Finite Group Databasebeta

← #6526701

Subgroups — #6526701

Order 512.

1944 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3236C211yesmaximal, characteristic, complemented
2C2×#3110C211yesmaximal, characteristic
3C2×#2566C211yesmaximal, characteristic
4#34213C211yesmaximal, characteristic, complemented
5#28123C212yesmaximal
6#28120C212yesmaximal
7#27653C212yesmaximal, complemented
8#27648C212yesmaximal, complemented
9#25462C214yesmaximal, complemented
10#25331C214yesmaximal, complemented
11#24824C214yesmaximal, complemented
12#24689C214yesmaximal, complemented
13#21955C211yesmaximal, characteristic, complemented
14#21838C211yesmaximal, characteristic, complemented
15#20738C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#3236—28no—
20C2×#501C2212yescomplemented
21C2×#491C2212yescomplemented
22#3110—28no—
23C2×#484C2212yes—
24C2×#484C2212yes—
25C2×#483C2212yes—
26C2×#479C2214yescomplemented
27C2×#474C2212yes—
28C2×#461C2212yes—
29C2×#450C2212yes—
30C2×#440C2214yes—
31C2×#432C2211yescharacteristic
32C2×#432C2211yescharacteristic
33#2566—22no—
34#2566—22no—
35#2566C2214yes—
36#2566C2214yes—
37C2×#396—21no—
38C2×#396—21no—
39#2370—22no—
40#2369C2212yescomplemented
41#2365C2212yescomplemented
42#2355—22no—
43#2355C2212yescomplemented
44Q8×C4⋊C4C2212yescomplemented
45#2329—21no—
46#2329—21no—
47#2317—21no—
48C4×C22⋊Q8C2212yes—
49C4×C22⋊Q8C2212yes—
50C2×#390C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#385—21no—
53C2×#385—21no—
54#2285C2211yescharacteristic
55C2×#384C2211yescharacteristic
56C2×#382C2211yescharacteristic
57#2274—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59#2085C2214yescomplemented
60#2083C2214yescomplemented
61#2074C2214yescomplemented
62#2072C2214yescomplemented
63#2067C2214yes—
64#2065C2214yes—
65#2055C2214yes—
66#2055C2214yes—
67#1992C2212yescomplemented
68#1992C2212yescomplemented
69#1989C2214yescomplemented
70#1949C2214yescomplemented
71#1947C2214yescomplemented
72#1940C2214yescomplemented
73#1940C2214yescomplemented
74#1926C2214yes—
75#1925C2214yes—
76#1880C2214yes—
77#1880C2214yes—
78#1824C2212yes—
79#1824C2212yes—
80#1823C2212yes—
81#1805C2214yescomplemented
82#1803C2214yescomplemented
83#1802C2212yes—
84#1802C2212yes—
85#1797C2212yes—
86#1797C2212yes—
87#1786C2212yes—
88#1762C2214yes—
89C23×D4C2311yescharacteristic
90C24×C4—21no—
91C22×SD16C2314yes—
92#533—24no—
93#533D418yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#125—21no—
97C2×#125—21no—
98C2×#125—21no—
99C2×#125—24no—
100C2×#125—24no—
101C2×#125C2312yes—
102C2×#125C2312yes—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8C2312yes—
105C2×C4×Q8—21no—
106C2×C4×Q8C2312yes—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—24no—
112C2×C4×D4—24no—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×#117—21no—
116C2×#117—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#501—28no—
135#491—28no—
136#484—28no—
137#484—28no—
138#483—28no—
139#479—216no—
140#474—28no—
141#461—28no—
142#450—28no—
143#440—216no—
144#432—24no—
145#432—24no—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#106C2314yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×#97C2312yes—
162C2×#97C2312yes—
163#396—24no—
164#396—24no—
165#396—24no—
166#396—24no—
167#391—22no—
168#391—22no—
169#391C2312yes—
170#391C2312yes—
171#390—22no—
172#390—22no—
173#390C2314yes—
174#390C2314yes—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389C2312yes—
180#389C2312yes—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#385—22no—
188#385—22no—
189#385—24no—
190#385—24no—
191#385—24no—
192#385—24no—
193#384—22no—
194#384—22no—
195#384C2314yes—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#382—22no—
199#382—22no—
200#382—21no—
201#382C2314yes—
202#380—21no—
203#380C2312yes—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C2×#95—21no—
221C2×#95—21no—
222C2×#95D411yescharacteristic
223C2×#95D411yescharacteristic
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232C25—21no—
233C22×Q8—21no—
234C22×Q8C2412yes—
235C22×D4—24no—
236C22×D4—21no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—42no—
240C22×D4C2412yes—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—44no—
247C23×C4—82no—
248C23×C4—82no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×SD16—28no—
257C2×SD16—28no—
258C2×SD16—44no—
259C22×C8C2414yes—
260C22×C8C2414yes—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8C2×D414yes—
264C4⋊Q8C2×D414yes—
265#125—28no—
266#125—24no—
267#125—24no—
268#125—24no—
269#125—24no—
270#125—44no—
271#125—44no—
272C22⋊Q8—28no—
273C22⋊Q8—24no—
274C22⋊Q8C2×D418yes—
275C4×Q8—28no—
