Finite Group Databasebeta

← #6463837

Subgroups — #6463837

Order 512.

2252 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3081C212yesmaximal, complemented
2#34272C211yesmaximal, characteristic, complemented
3#34210C211yesmaximal, characteristic
4#28214C214yesmaximal, complemented
5#28210C214yesmaximal, complemented
6#28207C212yesmaximal, complemented
7#28207C212yesmaximal, complemented
8#28202C212yesmaximal, complemented
9#28201C212yesmaximal, complemented
10#23657C211yesmaximal, characteristic, complemented
11#23656C211yesmaximal, characteristic, complemented
12#23505C211yesmaximal, characteristic
13C2×#2041C211yesmaximal, characteristic
14#21065C212yesmaximal, complemented
15#21063C212yesmaximal, complemented
16#20754C211yesmaximal, characteristic, complemented
17#20650C212yesmaximal, complemented
18C2×#544C2211yescharacteristic
19C22×#127C2211yescharacteristic
20C2×#518C2212yes—
21#3081—216no—
22C2×#478C2214yescomplemented
23C2×#478C2214yescomplemented
24C2×#476C2214yescomplemented
25C2×#474C2214yescomplemented
26C2×#428C2212yes—
27C2×#427C2212yes—
28C2×#424C2212yes—
29C2×#418C2212yes—
30C2×#418C2212yes—
31C22×#102C2212yes—
32C22×#102C2212yes—
33#2370—21no—
34#2370—21no—
35#2369—21no—
36#2368—22no—
37#2366—21no—
38#2366—21no—
39#2366—21no—
40#2366—21no—
41#2365—21no—
42#2360—22no—
43D4×C22⋊C4—22no—
44D4×C22⋊C4—22no—
45#2331C2212yescomplemented
46#2330C2212yescomplemented
47#2328C2214yes—
48#2317C2211yescharacteristic
49#2309C2212yes—
50C4×C4⋊Q8C2212yescomplemented
51C4×#127C2212yes—
52C4×#127C2212yes—
53C4×#125C2212yescomplemented
54#2296C2211yescharacteristic
55C2×#389C2211yescharacteristic
56#2285C2211yescharacteristic
57#2274C2211yescharacteristic
58#2274C2211yescharacteristic
59C4×#117C2211yescharacteristic
60#2085C2212yescomplemented
61#2085C2214yescomplemented
62#2085C2212yescomplemented
63#2084C2211yescharacteristic, complemented
64#2084C2212yescomplemented
65#2084C2211yescharacteristic, complemented
66#2083C2211yescharacteristic, complemented
67#2083C2212yescomplemented
68#2083C2211yescharacteristic, complemented
69#2041C2212yes—
70#2041C2212yes—
71#2041C2212yes—
72#2041C2212yes—
73#2041C2212yes—
74#2041C2214yes—
75#2041C2212yes—
76#1926C2214yescomplemented
77#1926C2214yescomplemented
78#1926C2214yescomplemented
79#1926C2214yescomplemented
80#1925C2214yescomplemented
81#1925C2214yescomplemented
82#1925C2214yescomplemented
83#1925C2214yescomplemented
84#1832C2214yescomplemented
85#1831C2212yescomplemented
86#1831C2212yescomplemented
87#1804C2214yescomplemented
88#1804C2214yescomplemented
89#1802C2214yescomplemented
90#1802C2214yescomplemented
91#1736C2212yes—
92#1735C2212yes—
93#1729C2212yes—
94#1728C2211yescharacteristic
95C2×#336C2211yescharacteristic
96C23×D4—21no—
97C23×D4—21no—
98C23×D4C2312yes—
99C22×M16C2312yes—
100C23×C8C2312yes—
101#544—24no—
102#544D418yescomplemented
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2312yes—
105C2×#127—28no—
106C2×#127—21no—
107C2×#127—21no—
108C2×#127C2312yes—
109C2×#127C2312yes—
110C2×#125—21no—
111C2×#125C2312yes—
112C2×C22⋊Q8—22no—
113#518—28no—
114#518—28no—
115C2×C4×Q8—22no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4C2314yes—
123C2×C4×D4C2314yes—
124C2×#117C2312yes—
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—24no—
133C22×C42C2311yescharacteristic
134#478—216no—
135#478—216no—
136#476—216no—
137#474—216no—
138#428—28no—
139#427—28no—
140C2×#107C2312yes—
141C2×#107C2314yes—
142C2×#107C2312yes—
143C2×#106C2312yes—
144C2×#106C2314yes—
145C2×#106C2312yes—
146#424—28no—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2314yes—
149#418—28no—
150#418—28no—
151C2×#102—28no—
152C2×#102—28no—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161#391—22no—
