Finite Group Databasebeta

← #763276

Subgroups — #763276

Order 512.

2974 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C2×#2296C211yesmaximal, characteristic, complemented
3#17008C211yesmaximal, characteristic, complemented
4#16990C211yesmaximal, characteristic, complemented
5#16888C211yesmaximal, characteristic, complemented
6#16811C211yesmaximal, characteristic, complemented
7#15203C214yesmaximal, complemented
8#15113C212yesmaximal, complemented
9#14905C212yesmaximal, complemented
10#14840C212yesmaximal, complemented
11#14764C212yesmaximal
12#14230C212yesmaximal, complemented
13#14229C212yesmaximal, complemented
14#14223C212yesmaximal, complemented
15#14176C212yesmaximal, complemented
16#14079C214yesmaximal, complemented
17#13753C211yesmaximal, characteristic, complemented
18C22×#121C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C42C2211yescharacteristic
24C2×#399C2212yes—
25C2×#398C2212yes—
26C2×#397C2212yes—
27C2×#394C2212yes—
28C4×#121—28no—
29#2296—28no—
30C2×#390C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#388C2214yes—
37C2×#388C2212yes—
38C2×#388C2214yes—
39C2×#387C2214yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2214yes—
44C2×#380C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×#376C2212yes—
51#1445C2212yes—
52#1445C2212yes—
53#1445C2212yes—
54#1444C2212yes—
55#1444C2212yes—
56#1444C2214yes—
57#1442C2212yes—
58#1442C2212yes—
59#1442C2214yes—
60#1441C2212yes—
61#1441C2212yes—
62#1439C2214yes—
63#1439C2212yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2214yes—
69#1438C2214yes—
70#1438C2212yes—
71#1437C2214yes—
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2214yes—
76#1434C2214yes—
77#1434C2211yescharacteristic
78#1434C2212yes—
79#1434C2214yes—
80#1434C2211yescharacteristic
81C22⋊#95C2212yes—
82C22⋊#95C2214yes—
83C22⋊#95C2212yes—
84C4×#95C2212yes—
85C23×D4C2311yescharacteristic
86C23×D4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C2×#121—24no—
92C2×#121—28no—
93C2×C4×D4—28no—
94C2×C4×D4—24no—
95C2×C4×D4—44no—
96C2×C4×D4—48no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117—44no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C42—44no—
129C22×C42D418yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2314yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#399—28no—
139#398—28no—
140#397—28no—
141#394—28no—
142#390—24no—
143#389—24no—
144#389—24no—
145#389—24no—
146#389—24no—
147#389—24no—
148#388—216no—
149#388—28no—
150#388—48no—
151#387—48no—
152#386—28no—
153#385—28no—
154#385—28no—
155#385—48no—
156#380—24no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—24no—
160C4×C22⋊C4—28no—
161C4×C22⋊C4—28no—
162#376—44no—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C25C2412yes—
189C25C2411yescharacteristic
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—28no—
194C22×D4—22no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—42no—
198C22×D4—44no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232#121—232no—
233C4×D4—416no—
234C4×D4—432no—
235#117—232no—
236#117—416no—
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—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4#4114yes—
270C2×C4⋊C4#4114yes—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
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#4118yes—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—28no—
311C2×C42—28no—
312C2×C42—28no—
313C2×C42—28no—
314C2×C42—416no—
315C2×C42—44no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42C2×D414yes—
321C2×C42C2×D414yes—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—44no—
339#95—42no—
340#95—42no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95#4118yes—
347C24—21no—
348C24—21no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—22no—
363C24—22no—
364C24C22×D412yes—
365C24C2×#4114yes—
366C242−1+414yes—
367C24C2511yescharacteristic
368C2×D4—28no—
369C2×D4—216no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—44no—
373C2×D4—416no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—44no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—28no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C2×#4114yes—
463C22×C4C2×#4114yes—
464C22×C4C2×#4118yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C4H(5,𝔽2)14yes—
471C22×C4H(5,𝔽2)14yes—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—416no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—88no—
486C22⋊C4—216no—
487C22⋊C4—216no—
488C22⋊C4—416no—
489C22⋊C4—416no—
490C22⋊C4—416no—
491C22⋊C4—416no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—88no—
498C42—216no—
499C42—28no—
500C42—28no—
501C42—44no—
502C42—44no—
503C42—416no—
504C42—48no—
505C42—48no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—28no—
515C23—28no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—28no—
520C23—28no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23#53614yes—
546C23C23×D411yescharacteristic
547C23C22×#4111yescharacteristic
548C23C22×#4112yes—
549C23C22×#4111yescharacteristic
550C23C22×#4112yes—
551C23C2×H(5,𝔽2)12yes—
552C23C2×H(5,𝔽2)12yes—
553C23C2×2−1+412yes—
554C23#58512yes—
555D4—416no—
556D4—432no—
557C2×C4—232no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—416no—
589C2×C4—416no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
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—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—816no—
633C2×C4#52514yes—
634C2×C4#52514yes—
635C2×C4#53214yes—
636C2×C4#53414yes—
637C2×C4#53514yes—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—42no—
642C22—42no—
643C22—48no—
644C22—42no—
645C22—42no—
646C22—48no—
647C22—44no—
648C22—48no—
649C22—44no—
650C22—44no—
651C22—48no—
652C22—82no—
653C22—84no—
654C22—82no—
655C22—84no—
656C22—82no—
657C22—82no—
658C22C2×#52511yescharacteristic
659C22C2×#52511yescharacteristic
660C22C2×#53211yescharacteristic
661C22C2×#53411yescharacteristic
662C22C2×#53511yescharacteristic
663C22C2×#53611yescharacteristic
664C22#344911yescharacteristic
665C22#347512yes—
666C22#348012yes—
667C22#348112yes—
668C22#348114yes—
669C22#348312yes—
670C22#349212yes—
671C22#349212yes—
672C22#349514yes—
673C22#352014yes—
674C22#354214yes—
675C4—24no—
676C4—24no—
677C4—24no—
678C4—24no—
679C4—24no—
680C4—42no—
681C4—44no—
682C4—48no—
683C4—44no—
684C4—44no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—84no—
690C2—44no—
691C2—84no—
692C2#3490911yescharacteristic
693C2#3495711yescharacteristic
694C2#3497711yescharacteristic
695C2#3505711yescharacteristic
696C2#3524011yescharacteristic
697C2#3532611yescharacteristic
698C2#3743711yescharacteristic
699C2#5959714yes—
700C2#5978114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2296C2semidirect
3. #17008C2semidirect
4. #16990C2semidirect
5. #16888C2semidirect
6. #16811C2semidirect
7. #15203C2semidirect
8. #15113C2semidirect
9. #14905C2semidirect
10. #14840C2semidirect
12. #14230C2semidirect
13. #14229C2semidirect
14. #14223C2semidirect
15. #14176C2semidirect
16. #14079C2semidirect
17. #13753C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C2×#22962
#170082
#169902
#168882
#168112
#152032
#151132
#149052
#148402
#147642
#142302
#142292
#142232
#141762
#140792
#137532