Finite Group Databasebeta

← #6529805

Subgroups — #6529805

Order 512.

1964 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3110C211yesmaximal, characteristic
2C2×#2983C211yesmaximal, characteristic, complemented
3#35179C211yesmaximal, characteristic, complemented
4C2×#2580C211yesmaximal, characteristic
5#28277C214yesmaximal, complemented
6#28200C214yesmaximal, complemented
7#25481C214yesmaximal, complemented
8#25308C214yesmaximal, complemented
9#24606C212yesmaximal, complemented
10#24602C212yesmaximal, complemented
11#24507C212yesmaximal
12#24501C212yesmaximal
13#21986C211yesmaximal, characteristic, complemented
14#21866C211yesmaximal, characteristic, complemented
15#20738C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#3110—28no—
20C2×#484C2212yes—
21C2×#474C2212yes—
22C2×#461C2212yes—
23C2×#450C2212yes—
24#2983—28no—
25C2×#446C2214yescomplemented
26C2×#446C2214yescomplemented
27C2×#440C2214yes—
28C2×#432C2211yescharacteristic
29C2×#432C2212yes—
30C2×#432C2211yescharacteristic
31C2×#431C2212yes—
32#2855C2212yes—
33#2855C2212yes—
34#2627—21no—
35#2627—21no—
36#2582C2212yescomplemented
37#2582C2212yescomplemented
38#2581—22no—
39#2580—22no—
40#2580—22no—
41#2580C2214yes—
42#2580C2214yes—
43#2568C2212yescomplemented
44#2567C2212yescomplemented
45#2565—22no—
46C2×#399—21no—
47C2×#393—21no—
48#2317—21no—
49#2310—21no—
50C2×#390C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#385—21no—
55C2×#385—21no—
56#2285C2211yescharacteristic
57C2×#382C2211yescharacteristic
58#2087C2214yescomplemented
59#2086C2214yescomplemented
60#2068C2214yescomplemented
61#2067C2214yes—
62#2066C2214yescomplemented
63#2065C2214yes—
64#2055C2214yes—
65#2055C2214yes—
66#1993C2212yescomplemented
67#1973C2214yes—
68#1968C2214yes—
69#1960C2212yescomplemented
70#1949C2214yescomplemented
71#1944C2214yescomplemented
72#1942C2214yescomplemented
73#1936C2214yes—
74#1935C2214yescomplemented
75#1934C2214yes—
76#1933C2214yescomplemented
77#1872C2212yes—
78#1872C2212yes—
79#1853C2212yes—
80#1852C2212yes—
81#1847C2212yescomplemented
82#1836C2212yescomplemented
83#1824C2212yes—
84#1805C2214yescomplemented
85#1802C2212yes—
86#1802C2212yes—
87#1786C2212yes—
88#1762C2214yes—
89C23×Q8D411yescharacteristic
90C23×D4C2311yescharacteristic
91C24×C4—21no—
92C22×SD16C2314yes—
93#533—24no—
94#533—24no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×#125—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8C2312yes—
100C2×C4×Q8—24no—
101C2×C4×Q8—21no—
102C2×C4×Q8—24no—
103C2×C4×Q8D412yes—
104C2×C4×Q8D412yes—
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2312yes—
107C2×C4×Q8C2312yes—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4—24no—
111C2×C4×D4—24no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×#117—21no—
115C2×#117—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#484—28no—
133#474—28no—
134#461—28no—
135#450—28no—
136#446—216no—
137#446—216no—
138#440—216no—
139#432—24no—
140#432—28no—
141#432D418yes—
142#431—28no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158C2×#97C2314yes—
159C2×#97C2312yes—
160C2×#97C2312yes—
161C2×C4×C8C2314yes—
162#400—22no—
163#400—22no—
164#399—24no—
165#399—24no—
166#398—22no—
167#398—22no—
168#398—22no—
169#398—22no—
170#395—22no—
171#395—22no—
172#393—24no—
173#393—24no—
174#391—22no—
175#391—22no—
176#391—22no—
177#391—22no—
178#391C2312yes—
179#391C2312yes—
180#391C2312yes—
181#391C2312yes—
182#390—22no—
183#390—22no—
184#390C2314yes—
185#390C2314yes—
186#389—22no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2314yes—
195#389C2314yes—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#385—22no—
201#385—22no—
202#385—24no—
203#385—24no—
204#385—24no—
205#385—24no—
206#384—22no—
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#382—22no—
210#382—21no—
211#382—21no—
212#382—21no—
213#382C2312yes—
214#382C2312yes—
215#381—22no—
216#381—22no—
217#380—21no—
218#380C2312yes—
219C4×C4⋊C4—21no—
220C4×C4⋊C4C2312yes—
221C2×#95—21no—
222C2×#95—21no—
223C2×#95—21no—
224C2×#95—21no—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231C25—21no—
232C22×Q8—24no—
233C22×Q8—24no—
234C22×Q8—21no—
235C22×Q8—42no—
236C22×Q8C2×D412yes—
237C22×Q8C2×D412yes—
238C22×Q8C2412yes—
239C22×D4—24no—
240C22×D4—21no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—42no—
244C22×D4C2412yes—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—44no—
251C23×C4—82no—
