Finite Group Databasebeta

← #6606571

Subgroups — #6606571

Order 512.

1925 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3340C212yesmaximal, complemented
2#36187C211yesmaximal, characteristic
3C4×#556C211yesmaximal, characteristic, complemented
4#33198C211yesmaximal, characteristic, complemented
5#33113C212yesmaximal, complemented
6#28687C211yesmaximal, characteristic, complemented
7#28279C211yesmaximal, characteristic, complemented
8#27649C212yesmaximal, complemented
9#25452C212yesmaximal, complemented
10#25203C214yesmaximal, complemented
11#24753C212yesmaximal, complemented
12#24679C212yesmaximal, complemented
13#24585C214yesmaximal, complemented
14D4×C4⋊C8C211yesmaximal, characteristic, complemented
15#22489C212yesmaximal, complemented
16C4×#460C211yesmaximal, characteristic, complemented
17#21169C212yesmaximal, complemented
18C2×#556C2211yescharacteristic
19C2×#555C2212yes—
20#3340—216no—
21C2×#498C2212yes—
22C2×#485C2214yes—
23C2×#482C2214yes—
24C2×#477C2212yes—
25C2×#474C2212yes—
26C2×#465C2214yes—
27C2×#460C2211yescharacteristic
28C2×C4×SD16C2212yes—
29C2×C8×D4C2212yes—
30C2×#427C2214yes—
31C22×C4⋊C8C2212yes—
32#2868C2211yescharacteristic
33#2856C2212yes—
34#2563C2212yes—
35#2537—21no—
36#2537—21no—
37#2500C2214yes—
38#2491—22no—
39#2491—22no—
40#2373—22no—
41#2367—21no—
42#2365C2211yescharacteristic
43#2362C2212yes—
44#2356—21no—
45#2355C2211yescharacteristic
46#2354—21no—
47#2354—21no—
48#2354C2211yescharacteristic
49#2347C2212yes—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×#126—22no—
53C4×#124—21no—
54C4×C22⋊Q8C2212yes—
55#2285C2212yes—
56C42×Q8—21no—
57C42×Q8C2211yescharacteristic
58C42×D4C2211yescharacteristic
59C4×#117C2212yes—
60#2085C2214yes—
61#2084C2214yes—
62#2074C2214yes—
63#2067C2214yes—
64#2065C2214yes—
65#2055C2214yes—
66#1990C2212yes—
67#1987C2212yes—
68#1982C2211yescharacteristic
69#1982C2212yes—
70#1949C2214yes—
71#1949C2214yes—
72#1948C2212yes—
73#1948C2212yes—
74#1948C2211yescharacteristic
75#1947C2212yes—
76#1947C2212yes—
77#1947C2212yes—
78#1939C2212yes—
79#1926C2212yes—
80#1926C2214yes—
81#1925C2214yes—
82#1893C2214yes—
83#1880C2212yes—
84#1875C2214yes—
85#1850C2214yes—
86#1842C2211yescharacteristic
87#1825C2214yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90#1803C2212yes—
91#1802C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95#1735C2214yes—
96C22×SD16C2312yes—
97C23×C8C2314yes—
98#556—24no—
99#556D418yescomplemented
100#555—28no—
101#555—28no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#126—22no—
108C2×#124—21no—
109C2×#124—24no—
110C2×C22⋊Q8C2312yes—
111C2×#121—22no—
112C2×#121—22no—
113C2×#121—22no—
114C2×#121C2314yes—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C2×#117C2314yes—
129C2×#117C2312yes—
130C2×#117C2312yes—
131C22×C42C2312yes—
132#498—28no—
133#485—216no—
134#482—216no—
135#477—28no—
136#474—28no—
137#465—216no—
138#460—24no—
139C4×SD16—28no—
140C8×D4—28no—
141#427—216no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×#106C2312yes—
147C2×C4⋊C8—28no—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2311yescharacteristic
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2311yescharacteristic
154C2×#103C2314yes—
155C2×#103C2312yes—
156C2×#102C2312yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2314yes—
161C2×C22⋊C8C2314yes—
162C2×C4×C8C2312yes—
163#394—24no—
164#394—24no—
165#393—24no—
166#391—21no—
167#391—21no—
168#391C2312yes—
169#390—21no—
170#390—21no—
171#390—21no—
172#390—22no—
173#390—22no—
174#390C2312yes—
175#390C2314yes—
176#389—21no—
177#389C2312yes—
178#389C2312yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—24no—
185#388—24no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#385—22no—
191#384—21no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#383—22no—
198#382—21no—
199#382C2314yes—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4D412yes—
229C4×C22⋊C4D412yes—
230C4×C22⋊C4C2312yes—
231#376—22no—
232C43—22no—
233C43—21no—
234C43C2311yescharacteristic
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#340C2314yes—
240#340C2314yes—
241#340C2312yes—
242#336C2314yes—
243#336C2312yes—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8C2411yescharacteristic
247C22×D4—24no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4C2×D411yescharacteristic
251C22×D4C2412yes—
252C23×C4—22no—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2412yes—
256C2×SD16—24no—
257C2×SD16—24no—
258C2×SD16—42no—
259C22×C8—28no—
260C22×C8—28no—
261C22×C8—44no—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—22no—
267C4⋊Q8—24no—
268C4⋊Q8—82no—
269C4⋊Q8—82no—
270C4⋊Q8C2×D414yes—
271#127—22no—
272#127—22no—
