Finite Group Databasebeta

← #6494006

Subgroups — #6494006

Order 512.

2466 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3215C212yesmaximal, complemented
2C2×#2858C211yesmaximal, characteristic
3C4×#533C211yesmaximal, characteristic, complemented
4#33165C214yesmaximal, complemented
5#29801C214yesmaximal, complemented
6#28754C212yesmaximal, complemented
7#28753C212yesmaximal, complemented
8#28451C212yesmaximal, complemented
9#28447C212yesmaximal, complemented
10#24217C214yesmaximal, complemented
11#24212C212yesmaximal, complemented
12#24211C212yesmaximal
13#21808C212yesmaximal, complemented
14#20810C211yesmaximal, characteristic, complemented
15C2×#533C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×#124C2212yes—
19C22×#124C2212yes—
20#3215—216no—
21C2×#497C2214yescomplemented
22C2×#489C2214yescomplemented
23C2×#464C2218yescomplemented
24C2×#464C2218yescomplemented
25C2×#462C2214yescomplemented
26C2×#460C2214yes—
27C2×#451C2214yes—
28C2×#450C2214yes—
29C2×#432C2212yes—
30#2858—21no—
31#2858—21no—
32#2858—21no—
33#2858—21no—
34#2858C2212yes—
35#2858C2212yes—
36#2858C2212yes—
37#2858C2212yes—
38C4×C4⋊Q8C2211yescharacteristic, complemented
39C4×C4⋊Q8C2211yescharacteristic, complemented
40C4×C4⋊Q8C2211yescharacteristic, complemented
41C4×C4⋊Q8C2211yescharacteristic, complemented
42C4×#125—21no—
43C4×#125—21no—
44C4×C22⋊Q8—21no—
45C4×C22⋊Q8—21no—
46C4×C22⋊Q8C2212yes—
47C4×C22⋊Q8C2212yes—
48#2296—21no—
49#2295—21no—
50#2295C2212yescomplemented
51C2×#390—21no—
52C2×#390—21no—
53C2×#390—21no—
54C2×#390—21no—
55#2285C2211yescharacteristic
56C42×Q8—21no—
57C42×Q8C2212yescomplemented
58C4×#117—21no—
59C2×C43C2211yescharacteristic
60#2043C2214yescomplemented
61#2042C2218yescomplemented
62#2041C2214yes—
63#2016C2214yescomplemented
64#2015C2218yescomplemented
65#2014C2214yes—
66#1983C2218yescomplemented
67#1983C2218yescomplemented
68#1982C2214yes—
69#1981C2214yescomplemented
70#1959C2214yescomplemented
71#1959C2214yescomplemented
72#1842C2212yescomplemented
73#1831C2212yescomplemented
74#1797C2212yes—
75#1797C2212yes—
76#1787C2214yes—
77#1786C2214yes—
78#1767C2212yescomplemented
79#1767C2212yescomplemented
80C23×Q8C2312yes—
81C23×D4—21no—
82C23×D4—21no—
83C23×D4C2312yes—
84C22×Q16C2318yes—
85C22×SD16C2318yes—
86#533—24no—
87#533D418yescomplemented
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8—44no—
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2312yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127—21no—
96C2×#127—21no—
97C2×#127—44no—
98C2×#127C2312yes—
99C2×#127C2312yes—
100C2×#125—21no—
101C2×#124—22no—
102C2×#124—22no—
103C2×#124—22no—
104C2×#124—22no—
105C2×#124—28no—
106C2×#124—28no—
107C2×#124—48no—
108C2×#124C2314yes—
109C2×#124C2314yes—
110C2×#124C2314yes—
111C2×#124C2314yes—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C2312yes—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×#117—21no—
121C2×#117—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C22⋊C4—22no—
125C22×C22⋊C4—22no—
126C22×C22⋊C4—22no—
127C22×C22⋊C4—22no—
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#497—216no—
136#489—216no—
137#464—232no—
138#464—416no—
139#462—48no—
140#460—48no—
141#451—216no—
142#450—216no—
143#432—28no—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2314yes—
146C2×C4⋊C8C2314yes—
147C2×#103C2318yes—
148C2×#103C2318yes—
149C2×#103C2318yes—
150C2×#102C2318yes—
151C2×#102C2318yes—
