Finite Group Databasebeta

← #401612

Subgroups — #401612

Order 512.

2022 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C22⋊C4C211yesmaximal, characteristic
2C2×#2072C211yesmaximal, characteristic, complemented
3C2×#1803C211yesmaximal, characteristic, complemented
4C2×#1787C212yesmaximal
5#15195C211yesmaximal, characteristic
6#11200C212yesmaximal
7#11190C212yesmaximal, complemented
8#11077C211yesmaximal, characteristic, complemented
9#11042C211yesmaximal, characteristic
10#10789C211yesmaximal, characteristic, complemented
11#10789C212yesmaximal, complemented
12C24×Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2212yes—
21Q8×C22⋊C4—24no—
22Q8×C22⋊C4—24no—
23C2×#391—21no—
24C2×#391—21no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#380—21no—
29C2×#380C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C22⋊C4—21no—
32C2×C4×C22⋊C4C2211yescharacteristic
33#2072—24no—
34#1803—24no—
35#1787—28no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1442—21no—
43#1442—21no—
44#1439—21no—
45#1439—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48C23×Q8—22no—
49C23×Q8—22no—
50C23×Q8—22no—
51C23×Q8—22no—
52C23×Q8—21no—
53C23×Q8—22no—
54C23×Q8—24no—
55C23×Q8—24no—
56C23×Q8—42no—
57C23×Q8—42no—
58C23×Q8D414yes—
59C23×Q8C2311yescharacteristic
60C23×Q8C2311yescharacteristic
61C24×C4—21no—
62C24×C4C2311yescharacteristic
63C23×C8C2312yes—
64C23×C8C2312yes—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—24no—
69C2×C4×Q8—24no—
70C2×C4×Q8—42no—
71C2×C4×Q8—42no—
72C2×C4×Q8D414yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C42—21no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×C4⋊C8—22no—
90C2×C4⋊C8—42no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—42no—
102C2×#103—42no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—42no—
105C2×C22⋊C8—42no—
106#391—28no—
107#391—28no—
108#391—22no—
109#391—22no—
110#391—22no—
111#391D414yes—
112#384—22no—
113#384—22no—
114#380—28no—
115#380—22no—
116#380—22no—
117#380—24no—
118#380—24no—
119C4×C22⋊C4—28no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95D411yescharacteristic
133C2×#95D411yescharacteristic
134#340—24no—
135#340—24no—
136#340—42no—
137#340—42no—
138#336—22no—
139#336—42no—
140C25C2×D411yescharacteristic
141C22×Q8—24no—
142C22×Q8—28no—
143C22×Q8—28no—
144C22×Q8—24no—
145C22×Q8—28no—
146C22×Q8—28no—
147C22×Q8—24no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—216no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—44no—
162C22×Q8—44no—
163C22×Q8—48no—
164C22×Q8—48no—
165C22×Q8—48no—
166C22×Q8—48no—
167C22×Q8—48no—
168C22×Q8—48no—
169C22×Q8—48no—
170C22×Q8—44no—
171C22×Q8—44no—
172C22×Q8—48no—
173C22×Q8—44no—
174C22×Q8—44no—
175C22×Q8—48no—
176C22×Q8C2×D412yes—
177C22×Q8C2×D412yes—
178C22×Q8C2×D412yes—
179C22×Q8C2×D412yes—
180C22×Q8#4112yes—
181C22×Q8#4112yes—
182C22×Q8#4114yes—
183C22×Q8#4114yes—
184C23×C4—21no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D412yes—
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4#4112yes—
210C23×C4C2411yescharacteristic
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C4×Q8—28no—
226C4×Q8—44no—
227C4×Q8—48no—
228C4×Q8—48no—
229C4×Q8—44no—
230C4×Q8—44no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
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—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—82no—
256C2×C4⋊C4—82no—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—82no—
269C2×C22⋊C4—82no—
270C2×C22⋊C4Q1614yes—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—44no—
280C2×C42—44no—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C4⋊C8—44no—
284#103—48no—
285#103—48no—
286#103—48no—
287#103—48no—
288C22⋊C8—48no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—22no—
296#95—42no—
297#95—44no—
298#95—42no—
299#95—82no—
300#95—82no—
301C24—21no—
302C24—21no—
303C24—22no—
304C24—22no—
305C24—21no—
306C24—21no—
307C24—22no—
308C24—22no—
309C24#12112yes—
310C24C2×Q1612yes—
311C24#13712yes—
312C24C22×D411yescharacteristic
