Finite Group Databasebeta

← #415883

Subgroups — #415883

Order 512.

2096 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#487C211yesmaximal, characteristic, complemented
2C2×#2857C211yesmaximal, characteristic
3C2×#2857C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic, complemented
5C2×#1956C211yesmaximal, characteristic, complemented
6C2×C4×C22⋊C8C211yesmaximal, characteristic
7#11067C214yesmaximal, complemented
8#10970C214yesmaximal
9#10841C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C22⋊Q8—21no—
13C22×C22⋊Q8C2212yes—
14C23×C42C2211yescharacteristic
15C2×#487—24no—
16C2×#487—24no—
17C2×#487—24no—
18C22×#103C2214yes—
19C22×#103C2214yes—
20C22×C22⋊C8C2212yes—
21C22×C4×C8C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2857—24no—
24#2857—24no—
25#2857—24no—
26#2857—24no—
27C2×#391—21no—
28C2×#391C2212yes—
29C2×#389—21no—
30C2×#389—21no—
31C2×#389—21no—
32C2×#389C2212yes—
33C2×#384C2211yescharacteristic
34C2×#384C2211yescharacteristic
35C2×#380—21no—
36C2×#380—21no—
37C2×#380—21no—
38C2×#380C2212yes—
39#1990—24no—
40#1956—24no—
41C4×C22⋊C8—24no—
42C2×#340C2214yes—
43C2×#340C2214yes—
44C2×#336C2212yes—
45C23×Q8—21no—
46C23×Q8C2312yes—
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8D414yes—
63C2×C4⋊Q8D414yes—
64C2×C22⋊Q8—28no—
65C2×C22⋊Q8—24no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8—28no—
68C2×C22⋊Q8—44no—
69C2×C22⋊Q8—44no—
70C2×C22⋊Q8—44no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C42—22no—
86C22×C42—22no—
87C22×C42—22no—
88C22×C42—22no—
89C22×C42—22no—
90C22×C42—22no—
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#487—232no—
96C2×#103—28no—
97C2×#103—28no—
98C2×#103—28no—
99C2×#103—28no—
100C2×#103—28no—
101C2×#103—44no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—24no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8D414yes—
111#391—28no—
112#391—24no—
113#391—24no—
114#389—28no—
115#389—28no—
116#389—28no—
117#389—24no—
118#389—24no—
119#384—22no—
120#384—22no—
121#384D414yes—
122#384D414yes—
123#380—28no—
124#380—28no—
125#380—28no—
126#380—24no—
127#380—24no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132#340—28no—
133#340—44no—
134#336—24no—
135C25C2×D411yescharacteristic
136C22×Q8—24no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—24no—
142C22×Q8—22no—
143C22×Q8—24no—
144C22×Q8—24no—
145C22×Q8—24no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8C2×D414yes—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4SD1614yes—
179C23×C4Q1614yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4#4114yes—
186C23×C4C2411yescharacteristic
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8C2×D414yes—
200C22×C8#4114yes—
201C4⋊Q8—28no—
202C4⋊Q8—28no—
203C4⋊Q8—28no—
204C4⋊Q8—28no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
207C4⋊Q8—44no—
208C4⋊Q8—44no—
209C22⋊Q8—416no—
210C22⋊Q8—416no—
211C22⋊Q8—432no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D414yes—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42#4112yes—
281#103—232no—
282#103—416no—
283C22⋊C8—216no—
284C4×C8—28no—
285C4×C8—44no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24#12412yes—
303C24C2×SD1612yes—
304C24C2×Q1612yes—
305C24C22×D411yescharacteristic
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—28no—
309C2×Q8—28no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—44no—
313C2×Q8—44no—
314C2×Q8—48no—
315C2×Q8—48no—
316C2×Q8—48no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—84no—
326C2×Q8—84no—
327C2×Q8—84no—
328C2×Q8—84no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
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—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4#12012yes—
406C22×C4#12012yes—
407C22×C4#12012yes—
408C22×C4#12012yes—
409C22×C4#12112yes—
410C22×C4#12112yes—
411C22×C4#12412yes—
412C22×C4C2×SD1612yes—
413C22×C4C2×SD1612yes—
414C22×C4C2×Q1612yes—
415C22×C4C2×Q1612yes—
416C22×C4#13512yes—
417C22×C4#13512yes—
418C22×C4AGL(1,ℤ/8)12yes—
419C22×C4#13712yes—
420C22×C4#13712yes—