152C2×#102C2318yes—
153C2×C22⋊C8C2314yes—
154C2×#97C2314yes—
155#391—21no—
156#391—21no—
157#391C2312yes—
158#391C2312yes—
159#390—22no—
160#390—22no—
161#390—22no—
162#390—22no—
163#390—22no—
164#390—22no—
165#390—22no—
166#390—22no—
167#390—22no—
168#390—22no—
169#390—22no—
170#390—22no—
171#390—22no—
172#390—22no—
173#390—22no—
174#390—22no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389C2312yes—
182#389C2312yes—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#385—22no—
188#385—22no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#382—21no—
192#380—21no—
193#380C2312yes—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212#376—22no—
213C43—22no—
214C43D411yescharacteristic
215C43D411yescharacteristic
216C43D411yescharacteristic
217C43D411yescharacteristic
218C43C2311yescharacteristic
219C43C2311yescharacteristic
220C43C2311yescharacteristic
221C43C2311yescharacteristic
222#336C2314yes—
223#336C2314yes—
224C25—22no—
225C22×Q8—21no—
226C22×Q8—22no—
227C22×Q8—24no—
228C22×Q8—24no—
229C22×Q8—44no—
230C22×Q8—44no—
231C22×Q8C2414yes—
232C22×Q8C2412yes—
233C22×D4—24no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—24no—
240C22×D4—44no—
241C22×D4—44no—
242C22×D4—44no—
243C22×D4—44no—
244C22×D4—44no—
245C22×D4—44no—
246C22×D4—44no—
247C22×D4—44no—
248C22×D4C2414yes—
249C23×C4—22no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2411yescharacteristic
257C2×Q16—216no—
258C2×Q16—48no—
259C2×Q16—48no—
260C2×SD16—216no—
261C2×SD16—48no—
262C2×SD16—48no—
263C22×C8C2418yes—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—28no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8C2×D414yescomplemented
271C4⋊Q8C2×D414yescomplemented
272#127—28no—
273#127—44no—
274#127—44no—
275#127—44no—
276#125—28no—
277#124—28no—
278#124—28no—
279#124—28no—
280#124—28no—
281#124—216no—
282#124—48no—
283#124—48no—
284#124—48no—
285#124—48no—
286#124—48no—
287C22⋊Q8—28no—
288C22⋊Q8—24no—
289C22⋊Q8C2×D418yes—
290C4×Q8—28no—
291C4×Q8—24no—
292C4×Q8—28no—
293C4×Q8—28no—
294C4×Q8—44no—
295C4×Q8—44no—
296C4×Q8C2×D418yescomplemented
297#117—28no—
298#117—44no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—48no—
332C2×C22⋊C4—48no—
333C2×C22⋊C4—48no—
334C2×C22⋊C4—48no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—42no—
345C2×C42—42no—
346C2×C42—42no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D414yes—
370C2×C42#4114yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C4⋊C8—28no—
378C4⋊C8—28no—
379C4⋊C8—28no—
380#103—216no—
381#103—48no—
382#103—48no—
383#102—216no—
384#102—48no—
385#102—48no—
386C22⋊C8—28no—
387#97—28no—
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—
397#95—22no—
398C24—24no—
399C24—44no—
400C24—44no—
401C24—44no—
402C24—44no—
403C24—44no—
404C24—44no—
405C24—44no—
406C24C22×D411yescharacteristic
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—28no—
414C2×Q8—28no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—88no—
421C2×Q8—88no—
422C2×Q8C22×D414yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—28no—
426C2×D4—28no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—88no—
443C2×D4—88no—
444C2×D4—816no—
445C2×D4—816no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4#12114yes—
472C22×C4#12114yes—
473C22×C4#13714yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4112yes—
489C22×C4C2511yescharacteristic
490Q16—816no—
491SD16—816no—
492C2×C8—28no—
493C2×C8—28no—
