Finite Group Databasebeta

← #732583

Subgroups — #732583

Order 512.

2614 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#17005C212yesmaximal, complemented
4#17005C212yesmaximal, complemented
5#15469C214yesmaximal, complemented
6#15371C214yesmaximal, complemented
7#15187C212yesmaximal, complemented
8#15139C212yesmaximal, complemented
9#15073C212yesmaximal, complemented
10#14843C212yesmaximal, complemented
11#14348C211yesmaximal, characteristic, complemented
12#13955C211yesmaximal, characteristic, complemented
13#13926C211yesmaximal, characteristic, complemented
14#13916C212yesmaximal
15#13916C212yesmaximal
16#13850C211yesmaximal, characteristic, complemented
17#13785C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22#2580—28no—
23C2×#400C2214yes—
24C2×#399C2212yes—
25C2×#393C2212yes—
26D4×C4⋊C4—28no—
27C2×#391C2212yes—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#390C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2211yescharacteristic
34C2×#385C2214yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#384C2211yescharacteristic
38C2×#382C2211yescharacteristic
39C2×#381C2214yes—
40C2×#381C2214yes—
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×#376C2211yescharacteristic
45#1446C2212yes—
46#1445C2214yes—
47#1445C2212yes—
48#1443C2214yes—
49#1443C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2212yes—
53#1442C2212yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2212yes—
57#1442C2212yes—
58#1441C2211yescharacteristic
59#1441C2211yescharacteristic
60#1441C2211yescharacteristic
61#1441C2211yescharacteristic
62#1441C2214yes—
63#1441C2212yes—
64#1441C2212yes—
65#1440C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1437C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78#1434C2214yes—
79C22⋊#95C2214yes—
80C22⋊#95C2214yes—
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2314yes—
88C24×C4C2312yes—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×D4—28no—
93C2×C4×D4—28no—
94C2×C4×D4—28no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4D418yes—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—48no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#400—48no—
130#399—28no—
131#393—28no—
132#391—44no—
133#390—24no—
134#390—24no—
135#390—28no—
136#389—24no—
137#389—24no—
138#389—28no—
139#385—28no—
140#385—28no—
141#385—216no—
142#384—24no—
143#382—24no—
144#381—48no—
145#381—48no—
146#380—28no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—24no—
149#376—24no—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C25C2412yes—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—41no—
189C22×Q8—41no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—24no—
193C22×D4C2×D414yes—
194C22×D4#4118yes—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C2×D418yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C4×Q8—416no—
224C4×D4—232no—
225C4×D4—432no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
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—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—432no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D414yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4C2×D418yes—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42C2×D414yes—
303C2×C42#4114yes—
304C2×C42#4114yes—
305#95—24no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—22no—
313#95—22no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—42no—
322#95—42no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—42no—
327#95—44no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—22no—
346C24C22×D414yes—
347C24C2×#4118yes—
348C24C2511yescharacteristic
349C2×Q8—28no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×D4—28no—
357C2×D4—28no—
358C2×D4—216no—
359C2×D4—216no—
360C2×D4#124116yes—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—216no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4114yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C4H(5,𝔽2)14yes—
469C22×C42−1+412yes—
470C22×C42−1+414yes—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—416no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C22⋊C4—216no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—416no—
490C22⋊C4—84no—
491C22⋊C4—84no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C42—44no—
495C42—44no—
496C42—48no—
497C42—48no—
498C42—84no—
499C42#123116yes—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—28no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23C2×#12418yes—
523C23C23×D411yescharacteristic
524C23C22×#4111yescharacteristic
525C23C22×#4111yescharacteristic
526C23C22×#4112yes—
527C23C2×H(5,𝔽2)12yes—
528C23C2×H(5,𝔽2)11yescharacteristic
529C23C2×2−1+412yes—
530C23C2×2−1+411yescharacteristic
531C23#58512yes—
532C23#58511yescharacteristic
533C23#58511yescharacteristic
534Q8—416no—
535D4—232no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—432no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
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—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—832no—
629C2×C4C2×#12314yes—
630C2×C4C2×#12314yes—
631C2×C4C2×#12314yes—
632C2×C4C2×#12414yes—
633C2×C4#53614yes—
634C2×C4#55218yes—
635C2×C4#55218yes—
636C22—28no—
637C22—28no—
638C22—28no—
639C22—48no—
640C22—44no—
641C22—48no—
642C22—48no—
643C22—48no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22C22×#12311yescharacteristic
648C22C22×#12411yescharacteristic
649C22C2×#53611yescharacteristic
650C22#346012yes—
651C22C2×#55212yes—
652C22C2×#55212yes—
653C22#348411yescharacteristic
654C22#348511yescharacteristic
655C22#348811yescharacteristic
656C22#348911yescharacteristic
657C22#349211yescharacteristic
658C22#349311yescharacteristic
659C22#349411yescharacteristic
660C22#349511yescharacteristic
661C22#350612yes—
662C22#351312yes—
663C22#351512yes—
664C22#351912yes—
665C22#352112yes—
666C22#355012yes—
667C22#355412yes—
668C22#355412yes—
669C22#355712yes—
670C4—24no—
671C4—24no—
672C4—44no—
673C4—44no—
674C4—42no—
675C4—42no—
676C4—84no—
677C4—82no—
678C4—81no—
679C4—81no—
680C4—84no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#250318yes—
686C2—48no—
687C2C2×#250311yescharacteristic
688C2#3515211yescharacteristic
689C2#3551611yescharacteristic
690C2#3590311yescharacteristic
691C2#3604211yescharacteristic
692C2#3822612yes—
693C2#3823012yes—
694C2#4248111yescharacteristic
695C2#4267811yescharacteristic
696C2#4491012yes—
697C2#4513212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17005C2semidirect
4. #17005C2semidirect
5. #15469C2semidirect
6. #15371C2semidirect
7. #15187C2semidirect
8. #15139C2semidirect
9. #15073C2semidirect
10. #14843C2semidirect
11. #14348C2semidirect
12. #13955C2semidirect
13. #13926C2semidirect
16. #13850C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
C2×D4×C4⋊C42
#170052
#154692
#153712
#151872
#151392
#150732
#148432
#143482
#139552
#139262
#139162
#138502
#137852