Finite Group Databasebeta

← #6524926

Subgroups — #6524926

Order 512.

1823 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3218C211yesmaximal, characteristic
2C22×#491C211yesmaximal, characteristic, complemented
3#35139C211yesmaximal, characteristic
4#34216C211yesmaximal, characteristic, complemented
5#28725C212yesmaximal, complemented
6#28723C212yesmaximal, complemented
7#28619C212yesmaximal, complemented
8#28619C212yesmaximal, complemented
9#25333C214yesmaximal, complemented
10#25331C214yesmaximal, complemented
11#24533C212yesmaximal
12#24532C212yesmaximal, complemented
13#24529C212yesmaximal, complemented
14#24528C212yesmaximal
15#21838C211yesmaximal, characteristic, complemented
16#21406C211yesmaximal, characteristic
17#21355C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2211yescharacteristic
20C22×#125C2211yescharacteristic
21#3218—28no—
22C2×#497C2211yescharacteristic
23C2×#497C2211yescharacteristic
24C2×#491—28no—
25C2×#491C2211yescharacteristic
26C2×#491C2212yes—
27C2×#491C2211yescharacteristic
28C2×#491C2212yes—
29C2×#491C2214yes—
30C2×#462C2212yes—
31C2×#461C2212yes—
32C2×#451C2212yes—
33C2×#448C2212yes—
34C2×#432C2211yescharacteristic
35C22×#103C2212yes—
36C22×#103C2212yes—
37C22×#97C2211yescharacteristic
38#2861C2212yes—
39#2859C2212yes—
40#2664—22no—
41#2611—22no—
42#2566C2212yes—
43#2565C2212yes—
44#2504C2211yescharacteristic
45#2503C2211yescharacteristic
46#2502C2211yescharacteristic
47#2501C2211yescharacteristic
48#2413—21no—
49#2413—21no—
50#2369—22no—
51#2365C2214yes—
52#2345—21no—
53#2345—21no—
54Q8×C22⋊C4C2212yes—
55Q8×C22⋊C4C2212yes—
56#2331—22no—
57#2322—21no—
58#2322—21no—
59#2316—21no—
60C4×C4⋊Q8C2212yes—
61C4×C4⋊Q8C2212yes—
62#2285C2211yescharacteristic
63#2274—21no—
64C4×#117C2211yescharacteristic
65#2074C2214yes—
66#2074C2214yes—
67#2072C2214yes—
68#2072C2214yes—
69#2056C2214yes—
70#2056C2214yes—
71#2055C2214yes—
72#2055C2214yes—
73#1992C2212yes—
74#1992C2212yes—
75#1992C2212yes—
76#1989C2214yes—
77#1959C2212yes—
78#1940C2214yes—
79#1893C2212yes—
80#1892C2212yes—
81#1880C2214yes—
82#1880C2214yes—
83#1878C2214yes—
84#1878C2214yes—
85#1850C2212yes—
86#1850C2212yes—
87#1824C2212yes—
88#1823C2212yes—
89#1805C2214yes—
90#1805C2214yes—
91#1803C2214yes—
92#1803C2214yes—
93#1767C2212yes—
94#1767C2212yes—
95#1763C2212yes—
96#1762C2212yes—
97C23×Q8C2312yes—
98C22×Q16C2312yes—
99C22×Q16C2312yes—
100C22×SD16C2312yes—
101C22×SD16C2312yes—
102C23×C8C2312yes—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—24no—
106C2×C4⋊Q8D418yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110#529—24no—
111#529—24no—
112C2×#126—21no—
113C2×#125—21no—
114C2×#125—21no—
115C2×#125—24no—
116C2×#125—24no—
117C2×#125C2312yes—
118C2×#125C2311yescharacteristic
119C2×#125C2312yes—
120C2×#124C2311yescharacteristic
121C2×#123—21no—
122C2×#123—21no—
123C2×#123—21no—
124C2×C22⋊Q8C2311yescharacteristic
125C2×#121C2311yescharacteristic
126C2×C4×Q8C2311yescharacteristic
127C2×C4×Q8C2314yes—
128C2×C4×Q8C2314yes—
129C2×C4×D4—21no—
130C2×C4×D4C2311yescharacteristic
131C2×#117—21no—
132C2×#117—21no—
133C2×#117D411yescharacteristic
134C2×#117D411yescharacteristic
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#497—24no—
143#497—24no—
144#491—216no—
145#491—28no—
146#491—28no—
147#491—24no—
148#491—24no—
149#462—28no—
150#461—28no—
151#451—28no—
152#448—28no—