494C2×C8—28no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4C22×D414yes—
516C22⋊C4—28no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C42—28no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—41no—
549C42—44no—
550C42—44no—
551C42—44no—
552C42—44no—
553C42—44no—
554C42#12114yes—
555C42#12114yes—
556C42C22×D412yes—
557C42C22×D412yes—
558C42C2×#4112yes—
559C42C2×#4112yes—
560C42H(5,𝔽2)14yes—
561C42H(5,𝔽2)14yes—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—88no—
574C23—88no—
575C23—88no—
576C23—88no—
577C23—88no—
578C23—88no—
579C23—88no—
580C23C2×#12112yes—
581C23C2×#13712yes—
582C23#57512yes—
583C23C23×D411yescharacteristic
584Q8—28no—
585Q8—48no—
586Q8—44no—
587Q8—48no—
588Q8—48no—
589Q8—88no—
590D4—48no—
591D4—48no—
592D4—88no—
593D4—88no—
594D4—88no—
595D4—88no—
596D4—88no—
597C2×C4—28no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—28no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—88no—
647C2×C4C2×#12112yes—
648C2×C4C2×#12112yes—
649C2×C4C2×#12112yes—
650C2×C4C2×#12112yes—
651C2×C4C2×#12112yes—
652C2×C4#53512yes—
653C2×C4#53512yes—
654C2×C4#53512yes—
655C2×C4#53512yes—
656C2×C4C2×#13712yes—
657C2×C4C2×#13712yes—
658C2×C4#57512yes—
659C2×C4#57512yes—
660C2×C4C23×D411yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C2×C4C2×H(5,𝔽2)12yes—
663C2×C4C2×H(5,𝔽2)12yes—
664C8—48no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—84no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—1616no—
679C22#303714yes—
680C22C22×#12111yescharacteristic
681C22C2×#53511yescharacteristic
682C22C2×#53511yescharacteristic
683C22C22×#13711yescharacteristic
684C22C2×#57511yescharacteristic
685C22C2×#57511yescharacteristic
686C22C2×#57511yescharacteristic
687C4—22no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—22no—
694C4—22no—
695C4—22no—
696C4—22no—
697C4—41no—
698C4—41no—
699C4—44no—
700C4—44no—
701C4—82no—
702C4—82no—
703C4—84no—
704C4#321514yescomplemented
705C4#349812yes—
706C4#350812yes—
707C2—24no—
708C2—164no—
709C2#3527211yescharacteristic
710C2C2×#303711yescharacteristic
711C2#6041511yescharacteristic
712C2C2×#321511yescharacteristic
713C2#6079011yescharacteristic
714C2#6079911yescharacteristic
715C2#6080411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3215C2semidirect
3. C4×#533C2semidirect
4. #33165C2semidirect
5. #29801C2semidirect
6. #28754C2semidirect
7. #28753C2semidirect
8. #28451C2semidirect
9. #28447C2semidirect
10. #24217C2semidirect
11. #24212C2semidirect
13. #21808C2semidirect
14. #20810C2semidirect
21. C2×#497C22semidirect
22. C2×#489C22semidirect
23. C2×#464C22semidirect
24. C2×#464C22semidirect
25. C2×#462C22semidirect
38. C4×C4⋊Q8C22semidirect
39. C4×C4⋊Q8C22semidirect
40. C4×C4⋊Q8C22semidirect
41. C4×C4⋊Q8C22semidirect
50. #2295C22semidirect
57. C42×Q8C22semidirect
60. #2043C22semidirect
61. #2042C22semidirect
63. #2016C22semidirect
64. #2015C22semidirect
66. #1983C22semidirect
67. #1983C22semidirect
69. #1981C22semidirect
70. #1959C22semidirect
71. #1959C22semidirect
72. #1842C22semidirect
73. #1831C22semidirect
78. #1767C22semidirect
79. #1767C22semidirect
87. #533D4semidirect
270. C4⋊Q8C2×D4semidirect
271. C4⋊Q8C2×D4semidirect
296. C4×Q8C2×D4semidirect
704. C4#3215semidirect

Maximal subgroups

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

subgroupindex
C2×#32152
C2×#28582
C4×#5332
#331652
#298012
#287542
#287532
#284512
#284472
#242172
#242122
#242112
#218082
#208102