Finite Group Databasebeta

← #6795015

Subgroups — #6795015

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36176C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic, complemented
3#32957C211yesmaximal, characteristic, complemented
4#32441C214yesmaximal, complemented
5#32357C214yesmaximal, complemented
6#29912C212yesmaximal
7#28529C212yesmaximal, complemented
8#27730C212yesmaximal, complemented
9#27589C212yesmaximal, complemented
10#25811C211yesmaximal, characteristic, complemented
11#25454C212yesmaximal, complemented
12#24681C212yesmaximal, complemented
13#24503C212yesmaximal
14#24298C211yesmaximal, characteristic
15#22599C211yesmaximal, characteristic, complemented
16#21983C212yesmaximal, complemented
17C4×#448C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#498C2212yes—
21C2×#477C2212yes—
22C2×#474C2212yes—
23C2×#448C2211yescharacteristic
24C2×C4×SD16C2212yes—
25C2×#431C2212yes—
26#2857C2211yescharacteristic
27#2854C2212yes—
28#2571C2211yescharacteristic
29#2568—21no—
30#2568—21no—
31#2568—22no—
32#2568C2212yes—
33#2568C2214yes—
34#2568C2212yes—
35#2563C2211yescharacteristic
36#2532—22no—
37#2513C2212yes—
38#2508C2212yes—
39#2495—21no—
40#2488—22no—
41#2486—21no—
42C2×#400—22no—
43#2369C2211yescharacteristic
44#2354C2211yescharacteristic
45#2352—21no—
46#2346—21no—
47C4×#121C2211yescharacteristic
48C2×#391—21no—
49C2×#391—21no—
50C2×#391C2212yes—
51#2283C2212yes—
52C2×#382C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2214yes—
55#2097C2214yes—
56#2092C2214yes—
57#2085C2214yes—
58#2074C2214yes—
59#2073C2214yes—
60#2066C2214yes—
61#2058C2214yes—
62#2057C2212yes—
63#2055C2214yes—
64#2023C2212yes—
65#1990C2212yes—
66#1988C2212yes—
67#1968C2214yes—
68#1949C2214yes—
69#1949C2214yes—
70#1939C2212yes—
71#1933C2214yes—
72#1928C2214yes—
73#1927C2214yes—
74#1916C2214yes—
75#1880C2212yes—
76#1874C2212yes—
77#1874C2211yescharacteristic
78#1865C2212yes—
79#1863C2214yes—
80#1853C2212yes—
81#1853C2212yes—
82#1853C2211yescharacteristic
83#1852C2212yes—
84#1852C2212yes—
85#1846C2214yes—
86#1835C2214yes—
87#1817C2212yes—
88#1803C2212yes—
89#1797C2211yescharacteristic
90#1787C2212yes—
91#1787C2211yescharacteristic
92C8×C4⋊C4C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×SD16C2312yes—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×#123—22no—
105C2×C22⋊Q8—21no—
106C2×#121C2311yescharacteristic
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—44no—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2312yes—
112C2×C4×D4—21no—
113C2×C4×D4C2311yescharacteristic
114C2×#117—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124#498—44no—
125#477—44no—
126#474—44no—
127#448—24no—
128C4×SD16—44no—
129#431—28no—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#106C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2311yescharacteristic
146C2×C22⋊C8C2311yescharacteristic
147C2×C4×C8C2314yes—
148C2×C4×C8C2312yes—
149#400—24no—
150#400—24no—
151#400—24no—
152#400—24no—
153#400—22no—
154#400—22no—
155#398—22no—
156#397—22no—
157#397—22no—
158#393—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391—24no—
164#391—24no—
165#391—22no—
166#391—22no—
167#391—21no—
168#391—22no—
169#391C2314yes—
170#391C2314yes—
171#390C2312yes—
172#390C2312yes—
173#390C2311yescharacteristic
174#389—21no—
175#389—21no—
176#389—21no—
177#389C2311yescharacteristic
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—21no—
185#388—22no—
186#388—22no—
187#387—22no—
188#386—22no—
189#386—21no—
190#384C2311yescharacteristic
191#383—22no—
192#382—21no—
193#382—22no—
194#382—22no—
195#382—22no—
196#382C2312yes—
197#382C2314yes—
198#382C2314yes—
199#382C2312yes—
200#381—22no—
201#380—21no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380C2312yes—
206#380C2311yescharacteristic
