Finite Group Databasebeta

← #393446

Subgroups — #393446

Order 512.

854 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21032C211yesmaximal, characteristic
2#21020C211yesmaximal, characteristic
3C4×#405C211yesmaximal, characteristic
4#10911C211yesmaximal, characteristic
5#10910C211yesmaximal, characteristic
6#10910C211yesmaximal, characteristic
7#10909C211yesmaximal, characteristic
8#10909C211yesmaximal, characteristic
9#10887C211yesmaximal, characteristic
10#10887C211yesmaximal, characteristic
11#10886C211yesmaximal, characteristic
12#10809C211yesmaximal, characteristic
13#10809C211yesmaximal, characteristic
14#10808C211yesmaximal, characteristic
15#10797C211yesmaximal, characteristic
16C22×#108C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C2×#405C2211yescharacteristic
21C2×#405C2211yescharacteristic
22C2×#405C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2296C2211yescharacteristic
26C4×#117C2211yescharacteristic
27#1921C411yescharacteristic, complemented
28#1921C411yescharacteristic, complemented
29#1920C411yescharacteristic, complemented
30#1920C411yescharacteristic, complemented
31#1919C411yescharacteristic, complemented
32#1919C411yescharacteristic, complemented
33#1918C411yescharacteristic, complemented
34#1918C411yescharacteristic, complemented
35#1916—21no—
36#1916—21no—
37#1916—21no—
38#1916—21no—
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41C2×#341C2211yescharacteristic
42#1735C2211yescharacteristic
43#1735C2211yescharacteristic
44#1735C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56#1728C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C2×#336C2211yescharacteristic
62#1726—21no—
63#1726—21no—
64C4×#97—21no—
65C42×C8—21no—
66C4×#95C2211yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#108—22no—
83C2×#108—22no—
84C2×#108—22no—
85#427—22no—
86#427—22no—
87C2×#107C2×C411yescharacteristic
88C2×#107C2×C411yescharacteristic
89C2×#106C2×C411yescharacteristic
90C2×#106C2×C411yescharacteristic
91#424—22no—
92#424—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95#405—22no—
96#405—22no—
97#405—22no—
98#405—22no—
99#405—22no—
100#405—22no—
101#405—22no—
102#405Q814yes—
103C4×M16—22no—
104C4×M16—22no—
105C2×#97—21no—
106C2×#97—21no—
107C2×#97—21no—
108C2×#97—21no—
109C2×#97—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—22no—
115C2×C4×C8—22no—
116C2×C4×C8—21no—
117C2×C4×C8D411yescharacteristic
118C2×C4×C8D411yescharacteristic
119C2×C4×C8D411yescharacteristic
120C2×C4×C8D411yescharacteristic
121#389C2×C411yescharacteristic
122#389C2×C411yescharacteristic
123#389C2×C411yescharacteristic
124#389C2×C411yescharacteristic
125#385—21no—
126#385—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135#376—21no—
136C2×#95—21no—
137C2×#95—21no—
138C43—21no—
139#341—22no—
140#341—22no—
141#341—22no—
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#340C2×C411yescharacteristic
154#339—22no—
155#339—22no—
156#339—22no—
157#339—22no—
158#339—22no—
159#339—22no—
160#339—22no—
161#339—22no—
162#336—21no—
163#336—21no—
164#336—21no—
165#336—21no—
166#336—21no—
167#336—21no—
168#336—22no—
169#336—22no—
170#336—22no—
171#336—22no—
172#336—21no—
173#336—21no—
174#336C2×C411yescharacteristic
175#336C2×C411yescharacteristic
176#336C2×C411yescharacteristic
177#336C2×C411yescharacteristic
178C23×C4—21no—
179C23×C4—21no—
180C23×C4#4111yescharacteristic
181C23×C4#4111yescharacteristic
182C23×C4C2411yescharacteristic
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16C2×Q812yes—
202C2×M16#4112yes—
203C2×M16#4112yes—
204C2×M16#4112yes—
205C2×M16#4112yes—
206C2×M16#4112yes—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8C22×C411yescharacteristic
232C22×C8C22×C411yescharacteristic
233C22×C8C22×C411yescharacteristic
234C22×C8C22×C411yescharacteristic
235C22×C8C2×D411yescharacteristic
236C22×C8C2×D411yescharacteristic
237C22×C8C2×D411yescharacteristic
238C22×C8C2×D411yescharacteristic
239C22×C8C2×Q812yes—
240C22×C8#4112yes—
241C22×C8#4112yes—
242C22×C8#4112yes—
243#117—21no—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117C2×Q812yes—
256#117#4112yes—
257#117#4112yes—
258#117#4112yes—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42C22×C411yescharacteristic
300C2×C42C22×C411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303#108—44no—
304#107—22no—
305#107—22no—
306#106—22no—
307#106—22no—
308C4⋊C8—42no—
309C4⋊C8—42no—
