Finite Group Databasebeta

← #6470955

Subgroups — #6470955

Order 512.

1936 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic, complemented
2C2×#2363C211yesmaximal, characteristic
3#33149C211yesmaximal, characteristic, complemented
4#33147C211yesmaximal, characteristic, complemented
5#32959C212yesmaximal, complemented
6#29973C212yesmaximal, complemented
7#29952C211yesmaximal, characteristic, complemented
8#29950C211yesmaximal, characteristic, complemented
9#26411C214yesmaximal, complemented
10#26409C214yesmaximal, complemented
11#26347C214yesmaximal, complemented
12#26345C214yesmaximal, complemented
13#21930C211yesmaximal, characteristic, complemented
14#21761C212yesmaximal
15#20843C212yesmaximal, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#478C2214yescomplemented
20C2×#427C2214yescomplemented
21C22×C22⋊C8C2212yes—
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2851C2212yes—
25#2626—21no—
26#2626—21no—
27#2608—22no—
28#2607—22no—
29#2441C2212yescomplemented
30#2441C2212yescomplemented
31#2439C2212yescomplemented
32#2439C2212yescomplemented
33#2363—22no—
34#2363—22no—
35#2363C2214yes—
36#2363C2214yes—
37#2329—21no—
38#2329—21no—
39C2×#390—21no—
40C2×#390C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#388—21no—
43C2×#386—21no—
44#2280C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4—21no—
47C2×C4×C22⋊C4C2211yescharacteristic
48#2101C2214yescomplemented
49#2094C2212yescomplemented
50#2093C2212yescomplemented
51#2088C2214yescomplemented
52#2086C2214yescomplemented
53#2067C2214yescomplemented
54#2065C2214yescomplemented
55#2059C2214yescomplemented
56#2023C2212yescomplemented
57#2023C2212yescomplemented
58#1988C2212yescomplemented
59#1987C2212yescomplemented
60#1948C2212yescomplemented
61#1947C2212yescomplemented
62#1921C2214yescomplemented
63#1919C2214yescomplemented
64#1893C2214yes—
65#1893C2214yes—
66#1875C2214yes—
67#1875C2214yes—
68#1864C2214yescomplemented
69#1862C2214yescomplemented
70#1854C2214yescomplemented
71#1854C2214yescomplemented
72#1853C2212yes—
73#1853C2212yes—
74#1852C2212yes—
75#1852C2212yes—
76#1850C2214yescomplemented
77#1850C2214yescomplemented
78#1824C2212yes—
79#1824C2212yes—
80#1804C2214yescomplemented
81#1797C2211yescharacteristic
82#1797C2212yes—
83#1797C2211yescharacteristic
84#1786C2212yes—
85#1786C2212yes—
86#1735C2214yescomplemented
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C23×C8C2314yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×#125C2312yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×#121—22no—
98C2×#121—24no—
99C2×#121Q818yescomplemented
100C2×#121C2314yes—
101C2×C4×D4—22no—
102C2×C4×D4—44no—
103C2×C4×D4C2314yes—
104C2×#117—22no—
105C2×#117—44no—
106C2×#117C2314yes—
107C2×#117C2312yes—
108C2×#117C2312yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4D411yescharacteristic
119C22×C22⋊C4D411yescharacteristic
120C22×C42—21no—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#478—48no—
125#427—48no—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×C22⋊C8—28no—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2314yes—
141#399—22no—
142#399—22no—
143#396—22no—
144#396—22no—
145#396—22no—
146#396—22no—
147#393—22no—
148#393—22no—
149#391—21no—
150#391—21no—
151#391—22no—
152#391—22no—
153#390—24no—
154#390—24no—
155#390—21no—
156#390—21no—
157#390C2312yes—
158#390C2312yes—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389C2314yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#388—24no—
170#388—24no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#386—24no—
178#386—24no—
179#385—22no—
180#385—22no—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#382C2312yes—
184#381—22no—
185#380—21no—
186#380—21no—
187#380C2312yes—
188#380C2312yes—
189#380C2312yes—
190#380C2312yes—
191#379—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—24no—
199C4×C22⋊C4—24no—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4C2312yes—
203C4×C22⋊C4C2312yes—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95D411yescharacteristic
207C2×#95D411yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2314yes—
217#336C2312yes—
218#336C2312yes—
219C25—21no—
220C25C2×D411yescharacteristic
221C22×Q8—21no—
222C22×Q8—21no—
223C22×D4—28no—
224C22×D4—24no—
225C22×D4—21no—
226C22×D4—22no—
227C22×D4—42no—
228C22×D4—42no—
229C22×D4—82no—
230C22×D4—82no—
231C22×D4C2×Q814yes—
232C22×D4C2412yes—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—24no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—82no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×Q814yes—
249C23×C4#4114yes—
250C23×C4C2411yescharacteristic
251C23×C4C2412yes—
252C22×C8—28no—
253C22×C8—44no—
254C22×C8—44no—
255C22×C8C2414yes—
256C22×C8C2414yes—
257C4⋊Q8—42no—
258C4⋊Q8—42no—
259#125—44no—
260C22⋊Q8—44no—
261C22⋊Q8—44no—
