Finite Group Databasebeta

← #6519283

Subgroups — #6519283

Order 512.

1544 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#34979C211yesmaximal, characteristic
3#29150C212yesmaximal, complemented
4#29131C212yesmaximal, complemented
5#29123C212yesmaximal, complemented
6#28721C212yesmaximal, complemented
7#27816C212yesmaximal, complemented
8#27768C212yesmaximal, complemented
9#27757C212yesmaximal, complemented
10#27736C212yesmaximal, complemented
11#27724C214yesmaximal, complemented
12#27272C214yesmaximal
13#21894C211yesmaximal, characteristic, complemented
14#21860C211yesmaximal, characteristic, complemented
15#20737C211yesmaximal, characteristic
16#20648C211yesmaximal, characteristic
17#20550C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#488C2214yes—
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#417C2212yes—
24#2859C2212yes—
25#2859C2212yes—
26#2647—22no—
27#2611—22no—
28#2571—21no—
29#2571C2212yes—
30#2568C2211yescharacteristic
31#2552—21no—
32#2552C2212yes—
33#2547C2211yescharacteristic
34#2546C2211yescharacteristic
35#2542C2211yescharacteristic
36#2499C2212yes—
37#2442—22no—
38#2442C2214yes—
39#2432C2212yes—
40#2415—21no—
41#2415—21no—
42#2395—22no—
43#2317—21no—
44#2316—21no—
45C4×C4⋊Q8C2212yes—
46C4×#124C2212yes—
47C4×C22⋊Q8—21no—
48C4×C22⋊Q8—21no—
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2101C2214yes—
53#2100C2214yes—
54#2074C2214yes—
55#2073C2214yes—
56#2043C2214yes—
57#2042C2214yes—
58#2026C2214yes—
59#2026C2214yes—
60#1991C2214yes—
61#1990C2212yes—
62#1988C2212yes—
63#1981C2212yes—
64#1960C2214yes—
65#1956C2212yes—
66#1936C2214yes—
67#1935C2214yes—
68#1933C2214yes—
69#1933C2214yes—
70#1894C2214yes—
71#1872C2214yes—
72#1836C2214yes—
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#533—24no—
98#533—24no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125—21no—
103C2×#125—22no—
104C2×#125C2314yes—
105C2×#124C2312yes—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8C2312yes—
110C4×#41—24no—
111C4×#41—24no—
112C2×C4×Q8—21no—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—21no—
121C2×#117—21no—
122C2×#117—21no—
123C2×#117D411yescharacteristic
124C2×#117D411yescharacteristic
125C2×#117C2311yescharacteristic
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2312yes—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131#488—48no—
132#431—24no—
133#423—24no—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2312yes—
138#417—44no—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×C22⋊C8C2312yes—
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2314yes—
153#400—22no—
154#400—24no—
155#400—24no—
156#399—22no—
157#398—22no—
158#398—24no—
159#398—24no—
160#397—22no—
161#397—22no—
162#397—22no—
163#394—22no—
164#394—22no—
165#391—21no—
166#391—21no—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#390C2312yes—
171#389—22no—
172#389—22no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389C2314yes—
184#389C2314yes—
185#388—22no—
186#388—22no—
187#387—22no—
188#387—22no—
189#385—24no—
190#385—22no—
191#385—22no—
192#384—22no—
193#384—21no—
194#384—21no—
195#384C2312yes—
196#384C2312yes—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2312yes—
200#382—22no—
201#382—22no—
202#382—21no—
203#382C2312yes—
204#382C2314yes—
205#380—21no—
206#380—21no—
207#380—21no—
208#380—24no—
209#380C2312yes—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222#376D412yes—
223C43C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#336C2314yes—
235#336C2314yes—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—41no—
241C2×#41—41no—
242C22×Q8—21no—
243C22×Q8C2412yes—
244C22×Q8C2411yescharacteristic
245C22×D4—21no—
246C22×D4C2411yescharacteristic
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×M16—24no—
