Finite Group Databasebeta

← #4389958

Subgroups — #4389958

Order 512.

3042 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2822C211yesmaximal, characteristic, complemented
2C2×#2562C211yesmaximal, characteristic, complemented
3C2×#2566C211yesmaximal, characteristic, complemented
4C2×#2535C211yesmaximal, characteristic
5C2×#2633C211yesmaximal, characteristic, complemented
6C2×#2633C211yesmaximal, characteristic, complemented
7#17062C212yesmaximal, complemented
8#16809C212yesmaximal, complemented
9#15526C214yesmaximal, complemented
10#15513C214yesmaximal, complemented
11#15505C214yesmaximal, complemented
12#15340C214yesmaximal, complemented
13#15020C212yesmaximal, complemented
14#15007C212yesmaximal, complemented
15#13781C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×#121C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2822—28no—
22#2633—28no—
23#2633—28no—
24#2566—28no—
25#2562—28no—
26#2535—28no—
27C2×#400C2212yes—
28C2×#400C2212yes—
29C2×#400C2212yes—
30C2×#399C2212yes—
31C2×#399C2212yes—
32C2×#399C2214yes—
33C2×#398C2212yes—
34C2×#398C2212yes—
35C2×#398C2212yes—
36C2×#398C2214yes—
37C2×#398C2212yes—
38C2×#397C2212yes—
39C2×#397C2212yes—
40C2×#397C2212yes—
41C2×#396C2214yes—
42C2×#394C2212yes—
43C2×#394C2212yes—
44C2×#394C2212yes—
45C2×#392C2212yes—
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#390C2214yes—
49C2×#388C2212yes—
50C2×#388C2212yes—
51C2×#386C2214yes—
52C2×#385C2214yes—
53C2×#385C2214yes—
54C2×#385C2214yes—
55C2×#384C2211yescharacteristic
56C2×#383C2211yescharacteristic
57C2×#382C2211yescharacteristic
58C2×#382C2211yescharacteristic
59C2×#380C2212yes—
60C2×#380C2212yes—
61C2×#380C2212yes—
62C2×C4×C4⋊C4C2211yescharacteristic
63#1445C2214yes—
64#1445C2214yes—
65#1444C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1438C2214yes—
72#1438C2214yes—
73#1438C2214yes—
74#1437C2214yes—
75#1437C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2214yes—
79#1434C2212yes—
80C4×#95C2212yes—
81C23×D4C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C23×D4C2314yes—
84C24×C4C2312yes—
85C24×C4C2312yes—
86C2×#125—24no—
87C2×#125—24no—
88C2×#125—24no—
89C2×#125—24no—
90C2×#125—24no—
91C2×#125—24no—
92C2×#121—28no—
93C2×#121—48no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C42C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#400—28no—
135#400—28no—
136#400—28no—
137#399—28no—
138#399—28no—
139#399—48no—
140#398—216no—
141#398—28no—
142#398—28no—
143#398—28no—
144#398—28no—
145#397—28no—
146#397—28no—
147#397—28no—
148#396—216no—
149#394—28no—
150#394—28no—
151#394—28no—
152#392—28no—
153#390—24no—
154#390—24no—
155#390—48no—
156#388—28no—
157#388—28no—
158#386—216no—
159#385—216no—
160#385—216no—
161#385—216no—
162#384—24no—
163#383D418yescomplemented
164#382—24no—
165#382D418yes—
166#380—28no—
167#380—28no—
168#380—28no—
169C4×C4⋊C4D418yescomplemented
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C25C2412yes—
191C25C2412yes—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—28no—
195C22×D4—28no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—28no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232#125—416no—
233#125—416no—
234#121—432no—
235C4×D4—416no—
236C4×D4—416no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4C2×D418yes—
282C2×C4⋊C4C2×D418yes—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—42no—
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⋊C4—44no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42—44no—
326C2×C42—44no—
327C2×C42—44no—
328C2×C42C2×D414yes—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95C2×D418yes—
348#95C2×D418yes—
349#95C2×D418yes—
350#95C2×D418yes—
351#95C2×D418yes—
352#95C2×D418yes—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—24no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—22no—
373C24H(5,𝔽2)14yes—
374C242−1+414yes—
375C24C2511yescharacteristic
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—416no—
385C2×D4—416no—
386C2×D4—416no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—88no—
390C2×D4—88no—
391C2×D4—88no—
392C2×D4—88no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
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—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—88no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4C22×D418yes—
461C22×C4C22×D414yes—
462C22×C4C22×D418yes—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—88no—
486C4⋊C4—88no—
487C4⋊C4—88no—
488C4⋊C4—88no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C22⋊C4—416no—
493C22⋊C4—416no—
494C22⋊C4—416no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C42—44no—
504C42—44no—
505C42—88no—
506C42#127116yes—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23C23×D411yescharacteristic
554C23C23×D411yescharacteristic
555C23C23×D411yescharacteristic
556C23C2×H(5,𝔽2)12yes—
557C23C2×2−1+412yes—
558C23#58514yes—
559C23#58512yes—
560C23#58512yes—
561D4—416no—
562D4—416no—
563D4—832no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
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—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—816no—
632C2×C4—816no—
633C2×C4C2×#12714yes—
634C2×C4C2×#12714yes—
635C2×C4C2×#12714yes—
636C22—48no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—48no—
641C22—48no—
642C22—44no—
643C22—44no—
644C22—48no—
645C22—44no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—84no—
651C22—82no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22C22×#12711yescharacteristic
659C22#344912yes—
660C22#348411yescharacteristic
661C22#348511yescharacteristic
662C22#348611yescharacteristic
663C22#348711yescharacteristic
664C22#348812yes—
665C22#348812yes—
666C22#348912yes—
667C22#348912yes—
668C22#350012yes—
669C22#350012yes—
670C22#353514yes—
671C22#355314yes—
672C22#356014yes—
673C22#356714yes—
674C4—24no—
675C4—24no—
676C4—24no—
677C4—48no—
678C4—44no—
679C4—44no—
680C4—48no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C2—84no—
688C2—84no—
689C2#3532911yescharacteristic
690C2#3556511yescharacteristic
691C2#3563411yescharacteristic
692C2#4142012yes—
693C2#4199112yes—
694C2#4768114yes—
695C2#5089712yes—
696C2#5121112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2535C2
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×#2822C2semidirect
2. C2×#2562C2semidirect
3. C2×#2566C2semidirect
5. C2×#2633C2semidirect
6. C2×#2633C2semidirect
7. #17062C2semidirect
8. #16809C2semidirect
9. #15526C2semidirect
10. #15513C2semidirect
11. #15505C2semidirect
12. #15340C2semidirect
13. #15020C2semidirect
14. #15007C2semidirect
15. #13781C2semidirect
163. #383D4semidirect
169. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28222
C2×#25622
C2×#25662
C2×#25352
C2×#26332
#170622
#168092
#155262
#155132
#155052
#153402
#150202
#150072
#137812