Finite Group Databasebeta

← #6491912

Subgroups — #6491912

Order 512.

1576 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#34127C211yesmaximal, characteristic
3#33199C212yesmaximal, complemented
4#33068C212yesmaximal, complemented
5#29996C212yesmaximal, complemented
6#29988C212yesmaximal, complemented
7#26107C212yesmaximal, complemented
8#26089C212yesmaximal, complemented
9#26045C212yesmaximal, complemented
10#26027C212yesmaximal, complemented
11#26013C214yesmaximal, complemented
12#25972C214yesmaximal
13#21981C211yesmaximal, characteristic, complemented
14#21974C211yesmaximal, characteristic, complemented
15#21964C211yesmaximal, characteristic, complemented
16#20737C211yesmaximal, characteristic
17#20648C211yesmaximal, characteristic
18C2×C4×#41C2211yescharacteristic
19C2×#478C2214yes—
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#417C2212yes—
23#2859C2212yes—
24#2859C2212yes—
25#2647—22no—
26#2571—21no—
27#2571C2212yes—
28#2552—21no—
29#2552C2212yes—
30#2442—22no—
31#2442C2214yes—
32#2355C2211yescharacteristic
33#2355C2211yescharacteristic
34#2352—21no—
35#2352—21no—
36D4×C4⋊C4C2211yescharacteristic
37D4×C4⋊C4C2211yescharacteristic
38#2347C2212yes—
39#2343—22no—
40#2339C2212yes—
41#2317—21no—
42#2316—21no—
43#2295C2212yes—
44#2290—22no—
45#2285C2211yescharacteristic
46#2285C2211yescharacteristic
47C2×#384C2211yescharacteristic
48#2283—21no—
49C42×D4C2212yes—
50C4×#117C2211yescharacteristic
51C2×C4×C4⋊C4—21no—
52#2101C2214yes—
53#2097C2214yes—
54#2088C2214yes—
55#2088C2214yes—
56#2066C2214yes—
57#2066C2214yes—
58#2059C2214yes—
59#2026C2214yes—
60#1989C2214yes—
61#1949C2214yes—
62#1948C2212yes—
63#1948C2212yes—
64#1948C2212yes—
65#1948C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1939C2212yes—
69#1939C2212yes—
70#1920C2214yes—
71#1920C2214yes—
72#1864C2214yes—
73#1864C2214yes—
74#1853C2214yes—
75#1851C2214yes—
76#1844C2212yes—
77#1842C2212yes—
78#1834C2212yes—
79#1831C2212yes—
80#1825C2212yes—
81#1809C2214yes—
82#1804C2214yes—
83#1803C2211yescharacteristic
84#1803C2211yescharacteristic
85#1803C2212yes—
86#1803C2212yes—
87#1802C2211yescharacteristic
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91#1788C2212yes—
92C4×#103C2212yes—
93C4×#102C2212yes—
94#1736C2212yes—
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97C2×#125—22no—
98C2×#125C2314yes—
99C4×#41—24no—
100C4×#41Q818yes—
101C2×C4×Q8C2311yescharacteristic
102C2×C4×Q8C2311yescharacteristic
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4C2312yes—
106C2×C4×D4C2312yes—
107C2×C4×D4C2311yescharacteristic
108C2×C4×D4C2311yescharacteristic
109C2×#117—21no—
110C2×#117—21no—
111C2×#117—21no—
112C2×#117D411yescharacteristic
113C2×#117D411yescharacteristic
114C2×#117C2311yescharacteristic
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4C2312yes—
120C22×C42—21no—
121C22×C42—22no—
122C22×C42C2311yescharacteristic
123#478—48no—
124#431—24no—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129#423—24no—
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2314yes—
136#417—44no—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C4×C8C2312yes—
145#400—24no—
146#400—24no—
147#398—24no—
148#398—24no—
149#391C2312yes—
150#391C2312yes—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389C2312yes—
160#389C2314yes—
161#389C2314yes—
162#389C2312yes—
163#388—22no—
164#388—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#386—22no—
170#386—22no—
171#385—24no—
172#385—22no—
173#385—22no—
174#384—22no—
175#384—21no—
176#384—21no—
177#384C2312yes—
178#384C2311yescharacteristic
179#384C2312yes—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#383—22no—
184#382—22no—
185#382—21no—
186#382—21no—
187#382—22no—
188#382—21no—
189#382—21no—
190#382C2314yes—
191#381—22no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380—21no—
196#380—24no—
197#380C2312yes—
198#380C2312yes—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218#376—22no—
219#376D412yes—
220C2×#95—22no—
221C43C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2314yes—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—41no—
238C2×#41—41no—
239C2×#41C2×Q814yes—
240C22×Q8C2411yescharacteristic
241C22×D4—21no—
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×M16—24no—
253C2×M16—42no—