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D412yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219#376—22no—
220C2×#95—22no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236C22×Q8—24no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—42no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8C2412yes—
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2411yescharacteristic
259C2×SD16—24no—
260C2×SD16—24no—
261C2×SD16—42no—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271#123—84no—
272C22⋊Q8—82no—
273#121—22no—
274#121—22no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—42no—
278C4×Q8—42no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×D4—24no—
282C4×D4—44no—
283#117—44no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—44no—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343#107—44no—
344#107—44no—
345#106—42no—
346#106—44no—
347#106—44no—
348C4⋊C8—28no—
349C4⋊C8—24no—
350#103—28no—
351#103—24no—
352#103—24no—
353#103—42no—
354#103—44no—
355#102—42no—
356#102—44no—
357C22⋊C8—24no—
358C22⋊C8—22no—
359C22⋊C8#4114yes—
360C4×C8—28no—
361C4×C8—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—21no—
388C24C22×D411yescharacteristic
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—81no—
407C2×Q8—81no—
408C2×Q8—816no—
409C2×Q82−1+414yes—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4#13514yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4112yes—
461C22×C42−1+412yes—
462C22×C4C2511yescharacteristic
463SD16—48no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8C2×#4114yes—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—81no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C42—22no—
513C42—22no—
514C42—24no—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—44no—
522C42—44no—
523C42—44no—
524C42—42no—
525C42—82no—
526C42H(5,𝔽2)14yes—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23#54712yes—
539C23C2×#13512yes—
540C23#57712yes—
541C23C23×D411yescharacteristic
542Q8—24no—
543Q8—24no—
544Q8—48no—
545Q8—42no—
546Q8—42no—
547Q8—88no—
548Q8—88no—
549Q8—162no—
550D4—42no—
551D4—42no—
552D4—84no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—88no—
607C2×C4—84no—
608C2×C4#53512yes—
609C2×C4#53512yes—
610C2×C4#54712yes—
611C2×C4C2×#13512yes—
612C2×C4C2×#13512yes—
613C2×C4#57512yes—
614C2×C4C22×#4111yescharacteristic
615C2×C4C2×H(5,𝔽2)12yes—
616C2×C4C2×2−1+412yes—
617C2×C4C2×2−1+411yescharacteristic
618C8—24no—
619C8—44no—
620C8—42no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22#334014yes—
635C22C2×#53511yescharacteristic
636C22C2×#54711yescharacteristic
637C22#348611yescharacteristic
638C22C22×#13511yescharacteristic
639C22C2×#57511yescharacteristic
640C22C2×#57711yescharacteristic
641C22#358611yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—44no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—41no—
651C4—81no—
652C4—81no—
653C4—84no—
654C4—81no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4#321114yes—
659C4#352012yes—
660C4#353812yes—
661C2—24no—
662C2—161no—
663C2#3891711yescharacteristic
664C2C2×#321111yescharacteristic
665C2#6081311yescharacteristic
666C2C2×#334011yescharacteristic
667C2#6125711yescharacteristic
668C2#6182211yescharacteristic
669C2#6183611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2568C2semidirect
3. #32957C2semidirect
4. #32441C2semidirect
5. #32357C2semidirect
7. #28529C2semidirect
8. #27730C2semidirect
9. #27589C2semidirect
10. #25811C2semidirect
11. #25454C2semidirect
12. #24681C2semidirect
15. #22599C2semidirect
16. #21983C2semidirect

Maximal subgroups

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

subgroupindex
#361762
C2×#25682
#329572
#324412
#323572
#299122
#285292
#277302
#275892
#258112
#254542
#246812
#245032
#242982
#225992
#219832
C4×#4482