Finite Group Databasebeta

← #6519305

Subgroups — #6519305

Order 512.

1664 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic, complemented
2#34979C211yesmaximal, characteristic
3#29141C212yesmaximal, complemented
4#29132C212yesmaximal, complemented
5#29122C212yesmaximal, complemented
6#28726C212yesmaximal, complemented
7#27823C212yesmaximal, complemented
8#27776C212yesmaximal, complemented
9#27749C212yesmaximal, complemented
10#27727C212yesmaximal, complemented
11#27706C214yesmaximal, complemented
12#27278C214yesmaximal
13#21894C211yesmaximal, characteristic, complemented
14#21860C211yesmaximal, characteristic, complemented
15#20737C211yesmaximal, characteristic
16#20648C211yesmaximal, characteristic
17#20550C211yesmaximal, characteristic, complemented
18C2×#544—21no—
19C2×#533C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#489C2214yes—
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×#417C2212yes—
25#2868C2212yes—
26#2859C2212yes—
27#2660—22no—
28#2611—22no—
29#2568C2211yescharacteristic
30#2552—21no—
31#2552C2212yes—
32#2551—21no—
33#2551C2212yes—
34#2547C2211yescharacteristic
35#2546C2211yescharacteristic
36#2542C2211yescharacteristic
37#2499C2212yes—
38#2440—22no—
39#2440C2214yes—
40#2432C2212yes—
41#2415—21no—
42#2415—21no—
43#2395—22no—
44#2317—21no—
45C4×C4⋊Q8C2212yes—
46C4×#124C2212yes—
47C4×C22⋊Q8—21no—
48C4×C22⋊Q8—21no—
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2101C2214yes—
53#2099C2214yes—
54#2072C2214yes—
55#2071C2214yes—
56#2043C2214yes—
57#2042C2214yes—
58#2024C2214yes—
59#2024C2214yes—
60#1993C2214yes—
61#1990C2212yes—
62#1988C2212yes—
63#1981C2212yes—
64#1956C2212yes—
65#1956C2214yes—
66#1935C2214yes—
67#1935C2214yes—
68#1934C2214yes—
69#1933C2214yes—
70#1894C2214yes—
71#1872C2214yes—
72#1834C2214yes—
73#1825C2212yes—
74#1803C2211yescharacteristic
75#1803C2211yescharacteristic
76#1803C2212yes—
77#1803C2212yes—
78#1803C2212yes—
79#1803C2212yes—
80#1803C2212yes—
81#1803C2212yes—
82#1802C2211yescharacteristic
83#1802C2211yescharacteristic
84#1802C2212yes—
85#1802C2212yes—
86#1788C2212yes—
87#1770C2212yes—
88C8×C4⋊C4C2212yes—
89#1767C2212yes—
90C4×C4⋊C8C2212yes—
91C4×#103C2212yes—
92C4×#102C2212yes—
93#1753C2214yes—
94#1736C2212yes—
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97#544—48no—
98#533—24no—
99#533—24no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2314yes—
107C2×#125—22no—
108C2×#125—22no—
109C2×#125—21no—
110C2×#124C2312yes—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8C2312yes—
116C4×#41—24no—
117C4×#41—24no—
118C2×C4×Q8—21no—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×#117—21no—
127C2×#117—21no—
128C2×#117—21no—
129C2×#117D411yescharacteristic
130C2×#117D411yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2312yes—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137#489—48no—
138#431—24no—
139#423—24no—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144#417—44no—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×C22⋊C8C2312yes—
156C2×#97C2314yes—
157C2×C4×C8C2312yes—
158C2×C4×C8C2314yes—
159#400—22no—
160#399—22no—
161#398—22no—
162#398—24no—
163#398—24no—
164#397—22no—
165#397—22no—
166#397—22no—
167#395—24no—
168#395—24no—
169#394—22no—
170#394—22no—
171#391—21no—
172#391—21no—
173#391C2312yes—
174#391C2312yes—
175#391C2312yes—
176#390C2312yes—
177#389—22no—
178#389—22no—
179#389—21no—
180#389—21no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389C2314yes—
186#389C2314yes—
187#388—22no—
188#388—22no—
189#387—22no—
190#387—22no—
191#385—24no—
192#385—22no—
193#385—22no—
194#384—22no—
195#384—22no—
196#384—21no—
197#384—21no—
198#384C2312yes—
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2311yescharacteristic
202#384C2312yes—
203#384C2314yes—
204#382—22no—
205#382—21no—
206#382C2312yes—
207#380—21no—
208#380—21no—
209#380—21no—
210#380C2312yes—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223#376D412yes—
224C43C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#336C2314yes—
236#336C2314yes—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—41no—
242C2×#41—41no—
243C22×Q8—21no—
244C22×Q8—24no—
245C22×Q8C2412yes—
246C22×Q8C2411yescharacteristic
247C22×D4—21no—
248C22×D4C2411yescharacteristic
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×M16—24no—
257C2×M16—42no—
258C2×M16—42no—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C4⋊Q8—24no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—42no—