252C23×C4—82no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×SD16—28no—
261C2×SD16—28no—
262C2×SD16—44no—
263C22×C8C2414yes—
264C22×C8C2414yes—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269#125—44no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272C22⋊Q8—44no—
273C4×Q8—28no—
274C4×Q8—28no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—44no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—44no—
287C4×D4—44no—
288#117—44no—
289#117—44no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—82no—
341C2×C22⋊C4—82no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D414yes—
353C2×C42#4114yes—
354C2×C42C2412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—28no—
359#107—44no—
360#106—44no—
361C4⋊C8—28no—
362C4⋊C8—28no—
363C4⋊C8—24no—
364C4⋊C8C2×D418yes—
365#103—28no—
366#103—44no—
367#103—44no—
368#102—28no—
369#102—28no—
370#102—44no—
371C22⋊C8—24no—
372C22⋊C8C2×D418yes—
373#97—28no—
374#97—24no—
375#97C2×D418yes—
376C4×C8—28no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397C24—24no—
398C24—22no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24C22×D411yescharacteristic
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—416no—
417C2×Q8—42no—
418C2×Q8—81no—
419C2×Q8—81no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D42−1+414yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4—816no—
471C22×C4#12114yes—
472C22×C4#12114yes—
473C22×C4#13714yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C2×#4112yes—
485C22×C4C2511yescharacteristic
486SD16—416no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8C22×D418yes—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—81no—
518C4⋊C4—81no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C22⋊C4—48no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—48no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—44no—
540C42—44no—
541C42#12114yes—
542C42#12114yes—
543C42C22×D412yes—
544C42C22×D412yes—
545C42C2×#4112yes—
546C42C2×#4112yes—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—24no—
552C23—48no—
553C23—48no—
554C23—48no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23C2×#12112yes—
566C23C2×#13712yes—
567C23#57512yes—
568C23C23×D411yescharacteristic
569Q8—48no—
570Q8—48no—
571Q8—44no—
572Q8—44no—
573Q8—84no—
574D4—24no—
575D4—24no—
576D4—48no—
577D4—42no—
578D4—42no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—44no—
605C2×C4—44no—
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—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—88no—
628C2×C4—88no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4C2×#12112yes—
635C2×C4C2×#12112yes—
636C2×C4C2×#12112yes—
637C2×C4C2×#12112yes—
638C2×C4C2×#12112yes—
639C2×C4C2×#13714yes—
640C2×C4C23×D411yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C2×2−1+412yes—
643C2×C4#58512yes—
644C8—28no—
645C8—44no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—816no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22#305614yes—
661C22C22×#12111yescharacteristic
662C22#348611yescharacteristic
663C22#348711yescharacteristic
664C22C22×#13711yescharacteristic
665C22C2×#57511yescharacteristic
666C22#358511yescharacteristic
667C22#358511yescharacteristic
668C4—22no—
669C4—22no—
670C4—24no—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4#348912yes—
685C4#349512yes—
686C2—24no—
687C2—84no—
688C2#3566111yescharacteristic
689C2C2×#305611yescharacteristic
690C2#6045411yescharacteristic
691C2#6182611yescharacteristic
692C2#6183111yescharacteristic
693C2#6189211yescharacteristic
694C2#6189511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2983C2semidirect
3. #35179C2semidirect
5. #28277C2semidirect
6. #28200C2semidirect
7. #25481C2semidirect
8. #25308C2semidirect
9. #24606C2semidirect
10. #24602C2semidirect
13. #21986C2semidirect
14. #21866C2semidirect
25. C2×#446C22semidirect
26. C2×#446C22semidirect
36. #2582C22semidirect
37. #2582C22semidirect
43. #2568C22semidirect
44. #2567C22semidirect
58. #2087C22semidirect
59. #2086C22semidirect
60. #2068C22semidirect
62. #2066C22semidirect
66. #1993C22semidirect
69. #1960C22semidirect
70. #1949C22semidirect
71. #1944C22semidirect
72. #1942C22semidirect
74. #1935C22semidirect
76. #1933C22semidirect
81. #1847C22semidirect
82. #1836C22semidirect
84. #1805C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31102
C2×#29832
#351792
C2×#25802
#282772
#282002
#254812
#253082
#246062
#246022
#245072
#245012
#219862
#218662
#207382