Finite Group Databasebeta

← #6508810

Subgroups — #6508810

Order 512.

2103 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3092C211yesmaximal, characteristic, complemented
2C2×#2973C211yesmaximal, characteristic, complemented
3#35031C211yesmaximal, characteristic
4C2×#2487C211yesmaximal, characteristic, complemented
5#28103C214yesmaximal, complemented
6#27668C214yesmaximal, complemented
7#25244C214yesmaximal, complemented
8#25240C214yesmaximal, complemented
9#24607C212yesmaximal, complemented
10#24606C212yesmaximal, complemented
11#24553C212yesmaximal, complemented
12#24552C212yesmaximal, complemented
13#21944C211yesmaximal, characteristic, complemented
14#21769C211yesmaximal, characteristic, complemented
15#20700C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#3092—28no—
20C2×#484C2212yes—
21C2×#481C2212yes—
22C2×#483C2212yescomplemented
23C2×#480C2212yescomplemented
24#2973—28no—
25C2×#435C2214yes—
26C2×#435C2214yes—
27C2×#432C2212yes—
28C2×#428C2212yescomplemented
29C2×#417C2212yescomplemented
30C2×#407C2211yescharacteristic
31C2×#407C2211yescharacteristic
32C22×C22⋊C8C2212yes—
33#2855C2211yescharacteristic
34#2854C2211yescharacteristic
35#2848C2211yescharacteristic
36#2847C2211yescharacteristic
37#2622—21no—
38#2613—21no—
39#2602—22no—
40#2600—22no—
41#2487—22no—
42#2487—22no—
43#2487C2214yes—
44#2487C2214yes—
45#2485C2212yes—
46#2415C2212yes—
47#2414C2212yes—
48#2407C2212yes—
49C2×#400—21no—
50C2×#399—21no—
51#2322—21no—
52#2321—21no—
53C2×#389C2212yes—
54C2×#386—21no—
55C2×#385—21no—
56#2280C2211yescharacteristic
57C2×#380C2211yescharacteristic
58C2×#380C2212yes—
59#2088C2214yescomplemented
60#2086C2214yescomplemented
61#2077C2214yescomplemented
62#2076C2214yescomplemented
63#2067C2214yescomplemented
64#2066C2214yescomplemented
65#2036C2214yescomplemented
66#2034C2214yescomplemented
67#1944C2214yescomplemented
68#1943C2214yescomplemented
69#1936C2214yescomplemented
70#1935C2214yescomplemented
71#1895C2212yescomplemented
72#1894C2212yescomplemented
73#1893C2214yescomplemented
74#1892C2214yescomplemented
75#1874C2212yes—
76#1872C2212yes—
77#1854C2214yescomplemented
78#1853C2212yes—
79#1852C2212yes—
80#1825C2212yescomplemented
81#1824C2212yes—
82#1824C2212yes—
83#1823C2214yescomplemented
84#1809C2214yescomplemented
85#1808C2214yescomplemented
86#1807C2214yescomplemented
87#1788C2212yescomplemented
88#1736C2214yescomplemented
89C22×#41C2311yescharacteristic
90C23×D4D411yescharacteristic
91C24×C4—21no—
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C22×M16C2312yes—
95C22×M16C2312yes—
96C23×C8C2314yes—
97#525—24no—
98#525—24no—
99C2×#123—21no—
100C2×#123—21no—
101C2×C22⋊Q8C2311yescharacteristic
102C2×C22⋊Q8C2311yescharacteristic
103C2×C22⋊Q8C2312yes—
104C2×#121C2311yescharacteristic
105C2×#121C2311yescharacteristic
106C2×#121C2312yes—
107C2×#120—21no—
108C2×#120—21no—
109C2×C4×D4—24no—
110C2×C4×D4—24no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4D412yes—
114C2×C4×D4D412yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×#117C2311yescharacteristic
118C2×#117C2312yes—
119C22×C4⋊C4—24no—
120C22×C4⋊C4—24no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—22no—
132C22×C22⋊C4D411yescharacteristic
133C22×C22⋊C4D411yescharacteristic
134C22×C42C2311yescharacteristic
135#484—28no—
136#483—28no—
137#481—28no—
138#480—28no—
139#435—216no—
140#435—216no—
141#432—28no—
142#428—28no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2314yes—
148#417—28no—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153#407—24no—