310#97—24no—
311#97—24no—
312#97—24no—
313#97—24no—
314#97—24no—
315C4×C8—24no—
316C4×C8—24no—
317C4×C8—24no—
318C4×C8—24no—
319C4×C8—24no—
320C4×C8—22no—
321C4×C8—22no—
322C4×C8—22no—
323C4×C8—22no—
324#95—24no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—42no—
334C24C2×#4111yescharacteristic
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—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4C4×D411yescharacteristic
366C22×C4C4×D411yescharacteristic
367C22×C4C4×D411yescharacteristic
368C22×C4C4×D411yescharacteristic
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×Q811yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383M16—22no—
384M16—22no—
385M16—22no—
386M16—22no—
387M16—22no—
388M16—22no—
389M16—41no—
390M16—41no—
391M16—41no—
392M16—41no—
393M16—41no—
394M16—41no—
395M16—41no—
396M16—41no—
397M16#12514yes—
398M16#12614yes—
399C2×C8—22no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8C22⋊Q812yes—
448C2×C8C22⋊Q812yes—
449C2×C8#12312yes—
450C2×C8#12312yes—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4#11712yes—
468C4⋊C4#11712yes—
469C4⋊C4C4×Q812yes—
470C4⋊C4C4×Q812yes—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—22no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—21no—
491C42—42no—
492C42—42no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—42no—
498C42—42no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23C2×C4×D411yescharacteristic
512C23#52911yescharacteristic
513C23#52911yescharacteristic
514C23#52911yescharacteristic
515C23#52911yescharacteristic
516C23#52911yescharacteristic
517C23#53311yescharacteristic
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
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—41no—
561C2×C4—41no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4#44312yes—
571C2×C4#44412yes—
572C2×C4C2×#11711yescharacteristic
573C2×C4C2×C4×Q811yescharacteristic
574C2×C4C4×#4111yescharacteristic
575C2×C4C4×#4111yescharacteristic
576C2×C4C4×#4111yescharacteristic
577C2×C4C4×#4111yescharacteristic
578C2×C4C2×C22⋊Q811yescharacteristic
579C2×C4C2×#12311yescharacteristic
580C2×C4#52511yescharacteristic
581C2×C4#52511yescharacteristic
582C2×C4#52511yescharacteristic
583C2×C4#52511yescharacteristic
584C2×C4C2×#12511yescharacteristic
585C2×C4C2×#12611yescharacteristic
586C2×C4#52911yescharacteristic
587C2×C4#52911yescharacteristic
588C2×C4#52911yescharacteristic
589C2×C4#52911yescharacteristic
590C2×C4#52911yescharacteristic
591C2×C4#52911yescharacteristic
592C2×C4#52911yescharacteristic
593C2×C4#52911yescharacteristic
594C2×C4#52911yescharacteristic
595C2×C4#52911yescharacteristic
596C2×C4#52911yescharacteristic
597C2×C4#53311yescharacteristic
598C2×C4#53311yescharacteristic
599C2×C4#53311yescharacteristic
600C8—24no—
601C8—24no—
602C8—22no—
603C8—22no—
604C8—22no—
605C8—22no—
606C8—41no—
607C8—41no—
608C8—41no—
609C8—41no—
610C8—81no—
611C8—81no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22#229011yescharacteristic
625C22C4×#12311yescharacteristic
626C22C4×C22⋊Q811yescharacteristic
627C22#286311yescharacteristic
628C22#286411yescharacteristic
629C22#286511yescharacteristic
630C22#286611yescharacteristic
631C22C2×#44311yescharacteristic
632C22C2×#44411yescharacteristic
633C22#295511yescharacteristic
634C22#295511yescharacteristic
635C4—22no—
636C4—22no—
637C4—41no—
638C4—41no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—82no—
644C4—82no—
645C4—81no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4#229511yescharacteristic
650C4#229811yescharacteristic
651C4C4×#12611yescharacteristic
652C4C4×#12511yescharacteristic
653C4#285111yescharacteristic
654C4#285211yescharacteristic
655C4#285311yescharacteristic
656C4#285511yescharacteristic
657C2—21no—
658C2—21no—
659C2—21no—
660C2—21no—
661C2#1470411yescharacteristic
662C2#2113711yescharacteristic
663C2#2113811yescharacteristic
664C2#2159211yescharacteristic
665C2#2159311yescharacteristic
666C2#2162711yescharacteristic
667C2#2162811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
27. #1921C4semidirect
28. #1921C4semidirect
29. #1920C4semidirect
30. #1920C4semidirect
31. #1919C4semidirect
32. #1919C4semidirect
33. #1918C4semidirect
34. #1918C4semidirect

Maximal subgroups

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

subgroupindex
#210322
#210202
C4×#4052
#109112
#109102
#109092
#108872
#108862
#108092
#108082
#107972