Finite Group Databasebeta

← #645678

Subgroups — #645678

Order 512.

2882 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×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×C4×#121C211yesmaximal, characteristic
3#17057C211yesmaximal, characteristic, complemented
4#16984C212yesmaximal, complemented
5#16805C212yesmaximal, complemented
6#16365C211yesmaximal, characteristic, complemented
7#16362C211yesmaximal, characteristic
8#16253C211yesmaximal, characteristic
9#15166C212yesmaximal, complemented
10#15163C212yesmaximal, complemented
11#14974C214yesmaximal, complemented
12#14733C214yesmaximal, complemented
13#14685C211yesmaximal, characteristic, complemented
14#14308C212yesmaximal, complemented
15#14300C212yesmaximal, complemented
16#14225C212yesmaximal, complemented
17#14172C212yesmaximal, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2212yes—
22C22×C4×D4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24C2×#398C2212yes—
25C2×#398C2214yes—
26C2×#397C2212yes—
27C2×#396C2214yes—
28C2×#394C2212yes—
29C2×#393C2212yes—
30C4×C22⋊Q8—28no—
31C4×#121—28no—
32C2×#391C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#388C2212yes—
37C2×#388C2214yes—
38C2×#387C2212yes—
39C2×#386C2212yes—
40C2×#385C2212yes—
41C2×#385C2214yes—
42C2×#381C2212yes—
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51#1446C2212yes—
52#1445C2212yes—
53#1445C2214yes—
54#1445C2212yes—
55#1444C2212yes—
56#1444C2212yes—
57#1444C2212yes—
58#1443C2214yes—
59#1442C2212yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2212yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1438C2212yes—
71#1438C2214yes—
72#1438C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79#1434C2214yes—
80#1434C2212yes—
81#1434C2212yes—
82C22⋊#95C2211yescharacteristic
83C22⋊#95C2211yescharacteristic
84C22⋊#95C2214yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C23×D4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—28no—
94C2×#121—24no—
95C2×#121—28no—
96C2×C4×Q8—24no—
97C2×C4×Q8—44no—
98C2×C4×D4—28no—
99C2×C4×D4—24no—
100C2×C4×D4—44no—
101C2×C4×D4—48no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C42—44no—
129C22×C42D418yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#398—28no—
138#398—48no—
139#397—28no—
140#396—48no—
141#394—28no—
142#393—28no—
143#391—24no—
144#390—24no—
145#389—24no—
146#389—24no—
147#388—28no—
148#388—48no—
149#387—28no—
150#386—28no—
151#385—28no—
152#385—48no—
153#381—44no—
154#380—24no—
155#380—24no—
156#380—44no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—24no—
160C4×C22⋊C4—28no—
161C4×C22⋊C4—28no—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C25C2412yes—
188C25C2411yescharacteristic
189C22×Q8—22no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—42no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—28no—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—42no—
201C22×D4—44no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—44no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C22⋊Q8—232no—
236#121—232no—
237C4×Q8—416no—
238C4×D4—416no—
239C4×D4—432no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
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#4118yes—
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—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4#4118yes—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—416no—
314C2×C42—44no—
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—24no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347C24—22no—
348C24—22no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—21no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—24no—
360C24—22no—
361C24—24no—
362C24—22no—
363C24—22no—
364C24C22×D412yes—
365C24C2×#4114yes—
366C242−1+414yes—
367C24C2511yescharacteristic
368C2×Q8—28no—
369C2×Q8—416no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×D4—28no—
373C2×D4—216no—
374C2×D4—416no—
375C2×D4—416no—
376C2×D4—44no—
377C2×D4—416no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—44no—
381C22×C4—22no—
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—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
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—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4114yes—
472C22×C4H(5,𝔽2)14yes—
473C22×C4H(5,𝔽2)14yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C42−1+412yes—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—216no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—88no—
487C4⋊C4—88no—
488C4⋊C4—88no—
489C22⋊C4—216no—
490C22⋊C4—216no—
491C22⋊C4—416no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—416no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—88no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—88no—
502C42—28no—
503C42—28no—
504C42—44no—
505C42—44no—
506C42—48no—
507C42—48no—
508C42—48no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—28no—
516C23—28no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23#53614yes—
549C23C23×D411yescharacteristic
550C23C22×#4112yes—
551C23C22×#4112yes—
552C23C2×H(5,𝔽2)12yes—
553C23C2×H(5,𝔽2)12yes—
554C23C2×H(5,𝔽2)11yescharacteristic
555C23C2×2−1+412yes—
556C23C2×2−1+411yescharacteristic
557C23#58512yes—
558Q8—416no—
559D4—416no—
560D4—432no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—416no—
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—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
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—44no—
615C2×C4—44no—
616C2×C4—44no—
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—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—816no—
637C2×C4#52514yes—
638C2×C4#52514yes—
639C2×C4#53414yes—
640C2×C4#53514yes—
641C2×C4#53714yes—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—48no—
649C22—48no—
650C22—42no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—48no—
655C22—48no—
656C22—82no—
657C22—82no—
658C22—84no—
659C22—82no—
660C22—84no—
661C22—82no—
662C22C2×#52511yescharacteristic
663C22C2×#52511yescharacteristic
664C22C2×#53411yescharacteristic
665C22C2×#53511yescharacteristic
666C22C2×#53611yescharacteristic
667C22C2×#53711yescharacteristic
668C22#344911yescharacteristic
669C22#346012yes—
670C22#346112yes—
671C22#348114yes—
672C22#350812yes—
673C22#351112yes—
674C22#351614yes—
675C22#352014yes—
676C22#353414yes—
677C22#353612yes—
678C22#353812yes—
679C4—24no—
680C4—24no—
681C4—24no—
682C4—24no—
683C4—24no—
684C4—42no—
685C4—44no—
686C4—44no—
687C4—44no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—84no—
692C4—84no—
693C4—82no—
694C2—44no—
695C2—84no—
696C2#3461411yescharacteristic
697C2#3495611yescharacteristic
698C2#3524011yescharacteristic
699C2#3524611yescharacteristic
700C2#3529211yescharacteristic
701C2#3744211yescharacteristic
702C2#3744611yescharacteristic
703C2#4036514yes—
704C2#4039714yes—
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
1. C2×C4×C22⋊Q8C2semidirect
3. #17057C2semidirect
4. #16984C2semidirect
5. #16805C2semidirect
6. #16365C2semidirect
9. #15166C2semidirect
10. #15163C2semidirect
11. #14974C2semidirect
12. #14733C2semidirect
13. #14685C2semidirect
14. #14308C2semidirect
15. #14300C2semidirect
16. #14225C2semidirect
17. #14172C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C2×C4×#1212
#170572
#169842
#168052
#163652
#163622
#162532
#151662
#151632
#149742
#147332
#146852
#143082
#143002
#142252
#141722