Finite Group Databasebeta

← #394429

Subgroups — #394429

Order 512.

2252 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#103C211yesmaximal, characteristic
2C2×#2442C211yesmaximal, characteristic
3C2×#2026C212yesmaximal, complemented
4C2×#1787C211yesmaximal, characteristic
5#13922C211yesmaximal, characteristic
6#11067C212yesmaximal, complemented
7#11067C212yesmaximal, complemented
8#11052C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10721C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#125C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#103—28no—
17C22×#103—24no—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23#2442—24no—
24#2442—24no—
25C2×#400—21no—
26C2×#398—21no—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#385—21no—
32C2×#382C2211yescharacteristic
33C2×#380—21no—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×C4×C22⋊C4C2211yescharacteristic
37#2026—28no—
38#1787—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1446—21no—
46#1446—21no—
47C22⋊#95—21no—
48C22⋊#95—21no—
49C23×Q8—21no—
50C23×Q8—22no—
51C23×Q8—24no—
52C23×Q8—24no—
53C23×Q8—22no—
54C23×Q8—22no—
55C23×Q8—22no—
56C23×Q8—22no—
57C23×Q8—42no—
58C23×Q8—42no—
59C23×Q8Q814yes—
60C23×Q8C2312yes—
61C24×C4—21no—
62C24×C4—21no—
63C24×C4C2311yescharacteristic
64C23×C8—22no—
65C23×C8—24no—
66C23×C8—22no—
67C23×C8—22no—
68C23×C8—42no—
69C23×C8C2312yes—
70C23×C8C2312yes—
71C2×#125—22no—
72C2×#125—22no—
73C2×#125—24no—
74C2×#125—42no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—42no—
83C22×C4⋊C4D414yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4D411yescharacteristic
92C22×C22⋊C4D411yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C2×#103—232no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—42no—
106C2×#103—42no—
107C2×#103D418yes—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111#400—28no—
112#398—28no—
113#391—24no—
114#391—24no—
115#391—42no—
116#391—42no—
117#389—22no—
118#389—22no—
119#389—22no—
120#389—22no—
121#385—28no—
122#382—22no—
123#382—22no—
124#380—28no—
125#380—22no—
126#380—22no—
127#380—22no—
128#380D414yes—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—21no—
141C2×#95D411yescharacteristic
142C2×#95D411yescharacteristic
143#340—24no—
144#340—24no—
145#340—42no—
146#340—42no—
147#336—24no—
148#336—42no—
149C25C2×D411yescharacteristic
150C22×Q8—24no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—28no—
155C22×Q8—216no—
156C22×Q8—28no—
157C22×Q8—216no—
158C22×Q8—28no—
159C22×Q8—28no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—44no—
169C22×Q8—44no—
170C22×Q8—48no—
171C22×Q8—48no—
172C22×Q8—48no—
173C22×Q8—48no—
174C22×Q8—48no—
175C22×Q8—48no—
176C22×Q8—48no—
177C22×Q8—44no—
178C22×Q8—44no—
179C22×Q8—48no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×Q8C2×D414yes—
185C22×Q8C2×Q814yes—
186C22×Q8#4114yes—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—21no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—82no—
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D412yes—
213C23×C4C2×D412yes—
214C23×C4C2×Q812yes—
215C23×C4#4112yes—
216C23×C4#4112yes—
217C23×C4#4112yes—
218C23×C4C2411yescharacteristic
219C22×C8—28no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
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—48no—
234C22×C8—48no—
235C22×C8—44no—
236C22×C8—44no—
237C22×C8C2×D414yes—
238#125—48no—
239#125—44no—
240#125—44no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—82no—
270C2×C4⋊C4—82no—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D412yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C42—22no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296#103—216no—
297#103—216no—
298#103—48no—
299#103—48no—
300C22⋊C8—44no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—22no—
307#95—22no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—42no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24#12012yes—
318C24C22⋊Q812yes—
319C24#12412yes—
320C24#13512yes—
321C24#13512yes—
322C24#13712yes—
323C24#13712yes—
324C24C22×D411yescharacteristic
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—232no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—416no—
