Finite Group Databasebeta

← #6475931

Subgroups — #6475931

Order 512.

2440 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#35310C211yesmaximal, characteristic, complemented
2C2×#2311C211yesmaximal, characteristic
3#32983C212yesmaximal, complemented
4#32883C212yesmaximal
5#29964C212yesmaximal, complemented
6#29726C212yesmaximal, complemented
7#29112C211yesmaximal, characteristic, complemented
8#29104C211yesmaximal, characteristic, complemented
9#28311C211yesmaximal, characteristic, complemented
10#28304C211yesmaximal, characteristic, complemented
11#28123C212yesmaximal, complemented
12#28121C212yesmaximal, complemented
13#27595C212yesmaximal, complemented
14#24515C214yesmaximal, complemented
15#23944C212yesmaximal, complemented
16#22057C214yesmaximal, complemented
17#20716C211yesmaximal, characteristic, complemented
18C23×#41C2211yescharacteristic
19C2×#541C2211yescharacteristic, complemented
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C2×#484C2212yes—
23C2×#484C2212yes—
24C2×#452C2214yes—
25C2×#452C2214yes—
26C2×#451C2212yes—
27C2×#450C2212yes—
28C2×#433C2212yescomplemented
29#2857C2211yescharacteristic
30#2851C2211yescharacteristic
31#2848C2212yes—
32#2629C2211yescharacteristic, complemented
33#2625C2212yescomplemented
34#2621C2211yescharacteristic, complemented
35#2480—22no—
36#2478—22no—
37#2475—22no—
38#2473—22no—
39#2323C2212yescomplemented
40#2314C2211yescharacteristic, complemented
41#2311—22no—
42#2311—22no—
43#2311C2212yes—
44#2311C2212yes—
45#2311C2212yes—
46#2311C2212yes—
47C4×C22⋊Q8C2212yes—
48C2×#391C2211yescharacteristic
49C2×#391C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#386—22no—
53C2×#380—22no—
54#2095C2212yescomplemented
55#2093C2212yescomplemented
56#2074C2212yescomplemented
57#2073C2212yescomplemented
58#2068C2214yescomplemented
59#2067C2214yescomplemented
60#2056C2214yescomplemented
61#2055C2214yescomplemented
62#2026C2212yescomplemented
63#2025C2212yescomplemented
64#2016C2212yescomplemented
65#2013C2212yescomplemented
66#1973C2214yescomplemented
67#1969C2214yescomplemented
68#1935C2214yescomplemented
69#1934C2214yescomplemented
70#1928C2214yescomplemented
71#1925C2214yescomplemented
72#1880C2214yescomplemented
73#1878C2214yescomplemented
74#1872C2214yescomplemented
75#1872C2214yescomplemented
76#1852C2214yescomplemented
77#1852C2214yescomplemented
78#1824C2212yes—
79#1824C2212yes—
80#1818C2212yescomplemented
81#1798C2212yescomplemented
82#1797C2212yes—
83#1788C2214yes—
84#1788C2214yes—
85#1787C2212yes—
86#1786C2212yes—
87#1779C2212yes—
88#1750C2212yescomplemented
89#1749C2211yescharacteristic
90C4×C22⋊C8C2211yescharacteristic
91C22×#41—28no—
92C22×#41—21no—
93C22×#41—21no—
94C22×#41—44no—
95C22×#41—44no—
96C22×#41C2312yes—
97C22×#41C2312yes—
98C23×Q8C2311yescharacteristic
99C23×D4—21no—
100C23×D4C2311yescharacteristic
101C24×C4—21no—
102C24×C4—21no—
103C24×C4C2311yescharacteristic
104C22×Q16C2314yes—
105C22×SD16C2314yes—
106#541—44no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#126—22no—
109C2×#125—22no—
110C2×#125C2311yescharacteristic
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×#117—44no—
116C2×#117D412yes—
117C2×#117D412yes—
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—44no—
126C22×C4⋊C4D412yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C22⋊C4—22no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4D412yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#484—44no—
137#484—44no—
138#452—216no—
139#452—48no—
140#451—44no—
141#450—44no—
142#433—44no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×C4⋊C8C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×#97C2312yes—
