Finite Group Databasebeta

← #6461901

Subgroups — #6461901

Order 512.

1809 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35312C211yesmaximal, characteristic
2#34254C211yesmaximal, characteristic, complemented
3#29571C214yesmaximal, complemented
4#29565C214yesmaximal, complemented
5#29563C214yesmaximal, complemented
6#29559C214yesmaximal, complemented
7#23556C211yesmaximal, characteristic, complemented
8#23510C211yesmaximal, characteristic
9#23457C211yesmaximal, characteristic, complemented
10#23425C211yesmaximal, characteristic
11#21237C211yesmaximal, characteristic
12C2×#1971C211yesmaximal, characteristic, complemented
13#21094C211yesmaximal, characteristic, complemented
14#21086C211yesmaximal, characteristic
15#21068C211yesmaximal, characteristic
16#21064C211yesmaximal, characteristic, complemented
17#20935C211yesmaximal, characteristic, complemented
18#20930C211yesmaximal, characteristic
19#20367C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C23×Q16C2211yescharacteristic
22C2×#544C2211yescharacteristic
23C2×#536C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C2×C4×M16C2211yescharacteristic
36C22×#97C2211yescharacteristic
37#2862C2212yes—
38#2861C2212yes—
39#2642C2212yes—
40#2640C2212yes—
41#2492—22no—
42#2489—22no—
43#2488—24no—
44#2356—22no—
45#2355—22no—
46#2343—24no—
47#2320C2214yes—
48#2317C2211yescharacteristic
49#2311C2211yescharacteristic
50#2310C2211yescharacteristic
51#2310C2211yescharacteristic
52#2297C2211yescharacteristic
53#2296C2211yescharacteristic
54#2295C2214yes—
55C2×#391C2211yescharacteristic
56C2×#391C2211yescharacteristic
57#2274C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2059C2214yes—
60#2058C2214yes—
61#2043C2214yes—
62#2042C2214yes—
63#2026C2214yes—
64#2024C2214yes—
65#2016C2214yes—
66#2015C2214yes—
67#1973C2212yes—
68#1971C2212yes—
69#1971C2214yes—
70#1971C2214yes—
71#1969C2212yes—
72#1968C2212yes—
73#1936C2212yes—
74#1936C2212yes—
75#1935C2212yes—
76#1935C2212yes—
77#1935C2214yes—
78#1933C2214yes—
79#1928C2214yes—
80#1927C2214yes—
81#1927C2214yes—
82#1926C2214yes—
83#1880C2214yes—
84#1878C2214yes—
85#1878C2214yes—
86#1878C2214yes—
87#1746C2214yes—
88#1745C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1729C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C2×#137—24no—
101C2×#137—24no—
102C2×#137—42no—
103C22×Q16—24no—
104C22×Q16—24no—
105C22×Q16—42no—
106C22×Q16C2312yes—
107C22×Q16C2314yes—
108C22×SD16C2312yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#544—42no—
114#544—42no—
115#536—42no—
116#536—42no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2312yes—
120C2×#125—21no—
121C2×#125C2312yes—
122C2×#124C2312yes—
123C2×#123—22no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—22no—
127#519—24no—
128#519—24no—
129C2×C4×Q8—22no—
130C2×C4×Q8C2314yes—
131C2×#117—22no—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#428—42no—
141C2×#107C2312yes—
142C2×#106—42no—
143C2×#106C2312yes—
144C2×#106C2314yes—
145#419—24no—
146#419—24no—
147#419—42no—
148#417—42no—
149#416—24no—
150#416—24no—
151C2×#103—42no—
152C2×#103—42no—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2312yes—
165C4×M16—24no—
166C2×#97—24no—
167C2×#97C2312yes—
168C2×#97C2312yes—
169C2×#97C2312yes—
170C2×C4×C8C2312yes—
171#400—24no—
172#400—24no—
173#397—24no—
174#393—24no—
175#391—21no—
176#391—21no—
177#391—22no—
178#391—24no—
179#391—22no—
180#391—22no—
181#391C2314yes—
182#391C2312yes—
183#391C2312yes—
184#391C2312yes—
185#391C2312yes—
186#391C2312yes—
187#391C2311yescharacteristic
188#391C2312yes—
189#391C2311yescharacteristic
190#391C2312yes—
191#390C2312yes—
192#390C2311yescharacteristic
193#390C2311yescharacteristic
194#389C2314yes—
195#389C2312yes—
196#389C2312yes—
197#388—24no—
198#388—24no—
199#387—24no—
200#387—24no—
201#387—24no—
202#387—24no—
203#386—22no—
204#386—22no—
205#385—22no—
206#385—22no—
207#384—21no—
208#383—24no—
209#382—21no—
210#382—24no—
211#382—24no—
212#381—22no—
213#381—22no—
214#380—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—24no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D412yes—
220C4×C4⋊C4C2314yes—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4D414yes—
224#376—24no—
225#376C2312yes—
226C2×#95—22no—
227C2×#95—22no—
228C43C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2314yes—
237#336C2314yes—
238#336C2314yes—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
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—22no—
259C22×Q8#4114yes—
260C22×Q8C2412yes—
261C22×Q8C2411yescharacteristic
262C22×Q8C2412yes—
263C22×D4—21no—
264C22×D4C2411yescharacteristic
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4C2×D412yes—
271C23×C4C2411yescharacteristic
272#137—416no—
273C2×Q16—28no—
274C2×Q16—24no—
275C2×Q16—28no—
276C2×Q16—24no—
277C2×Q16—44no—
