Finite Group Databasebeta

← #6986367

Subgroups — #6986367

Order 512.

1651 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
1#36445C211yesmaximal, characteristic
2#35280C211yesmaximal, characteristic
3#32771C212yesmaximal, complemented
4#30721C214yesmaximal, complemented
5#30577C214yesmaximal, complemented
6#30250C212yesmaximal, complemented
7#30116C212yesmaximal, complemented
8#29664C211yesmaximal, characteristic, complemented
9#29165C212yesmaximal, complemented
10#28232C212yesmaximal, complemented
11#27905C211yesmaximal, characteristic, complemented
12#27680C211yesmaximal, characteristic, complemented
13#25394C212yesmaximal, complemented
14#25060C212yesmaximal, complemented
15#24191C212yesmaximal, complemented
16#23571C211yesmaximal, characteristic, complemented
17#22473C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#483C2214yes—
22C2×#470C2214yes—
23C2×#470C2214yes—
24C2×#462C2212yes—
25C2×C4×Q16C2212yes—
26C2×#436C2212yes—
27#2857C2212yes—
28#2857C2211yescharacteristic
29#2857C2212yes—
30#2857C2212yes—
31#2629—22no—
32#2582—21no—
33#2582C2212yes—
34#2542C2212yes—
35#2542C2211yescharacteristic
36#2538—21no—
37#2538C2212yes—
38#2538C2211yescharacteristic
39#2538C2212yes—
40#2454—22no—
41#2454—21no—
42#2441—21no—
43#2441—22no—
44#2417—22no—
45#2417C2214yes—
46Q8×C4⋊C4—21no—
47Q8×C4⋊C4C2211yescharacteristic
48#2341—21no—
49#2316—21no—
50#2311—21no—
51C4×C22⋊Q8C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53C2×#384C2211yescharacteristic
54#2099C2212yes—
55#2096C2212yes—
56#2096C2212yes—
57#2096C2212yes—
58#2096C2214yes—
59#2084C2214yes—
60#2057C2211yescharacteristic
61#2043C2214yes—
62#2043C2211yescharacteristic
63#2036C2212yes—
64#2035C2214yes—
65#2035C2214yes—
66#2035C2214yes—
67#2024C2211yescharacteristic
68#2024C2212yes—
69#2016C2214yes—
70#2016C2211yescharacteristic
71#2016C2212yes—
72#2001C2212yes—
73#2001C2214yes—
74#2001C2214yes—
75#1984C2212yes—
76#1956C2212yes—
77#1948C2212yes—
78#1927C2214yes—
79#1920C2214yes—
80#1894C2214yes—
81#1878C2214yes—
82#1872C2212yes—
83#1870C2214yes—
84#1863C2214yes—
85#1853C2212yes—
86#1844C2212yes—
87#1817C2212yes—
88#1809C2214yes—
89#1803C2212yes—
90#1803C2212yes—
91#1797C2211yescharacteristic
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1728C2212yes—
95C22×#41D411yescharacteristic
96C22×Q16C2312yes—
97C22×Q16C2314yes—
98C22×M16C2314yes—
99#557—44no—
100#533—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#125—21no—
106C2×#125—22no—
107C2×#125—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2311yescharacteristic
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2314yes—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2312yes—
117C4×#41—44no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×D4D412yes—
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2312yes—
130C2×#117—21no—
131C2×#117D411yescharacteristic
132C2×#117D412yes—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#483—48no—
139#470—48no—
140#470—48no—
141#462—44no—
142C4×Q16—44no—
143#436—44no—
144C2×#107C2312yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2314yes—
161C2×C4×C8C2312yes—
162#399—22no—
163#399—22no—
164#399—22no—
165#398—24no—
166#398—24no—
167#396—24no—
168#396—24no—
169#396—22no—
170#396—22no—
171#396—22no—
172#391—22no—
173#391—22no—
174#391C2314yes—
175#391C2312yes—
176#391C2311yescharacteristic
177#390—22no—
178#390—21no—
179#390—22no—
180#390—21no—
181#390—22no—
182#389—22no—
183#389—22no—
184#389—21no—
185#389—22no—
186#389—21no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389—22no—
191#389—21no—
192#389—22no—
193#389—21no—
194#389—22no—
195#389C2311yescharacteristic
196#389C2314yes—
197#387—21no—
198#387—22no—
199#386—24no—
200#385—21no—
201#385—22no—
202#384—21no—
203#384—22no—
204#384—21no—
205#384—21no—
206#384C2311yescharacteristic
207#384C2312yes—
208#384C2312yes—
209#384C2312yes—
210#380—22no—
211#380—22no—
212#380—22no—
213#380—21no—
214#380—22no—
215#380—21no—
216#380—22no—
217#380—24no—
218#380C2312yes—
219#380C2314yes—
220#380C2311yescharacteristic
221C4×C4⋊C4—21no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C22⋊C4—22no—
226C4×C22⋊C4—21no—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2314yes—
236#336C2314yes—
237C2×#41—24no—
238C2×#41—24no—
239C2×#41—42no—
240C22×Q8—21no—
241C22×Q8C2×D411yescharacteristic
242C22×Q8C2412yes—
