Finite Group Databasebeta

← #645192

Subgroups — #645192

Order 512.

2706 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic, complemented
2C2×C4×#121C211yesmaximal, characteristic
3#17079C211yesmaximal, characteristic, complemented
4#17077C211yesmaximal, characteristic, complemented
5#17074C211yesmaximal, characteristic, complemented
6#17069C211yesmaximal, characteristic, complemented
7#17005C212yesmaximal, complemented
8#16557C212yesmaximal
9#15252C214yesmaximal, complemented
10#15050C212yesmaximal, complemented
11#15045C212yesmaximal, complemented
12#14842C212yesmaximal, complemented
13#14765C212yesmaximal, complemented
14#14344C214yesmaximal, complemented
15#14250C212yesmaximal, complemented
16#14190C212yesmaximal, complemented
17C4×#380C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2855—28no—
24C2×#400C2212yes—
25C2×#397C2212yes—
26C2×#394C2212yes—
27C2×#393C2212yes—
28C4×#121—28no—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#388C2212yes—
36C2×#388C2214yes—
37C2×#387C2212yes—
38C2×#387C2212yes—
39C2×#386C2212yes—
40C2×#385C2212yes—
41C2×#381C2212yes—
42C2×#380C2212yes—
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49#1446C2212yes—
50#1445C2212yes—
51#1445C2212yes—
52#1443C2214yes—
53#1442C2212yes—
54#1442C2214yes—
55#1442C2212yes—
56#1442C2212yes—
57#1442C2212yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2212yes—
62#1440C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2212yes—
70#1438C2214yes—
71#1438C2214yes—
72#1438C2212yes—
73#1437C2214yes—
74#1436C2214yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2211yescharacteristic
77#1434C2214yes—
78#1434C2214yes—
79#1434C2214yes—
80#1434C2212yes—
81C22⋊#95C2214yes—
82C22⋊#95C2214yes—
83C22⋊#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—44no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#121—24no—
97C2×C4×D4—28no—
98C2×C4×D4—216no—
99C2×C4×D4—24no—
100C2×C4×D4—44no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C42—44no—
126C22×C42D418yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#400—28no—
134#397—44no—
135#394—28no—
136#393—44no—
137#391—24no—
138#391—24no—
139#390—24no—
140#389—24no—
141#389—24no—
142#389—24no—
143#388—28no—
144#388—48no—
145#387—28no—
146#387—28no—
147#386—28no—
148#385—28no—
149#381—28no—
150#380—24no—
151#380—24no—
152#380—28no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—44no—
155C4×C22⋊C4—28no—
156C4×C22⋊C4—44no—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C25C2412yes—
186C25C2411yescharacteristic
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—28no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4—22no—
198C22×D4—42no—
199C22×D4—42no—
200C22×D4—44no—
201C22×D4#4114yes—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—42no—
210C23×C4—41no—
211C23×C4—41no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4#4118yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C22⋊Q8—416no—
238#121—232no—
239C4×D4—416no—
240C4×D4—432no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4#4114yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
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—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—44no—
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—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—28no—
306C2×C42—28no—
307C2×C42—416no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42C2×D414yes—
313C2×C42C2×D414yes—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—22no—
348C24—21no—
349C24—21no—
350C24—21no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—24no—
363C24C22×D412yes—
364C24C2×#4114yes—
365C24C2×#4114yes—
366C24C2511yescharacteristic
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—84no—
371C2×Q8—84no—
372C2×D4—216no—
373C2×D4—28no—
374C2×D4—48no—
375C2×D4—416no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—48no—
379C2×D4—416no—
380C2×D4—48no—
381C2×D4—48no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—88no—
467C22×C4—88no—
468C22×C4#124116yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4H(5,𝔽2)14yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C42−1+412yes—
483C4⋊C4—28no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C22⋊C4—216no—
496C22⋊C4—48no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C42—28no—
508C42—28no—
509C42—44no—
510C42—44no—
511C42—48no—
512C42—88no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—28no—
528C23—24no—
529C23—24no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23C2×#12414yes—
559C23C23×D411yescharacteristic
560C23C22×#4111yescharacteristic
561C23C22×#4111yescharacteristic
562C23C22×#4112yes—
563C23C2×H(5,𝔽2)12yes—
564C23C2×H(5,𝔽2)11yescharacteristic
565C23C2×2−1+411yescharacteristic
566C23#58512yes—
567C23#58512yes—
568C23#58511yescharacteristic
569C23#58511yescharacteristic
570Q8—88no—
571D4—416no—
572D4—432no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—28no—
598C2×C4—416no—
599C2×C4—416no—
600C2×C4—416no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
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—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
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—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—816no—
653C2×C4#52514yes—
654C2×C4C2×#12414yes—
655C2×C4C2×#12414yes—
656C2×C4#53414yes—
657C2×C4#53714yes—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—42no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—48no—
673C22—48no—
674C22—82no—
675C22—82no—
676C22—84no—
677C22—84no—
678C22—82no—
679C22—82no—
680C22C2×#52511yescharacteristic
681C22C22×#12411yescharacteristic
682C22C2×#53411yescharacteristic
683C22C2×#53711yescharacteristic
684C22#344911yescharacteristic
685C22#344911yescharacteristic
686C22#344911yescharacteristic
687C22#345912yes—
688C22#346012yes—
689C22#347512yes—
690C22#347512yes—
691C22#350512yes—
692C22#350512yes—
693C22#350612yes—
694C22#350712yes—
695C22#351411yescharacteristic
696C22#351511yescharacteristic
697C22#351711yescharacteristic
698C22#351811yescharacteristic
699C22#353412yes—
700C22#353512yes—
701C22#356712yes—
702C22#356812yes—
703C4—24no—
704C4—24no—
705C4—24no—
706C4—24no—
707C4—24no—
708C4—42no—
709C4—44no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—84no—
715C4—84no—
716C4—82no—
717C4—82no—
718C2—44no—
719C2—84no—
720C2#3456211yescharacteristic
721C2#3494611yescharacteristic
722C2#3501911yescharacteristic
723C2#3518011yescharacteristic
724C2#3518911yescharacteristic
725C2#3745211yescharacteristic
726C2#3754111yescharacteristic
727C2#3958012yes—
728C2#3958112yes—
729C2#3958212yes—
730C2#3958312yes—
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×#2855C2semidirect
3. #17079C2semidirect
4. #17077C2semidirect
5. #17074C2semidirect
6. #17069C2semidirect
7. #17005C2semidirect
9. #15252C2semidirect
10. #15050C2semidirect
11. #15045C2semidirect
12. #14842C2semidirect
13. #14765C2semidirect
14. #14344C2semidirect
15. #14250C2semidirect
16. #14190C2semidirect
17. C4×#380C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×C4×#1212
#170792
#170772
#170742
#170692
#170052
#165572
#152522
#150502
#150452
#148422
#147652
#143442
#142502
#141902
C4×#3802