278C2×Q16—42no—
279C2×Q16—416no—
280C2×SD16—24no—
281C2×SD16—24no—
282C2×SD16—42no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16#4114yes—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8#4114yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C4⋊Q8—42no—
300C4⋊Q8—42no—
301C4⋊Q8—42no—
302C4⋊Q8—42no—
303C4⋊Q8—44no—
304#125—42no—
305#125—42no—
306#125—44no—
307#124—42no—
308#124—42no—
309#123—48no—
310C22⋊Q8—48no—
311C22⋊Q8—44no—
312C22⋊Q8—44no—
313C4×Q8—216no—
314C4×Q8—28no—
315C4×Q8—28no—
316#117—216no—
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#4114yes—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C42—24no—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—24no—
368C2×C42C2×D412yes—
369C2×C42C2×D414yes—
370C2×C42C2×D414yes—
371C2×C42C2412yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—42no—
376#106—44no—
377#106—42no—
378#103—24no—
379#103—28no—
380#103—24no—
381#103—28no—
382#103—44no—
383#103—44no—
384#103—42no—
385#103—42no—
386#103—42no—
387#102—24no—
388#102—24no—
389#102—42no—
390#97—24no—
391#97—24no—
392#97—24no—
393C4×C8—24no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—22no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—22no—
403#95—22no—
404#95—24no—
405#95—22no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95C2×D414yes—
410#95C2×D414yes—
411C24—21no—
412C24C22×D411yescharacteristic
413#41—44no—
414#41—44no—
415#41—44no—
416#41—44no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—28no—
420C2×Q8—28no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—44no—
438C2×Q8—42no—
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×Q8—44no—
448C2×Q8—44no—
449C2×Q8—84no—
450C2×Q8C2×#4114yes—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
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×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D414yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
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—416no—
499Q16—48no—
500SD16—48no—
501M16—28no—
502M16—44no—
503C2×C8—28no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—44no—
515C2×C8C2×#4114yes—
516C2×C8C2×#4114yes—
517C4⋊C4—216no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—84no—
541C4⋊C4—82no—
542C4⋊C4C2×#4114yes—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—84no—
552C22⋊C4—82no—
553C42—216no—
554C42—24no—
555C42—24no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—44no—
565C42—44no—
566C42—42no—
567C42—42no—
568C42—84no—
569C23—22no—
570C23—22no—
571C23—42no—
572C23—42no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23#52512yes—
577C23#53212yes—
578C23#53512yes—
579C23C23×D411yescharacteristic
580Q8—24no—
581Q8—24no—
582Q8—48no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—41no—
587Q8—42no—
588Q8—41no—
589Q8—42no—
590Q8—42no—
591Q8—41no—
592Q8—41no—
593Q8—84no—
594Q8—164no—
595D4—41no—
596D4—41no—
597D4—41no—
598D4—41no—
599D4—84no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—48no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—162no—
641C2×C4—162no—
642C2×C4#52512yes—
643C2×C4#53212yes—
644C2×C4#53512yes—
645C2×C4#55114yes—
646C2×C4#55114yes—
647C2×C4#55514yes—
648C2×C4#55514yes—
649C2×C4C22×#4111yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C22×#4111yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×H(5,𝔽2)11yescharacteristic
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C8—24no—
656C8—24no—
657C8—42no—
658C8—42no—
659C22—24no—
660C22—22no—
661C22—22no—
662C22—24no—
663C22—82no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—82no—
668C22#261014yes—
669C22C2×#52511yescharacteristic
670C22C2×#53211yescharacteristic
671C22C2×#53511yescharacteristic
672C22#358612yes—
673C22#358612yes—
674C4—42no—
675C4—42no—
676C4—44no—
677C4—42no—
678C4—42no—
679C4—44no—
680C4—44no—
681C4—82no—
682C4—81no—
683C4—82no—
684C4—82no—
685C4—81no—
686C4—81no—
687C4—82no—
688C4#261014yes—
689C4C2×#55111yescharacteristic
690C4C2×#55111yescharacteristic
691C4C2×#55511yescharacteristic
692C4C2×#55511yescharacteristic
693C2—24no—
694C2—161no—
695C2C2×#261011yescharacteristic
696C2#6057712yes—
697C2#6071112yes—
698C2#6082611yescharacteristic
699C2#6082811yescharacteristic
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
2. #34254C2semidirect
3. #29571C2semidirect
4. #29565C2semidirect
5. #29563C2semidirect
6. #29559C2semidirect
7. #23556C2semidirect
9. #23457C2semidirect
12. C2×#1971C2semidirect
13. #21094C2semidirect
16. #21064C2semidirect
17. #20935C2semidirect
19. #20367C2semidirect

Maximal subgroups

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

subgroupindex
#353122
#342542
#295712
#295652
#295632
#295592
#235562
#235102
#234572
#234252
#212372
C2×#19712
#210942
#210862
#210682
#210642
#209352
#209302
#203672