313C2×Q8—216no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—24no—
322C2×Q8—22no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—416no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—416no—
340C2×Q8—432no—
341C2×Q8—416no—
342C2×Q8—416no—
343C2×Q8—432no—
344C2×Q8—416no—
345C2×Q8—416no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—42no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8#12114yes—
355C2×Q8#12114yes—
356C2×Q8#12318yes—
357C2×Q8#12314yes—
358C2×Q8#12414yes—
359C2×Q8#12618yes—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—216no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—416no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—416no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4#12012yes—
437C22×C4#12012yes—
438C22×C4#12112yes—
439C22×C4C2×Q1612yes—
440C22×C4C2×Q1612yes—
441C22×C4#13712yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4111yescharacteristic
450C22×C4H(5,𝔽2)12yes—
451C22×C4H(5,𝔽2)12yes—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C22⋊C4—28no—
485C22⋊C4—44no—
486C22⋊C4—48no—
487C22⋊C4—88no—
488C42—22no—
489C42—22no—
490C42—24no—
491C42—48no—
492C42—48no—
493C42—41no—
494C42—41no—
495C42—42no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23#46214yes—
519C23C2×#12011yescharacteristic
520C23C2×#12111yescharacteristic
521C23#52511yescharacteristic
522C23#54612yes—
523C23#54612yes—
524C23C22×Q1611yescharacteristic
525C23C2×#13711yescharacteristic
526C23C2×#13711yescharacteristic
527C23#57511yescharacteristic
528C23#57712yes—
529C23#57712yes—
530Q8—24no—
531Q8—24no—
532Q8—28no—
533Q8—28no—
534Q8—24no—
535Q8—28no—
536Q8—432no—
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—44no—
541Q8—42no—
542Q8#39718yes—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—432no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4#45114yes—
609C2×C4C2×#12111yescharacteristic
610C2×C4C2×#12111yescharacteristic
611C2×C4C2×#12312yes—
612C2×C4C2×#12311yescharacteristic
613C2×C4#52511yescharacteristic
614C2×C4#52511yescharacteristic
615C2×C4C2×#12411yescharacteristic
616C2×C4C2×#12612yes—
617C2×C4#53411yescharacteristic
618C2×C4#53412yes—
619C2×C4#53511yescharacteristic
620C2×C4#53812yes—
621C2×C4#54612yes—
622C2×C4#54612yes—
623C2×C4#55112yes—
624C2×C4#55112yes—
625C2×C4#55112yes—
626C2×C4#55112yes—
627C8—82no—
628C8—82no—
629C22—24no—
630C22—28no—
631C22—42no—
632C22—42no—
633C22—44no—
634C22—42no—
635C22—44no—
636C22—44no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22#240511yescharacteristic
641C22#241212yes—
642C22#243511yescharacteristic
643C22#248012yes—
644C22#284711yescharacteristic
645C22C2×#45111yescharacteristic
646C22#300711yescharacteristic
647C22#301011yescharacteristic
648C22#301011yescharacteristic
649C22C2×#46211yescharacteristic
650C22#303711yescharacteristic
651C22#304312yes—
652C22#310411yescharacteristic
653C22#311311yescharacteristic
654C22#312612yes—
655C22#329912yes—
656C22#329912yes—
657C22#331612yes—
658C22#331612yes—
659C22#332012yes—
660C22#332112yes—
661C22#332612yes—
662C22#332912yes—
663C4—28no—
664C4—24no—
665C4—42no—
666C4—42no—
667C4—44no—
668C4—42no—
669C4—81no—
670C4—82no—
671C4—82no—
672C4—162no—
673C4—162no—
674C4C2×#39711yescharacteristic
675C4#241212yes—
676C4#248012yes—
677C4#248211yescharacteristic
678C4#256111yescharacteristic
679C4#261711yescharacteristic
680C2—44no—
681C2#1482011yescharacteristic
682C2#2404211yescharacteristic
683C2#2405211yescharacteristic
684C2#2453212yes—
685C2#2454112yes—
686C2#2588811yescharacteristic
687C2#2589511yescharacteristic
688C2#2752412yes—
689C2#2753712yes—
690C2#3290611yescharacteristic
691C2#3291211yescharacteristic
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
2. C2×#2072C2semidirect
3. C2×#1803C2semidirect
7. #11190C2semidirect
8. #11077C2semidirect
10. #10789C2semidirect
11. #10789C2semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C22⋊C42
C2×#20722
C2×#18032
C2×#17872
#151952
#112002
#111902
#110772
#110422
#107892