Finite Group Databasebeta

← #6441439

Subgroups — #6441439

Order 512.

1838 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35313C211yesmaximal, characteristic, complemented
2#34268C211yesmaximal, characteristic
3#27088C214yesmaximal, complemented
4#27087C214yesmaximal, complemented
5#27085C214yesmaximal, complemented
6#27082C214yesmaximal, complemented
7#23738C211yesmaximal, characteristic
8#23545C211yesmaximal, characteristic
9#23512C211yesmaximal, characteristic
10#23425C211yesmaximal, characteristic
11C2×#1971C211yesmaximal, characteristic, complemented
12#21064C211yesmaximal, characteristic, complemented
13#21061C211yesmaximal, characteristic, complemented
14#21038C211yesmaximal, characteristic, complemented
15#21021C211yesmaximal, characteristic, complemented
16#20935C211yesmaximal, characteristic, complemented
17#20927C211yesmaximal, characteristic
18#20651C211yesmaximal, characteristic, complemented
19#20411C211yesmaximal, characteristic
20C22×#137C2211yescharacteristic
21C23×Q16C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×C4×#41C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C2×#416C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#97C2211yescharacteristic
37#2863—22no—
38#2645—22no—
39#2514C2214yes—
40#2511C2214yes—
41#2493—24no—
42#2367C2214yes—
43#2365C2214yes—
44#2359—24no—
45#2323—22no—
46#2317C2211yescharacteristic
47#2316C2211yescharacteristic
48#2311C2211yescharacteristic
49#2310C2211yescharacteristic
50#2310C2211yescharacteristic
51C4×C22⋊Q8—22no—
52#2296C2211yescharacteristic
53C2×#391C2211yescharacteristic
54C2×#391C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2212yes—
58#2096C2212yes—
59#2094C2212yes—
60#2092C2212yes—
61#2056C2214yes—
62#2055C2214yes—
63#2043C2214yes—
64#2042C2214yes—
65#2016C2212yes—
66#2016C2212yes—
67#2015C2212yes—
68#2015C2212yes—
69#1971C2214yes—
70#1971C2214yes—
71#1928C2214yes—
72#1928C2214yes—
73#1927C2214yes—
74#1927C2214yes—
75#1920C2214yes—
76#1919C2214yes—
77#1917C2214yes—
78#1917C2214yes—
79#1878C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1877C2214yes—
83#1805C2214yes—
84#1803C2214yes—
85#1763C2212yes—
86#1763C2212yes—
87#1762C2212yes—
88#1762C2212yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C2×#137—24no—
101C2×#137—42no—
102C2×#137—42no—
103C22×Q16—24no—
104C22×Q16—24no—
105C22×Q16—42no—
106C22×Q16C2312yes—
107C22×Q16C2312yes—
108C22×Q16C2312yes—
109C22×SD16C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114#536—24no—
115#536—24no—
116C2×C4⋊Q8C2312yes—
117C2×#124C2312yes—
118C2×#123—22no—
119C2×C22⋊Q8—22no—
120#519—24no—
121#519—42no—
122C4×#41—24no—
123C4×#41—42no—
124C2×C4×Q8—22no—
125C2×C4×Q8C2314yes—
126C2×C4×Q8C2312yes—
127C2×C4×D4—22no—
128C2×C4×D4C2312yes—
129C2×#117—22no—
130C2×#117—21no—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C42—21no—
139C22×C42C2311yescharacteristic
140#428—42no—
141C2×#107C2312yes—
142C2×#106—24no—
143C2×#106C2312yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146#424—42no—
147C2×C4⋊C8C2314yes—
148#419—24no—
149#419—24no—
150#419—42no—
151#417—42no—
152#416—24no—
153#416—24no—
154C2×#103—42no—
155C2×#103—42no—
156C2×#103C2312yes—
157C2×#103C2311yescharacteristic
158C2×#103C2311yescharacteristic
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2312yes—
163C2×#103C2314yes—
164C2×#103C2312yes—
165C2×#103C2312yes—
166C2×#102C2312yes—
167C2×#102C2311yescharacteristic
168C2×#102C2311yescharacteristic
169C2×#102C2312yes—
170C2×#97—42no—
171C2×#97C2312yes—
172C2×#97C2312yes—
173#401—24no—
174#400—24no—
175#400—24no—
176#400—24no—
177#391—22no—
178#391—22no—
179#391—21no—
180#391—21no—
181#391—21no—
182#391—21no—
183#391C2314yes—
184#391C2312yes—
185#391C2312yes—
186#391C2312yes—
187#391C2314yes—
188#391C2314yes—
189#390C2312yes—
190#389—22no—
191#389—22no—
192#389—24no—
193#389D412yes—
194#389D412yes—
195#389C2314yes—
196#388—24no—
197#388—24no—
198#387—24no—
199#387—24no—
200#387—24no—
201#387—24no—
202#387—24no—
203#386—22no—
204#385—24no—
205#385—22no—
206#385D414yes—
207#384C2312yes—
208#383—24no—
209#382C2312yes—
210#381—22no—
211#381—22no—
212#380—22no—
213#380—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—24no—
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2314yes—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—24no—
223#376—21no—
224C2×#95—22no—
225C2×#95—22no—
226C43—21no—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236#336C2312yes—
237#336C2312yes—
238#336C2314yes—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41#4114yes—
245C22×Q8—21no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8#4114yes—
259C22×Q8C2412yes—
260C22×Q8C2412yes—
261C22×Q8C2411yescharacteristic
262C22×D4—21no—
263C22×D4C2411yescharacteristic
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—22no—
270C23×C4C2×D412yes—
271C23×C4C2411yescharacteristic
