C4D4212yes—
641C2×C4#54612yes—
642C2×C4#54612yes—
643C2×C4#54612yes—
644C2×C4#55112yes—
645C2×C4#55112yes—
646C8—44no—
647C8—82no—
648C22—28no—
649C22—24no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—42no—
658C22—42no—
659C22#240311yescharacteristic
660C22#243511yescharacteristic
661C22#243712yes—
662C22#247212yes—
663C22#284811yescharacteristic
664C22C2×#44811yescharacteristic
665C22#300711yescharacteristic
666C22#301011yescharacteristic
667C22#301111yescharacteristic
668C22C2×#47511yescharacteristic
669C22#307811yescharacteristic
670C22#308212yes—
671C22#310311yescharacteristic
672C22#311311yescharacteristic
673C22#312112yes—
674C22#326612yes—
675C22#327012yes—
676C22#327712yes—
677C22D4×SD1612yes—
678C22#329812yes—
679C22#331612yes—
680C22#332112yes—
681C22#332812yes—
682C4—24no—
683C4—28no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—44no—
688C4—81no—
689C4—82no—
690C4—84no—
691C4—82no—
692C4—162no—
693C4C2×#39311yescharacteristic
694C4#240411yescharacteristic
695C4#243712yes—
696C4#247212yes—
697C4#256111yescharacteristic
698C4#262611yescharacteristic
699C2—44no—
700C2#1471711yescharacteristic
701C2#2385511yescharacteristic
702C2#2386711yescharacteristic
703C2#2588411yescharacteristic
704C2#2589511yescharacteristic
705C2#2595512yes—
706C2#2597112yes—
707C2#2722612yes—
708C2#2724112yes—
709C2#3295411yescharacteristic
710C2#3296411yescharacteristic
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
1. C22×#475C2semidirect
3. C2×#2024C2semidirect
6. #11122C2semidirect
8. #11077C2semidirect
10. #10789C2semidirect
11. #10789C2semidirect
60. C23×Q8Q8semidirect
536. C23#475semidirect
561. Q8#393semidirect

Maximal subgroups

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

subgroupindex
C22×#4752
C2×Q8×C22⋊C42
C2×#20242
C2×#17872
#151252
#111222
#110952
#110772
#110422
#107892