Finite Group Databasebeta

← #6657884

Subgroups — #6657884

Order 512.

1579 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#34947C211yesmaximal, characteristic
3#33148C212yesmaximal, complemented
4#32695C212yesmaximal, complemented
5#32645C212yesmaximal, complemented
6#32268C212yesmaximal, complemented
7#32037C212yesmaximal, complemented
8#31794C214yesmaximal, complemented
9#31334C214yesmaximal, complemented
10#29691C212yesmaximal, complemented
11#27628C212yesmaximal, complemented
12#24846C212yesmaximal, complemented
13#24711C212yesmaximal, complemented
14#24302C212yesmaximal, complemented
15#21853C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C2×#477C2212yes—
20C2×#477C2212yes—
21C2×#432C2211yescharacteristic
22C2×#423C2211yescharacteristic
23#2857C2212yes—
24#2857C2212yes—
25#2628—21no—
26#2628—21no—
27#2568—21no—
28#2568C2211yescharacteristic
29#2567C2211yescharacteristic
30#2567C2212yes—
31#2566—21no—
32#2565—21no—
33#2564—21no—
34#2564—21no—
35#2563C2211yescharacteristic
36#2563C2212yes—
37#2542—21no—
38#2542C2211yescharacteristic
39#2487C2212yes—
40#2487C2212yes—
41#2417C2212yes—
42#2417C2212yes—
43#2409—21no—
44#2409—21no—
45#2396—21no—
46#2396—21no—
47#2317—21no—
48#2311—21no—
49C4×C22⋊Q8C2212yes—
50C4×C22⋊Q8C2212yes—
51#2297—21no—
52#2285—21no—
53#2285C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55#2086C2214yes—
56#2077C2214yes—
57#2075C2214yes—
58#2066C2214yes—
59#2057C2214yes—
60#2057C2214yes—
61#2043C2214yes—
62#2036C2214yes—
63#2036C2214yes—
64#2035C2214yes—
65#2035C2214yes—
66#2024C2214yes—
67#1987C2212yes—
68#1987C2212yes—
69#1985C2212yes—
70#1955C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1944C2214yes—
74#1935C2214yes—
75#1927C2214yes—
76#1920C2214yes—
77#1920C2214yes—
78#1920C2214yes—
79#1894C2214yes—
80#1872C2214yes—
81#1870C2212yes—
82#1870C2212yes—
83#1853C2214yes—
84#1852C2214yes—
85#1823C2214yes—
86#1809C2214yes—
87#1803C2212yes—
88#1803C2212yes—
89#1797C2212yes—
90#1797C2212yes—
91C22×#41D411yescharacteristic
92C23×Q8—21no—
93C22×M16C2312yes—
94#533—42no—
95#533—42no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—42no—
99C2×C4⋊Q8—42no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#125—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2312yes—
111C4×#41—42no—
112C4×#41—42no—
113C2×C4×Q8—21no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—21no—
124C2×#117—21no—
125C2×#117D411yescharacteristic
126C2×#117D411yescharacteristic
127C2×#117D411yescharacteristic
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#477—44no—
136#477—44no—
137#432—42no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142#423—42no—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2312yes—
153C2×#97C2312yes—
154#400—21no—
155#400—21no—
156#400—22no—
157#400—22no—
158#399—21no—
159#399—21no—
160#399—22no—
161#399—22no—
162#398—22no—
163#398—22no—
164#396—22no—
165#396—22no—
166#395—21no—
167#395—21no—
168#395—22no—
169#395—22no—
170#393—21no—
171#393—21no—
172#393—22no—
173#393—22no—
174#391—21no—
175#391—21no—
176#391—22no—
177#391—22no—
178#391—22no—
179#391C2312yes—
180#391C2312yes—
181#390—21no—
182#390—22no—
183#390—21no—
184#389—22no—
185#389—22no—
186#389C2312yes—
187#389C2314yes—
188#389C2314yes—
189#389C2312yes—
190#387—22no—
191#387—22no—
192#386—22no—
193#386—22no—
194#386—22no—
195#386—22no—
196#385—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#384—21no—
201#384—21no—
202#384—22no—
203#384C2312yes—
204#382—21no—
205#382—22no—
206#382—21no—
207#382—21no—
208#380—22no—
209#380—22no—
210#380—21no—
211#380—22no—
212#380—22no—
213#380C2312yes—
214#380C2314yes—
215#380C2312yes—
216#380C2314yes—
217#380C2312yes—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C2×#95—21no—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#336C2314yes—
239#336C2314yes—
240C2×#41—24no—
241C2×#41—24no—
242C2×#41—42no—
243C22×Q8—22no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—44no—
247C22×Q8—84no—
248C22×Q8C2×D411yescharacteristic
249C22×Q8C2412yes—
250C22×D4C2×D411yescharacteristic
251C22×D4C2×D411yescharacteristic
252C22×D4C2×D411yescharacteristic
253C23×C4—22no—
254C23×C4—22no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2411yescharacteristic