159C2×C4×C8C2312yes—
160#400—22no—
161#398—22no—
162#398—22no—
163#398—22no—
164#397—22no—
165#397—22no—
166#397—22no—
167#396—22no—
168#391—22no—
169#391—22no—
170#391C2312yes—
171#391C2312yes—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#391C2312yes—
176#390—22no—
177#390—22no—
178#390C2312yes—
179#390C2312yes—
180#390C2312yes—
181#390C2312yes—
182#389—22no—
183#389—22no—
184#389C2312yes—
185#389C2312yes—
186#388—22no—
187#388—22no—
188#388—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#386—24no—
194#386—24no—
195#386—24no—
196#386—24no—
197#385—22no—
198#385—22no—
199#385—22no—
200#385—22no—
201#385—22no—
202#384C2311yescharacteristic
203#382C2311yescharacteristic
204#380—22no—
205#380—24no—
206#380—24no—
207#380—24no—
208#380—24no—
209#380C2312yes—
210#380C2312yes—
211#380C2312yes—
212#379—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—22no—
216C2×#95—22no—
217C2×#95—22no—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#336C2314yes—
223#336C2312yes—
224#336C2312yes—
225C25—21no—
226C25C2×D411yescharacteristic
227C2×#41—28no—
228C2×#41—28no—
229C2×#41—24no—
230C2×#41—24no—
231C2×#41—24no—
232C2×#41—28no—
233C2×#41—44no—
234C2×#41—44no—
235C2×#41—44no—
236C2×#41—464no—
237C2×#41—44no—
238C2×#41—44no—
239C2×#41—44no—
240C2×#41—44no—
241C2×#41#4118yes—
242C22×Q8—24no—
243C22×Q8—21no—
244C22×Q8—24no—
245C22×Q8—42no—
246C22×Q8C2412yes—
247C22×Q8C2412yes—
248C22×D4—28no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—24no—
252C22×D4—21no—
253C22×D4—24no—
254C22×D4—42no—
255C22×D4—44no—
256C22×D4—44no—
257C22×D4C2412yes—
258C23×C4—24no—
259C23×C4—24no—
260C23×C4—24no—
261C23×C4—28no—
262C23×C4—21no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—22no—
269C23×C4—44no—
270C23×C4—44no—
271C23×C4—44no—
272C23×C4—84no—
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D412yes—
277C23×C4C2×D412yes—
278C23×C4C2411yescharacteristic
279C23×C4C2411yescharacteristic
280C23×C4C2411yescharacteristic
281C2×Q16—28no—
282C2×Q16—44no—
283C2×Q16—44no—
284C2×SD16—28no—
285C2×SD16—44no—
286C2×SD16—44no—
287C22×C8C2414yes—
288C22×C8C2414yes—
289C4⋊Q8—42no—
290#126—48no—
291#125—42no—
292#125—48no—
293C22⋊Q8—28no—
294C22⋊Q8—28no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C4×Q8—44no—
298#117—24no—
299#117—24no—
300#117—42no—
301#117—42no—
302#117—44no—
303#117—44no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—84no—
346C2×C22⋊C4—84no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—42no—
354C2×C42—42no—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—44no—
362#107—44no—
363C4⋊C8—44no—
364#103—28no—
365#103—44no—
366#103—44no—
367#103—44no—
368#103—44no—
369#102—28no—
370#102—44no—
371#102—44no—
372C22⋊C8—28no—
373C22⋊C8—28no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376#97—42no—
377C4×C8—42no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
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—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—24no—
397C24—24no—
398C24—22no—
399C24—42no—
400C24—44no—
401C24—42no—
402C24—42no—
403C24—44no—
404C24C22×D411yescharacteristic
405C24C22×D411yescharacteristic
406C24C22×D411yescharacteristic
407#41—416no—
408#41—416no—
409#41—416no—
410#41—432no—
411#41—432no—
412#41—416no—
413C2×Q8—24no—
414C2×Q8—22no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—416no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8C2×#4114yes—
425C2×D4—24no—
426C2×D4—28no—
427C2×D4—28no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—416no—
432C2×D4—42no—
433C2×D4—432no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—42no—