421C22×C4#13712yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C42−1+412yes—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—416no—
451C4⋊C4—416no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—41no—
469C42—44no—
470C42—44no—
471C42#12014yes—
472C42#12114yes—
473C42#12114yes—
474C42#12714yes—
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—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23#48714yescomplemented
510C23C2×#12011yescharacteristic
511C23C2×#12011yescharacteristic
512C23C2×#12111yescharacteristic
513C23C2×#12411yescharacteristic
514C23#53211yescharacteristic
515C23#53511yescharacteristic
516C23#53611yescharacteristic
517C23C22×SD1611yescharacteristic
518C23C22×Q1611yescharacteristic
519C23C2×#13511yescharacteristic
520C23C2×#13511yescharacteristic
521C23C2×AGL(1,ℤ/8)11yescharacteristic
522C23C2×#13711yescharacteristic
523C23C2×#13711yescharacteristic
524C23C2×#13711yescharacteristic
525Q8—48no—
526Q8—48no—
527Q8—84no—
528Q8—816no—
529Q8—84no—
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—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—416no—
555C2×C4—416no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4#39214yes—
586C2×C4#39214yes—
587C2×C4#44814yes—
588C2×C4#44914yes—
589C2×C4#45014yes—
590C2×C4#45114yes—
591C2×C4#45114yes—
592C2×C4#45214yes—
593C2×C4#45214yes—
594C2×C4#45214yes—
595C2×C4#46514yes—
596C2×C4#46714yes—
597C2×C4#46814yes—
598C2×C4#47014yes—
599C2×C4C2×#12011yescharacteristic
600C2×C4C2×#12111yescharacteristic
601C2×C4C2×#12111yescharacteristic
602C2×C4#52512yes—
603C2×C4#52912yes—
604C2×C4C2×#12711yescharacteristic
605C2×C4#53712yes—
606C2×C4#54212yes—
607C2×C4#54612yes—
608C2×C4#54612yes—
609C2×C4#54612yes—
610C2×C4#54712yes—
611C2×C4#54812yes—
612C2×C4#54812yes—
613C2×C4#54812yes—
614C2×C4#55012yes—
615C8—44no—
616C8—44no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22C2×#39211yescharacteristic
633C22#243111yescharacteristic
634C22#243211yescharacteristic
635C22#243511yescharacteristic
636C22#243611yescharacteristic
637C22#262011yescharacteristic
638C22#285011yescharacteristic
639C22C2×#45011yescharacteristic
640C22C2×#44811yescharacteristic
641C22C2×#45111yescharacteristic
642C22C2×#45111yescharacteristic
643C22C2×#44911yescharacteristic
644C22C2×#45211yescharacteristic
645C22C2×#45211yescharacteristic
646C22C2×#45211yescharacteristic
647C22C2×#46511yescharacteristic
648C22C2×#46711yescharacteristic
649C22C2×#46811yescharacteristic
650C22C2×#47011yescharacteristic
651C22C2×#48711yescharacteristic
652C22#313111yescharacteristic
653C22#313911yescharacteristic
654C22#314111yescharacteristic
655C22#316711yescharacteristic
656C22#317011yescharacteristic
657C22#318211yescharacteristic
658C22#318511yescharacteristic
659C22#320611yescharacteristic
660C22#320711yescharacteristic
661C22#320911yescharacteristic
662C22#321011yescharacteristic
663C22#322911yescharacteristic
664C22#323011yescharacteristic
665C22#323111yescharacteristic
666C22#323111yescharacteristic
667C4—42no—
668C4—42no—
669C4—42no—
670C4—84no—
671C4—84no—
672C4—84no—
673C4—84no—
674C4#200018yes—
675C4#200118yes—
676C4#240712yes—
677C4#242612yes—
678C4#242712yes—
679C4#254012yes—
680C2—44no—
681C2#1521511yescharacteristic
682C2C2×#200011yescharacteristic
683C2C2×#200111yescharacteristic
684C2#2559711yescharacteristic
685C2#2560011yescharacteristic
686C2#2567211yescharacteristic
687C2#2567411yescharacteristic
688C2#2585611yescharacteristic
689C2#2585711yescharacteristic
690C2#2590211yescharacteristic
691C2#2590411yescharacteristic
692C2#2984411yescharacteristic
693C2#2984511yescharacteristic
694C2#3304311yescharacteristic
695C2#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
1. C22×#487C2semidirect
4. C2×#1990C2semidirect
5. C2×#1956C2semidirect
7. #11067C2semidirect
9. #10841C2semidirect
509. C23#487semidirect

Maximal subgroups

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

subgroupindex
C22×#4872
C2×#28572
C2×#19902
C2×#19562
C2×C4×C22⋊C82
#110672
#109702
#108412