270C4⋊Q8—48no—
271C4⋊Q8—48no—
272C4⋊Q8—84no—
273#125—48no—
274#125—48no—
275#125—44no—
276#124—24no—
277#124—42no—
278C22⋊Q8—24no—
279C22⋊Q8—24no—
280C22⋊Q8—416no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—44no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—44no—
296C4×D4—44no—
297#117—24no—
298#117—24no—
299#117—22no—
300#117—22no—
301#117—44no—
302#117—44no—
303#117—44no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—82no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—21no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—44no—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42#4114yes—
367C2×C42#4114yes—
368C2×C42C2412yes—
369C2×C42C2412yes—
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372C4⋊C8—24no—
373C4⋊C8—24no—
374C4⋊C8—22no—
375C4⋊C8—22no—
376#103—24no—
377#103—44no—
378#103—44no—
379#103—42no—
380#103—44no—
381#103—44no—
382#103—44no—
383#102—28no—
384#102—24no—
385#102—42no—
386C22⋊C8—24no—
387#97—44no—
388C4×C8—24no—
389C4×C8—44no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—24no—
401#95—24no—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—21no—
407C24C22×D411yescharacteristic
408#41—28no—
409#41—44no—
410#41—44no—
411#41—44no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—48no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—84no—
428C2×Q8—84no—
429C2×Q82−1+412yes—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4H(5,𝔽2)12yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4#12414yes—
468C22×C4#12414yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2511yescharacteristic
481M16—84no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—48no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4AGL(1,ℤ/8)14yes—
531C4⋊C4#13714yes—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—42no—
538C22⋊C4—44no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C42—24no—
544C42—24no—
545C42—24no—
546C42—24no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—42no—
552C42—42no—
553C42—42no—
554C42—44no—
555C42—44no—
556C42#12414yes—
557C42#12414yes—
558C42C2×#4112yes—
559C42C2×#4112yes—
560C42C2×#4112yes—
561C42C2×#4112yes—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23C2×#12412yes—
576C23C23×D411yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—41no—
580Q8—41no—
581Q8—42no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—84no—
586Q8—168no—
587D4—22no—
588D4—22no—
589D4—48no—
590D4—41no—
591D4—41no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—48no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—164no—
642C2×C4C2×#12412yes—
643C2×C4C2×#12412yes—
644C2×C4C2×#12412yes—
645C2×C4C2×#12412yes—
646C2×C4C2×#12412yes—
647C2×C4C2×AGL(1,ℤ/8)14yes—
648C2×C4C2×AGL(1,ℤ/8)12yes—
649C2×C4C2×#13714yes—
650C2×C4C2×#13712yes—
651C2×C4#57712yes—
652C2×C4#57812yes—
653C2×C4C22×#4111yescharacteristic
654C2×C4C22×#4111yescharacteristic
655C2×C4C2×H(5,𝔽2)11yescharacteristic
656C2×C4C2×2−1+411yescharacteristic
657C2×C4#58511yescharacteristic
658C2×C4#58511yescharacteristic
659C8—42no—
660C8—42no—
661C8—44no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22C22×#12411yescharacteristic
677C22#348811yescharacteristic
678C22#348911yescharacteristic
679C22C22×AGL(1,ℤ/8)11yescharacteristic
680C22C22×#13711yescharacteristic
681C22C2×#57711yescharacteristic
682C22C2×#57811yescharacteristic
683C4—24no—
684C4—24no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—84no—
692C4—82no—
693C4—82no—
694C4—81no—
695C4—81no—
696C4—82no—
697C4—82no—
698C4#313714yes—
699C4#314114yes—
700C4#349211yescharacteristic
701C4#349311yescharacteristic
702C4#349411yescharacteristic
703C4#349511yescharacteristic
704C2—42no—
705C2—82no—
706C2#3554211yescharacteristic
707C2C2×#313711yescharacteristic
708C2C2×#314111yescharacteristic
709C2#6199911yescharacteristic
710C2#6200011yescharacteristic
711C2#6213111yescharacteristic
712C2#6213211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35275C2semidirect
3. #29141C2semidirect
4. #29132C2semidirect
5. #29122C2semidirect
6. #28726C2semidirect
7. #27823C2semidirect
8. #27776C2semidirect
9. #27749C2semidirect
10. #27727C2semidirect
11. #27706C2semidirect
13. #21894C2semidirect
14. #21860C2semidirect
17. #20550C2semidirect

Maximal subgroups

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

subgroupindex
#352752
#349792
#291412
#291322
#291222
#287262
#278232
#277762
#277492
#277272
#277062
#272782
#218942
#218602
#207372
#206482
#205502