162#391—22no—
163#391—22no—
164#390—22no—
165#390—22no—
166#390—22no—
167#390—22no—
168#390—22no—
169#390—22no—
170#390C2314yes—
171#390C2314yes—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389C2311yescharacteristic
180#389C2312yes—
181#389C2311yescharacteristic
182#389C2311yescharacteristic
183#389C2312yes—
184#389C2311yescharacteristic
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#386—24no—
192#386—24no—
193#386—24no—
194#386—24no—
195#385—21no—
196#385—21no—
197#385—22no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2312yes—
201#382C2312yes—
202#382C2312yes—
203#380—21no—
204#380—21no—
205#379—24no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4D412yes—
229#376C2312yes—
230C2×#95—21no—
231C43C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#340C2312yes—
238#336C2311yescharacteristic
239#336C2312yes—
240#336C2311yescharacteristic
241#336C2312yes—
242#336C2312yes—
243#336C2314yes—
244#336C2314yes—
245C25—22no—
246C22×Q8—22no—
247C22×D4—28no—
248C22×D4—28no—
249C22×D4—24no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—24no—
256C22×D4—24no—
257C22×D4—24no—
258C22×D4—24no—
259C22×D4—82no—
260C22×D4—82no—
261C22×D4—82no—
262C22×D4—82no—
263C22×D4#4118yes—
264C22×D4C2414yes—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—24no—
268C23×C4—24no—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2411yescharacteristic
272C2×M16—24no—
273C2×M16—24no—
274C2×M16—24no—
275C22×C8—24no—
276C22×C8—24no—
277C22×C8—24no—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2414yes—
281C4⋊Q8—28no—
282C4⋊Q8—24no—
283C4⋊Q8C2×D418yescomplemented
284#127—28no—
285#127—28no—
286#127—28no—
287#127—28no—
288#125—28no—
289#125—24no—
290#125C2×D418yescomplemented
291C22⋊Q8—216no—
292C4×Q8—84no—
293C4×D4—28no—
294C4×D4—28no—
295C4×D4—28no—
296C4×D4—28no—
297C4×D4—48no—
298C4×D4—48no—
299C4×D4—84no—
300C4×D4—84no—
301C4×D4—84no—
302C4×D4—84no—
303#117—22no—
304#117—24no—
305#117—24no—
306#117—24no—
307#117—24no—
308#117C2×D414yes—
309#117C2×D414yes—
310#117#4114yes—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D414yes—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—416no—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—21no—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D412yes—
369C2×C42C2×D412yes—
370C2×C42C2412yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2412yes—
374#107—24no—
375#107—24no—
376#107—28no—
377#106—24no—
378#106—24no—
379#106—28no—
380C4⋊C8—28no—
381C4⋊C8—28no—
382#102—28no—
383#102—28no—
384#102—28no—
385#102—28no—
386#102—28no—
387#102—28no—
388#102—28no—
389#102—28no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—21no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—21no—
403#95—21no—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408C24—24no—
409C24—28no—
410C24—44no—
411C24—44no—
412C24—82no—
413C24—82no—
414C24—82no—
415C24—82no—
416C24C22×D411yescharacteristic
417C2×Q8—28no—
418C2×Q8—82no—
419C2×Q8—82no—
420C2×D4—28no—
421C2×D4—28no—
422C2×D4—28no—
423C2×D4—28no—
424C2×D4—28no—
425C2×D4—216no—
426C2×D4—216no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—48no—
437C2×D4—48no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4—816no—
445C2×D4—816no—
446C2×D4C2×#4118yes—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—42no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—84no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4114yes—
485C22×C42−1+414yes—
486C22×C4C2511yescharacteristic
487M16—48no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—48no—
498C4⋊C4—24no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—82no—
523C4⋊C4—81no—