243C22×Q8C2411yescharacteristic
244C22×Q8C2412yes—
245C22×D4C2×D411yescharacteristic
246C22×D4C2×D412yes—
247C23×C4—22no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2411yescharacteristic
253C23×C4C2412yes—
254C2×Q16—24no—
255C2×Q16—42no—
256C2×Q16—42no—
257C2×Q16—44no—
258C2×Q16—44no—
259C2×Q16—44no—
260C2×M16—28no—
261C2×M16—44no—
262C2×M16—44no—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C4⋊Q8—24no—
267C4⋊Q8—42no—
268C4⋊Q8—42no—
269C4⋊Q8—44no—
270#125—44no—
271#125—48no—
272#125—44no—
273C22⋊Q8—22no—
274C22⋊Q8—24no—
275C22⋊Q8—41no—
276C22⋊Q8—42no—
277C22⋊Q8—42no—
278C22⋊Q8—42no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C22⋊Q8—44no—
285C22⋊Q8—84no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—42no—
291C4×Q8—44no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×D4—24no—
296C4×D4—42no—
297C4×D4—44no—
298C4×D4—44no—
299#117—44no—
300#117—42no—
301#117—44no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—82no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—44no—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370#107—42no—
371#106—28no—
372#106—44no—
373#106—44no—
374C4⋊C8—24no—
375#103—24no—
376#103—28no—
377#103—24no—
378#103—42no—
379#103—42no—
380#103—44no—
381#103—44no—
382#103—44no—
383#103—44no—
384C22⋊C8—24no—
385C22⋊C8—42no—
386C22⋊C8—44no—
387C4×C8—24no—
388#95—21no—
389#95—22no—
390#95—21no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—22no—
396#95—21no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401C24C22×D412yes—
402C24C22×D411yescharacteristic
403#41—416no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8#13714yes—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—42no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—42no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4#13714yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2511yescharacteristic
471Q16—48no—
472Q16—88no—
473M16—88no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8H(5,𝔽2)14yes—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—81no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C42−1+412yes—
524C22⋊C4—24no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—81no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—24no—
541C42—42no—
542C42—42no—
543C42—44no—
544C42—44no—
545C42—42no—
546C42—44no—
547C42—42no—
548C42H(5,𝔽2)14yes—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23C2×#13712yes—
563C23C23×D411yescharacteristic
564Q8—28no—
565Q8—42no—
566Q8—42no—
567Q8—42no—
568Q8—44no—
569Q8—44no—
570Q8—42no—
571Q8—42no—
572Q8—42no—
573Q8—84no—
574D4—48no—
575D4—44no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—41no—
627C2×C4—84no—
628C2×C4—164no—
629C2×C4#53412yes—
630C2×C4#53412yes—
631C2×C4C2×#13714yes—
632C2×C4C2×#13712yes—
633C2×C4C2×#13712yes—
634C2×C4C2×#13712yes—
635C2×C4#57712yes—
636C2×C4C2×H(5,𝔽2)12yes—
637C2×C4C2×H(5,𝔽2)12yes—
638C2×C4C2×2−1+411yescharacteristic
639C2×C4#58511yescharacteristic
640C8—24no—
641C8—44no—
642C8—42no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22C2×#53411yescharacteristic
658C22#348911yescharacteristic
659C22#349811yescharacteristic
660C22C22×#13711yescharacteristic
661C22C22×#13711yescharacteristic
662C22C2×#57711yescharacteristic
663C22#358611yescharacteristic
664C4—22no—
665C4—24no—
666C4—22no—
667C4—22no—
668C4—42no—
669C4—41no—
670C4—42no—
671C4—42no—
672C4—81no—
673C4—82no—
674C4—82no—
675C4—81no—
676C4—82no—
677C4—84no—
678C4—81no—
679C4—82no—
680C4#319514yes—
681C4#354212yes—
682C4#354812yes—
683C2—42no—
684C2—82no—
685C2#4290911yescharacteristic
686C2C2×#319511yescharacteristic
687C2#6076311yescharacteristic
688C2#6211311yescharacteristic
689C2#6214711yescharacteristic
690C2#6242211yescharacteristic
691C2#6245411yescharacteristic
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. #32771C2semidirect
4. #30721C2semidirect
5. #30577C2semidirect
6. #30250C2semidirect
7. #30116C2semidirect
8. #29664C2semidirect
9. #29165C2semidirect
10. #28232C2semidirect
11. #27905C2semidirect
12. #27680C2semidirect
13. #25394C2semidirect
14. #25060C2semidirect
15. #24191C2semidirect
16. #23571C2semidirect

Maximal subgroups

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

subgroupindex
#364452
#352802
#327712
#307212
#305772
#302502
#301162
#296642
#291652
#282322
#279052
#276802
#253942
#250602
#241912
#235712
#224732