Finite Group Databasebeta

← #4456247

Subgroups — #4456247

Order 512.

2528 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2562C211yesmaximal, characteristic, complemented
2C2×#2611C211yesmaximal, characteristic, complemented
3C2×#2598C211yesmaximal, characteristic
4C2×#2319C211yesmaximal, characteristic, complemented
5#17069C212yesmaximal, complemented
6#17022C212yesmaximal
7#17010C212yesmaximal, complemented
8#16899C212yesmaximal, complemented
9#16399C212yesmaximal, complemented
10#16398C212yesmaximal, complemented
11#16395C212yesmaximal, complemented
12#16394C212yesmaximal, complemented
13#14988C214yesmaximal, complemented
14#14755C214yesmaximal, complemented
15#14289C211yesmaximal, characteristic, complemented
16#13952C211yesmaximal, characteristic, complemented
17#13852C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2211yescharacteristic
21#2611—28no—
22#2598—28no—
23#2562—28no—
24C2×#400C2212yes—
25C2×#400C2211yescharacteristic
26C2×#400C2211yescharacteristic
27C2×#399C2211yescharacteristic
28C2×#399C2211yescharacteristic
29C2×#398C2211yescharacteristic
30C2×#397C2211yescharacteristic
31C2×#397C2214yes—
32C2×#394C2211yescharacteristic
33C2×#394C2212yes—
34C2×#392C2211yescharacteristic
35#2319—28no—
36C2×#390C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#388C2212yes—
40C2×#388C2214yes—
41C2×#388C2212yes—
42C2×#388C2214yes—
43C2×#387C2212yes—
44C2×#386C2212yes—
45C2×#386C2212yes—
46C2×#385C2212yes—
47C2×#385C2212yes—
48C2×#383C2211yescharacteristic
49C2×#381C2212yes—
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×#380C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2212yes—
56C2×#376C2211yescharacteristic
57#1445C2212yes—
58#1445C2212yes—
59#1444C2212yes—
60#1443C2214yes—
61#1443C2214yes—
62#1442C2212yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2212yes—
71#1440C2214yes—
72#1440C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2212yes—
76#1439C2214yes—
77#1439C2212yes—
78#1439C2214yes—
79#1439C2214yes—
80#1438C2214yes—
81#1438C2214yes—
82#1436C2214yes—
83#1436C2214yes—
84C4⋊#95C2212yes—
85#1434C2212yes—
86C4×#95C2212yes—
87C23×D4C2312yes—
88C23×D4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×#125—24no—
92C2×#125—24no—
93C2×#125—24no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4D418yes—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117—24no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#400—28no—
134#400—24no—
135#400—24no—
136#399—24no—
137#399—24no—
138#398—24no—
139#397—24no—
140#397—48no—
141#394—28no—
142#394—24no—
143#392—24no—
144#390—28no—
145#389—28no—
146#389—28no—
147#388—216no—
148#388—28no—
149#388—28no—
150#388—216no—
151#387—28no—
152#386—28no—
153#386—28no—
154#385—28no—
155#385—28no—
156#383—24no—
157#381—28no—
158#380—28no—
159#380—28no—
160#380—28no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—24no—
163C4×C22⋊C4—44no—
164#376D418yescomplemented
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C2×#95C2312yes—
193C25C2412yes—
194C25C2411yescharacteristic
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—44no—
200C22×D4—42no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4C2×D414yes—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—41no—
212C23×C4—41no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C2×D418yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231#125—416no—
232C4×D4—232no—
233#117—416no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C2×D414yes—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4C2×D418yes—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—44no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42C2×D418yes—
321C2×C42C2×D414yes—
322C2×C42C2×D414yes—
323C2×C42#4114yes—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—24no—
330#95—24no—
331#95—22no—
332#95—22no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95C2×D414yes—
350#95C2×D414yes—
351#95C2×D418yes—
352C24—21no—
353C24—22no—
354C24—21no—
355C24—21no—
356C24—22no—
357C24—22no—
358C24—21no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—21no—
363C24—24no—
364C24—22no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—21no—
370C24C22×D414yes—
371C242−1+412yes—
372C24C2511yescharacteristic
373C2×D4—28no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—84no—
380C2×D4—88no—
381C2×D4—88no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—216no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—88no—
461C22×C4—88no—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4H(5,𝔽2)12yes—
475C4⋊C4—28no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—84no—
487C4⋊C4—88no—
488C4⋊C4—88no—
489C22⋊C4—216no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—416no—
494C22⋊C4—48no—
495C22⋊C4—416no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—84no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C22⋊C4—84no—
502C22⋊C4—88no—
503C42—48no—
504C42—44no—
505C42—84no—
506C42—84no—
507C42#121116yes—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—28no—
519C23—41no—
520C23—42no—
521C23—44no—
522C23—42no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—44no—
534C23—42no—
535C23—44no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23C23×D411yescharacteristic
549C23C23×D411yescharacteristic
550C23C22×#4111yescharacteristic
551C23C2×H(5,𝔽2)11yescharacteristic
552C23C2×2−1+411yescharacteristic
553C23#58514yes—
554C23#58514yes—
555C23#58511yescharacteristic
556C23#58511yescharacteristic
557D4—416no—
558D4—816no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
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—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—416no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—816no—
639C2×C4C2×#12114yes—
640C2×C4C2×#12114yes—
641C2×C4C2×#12114yes—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—42no—
651C22—44no—
652C22—44no—
653C22—82no—
654C22—81no—
655C22—84no—
656C22—81no—
657C22—84no—
658C22—82no—
659C22—84no—
660C22—82no—
661C22—84no—
662C22C22×#12111yescharacteristic
663C22#344912yes—
664C22#344912yes—
665C22#345512yes—
666C22#348512yes—
667C22#348811yescharacteristic
668C22#348811yescharacteristic
669C22#348911yescharacteristic
670C22#348911yescharacteristic
671C22#349211yescharacteristic
672C22#349311yescharacteristic
673C22#350012yes—
674C22#351012yes—
675C22#353914yes—
676C22#356414yes—
677C22#356614yes—
678C22#356614yes—
679C4—24no—
680C4—24no—
681C4—24no—
682C4—44no—
683C4—44no—
684C4—42no—
685C4—44no—
686C4—82no—
687C4—84no—
688C4—81no—
689C4—81no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—84no—
694C2—84no—
695C2—82no—
696C2#3463311yescharacteristic
697C2#3551811yescharacteristic
698C2#3561711yescharacteristic
699C2#3683112yes—
700C2#3702512yes—
701C2#4195112yes—
702C2#4199012yes—
703C2#4217312yes—
704C2#4230212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2562C2semidirect
2. C2×#2611C2semidirect
4. C2×#2319C2semidirect
5. #17069C2semidirect
7. #17010C2semidirect
8. #16899C2semidirect
9. #16399C2semidirect
10. #16398C2semidirect
11. #16395C2semidirect
12. #16394C2semidirect
13. #14988C2semidirect
14. #14755C2semidirect
15. #14289C2semidirect
16. #13952C2semidirect
17. #13852C2semidirect
164. #376D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25622
C2×#26112
C2×#25982
C2×#23192
#170692
#170222
#170102
#168992
#163992
#163982
#163952
#163942
#149882
#147552
#142892
#139522
#138522