439C2×D4C2×#4114yes—
440C22×C4—232no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—28no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—28no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—432no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—84no—
486C22×C4—84no—
487C22×C4—816no—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C2×#4114yes—
496C22×C4H(5,𝔽2)14yes—
497C22×C4H(5,𝔽2)14yes—
498C22×C4C2511yescharacteristic
499Q16—88no—
500SD16—88no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C22⋊C4—84no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—42no—
549C42—42no—
550C42—44no—
551C42—82no—
552C23—22no—
553C23—21no—
554C23—21no—
555C23—28no—
556C23—22no—
557C23—22no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—22no—
564C23—24no—
565C23—48no—
566C23—48no—
567C23—44no—
568C23—48no—
569C23—416no—
570C23—44no—
571C23—44no—
572C23—42no—
573C23—42no—
574C23—44no—
575C23—88no—
576C23—84no—
577C23—84no—
578C23#52512yes—
579C23#53412yes—
580C23#53512yes—
581C23#57814yes—
582C23C23×D411yescharacteristic
583Q8—28no—
584Q8—48no—
585Q8—44no—
586Q8—44no—
587Q8—44no—
588D4—28no—
589D4—48no—
590D4—416no—
591D4—416no—
592D4—44no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—416no—
619C2×C4—416no—
620C2×C4—48no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—88no—
644C2×C4—88no—
645C2×C4#52512yes—
646C2×C4#53412yes—
647C2×C4#53512yes—
648C2×C4C22×#4112yes—
649C2×C4C2×H(5,𝔽2)12yes—
650C2×C4C2×H(5,𝔽2)12yes—
651C8—44no—
652C8—44no—
653C22—24no—
654C22—22no—
655C22—24no—
656C22—22no—
657C22—24no—
658C22—24no—
659C22—22no—
660C22—22no—
661C22—416no—
662C22—42no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—42no—
668C22—48no—
669C22—44no—
670C22—816no—
671C22—84no—
672C22—82no—
673C22—82no—
674C22#322314yes—
675C22C2×#52511yescharacteristic
676C22C2×#53411yescharacteristic
677C22C2×#53511yescharacteristic
678C22C2×#57812yes—
679C22#358512yes—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—44no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4#349014yes—
695C2—24no—
696C2—42no—
697C2—42no—
698C2—84no—
699C2#3505011yescharacteristic
700C2#6058912yes—
701C2#6076412yes—
702C2C2×#322311yescharacteristic
703C2#6080611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35310C2semidirect
3. #32983C2semidirect
5. #29964C2semidirect
6. #29726C2semidirect
7. #29112C2semidirect
8. #29104C2semidirect
9. #28311C2semidirect
10. #28304C2semidirect
11. #28123C2semidirect
12. #28121C2semidirect
13. #27595C2semidirect
14. #24515C2semidirect
15. #23944C2semidirect
16. #22057C2semidirect
17. #20716C2semidirect
19. C2×#541C22semidirect
28. C2×#433C22semidirect
32. #2629C22semidirect
33. #2625C22semidirect
34. #2621C22semidirect
39. #2323C22semidirect
40. #2314C22semidirect
54. #2095C22semidirect
55. #2093C22semidirect
56. #2074C22semidirect
57. #2073C22semidirect
58. #2068C22semidirect
59. #2067C22semidirect
60. #2056C22semidirect
61. #2055C22semidirect
62. #2026C22semidirect
63. #2025C22semidirect
64. #2016C22semidirect
65. #2013C22semidirect
66. #1973C22semidirect
67. #1969C22semidirect
68. #1935C22semidirect
69. #1934C22semidirect
70. #1928C22semidirect
71. #1925C22semidirect
72. #1880C22semidirect
73. #1878C22semidirect
74. #1872C22semidirect
75. #1872C22semidirect
76. #1852C22semidirect
77. #1852C22semidirect
80. #1818C22semidirect
81. #1798C22semidirect
88. #1750C22semidirect

Maximal subgroups

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

subgroupindex
#353102
C2×#23112
#329832
#328832
#299642
#297262
#291122
#291042
#283112
#283042
#281232
#281212
#275952
#245152
#239442
#220572
#207162