255C2×M16—42no—
256C2×M16—42no—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—42no—
266#125—44no—
267#125—44no—
268#125—44no—
269#125—84no—
270#124—24no—
271#124—42no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—44no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—44no—
289C4×D4—44no—
290#117—22no—
291#117—22no—
292#117—42no—
293#117—42no—
294#117—44no—
295#117—44no—
296#117—44no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—82no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—44no—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42#4114yes—
359C2×C42#4114yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C4⋊C8—24no—
365C4⋊C8—24no—
366C4⋊C8—22no—
367C4⋊C8—22no—
368#103—24no—
369#103—44no—
370#103—44no—
371#103—42no—
372#103—44no—
373#103—44no—
374#103—44no—
375#102—28no—
376#102—24no—
377#102—42no—
378C22⋊C8—24no—
379#97—44no—
380C4×C8—24no—
381C4×C8—44no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—21no—
401C24C22×D411yescharacteristic
402#41—28no—
403#41—44no—
404#41—44no—
405#41—44no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q82−1+412yes—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4H(5,𝔽2)12yes—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4#12414yes—
459C22×C4#12414yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2511yescharacteristic
472M16—84no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—42no—
540C42—42no—
541C42—42no—
542C42—42no—
543C42—42no—
544C42—44no—
545C42—44no—
546C42#12414yes—
547C42#12414yes—
548C42C2×#4112yes—
549C42C2×#4112yes—
550C42C2×#4112yes—
551C42C2×#4112yes—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—22no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23C2×#12412yes—
566C23C23×D411yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—41no—
570Q8—41no—
571Q8—42no—
572Q8—42no—
573Q8—42no—
574Q8—42no—
575Q8—84no—
576D4—22no—
577D4—22no—
578D4—48no—
579D4—41no—
580D4—41no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—48no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—164no—
633C2×C4C2×#12412yes—
634C2×C4C2×#12412yes—
635C2×C4C2×#12412yes—
636C2×C4C2×#12412yes—
637C2×C4C2×#12412yes—
638C2×C4#57512yes—
639C2×C4#57512yes—
640C2×C4#57712yes—
641C2×C4#57812yes—
642C2×C4C22×#4111yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4C2×2−1+411yescharacteristic
646C2×C4#58511yescharacteristic
647C2×C4#58511yescharacteristic
648C8—42no—
649C8—42no—
650C8—44no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22C22×#12411yescharacteristic
666C22#348811yescharacteristic
667C22#348911yescharacteristic
668C22C2×#57511yescharacteristic
669C22C2×#57511yescharacteristic
670C22C2×#57711yescharacteristic
671C22C2×#57811yescharacteristic
672C4—22no—
673C4—22no—
674C4—22no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—82no—
681C4—84no—
682C4—82no—
683C4—81no—
684C4—81no—
685C4—82no—
686C4—82no—
687C4#313714yes—
688C4#314214yes—
689C4#349211yescharacteristic
690C4#349311yescharacteristic
691C4#349411yescharacteristic
692C4#349511yescharacteristic
693C2—42no—
694C2—82no—
695C2#3554211yescharacteristic
696C2C2×#313711yescharacteristic
697C2C2×#314211yescharacteristic
698C2#6198511yescharacteristic
699C2#6198611yescharacteristic
700C2#6214511yescharacteristic
701C2#6214611yescharacteristic
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. #35286C2semidirect
3. #29150C2semidirect
4. #29131C2semidirect
5. #29123C2semidirect
6. #28721C2semidirect
7. #27816C2semidirect
8. #27768C2semidirect
9. #27757C2semidirect
10. #27736C2semidirect
11. #27724C2semidirect
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
#352862
#349792
#291502
#291312
#291232
#287212
#278162
#277682
#277572
#277362
#277242
#272722
#218942
#218602
#207372
#206482
#205502