264C2×M16—44no—
265C2×M16—42no—
266C22×C8C2414yes—
267C22×C8C2414yes—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—42no—
271C4⋊Q8—42no—
272C4⋊Q8—41no—
273C4⋊Q8—41no—
274C4⋊Q8—41no—
275C4⋊Q8—41no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278#125—44no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—42no—
282C22⋊Q8—42no—
283C22⋊Q8—42no—
284C22⋊Q8—42no—
285C22⋊Q8—44no—
286C22⋊Q8—84no—
287C22⋊Q8—84no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—42no—
291C4×Q8—41no—
292C4×Q8—41no—
293C4×Q8—44no—
294C4×D4—42no—
295C4×D4—42no—
296C4×D4—42no—
297C4×D4—41no—
298C4×D4—41no—
299C4×D4—41no—
300C4×D4—41no—
301C4×D4—41no—
302C4×D4—41no—
303#117—42no—
304#117—42no—
305#117—42no—
306#117—44no—
307#117—44no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2412yes—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4C2×D411yescharacteristic
350C2×C22⋊C4C2×D411yescharacteristic
351C2×C22⋊C4C2×D411yescharacteristic
352C2×C22⋊C4C2×D411yescharacteristic
353C2×C22⋊C4C2×D411yescharacteristic
354C2×C22⋊C4C2×D411yescharacteristic
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—44no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377#107—44no—
378#107—44no—
379#106—28no—
380#106—44no—
381C4⋊C8—24no—
382C4⋊C8—24no—
383C4⋊C8—42no—
384#103—44no—
385#103—44no—
386#103—44no—
387#103—44no—
388C22⋊C8—44no—
389C22⋊C8—44no—
390C22⋊C8—42no—
391#97—42no—
392#95—22no—
393#95—21no—
394#95—21no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409C24C22×D411yescharacteristic
410C24C22×D411yescharacteristic
411C24C22×D411yescharacteristic
412#41—416no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—41no—
422C2×Q8—42no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—82no—
428C2×Q8—84no—
429C2×Q8—816no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—41no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—82no—
474C22×C4—84no—
475C22×C4#13714yes—
476C22×C4#13714yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2511yescharacteristic
489M16—84no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C42−1+414yes—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—44no—
556C42—42no—
557C42—44no—
558C42—42no—
559C42—44no—
560C42H(5,𝔽2)12yes—
561C42H(5,𝔽2)12yes—
562C23—22no—
563C23—22no—
564C23—21no—
565C23—21no—
566C23—22no—
567C23—22no—
568C23—21no—
569C23—21no—
570C23—22no—
571C23—21no—
572C23—22no—
573C23—21no—
574C23—21no—
575C23C2×#13712yes—
576C23C2×#13712yes—
577C23C23×D411yescharacteristic
578Q8—44no—
579Q8—42no—
580Q8—42no—
581Q8—42no—
582Q8—42no—
583Q8—88no—
584Q8—84no—
585Q8—88no—
586D4—44no—
587D4—44no—
588D4—44no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—88no—
643C2×C4—88no—
644C2×C4#53412yes—
645C2×C4#53412yes—
646C2×C4C2×#13712yes—
647C2×C4C2×#13712yes—
648C2×C4C2×#13712yes—
649C2×C4C2×#13712yes—
650C2×C4C2×H(5,𝔽2)11yescharacteristic
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C2×C4C2×2−1+412yes—
653C2×C4#58512yes—
654C8—44no—
655C8—44no—
656C22—22no—
657C22—22no—
658C22—24no—
659C22—24no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—42no—
665C22—42no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—42no—
670C22—42no—
671C22C2×#53411yescharacteristic
672C22#348912yes—
673C22C22×#13711yescharacteristic
674C22C22×#13711yescharacteristic
675C22#358512yes—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—44no—
687C4—82no—
688C4—84no—
689C4—82no—
690C4—82no—
691C4—84no—
692C4#319514yes—
693C4#354712yes—
694C4#355011yescharacteristic
695C4#355111yescharacteristic
696C2—42no—
697C2—42no—
698C2—81no—
699C2#3737611yescharacteristic
700C2C2×#319511yescharacteristic
701C2#6075311yescharacteristic
702C2#6210812yes—
703C2#6211412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33148C2semidirect
4. #32695C2semidirect
5. #32645C2semidirect
6. #32268C2semidirect
7. #32037C2semidirect
8. #31794C2semidirect
9. #31334C2semidirect
10. #29691C2semidirect
11. #27628C2semidirect
12. #24846C2semidirect
13. #24711C2semidirect
14. #24302C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#349472
#331482
#326952
#326452
#322682
#320372
#317942
#313342
#296912
#276282
#248462
#247112
#243022
#218532