262#121—216no—
263#121—28no—
264#121C2×Q8116yes—
265C4×D4—28no—
266C4×D4—44no—
267C4×D4—48no—
268#117—28no—
269#117—44no—
270#117—48no—
271#117—44no—
272#117—44no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4C2×Q814yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—82no—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—44no—
333C2×C42—82no—
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340#107—28no—
341#107—44no—
342#106—28no—
343#106—44no—
344C4⋊C8—42no—
345C4⋊C8—42no—
346C4⋊C8—44no—
347#102—44no—
348#102—44no—
349#102—44no—
350#102—44no—
351C22⋊C8—24no—
352C22⋊C8—24no—
353C22⋊C8—28no—
354#95—24no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95C2×D412yes—
368#95C2×D412yes—
369#95C2×D412yes—
370#95C2×D412yes—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—22no—
375C24—41no—
376C24—41no—
377C24—41no—
378C24—41no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24C22⋊Q814yes—
386C24C22×D412yes—
387C24C22×D411yescharacteristic
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—81no—
391C2×Q8—81no—
392C2×Q8—81no—
393C2×Q8—81no—
394C2×D4—216no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—48no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—48no—
403C2×D4—84no—
404C2×D4—84no—
405C2×D4—816no—
406C2×D4C22×Q814yes—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—84no—
447C22×C4—816no—
448C22×C4C22⋊Q814yes—
449C22×C4C22⋊Q818yes—
450C22×C4#13514yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×Q812yes—
460C22×C4C22×Q814yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4114yes—
463C22×C4C2511yescharacteristic
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—416no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—84no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4C22×Q814yes—
495C4⋊C4H(5,𝔽2)14yes—
496C22⋊C4—216no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—41no—
517C42—41no—
518C42—42no—
519C42—48no—
520C42—82no—
521C42—82no—
522C42—82no—
523C42—82no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—28no—
535C23—24no—
536C23—44no—
537C23—44no—
538C23—48no—
539C23—48no—
540C23—42no—
541C23—42no—
542C23—48no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—82no—
548C23—82no—
549C23—82no—
550C23—82no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23C2×C22⋊Q814yes—
556C23C2×C22⋊Q812yes—
557C23C2×#13512yes—
558C23#57512yes—
559C23C23×D411yescharacteristic
560Q8—162no—
561Q8—162no—
562D4—28no—
563D4—48no—
564D4—44no—
565D4—816no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—816no—
620C2×C4—84no—
621C2×C4C2×C22⋊Q814yes—
622C2×C4C2×C22⋊Q812yes—
623C2×C4C2×#13512yes—
624C2×C4C2×#13512yes—
625C2×C4C23×Q811yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C2×H(5,𝔽2)12yes—
628C2×C4#58512yes—
629C8—44no—
630C8—44no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—88no—
648C22—816no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22#307614yes—
654C22C22×C22⋊Q811yescharacteristic
655C22#348411yescharacteristic
656C22#348511yescharacteristic
657C22C22×#13511yescharacteristic
658C22C2×#57511yescharacteristic
659C22#358511yescharacteristic
660C22#358511yescharacteristic
661C4—22no—
662C4—22no—
663C4—24no—
664C4—22no—
665C4—41no—
666C4—41no—
667C4—44no—
668C4—42no—
669C4—42no—
670C4—44no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4#345912yes—
677C4#350112yes—
678C2—24no—
679C2—44no—
680C2—84no—
681C2#3498511yescharacteristic
682C2C2×#307611yescharacteristic
683C2#6049011yescharacteristic
684C2#6156611yescharacteristic
685C2#6157611yescharacteristic
686C2#6169911yescharacteristic
687C2#6171011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35033C2semidirect
3. #33149C2semidirect
4. #33147C2semidirect
5. #32959C2semidirect
6. #29973C2semidirect
7. #29952C2semidirect
8. #29950C2semidirect
9. #26411C2semidirect
10. #26409C2semidirect
11. #26347C2semidirect
12. #26345C2semidirect
13. #21930C2semidirect
15. #20843C2semidirect
19. C2×#478C22semidirect
20. C2×#427C22semidirect
29. #2441C22semidirect
30. #2441C22semidirect
31. #2439C22semidirect
32. #2439C22semidirect
48. #2101C22semidirect
49. #2094C22semidirect
50. #2093C22semidirect
51. #2088C22semidirect
52. #2086C22semidirect
53. #2067C22semidirect
54. #2065C22semidirect
55. #2059C22semidirect
56. #2023C22semidirect
57. #2023C22semidirect
58. #1988C22semidirect
59. #1987C22semidirect
60. #1948C22semidirect
61. #1947C22semidirect
62. #1921C22semidirect
63. #1919C22semidirect
68. #1864C22semidirect
69. #1862C22semidirect
70. #1854C22semidirect
71. #1854C22semidirect
76. #1850C22semidirect
77. #1850C22semidirect
80. #1804C22semidirect
86. #1735C22semidirect
99. C2×#121Q8semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#23632
#331492
#331472
#329592
#299732
#299522
#299502
#264112
#264092
#263472
#263452
#219302
#217612
#208432