342C2×Q8—432no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—416no—
346C2×Q8—416no—
347C2×Q8—416no—
348C2×Q8—416no—
349C2×Q8—432no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—48no—
363C2×Q8C22⋊Q818yes—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—416no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—416no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—48no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—48no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—84no—
442C22×C4—88no—
443C22×C4—88no—
444C22×C4#12012yes—
445C22×C4C22⋊Q814yes—
446C22×C4C22⋊Q814yes—
447C22×C4C22⋊Q812yes—
448C22×C4#12414yes—
449C22×C4#12414yes—
450C22×C4#12412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×Q811yescharacteristic
455C22×C4C2×#4111yescharacteristic
456C22×C4C2×#4112yes—
457C22×C4C2×#4111yescharacteristic
458C22×C4C2×#4111yescharacteristic
459C22×C4H(5,𝔽2)12yes—
460C22×C4H(5,𝔽2)12yes—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—416no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C4⋊C4#12014yes—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—48no—
500C22⋊C4—84no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—42no—
506C42—42no—
507C23—24no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—22no—
514C23—24no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23#39314yes—
524C23#45214yes—
525C23#45214yes—
526C23#47914yes—
527C23#47914yes—
528C23#48814yes—
529C23#49114yes—
530C23C2×#12011yescharacteristic
531C23C2×C22⋊Q811yescharacteristic
532C23C2×#12411yescharacteristic
533C23D4212yes—
534C23#54612yes—
535C23C2×#13511yescharacteristic
536C23C2×#13511yescharacteristic
537C23C2×#13711yescharacteristic
538C23C2×#13711yescharacteristic
539C23#57712yes—
540C23#57812yes—
541Q8—28no—
542Q8—28no—
543Q8—432no—
544Q8—44no—
545Q8—44no—
546Q8—44no—
547Q8—44no—
548Q8—44no—
549Q8—44no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—432no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—816no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4#39314yes—
617C2×C4C2×#12011yescharacteristic
618C2×C4C2×C22⋊Q811yescharacteristic
619C2×C4C2×C22⋊Q812yes—
620C2×C4C2×C22⋊Q811yescharacteristic
621C2×C4#52511yescharacteristic
622C2×C4#52512yes—
623C2×C4C2×#12411yescharacteristic
624C2×C4C2×#12411yescharacteristic
625C2×C4#53411yescharacteristic
626C2×C4#53511yescharacteristic
627C2×C4#53812yes—
628C2×C4#54312yes—
629C2×C4D4212yes—
630C2×C4#54612yes—
631C2×C4#54612yes—
632C2×C4#54612yes—
633C2×C4#55112yes—
634C2×C4#55112yes—
635C8—44no—
636C8—82no—
637C22—24no—
638C22—24no—
639C22—28no—
640C22—28no—
641C22—28no—
642C22—24no—
643C22—28no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22#202618yescomplemented
648C22C2×#39311yescharacteristic
649C22#240311yescharacteristic
650C22#240411yescharacteristic
651C22#243712yes—
652C22#247212yes—
653C22C2×#45211yescharacteristic
654C22C2×#45211yescharacteristic
655C22#300911yescharacteristic
656C22#301111yescharacteristic
657C22C2×#47911yescharacteristic
658C22C2×#47911yescharacteristic
659C22#308212yes—
660C22C2×#48811yescharacteristic
661C22C2×#49111yescharacteristic
662C22#313712yes—
663C22#326612yes—
664C22#327012yes—
665C22#328212yes—
666C22#328512yes—
667C22#330012yes—
668C22#330112yes—
669C22#331212yes—
670C22#332012yes—
671C4—24no—
672C4—44no—
673C4—48no—
674C4—81no—
675C4—82no—
676C4—81no—
677C4—82no—
678C4—81no—
679C4—82no—
680C4—81no—
681C4—162no—
682C4#243511yescharacteristic
683C4#243712yes—
684C4#247212yes—
685C4#256111yescharacteristic
686C4#262611yescharacteristic
687C4#284811yescharacteristic
688C2—28no—
689C2#1471711yescharacteristic
690C2C2×#202611yescharacteristic
691C2#2345711yescharacteristic
692C2#2385211yescharacteristic
693C2#2385911yescharacteristic
694C2#2396411yescharacteristic
695C2#2396711yescharacteristic
696C2#2595712yes—
697C2#2596512yes—
698C2#2726012yes—
699C2#2726812yes—
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
3. C2×#2026C2semidirect
6. #11067C2semidirect
7. #11067C2semidirect
8. #11052C2semidirect
647. C22#2026semidirect

Maximal subgroups

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

subgroupindex
C23×#1032
C2×#24422
C2×#20262
C2×#17872
#139222
#110672
#110522
#107302
#107212
#107142