Finite Group Databasebeta

← #4416612

Subgroups — #4416612

Order 512.

2720 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×#2817C211yesmaximal, characteristic, complemented
2C2×#2816C211yesmaximal, characteristic
3C2×#2850C211yesmaximal, characteristic
4C2×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
5#16901C212yesmaximal, complemented
6#16899C212yesmaximal, complemented
7#16806C212yesmaximal, complemented
8#16802C212yesmaximal, complemented
9#16798C214yesmaximal, complemented
10#15176C212yesmaximal, complemented
11#15175C212yesmaximal, complemented
12#15085C212yesmaximal, complemented
13#15070C212yesmaximal, complemented
14#15002C214yesmaximal, complemented
15#14685C211yesmaximal, characteristic, complemented
16#14660C211yesmaximal, characteristic, complemented
17#14659C211yesmaximal, characteristic
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2850—28no—
25#2817—28no—
26#2816—28no—
27C2×#400C2212yes—
28C2×#400C2212yes—
29C2×#399C2212yes—
30C2×#398C2212yes—
31C2×#397C2214yes—
32C2×#395C2212yes—
33C2×#395C2212yes—
34C2×#394C2214yes—
35C2×#394C2212yes—
36C2×#393C2212yes—
37C4×C22⋊Q8—28no—
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#388C2212yes—
43C2×#388C2212yes—
44C2×#388C2214yes—
45C2×#387C2212yes—
46C2×#386C2212yes—
47C2×#386C2212yes—
48C2×#385C2212yes—
49C2×#382C2211yescharacteristic
50C2×#381C2211yescharacteristic
51C2×#381C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×#380C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C22⋊C4C2212yes—
57#1446C2212yes—
58#1445C2212yes—
59#1445C2212yes—
60#1443C2214yes—
61#1442C2214yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1440C2214yes—
69#1440C2214yes—
70#1439C2212yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1438C2214yes—
78#1438C2212yes—
79#1437C2214yes—
80#1436C2214yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83#1434C2212yes—
84#1434C2214yes—
85#1434C2212yes—
86C22⋊#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×D4C2311yescharacteristic
90C23×D4C2312yes—
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C2×#124—24no—
95C2×#124—44no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8D418yes—
102C2×#121—24no—
103C2×#121—24no—
104C2×#121—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8—44no—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C42—44no—
134C22×C42D418yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#400—28no—
142#400—28no—
143#399—28no—
144#398—28no—
145#397—216no—
146#395—28no—
147#395—28no—
148#394—28no—
149#394—216no—
150#393—28no—
151#391—24no—
152#391—24no—
153#390—24no—
154#389—24no—
155#388—28no—
156#388—28no—
157#388—48no—
158#387—28no—
159#386—28no—
160#386—28no—
161#385—28no—
162#382—24no—
163#381—24no—
164#381—24no—
165#380—24no—
166#380—44no—
167C4×C4⋊C4—28no—
168C4×C4⋊C4—24no—
169C4×C22⋊C4—28no—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C25C2412yes—
197C25C2411yescharacteristic
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×Q8—42no—
207C22×Q8C2×D414yes—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—42no—
215C22×D4—42no—
216C22×D4—44no—
217C22×D4—44no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4#4114yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246#124—416no—
247C22⋊Q8—232no—
248C22⋊Q8—416no—
249#121—416no—
250C4×Q8—416no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
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⋊C4—44no—
282C2×C4⋊C4C2×D418yes—
283C2×C4⋊C4C2×D414yes—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
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—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
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—42no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4C2×D418yes—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—28no—
328C2×C42—416no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335C2×C42C2×D414yes—
336C2×C42C2×D414yes—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—24no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—24no—
378C24—24no—
379C24—24no—
380C24C22×D412yes—
381C24C22×D412yes—
382C24C2×#4112yes—
383C24C2511yescharacteristic
384C2×Q8—28no—
385C2×Q8—48no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—48no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—84no—
393C2×Q8—84no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—44no—
400C2×D4—84no—
401C2×D4—84no—
402C2×D4—88no—
403C2×D4—88no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—48no—
479C22×C4—416no—
480C22×C4—416no—
481C22×C4#121116yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4H(5,𝔽2)14yes—
495C22×C42−1+414yes—
496C4⋊C4—28no—
497C4⋊C4—216no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—88no—
506C4⋊C4—88no—
507C4⋊C4—88no—
508C22⋊C4—216no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—416no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—88no—
518C22⋊C4—88no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—88no—
522C42—28no—
523C42—28no—
524C42—44no—
525C42—44no—
526C42—48no—
527C42—48no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—42no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23C2×#12114yes—
568C23C23×D411yescharacteristic
569C23C23×D411yescharacteristic
570C23C22×#4111yescharacteristic
571C23C2×H(5,𝔽2)12yes—
572C23C2×2−1+412yes—
573C23#58512yes—
574C23#58512yes—
575C23#58512yes—
576C23#58512yes—
577Q8—416no—
578Q8—88no—
579D4—88no—
580D4—816no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—416no—
607C2×C4—416no—
608C2×C4—416no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
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—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4C2×#12114yes—
660C2×C4C2×#12114yes—
661C2×C4#53414yes—
662C2×C4#53614yes—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—42no—
667C22—42no—
668C22—48no—
669C22—42no—
670C22—42no—
671C22—48no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—84no—
676C22—84no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—84no—
681C22—84no—
682C22—82no—
683C22C22×#12111yescharacteristic
684C22C2×#53411yescharacteristic
685C22C2×#53611yescharacteristic
686C22#344911yescharacteristic
687C22#344911yescharacteristic
688C22#344911yescharacteristic
689C22#344911yescharacteristic
690C22#349612yes—
691C22#349712yes—
692C22#350012yes—
693C22#350512yes—
694C22#350712yes—
695C22#351012yes—
696C22#352914yes—
697C22#353514yes—
698C22#356514yes—
699C22#356614yes—
700C4—24no—
701C4—24no—
702C4—24no—
703C4—24no—
704C4—42no—
705C4—42no—
706C4—44no—
707C4—44no—
708C4—82no—
709C4—82no—
710C4—84no—
711C4—84no—
712C4—82no—
713C4—82no—
714C4—82no—
715C2—44no—
716C2—162no—
717C2#3500011yescharacteristic
718C2#3517811yescharacteristic
719C2#3532711yescharacteristic
720C2#3665211yescharacteristic
721C2#3744711yescharacteristic
722C2#3755211yescharacteristic
723C2#3756011yescharacteristic
724C2#4548114yes—
725C2#4552114yes—
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×#2817C2semidirect
4. C2×C4×C22⋊Q8C2semidirect
5. #16901C2semidirect
6. #16899C2semidirect
7. #16806C2semidirect
8. #16802C2semidirect
9. #16798C2semidirect
10. #15176C2semidirect
11. #15175C2semidirect
12. #15085C2semidirect
13. #15070C2semidirect
14. #15002C2semidirect
15. #14685C2semidirect
16. #14660C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28172
C2×#28162
C2×#28502
C2×C4×C22⋊Q82
#169012
#168992
#168062
#168022
#167982
#151762
#151752
#150852
#150702
#150022
#146852
#146602
#146592