273#126—216no—
274#124—28no—
275#124—416no—
276C22⋊Q8—24no—
277C22⋊Q8C2×D418yes—
278#121—28no—
279#121—28no—
280#121—48no—
281#121—84no—
282#121—84no—
283C4×Q8—28no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—41no—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×Q8C2×D414yes—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—42no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4C2×D414yes—
299C4×D4#4114yes—
300#117—24no—
301#117—28no—
302#117—28no—
303#117—24no—
304#117—42no—
305#117—48no—
306#117C2×D418yes—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107—28no—
372#107—28no—
373#106—24no—
374#106—28no—
375#106—28no—
376C4⋊C8—28no—
377C4⋊C8—24no—
378C4⋊C8—24no—
379C4⋊C8—22no—
380C4⋊C8—42no—
381C4⋊C8#4114yes—
382#103—24no—
383#103—28no—
384#102—24no—
385#102—24no—
386#102—28no—
387#102—24no—
388#102—44no—
389C22⋊C8—28no—
390C4×C8—24no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—21no—
400#95—21no—
401#95—24no—
402#95—24no—
403#95—22no—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407C24—22no—
408C24C22×D412yes—
409C2×Q8—24no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—44no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—82no—
419C2×Q8—82no—
420C2×Q8C22×D412yes—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—48no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—84no—
437C2×D4—84no—
438C2×D4#12114yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D414yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4114yes—
471C22×C4C2511yescharacteristic
472SD16—48no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—416no—
482C2×C8C2×#4114yes—
483C4⋊C4—28no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—81no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4#12114yes—
513C4⋊C4C22×D414yes—
514C4⋊C4C22×D412yes—
515C4⋊C42−1+412yes—
516C22⋊C4—28no—
517C22⋊C4—28no—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—82no—
529C22⋊C4#12118yes—
530C42—28no—
531C42—24no—
532C42—24no—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—21no—
539C42—21no—
540C42—41no—
541C42—44no—
542C42—42no—
543C42—44no—
544C42—48no—
545C42—42no—
546C42—44no—
547C42—42no—
548C42—42no—
549C42—81no—
550C42—82no—
551C42#13514yes—
552C42C22×D412yes—
553C42C2×#4112yes—
554C42C2×#4112yes—
555C422−1+412yes—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23C2×#12114yes—
570C23C23×D411yescharacteristic
571Q8—22no—
572Q8—22no—
573Q8—41no—
574Q8—41no—
575Q8—48no—
576Q8—164no—
577D4—28no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—84no—
583D4—84no—
584D4—168no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—48no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—84no—
632C2×C4—88no—
633C2×C4C2×#12114yes—
634C2×C4C2×#12112yes—
635C2×C4C2×#12112yes—
636C2×C4#54712yes—
637C2×C4#54712yes—
638C2×C4C2×#13512yes—
639C2×C4C2×#13512yes—
640C2×C4C2×#13512yes—
641C2×C4#57512yes—
642C2×C4#57712yes—
643C2×C4C23×D411yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C2×2−1+411yescharacteristic
647C2×C4C2×2−1+411yescharacteristic
648C2×C4#58511yescharacteristic
649C8—24no—
650C8—44no—
651C8—42no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22C22×#12111yescharacteristic
667C22C2×#54711yescharacteristic
668C22#348911yescharacteristic
669C22C22×#13511yescharacteristic
670C22C2×#57511yescharacteristic
671C22C2×#57711yescharacteristic
672C22#358511yescharacteristic
673C4—22no—
674C4—22no—
675C4—22no—
676C4—24no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—41no—
682C4—42no—
683C4—44no—
684C4—84no—
685C4—81no—
686C4—82no—
687C4—82no—
688C4#303514yes—
689C4#334014yescomplemented
690C4D4×#4111yescharacteristic
691C4#348211yescharacteristic
692C4#349911yescharacteristic
693C4#351511yescharacteristic
694C2—44no—
695C2—162no—
696C2#3625911yescharacteristic
697C2C2×#303511yescharacteristic
698C2#6043111yescharacteristic
699C2C2×#334011yescharacteristic
700C2#6125611yescharacteristic
701C2#6210211yescharacteristic
702C2#6213411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3340C2semidirect
3. C4×#556C2semidirect
4. #33198C2semidirect
5. #33113C2semidirect
6. #28687C2semidirect
7. #28279C2semidirect
8. #27649C2semidirect
9. #25452C2semidirect
10. #25203C2semidirect
11. #24753C2semidirect
12. #24679C2semidirect
13. #24585C2semidirect
14. D4×C4⋊C8C2semidirect
15. #22489C2semidirect
16. C4×#460C2semidirect
17. #21169C2semidirect
99. #556D4semidirect
689. C4#3340semidirect

Maximal subgroups

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

subgroupindex
C2×#33402
#361872
C4×#5562
#331982
#331132
#286872
#282792
#276492
#254522
#252032
#247532
#246792
#245852
D4×C4⋊C82
#224892
C4×#4602
#211692