254C2×M16—42no—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258#125—44no—
259#125—44no—
260#125—84no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8C2×Q814yes—
264C4×Q8C2×Q814yes—
265C4×D4—28no—
266C4×D4—28no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—42no—
272C4×D4—42no—
273C4×D4C2×Q814yes—
274C4×D4C2×Q814yes—
275#117—22no—
276#117—28no—
277#117—42no—
278#117—42no—
279#117—44no—
280#117—82no—
281#117C2×Q814yes—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—28no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—84no—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×Q814yes—
345C2×C42#4114yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—44no—
351#106—44no—
352#106—44no—
353#106—44no—
354C4⋊C8—24no—
355C4⋊C8—24no—
356C4⋊C8—22no—
357C4⋊C8—22no—
358C4⋊C8—44no—
359C4⋊C8—44no—
360#103—42no—
361#103—42no—
362#103—44no—
363#102—28no—
364#102—24no—
365#102—24no—
366C22⋊C8—24no—
367C4×C8—24no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24—21no—
389C24C22×D411yescharacteristic
390#41—28no—
391#41—44no—
392#41—44no—
393#41—44no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8C22×Q812yes—
400C2×D4—24no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4C22×Q812yes—
407C2×D4H(5,𝔽2)12yes—
408C2×D42−1+412yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4C22⋊Q814yes—
444C22×C4C22⋊Q814yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×Q812yes—
455C22×C4C2×#4112yes—
456C22×C4C2511yescharacteristic
457M16—84no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4C22×Q812yes—
505C4⋊C4C22×Q812yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—82no—
515C42—24no—
516C42—24no—
517C42—24no—
518C42—24no—
519C42—24no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—42no—
525C42—42no—
526C42—44no—
527C42—44no—
528C42—82no—
529C42—82no—
530C42C22⋊Q814yes—
531C42C22⋊Q814yes—
532C42C22×Q812yes—
533C42C22×Q812yes—
534C42C2×#4112yes—
535C42C2×#4112yes—
536C23—22no—
537C23—22no—
538C23—21no—
539C23—22no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C2×C22⋊Q812yes—
550C23C23×D411yescharacteristic
551Q8—22no—
552Q8—22no—
553Q8—41no—
554Q8—41no—
555D4—22no—
556D4—22no—
557D4—22no—
558D4—48no—
559D4#54414yes—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—48no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—88no—
612C2×C4—88no—
613C2×C4—164no—
614C2×C4C2×C22⋊Q812yes—
615C2×C4C2×C22⋊Q812yes—
616C2×C4C2×C22⋊Q812yes—
617C2×C4C2×C22⋊Q812yes—
618C2×C4C2×C22⋊Q812yes—
619C2×C4#57512yes—
620C2×C4#57512yes—
621C2×C4#57712yes—
622C2×C4#57712yes—
623C2×C4C23×Q811yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4C2×H(5,𝔽2)11yescharacteristic
626C2×C4C2×2−1+411yescharacteristic
627C2×C4#58511yescharacteristic
628C2×C4#58511yescharacteristic
629C8—42no—
630C8—44no—
631C8—42no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—24no—
636C22—41no—
637C22—42no—
638C22—42no—
639C22—41no—
640C22—42no—
641C22—42no—
642C22—41no—
643C22—42no—
644C22—42no—
645C22—41no—
646C22C22×C22⋊Q811yescharacteristic
647C22#348811yescharacteristic
648C22#348911yescharacteristic
649C22C2×#57511yescharacteristic
650C22C2×#57511yescharacteristic
651C22C2×#57711yescharacteristic
652C22C2×#57711yescharacteristic
653C4—22no—
654C4—22no—
655C4—22no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—84no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4#308014yes—
669C4#308414yes—
670C4C2×#54411yescharacteristic
671C4#345811yescharacteristic
672C4#349411yescharacteristic
673C4#349511yescharacteristic
674C2—42no—
675C2—44no—
676C2#3525311yescharacteristic
677C2C2×#308011yescharacteristic
678C2C2×#308411yescharacteristic
679C2#6198611yescharacteristic
680C2#6199611yescharacteristic
681C2#6213411yescharacteristic
682C2#6214411yescharacteristic
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
1. #35286C2semidirect
3. #33199C2semidirect
4. #33068C2semidirect
5. #29996C2semidirect
6. #29988C2semidirect
7. #26107C2semidirect
8. #26089C2semidirect
9. #26045C2semidirect
10. #26027C2semidirect
11. #26013C2semidirect
13. #21981C2semidirect
14. #21974C2semidirect
15. #21964C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#341272
#331992
#330682
#299962
#299882
#261072
#260892
#260452
#260272
#260132
#259722
#219812
#219742
#219642
#207372
#206482