154#407D418yescomplemented
155C2×C22⋊C8—28no—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×#97C2314yes—
162#400—24no—
163#400—24no—
164#399—24no—
165#399—24no—
166#397—22no—
167#397—22no—
168#394—22no—
169#394—22no—
170#394—22no—
171#394—22no—
172#392—22no—
173#392—22no—
174#391C2312yes—
175#390—22no—
176#390C2312yes—
177#389—22no—
178#389—22no—
179#389—22no—
180#389C2314yes—
181#389C2314yes—
182#389C2312yes—
183#389C2312yes—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#387—22no—
189#387—22no—
190#386—24no—
191#386—24no—
192#386—21no—
193#386—21no—
194#386—22no—
195#386—22no—
196#385—21no—
197#385—21no—
198#385—22no—
199#385—22no—
200#385—24no—
201#385—24no—
202#381—21no—
203#381—21no—
204#380—22no—
205#380—22no—
206#380—21no—
207#380—21no—
208#380C2312yes—
209#380C2314yes—
210#380C2312yes—
211#380C2314yes—
212#380C2312yes—
213#380C2312yes—
214#380C2311yescharacteristic
215#380C2311yescharacteristic
216#380C2312yes—
217#379—21no—
218#379—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C2×#95—21no—
222C2×#95—22no—
223C2×#95—22no—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232C25C2×D411yescharacteristic
233C25C2×D411yescharacteristic
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—44no—
237C22×Q8C2411yescharacteristic
238C22×D4—24no—
239C22×D4—24no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—42no—
245C22×D4C2×D412yes—
246C22×D4C2×D412yes—
247C22×D4C2411yescharacteristic
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4—84no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D414yes—
267C23×C4#4114yes—
268C23×C4C2412yes—
269C23×C4C2411yescharacteristic
270C23×C4C2411yescharacteristic
271C23×C4C2411yescharacteristic
272C2×M16—24no—
273C2×M16—44no—
274C2×M16—44no—
275C2×M16C2×D418yes—
276C22×C8—28no—
277C22×C8—44no—
278C22×C8—44no—
279C22×C8C2414yes—
280C22×C8C2414yes—
281#123—44no—
282#123—44no—
283C22⋊Q8—22no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8—44no—
288#121—22no—
289#121—22no—
290#121—22no—
291#121—22no—
292#121—44no—
293#120—44no—
294#120—44no—
295C4×D4—28no—
296C4×D4—28no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—44no—
302C4×D4—44no—
303#117—22no—
304#117—22no—
305#117—44no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—84no—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—44no—
359C2×C22⋊C4—84no—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—21no—
367C2×C42—22no—
368C2×C42C2×D412yes—
369C2×C42C2412yes—
370#107—28no—
371#107—44no—
372#106—28no—
373#106—44no—
374C4⋊C8—28no—
375#103—28no—
376#103—44no—
377#102—28no—
378#102—44no—
379C22⋊C8—24no—
380C22⋊C8—24no—
381C22⋊C8—28no—
382C22⋊C8C2×D418yes—
383C22⋊C8C2×D418yes—
384#97—28no—
385#95—24no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95—21no—
392#95—22no—
393#95—21no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—21no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24#12114yes—
410C24#13714yes—
411C24C22×D411yescharacteristic
412C24C22×D411yescharacteristic
413C24C22×D412yes—
414C24C22×D411yescharacteristic
415#41—48no—
416#41—84no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—24no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—416no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4—82no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4—82no—
481C22×C4—84no—
482C22×C4—84no—
483C22×C4—816no—
484C22×C4#12114yes—
485C22×C4#12114yes—
486C22×C4#12114yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4C2511yescharacteristic
500M16—48no—