276C4×Q8—24no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8C2×D418yes—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—44no—
285C4×D4—44no—
286#117—28no—
287#117—44no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4—82no—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—82no—
344C2×C22⋊C4—82no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—42no—
352C2×C42—42no—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D414yes—
358C2×C42#4114yes—
359C2×C42C2412yes—
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—28no—
364#107—28no—
365#106—28no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—44no—
369C4⋊C8—44no—
370#103—28no—
371#103—28no—
372#103—28no—
373#102—28no—
374#102—28no—
375#102—44no—
376C22⋊C8—24no—
377C22⋊C8—24no—
378#97—24no—
379#97—24no—
380#95—24no—
381#95—24no—
382#95—22no—
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#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—24no—
397C24—22no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24C22×D411yescharacteristic
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—81no—
410C2×Q8—81no—
411C2×Q8C22×D414yes—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D42−1+414yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—816no—
463C22×C4#12114yes—
464C22×C4#12114yes—
465C22×C4#13714yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C2×#4112yes—
477C22×C4C2511yescharacteristic
478SD16—416no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—81no—
513C4⋊C4—81no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4C22×D414yes—
521C22⋊C4—28no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—48no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C42—28no—
530C42—24no—
531C42—24no—
532C42—42no—
533C42—42no—
534C42—44no—
535C42—82no—
536C42—82no—
537C42—82no—
538C42—82no—
539C42#12114yes—
540C42#12114yes—
541C42C22×D412yes—
542C42C22×D412yes—
543C42C2×#4112yes—
544C42C2×#4112yes—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—24no—
550C23—48no—
551C23—48no—
552C23—48no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23C2×#12112yes—
564C23C2×#13712yes—
565C23#57512yes—
566C23C23×D411yescharacteristic
567Q8—28no—
568Q8—44no—
569Q8—84no—
570D4—24no—
571D4—24no—
572D4—48no—
573D4—42no—
574D4—42no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—28no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
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—44no—
602C2×C4—44no—
603C2×C4—44no—
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—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—88no—
624C2×C4—88no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4C2×#12112yes—
631C2×C4C2×#12112yes—
632C2×C4C2×#12112yes—
633C2×C4C2×#12112yes—
634C2×C4C2×#12112yes—
635C2×C4C2×#13714yes—
636C2×C4C23×D411yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C2×2−1+412yes—
639C2×C4#58512yes—
640C8—44no—
641C8—44no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—816no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22#303714yes—
657C22C22×#12111yescharacteristic
658C22#348511yescharacteristic
659C22#348611yescharacteristic
660C22C22×#13711yescharacteristic
661C22C2×#57511yescharacteristic
662C22#358511yescharacteristic
663C22#358511yescharacteristic
664C4—22no—
665C4—22no—
666C4—24no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—44no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—44no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4#348912yes—
681C4#349512yes—
682C2—24no—
683C2—84no—
684C2#3563311yescharacteristic
685C2C2×#303711yescharacteristic
686C2#6042611yescharacteristic
687C2#6170811yescharacteristic
688C2#6171311yescharacteristic
689C2#6182611yescharacteristic
690C2#6183111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
1. C2×#3236C2semidirect
4. #34213C2semidirect
7. #27653C2semidirect
8. #27648C2semidirect
9. #25462C2semidirect
10. #25331C2semidirect
11. #24824C2semidirect
12. #24689C2semidirect
13. #21955C2semidirect
14. #21838C2semidirect
20. C2×#501C22semidirect
21. C2×#491C22semidirect
26. C2×#479C22semidirect
40. #2369C22semidirect
41. #2365C22semidirect
43. #2355C22semidirect
44. Q8×C4⋊C4C22semidirect
59. #2085C22semidirect
60. #2083C22semidirect
61. #2074C22semidirect
62. #2072C22semidirect
67. #1992C22semidirect
68. #1992C22semidirect
69. #1989C22semidirect
70. #1949C22semidirect
71. #1947C22semidirect
72. #1940C22semidirect
73. #1940C22semidirect
81. #1805C22semidirect
82. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32362
C2×#31102
C2×#25662
#342132
#281232
#281202
#276532
#276482
#254622
#253312
#248242
#246892
#219552
#218382
#207382