Finite Group Databasebeta

← #4389628

Subgroups — #4389628

Order 512.

3196 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2500C211yesmaximal, characteristic, complemented
2C2×#2624C211yesmaximal, characteristic, complemented
3C2×#2656C211yesmaximal, characteristic
4C2×#2333C211yesmaximal, characteristic, complemented
5#16801C212yesmaximal, complemented
6#16701C212yesmaximal, complemented
7#15365C214yesmaximal, complemented
8#15316C214yesmaximal, complemented
9#15307C214yesmaximal, complemented
10#15274C214yesmaximal, complemented
11#15160C212yesmaximal, complemented
12#15135C212yesmaximal, complemented
13#14659C211yesmaximal, characteristic, complemented
14#13965C211yesmaximal, characteristic, complemented
15#13958C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#121C2214yes—
19C22×#117C2211yescharacteristic
20#2656—28no—
21#2624—28no—
22#2500—28no—
23C2×#399C2212yes—
24C2×#399C2212yes—
25C2×#399C2212yes—
26C2×#397C2212yes—
27C2×#396C2214yes—
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#394C2214yes—
31C2×#394C2214yes—
32C2×#392C2212yes—
33C2×#392C2212yes—
34#2333—28no—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#389C2212yes—
40C2×#389C2214yes—
41C2×#388C2214yes—
42C2×#388C2212yes—
43C2×#387C2212yes—
44C2×#386C2212yes—
45C2×#386C2212yes—
46C2×#386C2214yes—
47C2×#385C2214yes—
48C2×#384C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#381C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2214yes—
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C22×#95C2211yescharacteristic
56#1445C2212yes—
57#1445C2212yes—
58#1445C2212yes—
59#1444C2214yes—
60#1444C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2212yes—
70#1438C2212yes—
71#1438C2212yes—
72#1438C2214yes—
73#1437C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76#1434C2214yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×D4C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2314yes—
82C23×D4C2311yescharacteristic
83C23×D4C2312yes—
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C2×#121—24no—
88C2×#121—24no—
89C2×#121—216no—
90C2×#121—24no—
91C2×#121—24no—
92C2×#121—24no—
93C2×#121—48no—
94C2×#121—48no—
95C2×#121D418yes—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117—24no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#399—28no—
133#399—28no—
134#399—28no—
135#397—28no—
136#396—48no—
137#394—216no—
138#394—216no—
139#394—48no—
140#394—48no—
141#392—28no—
142#392—28no—
143#390—28no—
144#390—28no—
145#390—24no—
146#390—24no—
147#389—28no—
148#389—48no—
149#388—28no—
150#388—216no—
151#387—28no—
152#386—28no—
153#386—28no—
154#386—48no—
155#385—48no—
156#384—24no—
157#382—24no—
158#381—24no—
159#380—24no—
160#380—48no—
161C4×C22⋊C4—28no—
162C4×C22⋊C4D418yescomplemented
163C2×#95—24no—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C25C2412yes—
183C25C2412yes—
184C25C2411yescharacteristic
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—28no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—28no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—42no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4C2×D418yes—
211C22×D4C2×D414yes—
212C23×C4—24no—
213C23×C4—28no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—44no—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4#4114yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239#121—232no—
240#121—464no—
241#121—416no—
242#117—416no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×D414yes—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4C2×D418yes—
323C2×C22⋊C4C2×D418yes—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—44no—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95C2×D418yes—
351C24—22no—
352C24—24no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—24no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—24no—
365C24—24no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—24no—
375C24C22×D414yes—
376C24C22×D412yes—
377C24C22×D412yes—
378C24C2×#4112yes—
379C24C2511yescharacteristic
380C2×D4—216no—
381C2×D4—28no—
382C2×D4—416no—
383C2×D4—48no—
384C2×D4—416no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—416no—
388C2×D4—416no—
389C2×D4—48no—
390C2×D4—44no—
391C2×D4—82no—
392C2×D4—82no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4—84no—
396C2×D4—88no—
397C2×D4—88no—
398C2×D4—84no—
399C2×D4—84no—
400C2×D4—88no—
401C2×D4—88no—
402C2×D4—88no—
403C2×D4—84no—
404C2×D4—84no—
405C2×D4—88no—
406C2×D4—88no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—216no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—416no—
465C22×C4—48no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4—88no—
471C22×C4—84no—
472C22×C4—84no—
473C22×C4#121116yes—
474C22×C4C22×D414yes—
475C22×C4C22×D414yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D414yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4H(5,𝔽2)18yes—
484C22×C4H(5,𝔽2)14yes—
485C22×C42−1+414yes—
486C4⋊C4—28no—
487C4⋊C4—416no—
488C4⋊C4—416no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—88no—
494C4⋊C4—88no—
495C4⋊C4—84no—
496C4⋊C4—88no—
497C22⋊C4—216no—
498C22⋊C4—216no—
499C22⋊C4—416no—
500C22⋊C4—416no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—416no—
505C22⋊C4—84no—
506C22⋊C4—88no—
507C22⋊C4—84no—
508C22⋊C4—88no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—84no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C22⋊C4—88no—
515C42—44no—
516C42—44no—
517C42—48no—
518C42—84no—
519C23—24no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23—24no—
534C23—24no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—42no—
561C23—42no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23C2×#12114yes—
579C23C23×D411yescharacteristic
580C23C23×D411yescharacteristic
581C23C22×#4111yescharacteristic
582C23C2×H(5,𝔽2)12yes—
583C23C2×H(5,𝔽2)14yes—
584C23C2×2−1+412yes—
585C23#58512yes—
586C23#58512yes—
587D4—88no—
588D4—88no—
589D4—816no—
590D4—832no—
591D4—816no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—416no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—816no—
655C2×C4—816no—
656C2×C4C2×#12114yes—
657C2×C4C2×#12114yes—
658C2×C4#53718yes—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—48no—
667C22—48no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—48no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—82no—
680C22—84no—
681C22—84no—
682C22—82no—
683C22—82no—
684C22—84no—
685C22—84no—
686C22—84no—
687C22C22×#12111yescharacteristic
688C22C2×#53712yes—
689C22#348412yes—
690C22#348512yes—
691C22#348511yescharacteristic
692C22#348511yescharacteristic
693C22#349112yes—
694C22#349812yes—
695C22#349811yescharacteristic
696C22#349911yescharacteristic
697C22#350812yes—
698C22#351112yes—
699C22#353414yes—
700C22#354114yes—
701C22#354214yes—
702C22#354714yes—
703C4—24no—
704C4—24no—
705C4—28no—
706C4—44no—
707C4—44no—
708C4—48no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—84no—
715C2—44no—
716C2—84no—
717C2—162no—
718C2#3527311yescharacteristic
719C2#3544711yescharacteristic
720C2#3552811yescharacteristic
721C2#4136812yes—
722C2#4264612yes—
723C2#4681312yes—
724C2#4721412yes—
725C2#4888714yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2656C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2500C2semidirect
2. C2×#2624C2semidirect
4. C2×#2333C2semidirect
5. #16801C2semidirect
6. #16701C2semidirect
7. #15365C2semidirect
8. #15316C2semidirect
9. #15307C2semidirect
10. #15274C2semidirect
11. #15160C2semidirect
12. #15135C2semidirect
13. #14659C2semidirect
14. #13965C2semidirect
15. #13958C2semidirect
162. C4×C22⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25002
C2×#26242
C2×#26562
C2×#23332
#168012
#167012
#153652
#153162
#153072
#152742
#151602
#151352
#146592
#139652
#139582