501M16—84no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—88no—
508C2×C8C22×D418yes—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—24no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—82no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4—82no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—44no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C42—24no—
556C42—22no—
557C42—22no—
558C42—42no—
559C42—82no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—28no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—22no—
577C23—28no—
578C23—44no—
579C23—44no—
580C23—48no—
581C23—48no—
582C23—48no—
583C23—42no—
584C23—41no—
585C23—42no—
586C23—41no—
587C23—41no—
588C23—82no—
589C23—84no—
590C23—82no—
591C23C2×#12112yes—
592C23C2×#12112yes—
593C23C2×#12112yes—
594C23C2×#13714yes—
595C23C2×#13712yes—
596C23#57512yes—
597C23C23×D411yescharacteristic
598Q8—42no—
599Q8—42no—
600D4—48no—
601D4—48no—
602D4—42no—
603D4—42no—
604D4—84no—
605D4—84no—
606D4—84no—
607D4—164no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—41no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—82no—
654C2×C4—82no—
655C2×C4—88no—
656C2×C4—88no—
657C2×C4—88no—
658C2×C4—88no—
659C2×C4C2×#12112yes—
660C2×C4C2×#12112yes—
661C2×C4C2×#12112yes—
662C2×C4C23×D411yescharacteristic
663C2×C4C22×#4111yescharacteristic
664C2×C4#58514yes—
665C8—28no—
666C8—44no—
667C22—24no—
668C22—24no—
669C22—22no—
670C22—22no—
671C22—24no—
672C22—24no—
673C22—22no—
674C22—22no—
675C22—24no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—416no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—88no—
690C22—82no—
691C22—82no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22#305614yes—
696C22C22×#12111yescharacteristic
697C22#348512yes—
698C22C22×#13711yescharacteristic
699C22C2×#57511yescharacteristic
700C22#358512yes—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—42no—
705C4—42no—
706C4—42no—
707C4—82no—
708C4—82no—
709C4—84no—
710C4—81no—
711C4—84no—
712C4—81no—
713C4—81no—
714C4#348512yes—
715C4#349112yes—
716C2—24no—
717C2—42no—
718C2—42no—
719C2—44no—
720C2—161no—
721C2#3544711yescharacteristic
722C2C2×#305611yescharacteristic
723C2#6045411yescharacteristic
724C2#6170912yes—
725C2#6171312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35031C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3092C2semidirect
2. C2×#2973C2semidirect
4. C2×#2487C2semidirect
5. #28103C2semidirect
6. #27668C2semidirect
7. #25244C2semidirect
8. #25240C2semidirect
9. #24607C2semidirect
10. #24606C2semidirect
11. #24553C2semidirect
12. #24552C2semidirect
13. #21944C2semidirect
14. #21769C2semidirect
15. #20700C2semidirect
22. C2×#483C22semidirect
23. C2×#480C22semidirect
28. C2×#428C22semidirect
29. C2×#417C22semidirect
59. #2088C22semidirect
60. #2086C22semidirect
61. #2077C22semidirect
62. #2076C22semidirect
63. #2067C22semidirect
64. #2066C22semidirect
65. #2036C22semidirect
66. #2034C22semidirect
67. #1944C22semidirect
68. #1943C22semidirect
69. #1936C22semidirect
70. #1935C22semidirect
71. #1895C22semidirect
72. #1894C22semidirect
73. #1893C22semidirect
74. #1892C22semidirect
77. #1854C22semidirect
80. #1825C22semidirect
83. #1823C22semidirect
84. #1809C22semidirect
85. #1808C22semidirect
86. #1807C22semidirect
87. #1788C22semidirect
88. #1736C22semidirect
154. #407D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30922
C2×#29732
#350312
C2×#24872
#281032
#276682
#252442
#252402
#246072
#246062
#245532
#245522
#219442
#217692
#207002