524C4⋊C4—81no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4#12114yes—
528C4⋊C4#12114yes—
529C4⋊C4C22×D414yes—
530C4⋊C4C22×D412yes—
531C4⋊C4C22×D412yes—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—44no—
535C22⋊C4—42no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C22⋊C4—84no—
542C22⋊C4#12118yes—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—24no—
547C42—24no—
548C42—22no—
549C42—22no—
550C42—24no—
551C42—42no—
552C42—42no—
553C42—44no—
554C42—42no—
555C42—82no—
556C42—82no—
557C42—82no—
558C42—82no—
559C42C22×D414yes—
560C42C2×#4114yes—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—28no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—416no—
573C23—88no—
574C23—88no—
575C23—164no—
576C23—164no—
577C23—164no—
578C23—164no—
579C23C2×#12112yes—
580C23#52512yes—
581C23#53612yes—
582C23C23×D411yescharacteristic
583Q8—88no—
584D4—28no—
585D4—28no—
586D4—416no—
587D4—416no—
588D4—44no—
589D4—44no—
590D4—416no—
591D4—88no—
592D4—88no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—84no—
640C2×C4—168no—
641C2×C4—168no—
642C2×C4—162no—
643C2×C4C2×#12114yes—
644C2×C4C2×#12114yes—
645C2×C4C2×#12112yes—
646C2×C4#52512yes—
647C2×C4#53612yes—
648C2×C4#54818yes—
649C2×C4#55218yes—
650C2×C4#57612yes—
651C2×C4#57712yes—
652C2×C4C23×D411yescharacteristic
653C2×C4C22×#4112yes—
654C2×C4C22×#4111yescharacteristic
655C2×C4C2×2−1+412yes—
656C8—42no—
657C8—42no—
658C8—44no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—48no—
666C22—48no—
667C22—48no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22—84no—
672C22—1616no—
673C22#253614yes—
674C22C22×#12111yescharacteristic
675C22C2×#52511yescharacteristic
676C22C2×#53611yescharacteristic
677C22C2×#57611yescharacteristic
678C22C2×#57711yescharacteristic
679C22#358511yescharacteristic
680C22#358511yescharacteristic
681C4—24no—
682C4—24no—
683C4—22no—
684C4—22no—
685C4—44no—
686C4—42no—
687C4—42no—
688C4—41no—
689C4—41no—
690C4—42no—
691C4—42no—
692C4—82no—
693C4—82no—
694C4—81no—
695C4—82no—
696C4—81no—
697C4#253614yes—
698C4C2×#54812yes—
699C4C2×#55212yes—
700C2—24no—
701C2—88no—
702C2C2×#253611yescharacteristic
703C2#6042811yescharacteristic
704C2#6043111yescharacteristic
705C2#6057811yescharacteristic
706C2#6057911yescharacteristic
707C2#6087711yescharacteristic
708C2#6088311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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
1. C2×#3081C2semidirect
2. #34272C2semidirect
4. #28214C2semidirect
5. #28210C2semidirect
6. #28207C2semidirect
7. #28207C2semidirect
8. #28202C2semidirect
9. #28201C2semidirect
10. #23657C2semidirect
11. #23656C2semidirect
14. #21065C2semidirect
15. #21063C2semidirect
16. #20754C2semidirect
17. #20650C2semidirect
22. C2×#478C22semidirect
23. C2×#478C22semidirect
24. C2×#476C22semidirect
25. C2×#474C22semidirect
45. #2331C22semidirect
46. #2330C22semidirect
50. C4×C4⋊Q8C22semidirect
53. C4×#125C22semidirect
60. #2085C22semidirect
61. #2085C22semidirect
62. #2085C22semidirect
63. #2084C22semidirect
64. #2084C22semidirect
65. #2084C22semidirect
66. #2083C22semidirect
67. #2083C22semidirect
68. #2083C22semidirect
76. #1926C22semidirect
77. #1926C22semidirect
78. #1926C22semidirect
79. #1926C22semidirect
80. #1925C22semidirect
81. #1925C22semidirect
82. #1925C22semidirect
83. #1925C22semidirect
84. #1832C22semidirect
85. #1831C22semidirect
86. #1831C22semidirect
87. #1804C22semidirect
88. #1804C22semidirect
89. #1802C22semidirect
90. #1802C22semidirect
102. #544D4semidirect
283. C4⋊Q8C2×D4semidirect
290. #125C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30812
#342722
#342102
#282142
#282102
#282072
#282022
#282012
#236572
#236562
#235052
C2×#20412
#210652
#210632
#207542
#206502