153#432—24no—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2314yes—
156C2×C4⋊C8C2314yes—
157C2×#103—28no—
158C2×#103—28no—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#102C2314yes—
169C2×C22⋊C8C2312yes—
170C2×#97—24no—
171C2×#97C2312yes—
172C2×#97C2312yes—
173C2×#97C2311yescharacteristic
174C2×#97C2311yescharacteristic
175#400—22no—
176#399—22no—
177#398—22no—
178#398—22no—
179#398—22no—
180#397—22no—
181#396—22no—
182#396—22no—
183#391—24no—
184#391—24no—
185#391C2314yes—
186#391C2314yes—
187#391C2312yes—
188#391C2312yes—
189#390C2312yes—
190#390C2312yes—
191#389—24no—
192#389—22no—
193#389—22no—
194#389—22no—
195#389—22no—
196#389—22no—
197#389—22no—
198#389—22no—
199#389—22no—
200#388—22no—
201#388—22no—
202#388—22no—
203#388—22no—
204#387—22no—
205#387—22no—
206#387—24no—
207#387—24no—
208#386—21no—
209#386—21no—
210#385—21no—
211#385—21no—
212#385—22no—
213#385—22no—
214#384C2312yes—
215#384C2312yes—
216#384C2312yes—
217#382—21no—
218#382C2312yes—
219#382C2312yes—
220#381—21no—
221#381—21no—
222#380—21no—
223#380—22no—
224#380—22no—
225#380C2312yes—
226C4×C4⋊C4—24no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4D412yes—
233C4×C4⋊C4C2312yes—
234C4×C4⋊C4C2312yes—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4D411yescharacteristic
238C4×C22⋊C4D411yescharacteristic
239C4×C22⋊C4C2311yescharacteristic
240C4×C22⋊C4C2311yescharacteristic
241#376D412yes—
242C43C2312yes—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#336C2314yes—
250#336C2314yes—
251C22×Q8—22no—
252C22×Q8—24no—
253C22×Q8—44no—
254C22×Q8—44no—
255C22×Q8C2×D418yes—
256C22×Q8C2411yescharacteristic
257C22×Q8C2412yes—
258C22×Q8C2412yes—
259C22×D4—21no—
260C22×D4C2411yescharacteristic
261C23×C4—22no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2411yescharacteristic
268C2×Q16—24no—
269C2×Q16—24no—
270C2×Q16—44no—
271C2×Q16—44no—
272C2×SD16—24no—
273C2×SD16—24no—
274C2×SD16—44no—
275C2×SD16—44no—
276C22×C8—24no—
277C22×C8—44no—
278C22×C8C2414yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C4⋊Q8—28no—
282C4⋊Q8—28no—
283C4⋊Q8—24no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8C2×D418yes—
287C4⋊Q8C2×D414yes—
288C4⋊Q8C2×D414yes—
289#126—44no—
290#125—24no—
291#125—24no—
292#125—22no—
293#125—24no—
294#125—24no—
295#125—22no—
296#125—44no—
297#125—44no—
298#124—22no—
299#124—22no—
300#123—44no—
301#123—44no—
302#123—44no—
303C22⋊Q8—22no—
304C22⋊Q8—22no—
305#121—22no—
306#121—22no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—44no—
310C4×Q8—44no—
311C4×Q8—44no—
312C4×Q8—44no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—44no—
316#117—24no—
317#117—42no—
318#117—44no—
319#117—44no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4—44no—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2414yes—
358C2×C4⋊C4C2411yescharacteristic
359C2×C22⋊C4—22no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4C2×D411yescharacteristic
370C2×C22⋊C4C2×D411yescharacteristic
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—24no—
376C2×C42—24no—
377C2×C42C2×D412yes—
378C2×C42C2×D412yes—
379C2×C42C2×D412yes—
380C2×C42C2×D412yes—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D414yes—
384C2×C42#4114yes—
385C2×C42C2412yes—
386C2×C42C2412yes—
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C4⋊C8—24no—
390C4⋊C8—44no—
391C4⋊C8—44no—
392#103—28no—
393#103—28no—
394#103—28no—
395#103—28no—