272#137—416no—
273C2×Q16—24no—
274C2×Q16—24no—
275C2×Q16—24no—
276C2×Q16—24no—
277C2×Q16—24no—
278C2×Q16—42no—
279C2×Q16—42no—
280C2×Q16—42no—
281C2×Q16—42no—
282C2×Q16—416no—
283C2×SD16—24no—
284C2×SD16—42no—
285C2×SD16—42no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8—22no—
297C22×C8#4114yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2411yescharacteristic
302C22×C8C2411yescharacteristic
303C4⋊Q8—24no—
304C4⋊Q8—24no—
305#124—24no—
306#124—24no—
307#123—216no—
308C22⋊Q8—216no—
309C4×Q8—24no—
310C4×Q8—28no—
311C4×Q8—42no—
312C4×Q8—44no—
313C4×Q8—48no—
314C4×D4—24no—
315C4×D4—42no—
316C4×D4—48no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—44no—
327#117—48no—
328#117#4114yes—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D414yes—
343C2×C4⋊C4C2×D414yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—24no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—24no—
368C2×C42—44no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2412yes—
372C2×C42C2412yes—
373C2×C42C2412yes—
374#107—42no—
375#106—24no—
376#106—24no—
377#106—42no—
378C4⋊C8—44no—
379#103—24no—
380#103—28no—
381#103—22no—
382#103—22no—
383#103—28no—
384#103—44no—
385#103—44no—
386#103—42no—
387#103—42no—
388#103—42no—
389#102—24no—
390#102—22no—
391#102—22no—
392#102—42no—
393#97—42no—
394#97—42no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95C2×D414yes—
410#95C2×D414yes—
411C24—21no—
412C24C22×D411yescharacteristic
413#41—28no—
414#41—28no—
415#41—44no—
416#41—44no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—21no—
420C2×Q8—28no—
421C2×Q8—28no—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8—21no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—44no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8C2×#4114yes—
448C2×Q8C2×#4112yes—
449C2×D4—24no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4C2×#4112yes—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D414yes—
489C22×C4C22×D414yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C4C2511yescharacteristic
498Q16—48no—
499Q16—48no—
500Q16—48no—
501SD16—48no—
502M16—44no—
503M16—44no—
504C2×C8—28no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—44no—
519C2×C8C2×#4112yes—
520C2×C8C2×#4112yes—
521C4⋊C4—216no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—48no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—84no—
542C4⋊C4—84no—
543C4⋊C4C2×#4114yes—
544C22⋊C4—216no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—44no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—84no—
552C42—22no—
553C42—22no—
554C42—24no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—44no—
562C42—48no—
563C42—44no—
564C42—42no—
565C42—42no—
566C42—84no—
567C42—82no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—42no—
572C23—42no—
573C23—41no—
574C23—41no—
575C23#52512yes—
576C23#54612yes—
577C23#55012yes—
578C23C23×D411yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—24no—
582Q8—24no—
583Q8—42no—
584Q8—42no—
585Q8—41no—
586Q8—41no—
587Q8—48no—
588Q8—42no—
589Q8—42no—
590Q8—42no—
591Q8—42no—
592Q8—84no—
593D4—22no—
594D4—22no—
595D4—48no—
596D4—41no—
597D4—41no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—84no—
636C2×C4—88no—
637C2×C4—84no—
638C2×C4—162no—
639C2×C4—162no—
640C2×C4#52512yes—
641C2×C4#52914yes—
642C2×C4#53814yes—
643C2×C4#54612yes—
644C2×C4#55012yes—
645C2×C4#55114yes—
646C2×C4#55314yes—
647C2×C4C22×#4111yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C2×C4C2×H(5,𝔽2)11yescharacteristic
653C8—22no—
654C8—22no—
655C8—42no—
656C8—42no—
657C8—42no—
658C22—24no—
659C22—22no—
660C22—22no—
661C22—24no—
662C22—42no—
663C22—44no—
664C22—81no—
665C22—81no—
666C22—82no—
667C22#246714yes—
668C22C2×#52511yescharacteristic
669C22C2×#54611yescharacteristic
670C22C2×#55011yescharacteristic
671C22#358612yes—
672C22#358612yes—
673C4—42no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—42no—
678C4—42no—
679C4—44no—
680C4—82no—
681C4—82no—
682C4—81no—
683C4—81no—
684C4—82no—
685C4—82no—
686C4#246714yes—
687C4C2×#52911yescharacteristic
688C4C2×#53811yescharacteristic
689C4C2×#55111yescharacteristic
690C4C2×#55311yescharacteristic
691C2—24no—
692C2—82no—
693C2C2×#246711yescharacteristic
694C2#6057712yes—
695C2#6118112yes—
696C2#6141811yescharacteristic
697C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35313C2semidirect
3. #27088C2semidirect
4. #27087C2semidirect
5. #27085C2semidirect
6. #27082C2semidirect
11. C2×#1971C2semidirect
12. #21064C2semidirect
13. #21061C2semidirect
14. #21038C2semidirect
15. #21021C2semidirect
16. #20935C2semidirect
18. #20651C2semidirect

Maximal subgroups

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

subgroupindex
#353132
#342682
#270882
#270872
#270852
#270822
#237382
#235452
#235122
#234252
C2×#19712
#210642
#210612
#210382
#210212
#209352
#209272
#206512
#204112