396#103—24no—
397#103—24no—
398#103—24no—
399#103—24no—
400#103—44no—
401#102—44no—
402C22⋊C8—24no—
403#97—24no—
404#97—24no—
405#97—22no—
406#97—22no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95—24no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95C2×D412yes—
422#95C2×D412yes—
423#95C2×D412yes—
424#95C2×D412yes—
425C24—21no—
426C24C22×D411yescharacteristic
427C2×Q8—28no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—24no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—48no—
438C2×Q8—48no—
439C2×Q8—84no—
440C2×Q8—88no—
441C2×Q8C22×D414yes—
442C2×Q8C22×D414yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—24no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—28no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—84no—
470C22×C4#12114yes—
471C22×C4#12114yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C2×#4112yes—
483C22×C4C2511yescharacteristic
484Q16—84no—
485Q16—84no—
486SD16—84no—
487SD16—84no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—48no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—44no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—44no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C22⋊C4#13714yes—
546C42—24no—
547C42—24no—
548C42—24no—
549C42—42no—
550C42—42no—
551C42—42no—
552C42—44no—
553C42—42no—
554C42—84no—
555C42—84no—
556C42#12114yes—
557C42#12114yes—
558C42C22×D412yes—
559C42C22×D412yes—
560C42C2×#4112yes—
561C42C2×#4112yes—
562C23—22no—
563C23—22no—
564C23—42no—
565C23—41no—
566C23—42no—
567C23—41no—
568C23—41no—
569C23C2×#12112yes—
570C23C2×#13712yes—
571C23#57512yes—
572C23C23×D411yescharacteristic
573Q8—24no—
574Q8—24no—
575Q8—24no—
576Q8—24no—
577Q8—44no—
578Q8—44no—
579Q8—42no—
580Q8—42no—
581Q8—88no—
582D4—42no—
583D4—42no—
584D4—84no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—84no—
628C2×C4—88no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4C2×#12112yes—
632C2×C4C2×#12112yes—
633C2×C4C2×#12112yes—
634C2×C4C2×#12112yes—
635C2×C4C2×#12112yes—
636C2×C4C2×#13714yes—
637C2×C4C23×D411yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4#58514yes—
640C8—44no—
641C8—42no—
642C8—42no—
643C22—22no—
644C22—22no—
645C22—24no—
646C22—24no—
647C22—81no—
648C22—81no—
649C22—82no—
650C22—82no—
651C22—81no—
652C22#303714yes—
653C22C22×#12111yescharacteristic
654C22#350012yes—
655C22C22×#13711yescharacteristic
656C22C2×#57511yescharacteristic
657C22#358612yes—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—24no—
662C4—22no—
663C4—22no—
664C4—44no—
665C4—44no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—82no—
670C4—81no—
671C4—81no—
672C4—84no—
673C4—82no—
674C4—81no—
675C4#348512yes—
676C4#351012yes—
677C2—24no—
678C2—161no—
679C2#3561711yescharacteristic
680C2C2×#303711yescharacteristic
681C2#6042411yescharacteristic
682C2#6257712yes—
683C2#6257912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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. C22×#491C2semidirect
4. #34216C2semidirect
5. #28725C2semidirect
6. #28723C2semidirect
7. #28619C2semidirect
8. #28619C2semidirect
9. #25333C2semidirect
10. #25331C2semidirect
12. #24532C2semidirect
13. #24529C2semidirect
15. #21838C2semidirect
17. #21355C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32182
C22×#4912
#351392
#342162
#287252
#287232
#286192
#253332
#253312
#245332
#245322
#245292
#245282
#218382
#214062
#213552